{"properties":[{"name":"$T_0$","aliases":["Kolmogorov","T0"],"uid":"P000001","description":"Given any two distinct points, there is an open set containing one but not the other.\n\nDefined on page 11 of {{zb:0386.54001}}.\n\n----\n#### Meta-properties\n\n- This property is hereditary.\n- This property is preserved by arbitrary disjoint unions.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"$T_1$","aliases":["Fréchet","T1"],"uid":"P000002","description":"Given any two distinct points, each has a neighborhood not containing the other point.\n\nEquivalently, any of the following equivalent properties holds:\n\n- Points are closed in $X$ (that is, for each $x\\in X$ the singleton $\\{x\\}$ is closed).\n- {P78} subspaces are {P52}.\n- Each point is an intersection of open sets.\n- Each point is an intersection of its neighborhoods.\n\nSee {{wikipedia:T1_space}} for a more extensive list.\n\nDefined on page 11 of {{zb:0386.54001}}.\n\n----\n#### Meta-properties\n\n- The Kolmogorov quotient $\\text{Kol}(X)$ is {P2} iff $X$ is {P135}.\n- This property is hereditary.","refs":[{"name":"T1 space on Wikipedia","wikipedia":"T1_space"},{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"$T_2$","aliases":["Hausdorff","T2"],"uid":"P000003","description":"Given any two distinct points $a$ and $b$, there are disjoint open sets $O_a$ and $O_b$ containing $a$ and $b$, respectively.\n\nDefined on page 11 of {{zb:0386.54001}}.\n\n----\n#### Meta-properties\n\n- This property is hereditary.\n- The Kolmogorov quotient $\\text{Kol}(X)$ is {P3} iff $X$ is {P134}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"$T_{2 \\frac{1}{2}}$","aliases":["Urysohn","Completely Hausdorff","T2.5"],"uid":"P000004","description":"Distinct points are separated by closed neighborhoods.\nIn other words, given any two distinct points $a$ and $b$, there are open sets $O_a$ and $O_b$ containing $a$ and $b$,\nrespectively, such that $cl(O_a)$ and $cl(O_b)$ are disjoint.\n\nDefined as \"Urysohn\" in 14F of {{zb:1052.54001}}.\nDefined as \"completely Hausdorff\" and $T_{2\\frac{1}{2}}$ on page 13 of\n{{zb:0386.54001}}.\n\n----\n#### Meta-properties\n\n- This property is hereditary.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"},{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Urysohn and completely Hausdorff spaces","wikipedia":"Urysohn_and_completely_Hausdorff_spaces"}]},{"name":"$T_3$","aliases":["Regular Hausdorff","T3"],"uid":"P000005","description":"A space which is both {P11} and {P3}. \nEquivalently, a space that is both {P11} and {P1}.\n\nDefined in 14.1 of {{zb:1052.54001}}.\n\n----\n#### Meta-properties\n\n- The Kolmogorov quotient $\\text{Kol}(X)$ is {P5} iff $X$ is {P11}.\n- This property is hereditary.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"name":"$T_{3 \\frac{1}{2}}$","aliases":["Tychonoff","Completely regular Hausdorff","T3.5"],"uid":"P000006","description":"A space which is both {P12} and {P3}.\n\nEquivalently, a space that is both {P12} and {P1}.\n\n----\n#### Meta-properties\n\n- The Kolmogorov quotient $\\text{Kol}(X)$ is {P6} iff $X$ is {P12}.\n- This property is hereditary.\n- This property is preserved by arbitrary products.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"},{"name":"Tychonoff space","wikipedia":"Tychonoff_space"}]},{"name":"$T_4$","aliases":["Normal Hausdorff","T4"],"uid":"P000007","description":"A space which is both {P13} and {P3}. \nEquivalently, a space that is both {P13} and {P2}.\n\nDefined in 15.1 of {{zb:1052.54001}}.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"name":"$T_5$","aliases":["Completely normal Hausdorff","T5"],"uid":"P000008","description":"A space which is both {P14} and {P3}. \nEquivalently, a space that is both {P14} and {P2}.\n\nDefined on page 12 of {{zb:0386.54001}} as \"completely normal\".\n\n----\n#### Meta-properties\n\n- This property is hereditary.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Functionally Hausdorff","aliases":["Completely Hausdorff","Urysohn","Completely T2"],"uid":"P000009","description":"Given any two distinct $a,b \\in X$ there is a continuous function $f:X \\rightarrow [0,1]$ such that $f(a) = 0$ and $f(b)=1$.\n\nDefined as \"completely Hausdorff\"/\"functionally Hausdorff\" in 14G of {{zb:1052.54001}}.\nDefined as \"Urysohn\" on page 16 of {{zb:0386.54001}}.\n\n----\n#### Meta-properties\n\n- This property is hereditary.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"},{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Urysohn and completely Hausdorff spaces","wikipedia":"Urysohn_and_completely_Hausdorff_spaces"}]},{"name":"Semiregular","aliases":[],"uid":"P000010","description":"A space with a base $\\mathcal{B}$ of regular open sets, that is,\n$\\operatorname{int}(\\operatorname{cl}(O)) = O$ for every $O \\in \\mathcal{B}$.\n\nDefined in 14E of {{zb:1052.54001}}.\n\n----\n### Meta-properties\n\n- This property is hereditary with respect to open sets.\n- This property is not hereditary with respect to closed sets\n(Example: {S11|P10}\nand {S10|P10}.)","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"name":"Regular","aliases":[],"uid":"P000011","description":"A space in which a closed set and a point not contained in it\ncan be separated by open neighborhoods.\nIn other words, given a closed set $A$ and a point\n$b \\notin A$, there are disjoint open sets $O_A$ and $O_b$ containing $A$ and\n$b$, respectively.\n\nEquivalently, every point has a neighborhood base consisting of closed sets.\n\nSee Definition 14.1 and Theorem 14.3 in {{zb:1052.54001}}.\n\n----\n#### Meta-properties\n\n- This property is hereditary.\n- $X$ is {P11} iff its Kolmogorov quotient $\\text{Kol}(X)$ is {P5}.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"name":"Completely regular","aliases":["Uniformizable"],"uid":"P000012","description":"A space in which points and closed sets can be separated by a function. \nNamely, given a closed set $A$ and a point $b \\notin A$, there is a continuous function $f:X \\rightarrow [0,1]$ such that $f(A) = \\{0\\}$ and $f(b)=1$.\n\nEquivalently, the cozero sets form a base for the topology.\n\nDefined in 14.8 of {{zb:1052.54001}}.\n\nEquivalently, the topology is induced by a uniformity. See Theorem 38.2 in {{zb:1052.54001}} and {{wikipedia:Uniform_space}}.\n\n----\n#### Meta-properties\n\n- $X$ is {P12} iff its Kolmogorov quotient $\\text{Kol}(X)$ is {P6}.\n- This property is hereditary.\n- This property is preserved by arbitrary products.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"},{"name":"Uniform space on Wikipedia","wikipedia":"Uniform_space"}]},{"name":"Normal","aliases":[],"uid":"P000013","description":"A space in which disjoint closed sets can be separated by open neighborhoods.\nIn other words, given disjoint closed sets $A$ and $B$, there are disjoint open sets $O_A$ and $O_B$ containing $A$ and $B$, respectively.\n\nDefined in 15.1 of {{zb:1052.54001}}.\n\n----\n#### Meta-properties\n\n- $X$ satisfies this property iff its Kolmogorov quotient $\\text{Kol}(X)$ does.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"name":"Completely normal","aliases":["Hereditarily normal"],"uid":"P000014","description":"Every subspace of $X$ is {P13}.\n\nEquivalently, every open subspace of $X$ is {P13}.\n\nEquivalently, any two *separated sets* (sets for which each misses the other's closure)\ncan be placed into disjoint open sets.\n\nDefined on page 11 of {{zb:0386.54001}} (as $T_5$).\n\n----\n#### Meta-properties\n\n- This property is hereditary.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Normal space on Wikipedia","wikipedia":"Normal_space"}]},{"name":"Perfectly normal","aliases":[],"uid":"P000015","description":"For any disjoint closed sets $A$ and $B$ in $X$,\nthere is a continuous function $f: X \\rightarrow [0,1]$\nsuch that $f^{-1}(\\{0\\}) = A$ and $f^{-1}(\\{1\\}) = B$.\n\nThis is equivalent to each of the following:\n\n- $X$ is {P13} and a {P132} (i.e., every closed set is a $G_\\delta$ set).\n\n- Every closed set in $X$ is a zero set (i.e., $f^{-1}(0)$ for some continuous $f:X\\to[0,1]$).\n\n- Every open set in $X$ is a cozero set (i.e., $f^{-1}((0,1])$ for some continuous $f:X\\to[0,1]$).\n\nThe equivalence between these conditions (Vedenissoff's theorem) is shown for example\nin Theorem 1.5.19 of {{zb:0684.54001}} (under the assumption of a {P2} space,\nbut the proof does not use that assumption). See also {{mathse:72138}}.\n\nDefined on page 16 of {{zb:0386.54001}} (as perfectly $T_4$).\n\n----\n#### Meta-properties\n\n- This property is hereditary (see Theorem 2.1.6 in {{zb:0684.54001}}).","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"},{"name":"Why are these two definitions of a perfectly normal space equivalent?","mathse":72138}]},{"name":"Compact","aliases":["Quasi-compact"],"uid":"P000016","description":"Every open cover of the space has a finite subcover.\n\nEquivalently, every open cover of the space has an [open] finite refinement.\n\nDefined on page 18 of {{zb:0386.54001}}.\n\nSome authors call this property \"quasi-compact\", and use \"compact\" to mean quasi-compact and {P3}.\n\n----\n#### Meta-properties\n\n- $X$ satisfies this property iff its Kolmogorov quotient $\\text{Kol}(X)$ does.\n- This property is hereditary with respect to closed sets.\n- This property is preserved by finite disjoint unions.\n- This property is preserved by arbitrary products. ([Tychonoff's theorem](https://en.wikipedia.org/wiki/Tychonoff%27s_theorem))\n- This property is preserved in any coarser topology.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Compact space on Wikipedia","wikipedia":"Compact_space"}]},{"name":"$\\sigma$-compact","aliases":[],"uid":"P000017","description":"A space which is the union of countably many {P16} subsets.\n\nDefined on page 19 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Lindelöf","aliases":[],"uid":"P000018","description":"Every open cover of $X$ has a countable subcover.\n\nDefined on page 19 of {{zb:0386.54001}}.\n\n----\n#### Meta-properties\n\n- $X$ satisfies this property iff its Kolmogorov quotient $\\text{Kol}(X)$ does.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Countably compact","aliases":[],"uid":"P000019","description":"Every countable open cover of $X$ has a finite subcover.\n\nEquivalently, every infinite set in $X$ has an $\\omega$-accumulation point in $X$.\n\nEquivalently, every sequence in $X$ has an accumulation point in $X$ (that is, a point such that each of its neighborhoods contains infinitely many terms of the sequence).\n\nDefined on page 19 of {{zb:0386.54001}}. See for example {{wikipedia:Countably_compact_space}} for a proof of the equivalences above.\n\n----\n#### Meta-properties\n\n- This property is hereditary with respect to closed sets.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Countably compact space on Wikipedia","wikipedia":"Countably_compact_space"}]},{"name":"Sequentially compact","aliases":[],"uid":"P000020","description":"Every sequence in $X$ has a convergent subsequence.\n\nDefined on page 19 of {{zb:0386.54001}}.\n\n---\n### Meta-properties\n\n- This property is hereditary with respect to closed sets.\n- This property is preserved by countable products (see Theorem 3.10.35 in {{zb:0684.54001}}).","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"name":"Weakly countably compact","aliases":["Limit point compact"],"uid":"P000021","description":"Every infinite set in $X$ has a limit point.\n\nEquivalently, every closed discrete subset of $X$ is finite.\n\nDefined on page 19 of {{zb:0386.54001}}.\n\n----\n#### Meta-properties\n\n- This property is hereditary with respect to closed sets.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Limit point compact on Wikipedia","wikipedia":"Limit_point_compact"}]},{"name":"Pseudocompact","aliases":[],"uid":"P000022","description":"Every continuous function from $X$ to {S25} is bounded.\n\nDefined on page 20 of {{zb:0386.54001}}.\n\n----\n#### Meta-properties\n\n- $X$ satisfies this property iff its Kolmogorov quotient $\\mathrm{Kol}(X)$ does.\n- This property is preserved in any coarser topology.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Weakly locally compact","aliases":["WLC"],"uid":"P000023","description":"Each point has a {P16} neighborhood.\n\nDefined as \"locally compact\" on page 20 of {{zb:0386.54001}}.\n\n----\n#### Meta-properties\n\n- $X$ satisfies this property iff its Kolmogorov quotient $\\text{Kol}(X)$ does.\n- This property is hereditary with respect to closed sets.\n- This property is preserved by finite products.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Locally relatively compact","aliases":["Weakly locally closed-and-compact"],"uid":"P000024","description":"Each point has a local base of neighborhoods with compact closure.\n\nEquivalently (see Condition 2 of {{wikipedia:Locally_compact_space}}), every point in $X$ has\na closed and compact neighborhood.\n\nDefined on page 20 of {{zb:0386.54001}} as \"strongly locally compact\".\n\nCompare with {P130}.\n\n----\n#### Meta-properties\n\n- $X$ satisfies this property iff its Kolmogorov quotient $\\text{Kol}(X)$ does.\n- This property is hereditary with respect to closed sets.\n- This property is preserved by finite products.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Locally compact space on Wikipedia","wikipedia":"Locally_compact_space"}]},{"name":"Exhaustible by compacts","aliases":["$\\sigma$-locally compact"],"uid":"P000025","description":"A space that is {P17} and {P23}.\n\nEquivalently, $X=\\bigcup_{n<\\omega}K_n$ where for each $x\\in X$, there exists\n$n<\\omega$ such that $K_n$ is a compact neighborhood of $x$.\n\nEquivalently, $X=\\bigcup_{n<\\omega}K_n$ for $K_n$ compact, and such that $K_n\\subseteq int(K_{n+1})$.\n\nThe equivalences above can be shown using the ideas in {{mathse:4568032}}.\n\nDefined on page 21 of {{zb:0386.54001}} as \"$\\sigma$-locally compact\".","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Exhaustion by compact sets on Wikipedia","wikipedia":"Exhaustion_by_compact_sets"},{"name":"Answer to \"A question about local compactness and σ-compactness\"","mathse":4568032}]},{"name":"Separable","aliases":["Density $\\leq\\aleph_0$"],"uid":"P000026","description":"A space with a countable dense subset.\n\nDefined on page 7 of {{zb:0386.54001}}.\n\n----\n#### Meta-properties\n\n- $X$ satisfies this property iff its Kolmogorov quotient $\\text{Kol}(X)$ does.\n- This property is preserved in any coarser topology.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Second countable","aliases":[],"uid":"P000027","description":"A space with a countable basis.\n\nDefined on page 7 of {{zb:0386.54001}}.\n\n----\n#### Meta-properties\n\n- This property is hereditary.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"First countable","aliases":[],"uid":"P000028","description":"A space with a countable local basis at every point.\n\nDefined on page 7 of {{zb:0386.54001}}.\n\n----\n#### Meta-properties\n\n- This property is hereditary.\n- If each point has a neighborhood with the property, $X$ also has the property.\n- $X$ satisfies this property iff its Kolmogorov quotient $\\text{Kol}(X)$ does.\n- This property is preserved by countable products. (Corollary 2.3.14 in {{zb:0684.54001}})","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"name":"Countable chain condition","aliases":["CCC","Suslin property"],"uid":"P000029","description":"A space in which every collection of pairwise-disjoint nonempty open sets is countable.\n\nDefined on page 22 of {{zb:0386.54001}}.\nAlso referred to as the \"Suslin/Souslin property\" as in e.g. 1.7.12 of {{zb:0684.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"},{"name":"Countable chain condition on Wikipedia","wikipedia":"Countable chain condition"}]},{"name":"Paracompact","aliases":[],"uid":"P000030","description":"Every open cover of the space has a locally finite open refinement.\n\nDefined on page 23 of {{zb:0386.54001}}.\n\n----\n#### Meta-properties\n\n- $X$ satisfies this property iff its Kolmogorov quotient $\\text{Kol}(X)$ does.\n- This property is preserved by arbitrary disjoint unions.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Metacompact","aliases":["Weakly paracompact","Pointwise paracompact"],"uid":"P000031","description":"Every open cover of the space has a point-finite open refinement.\n\nDefined on page 23 of {{zb:0386.54001}}.\n\n----\n#### Meta-properties\n\n- This property is hereditary with respect to closed sets.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"name":"Countably paracompact","aliases":[],"uid":"P000032","description":"Every countable open cover of the space has a locally finite open refinement.\n\nDefined on page 23 of {{zb:0386.54001}}.\n\n----\n#### Meta-properties\n\n- This property is hereditary with respect to closed sets.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Countably metacompact","aliases":[],"uid":"P000033","description":"Every countable open cover of the space has a point-finite open refinement.\n\nEquivalently (*Ishikawa's characterization*),\nfor any nested sequence of closed sets $F_0 \\supseteq F_1 \\supseteq \\cdots$ with empty intersection,\nthere exist open sets $U_n \\supseteq F_n$ for each $n$ such that the $U_n$ have empty intersection.\n\n(One can choose the $U_n$ to be nested if desired.)\n\nFor a proof of the equivalence, see the Corollary in {{zb:0066.41001}}.\n\nDefined on page 23 of {{zb:0386.54001}}.\n\n----\n#### Meta-properties\n\n- This property is hereditary with respect to closed sets.\n- $X$ satisfies this property iff its Kolmogorov quotient $\\text{Kol}(X)$ does.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"On Countably Paracompact Spaces (F. Ishikawa)","zb":"0066.41001"}]},{"name":"Fully normal","aliases":[],"uid":"P000034","description":"Every open cover of $X$ is normal.\n\nThis is equivalent to each of the following:\n\n- Every open cover has an open star-refinement.\n\n- Every open cover has an open barycentric refinement.\n\n- Every open cover has a subordinate partition of unity.\n\n- Every open cover has a locally finite subordinate partition of unity.\n\n- $X$ is {P13} and {P30}.\n\nThe *star* $\\text{St}(A, \\mathcal{B})$ for $A\\subseteq X$ and $\\mathcal{B}$ a family of subsets of $X$, is the set $\\bigcup\\{B\\in \\mathcal{B} : B\\cap A\\neq\\emptyset\\}$. We say $\\mathcal{V}$ is a *star-refinement* of $\\mathcal{U}$ if $\\mathcal{V}^\\ast = \\{\\text{St}(V, \\mathcal{V}): V\\in\\mathcal{V}\\}$ is a refinement of $\\mathcal{U}$. We say $\\mathcal{V}$ is a *barycentric refinement* of $\\mathcal{U}$ if $\\mathcal{V}^\\Delta = \\{\\text{St}(x, \\mathcal{V}): x\\in X\\}$ is a refinement of $\\mathcal{U}$.\nWe say $\\mathcal{U}$ is a *normal cover* if there exists a sequence of open covers $(\\mathcal{U}_n)$ such that $\\mathcal U=\\mathcal U_1$ and each $\\mathcal{U}_{n+1}$ is a star-refinement of $\\mathcal{U}_n$.\nWe say that a family of functions $(f_i)$ is a *partition of unity* if the $f_i:X\\to [0, 1]$ are continuous and $\\sum_i f_i = 1$. Here $(\\sum_i f_i)(x) = \\sum_i f_i(x)$ is defined as supremum over finite sums $f_{i_1}(x)+ \\dots +f_{i_k}(x)$. We say that a partition of unity $(f_i)$ is *subordinate* to a cover $(U_i)$ of $X$ if $\\{x\\in X : f_i(x) > 0\\}\\subseteq U_i$ for all $i$. We say that a partition of unity $(f_i)$ is *locally finite* when the cover consisting of sets $\\{x\\in X : f_i(x) > 0\\}$ is locally finite.\n\nSee {{mathse:4862626}} for proof of the equivalences. \n\nAlso see [Henno Brandsma: On paracompactness, full normality and the like](http://at.yorku.ca/p/a/c/a/02.pdf)\n(available [here](https://web.archive.org/web/20250119093147/http://at.yorku.ca/p/a/c/a/02.pdf)).\n\n(Note: {{zb:0386.54001}} defines this property on page 23 as \"fully $T_4$\".\nMisleadingly, it also calls \"star refinement\" what all other major sources call a barycentric refinement.)","refs":[{"name":"Star refinement on Wikipedia","wikipedia":"Star_refinement"},{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Answer to \"Fully normal implies paracompact without a $T_1$ assumption?\"","mathse":4862626}]},{"name":"Fully $T_4$","aliases":["Fully T4"],"uid":"P000035","description":"A space which is {P2} and {P34}.\n\nDefined as \"fully normal\" on page 23 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Connected","aliases":[],"uid":"P000036","description":"No pair of disjoint nonempty open sets covers the space.\n\nDefined on page 7 of {{zb:0386.54001}}.\n\n----\n#### Meta-properties\n\n- $X$ satisfies this property iff its Kolmogorov quotient $\\text{Kol}(X)$ does.\n- An arbitrary product of nonempty spaces satisfies this property iff each of its factors does.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Path connected","aliases":[],"uid":"P000037","description":"Given any two points $x,y\\in X$, there is a *path* in $X$ from $x$ to $y$;\nthat is, a continuous map $f:[0,1]\\to X$ with $f(0)=x$ and $f(1)=y$.\n\nDefined on page 29 of {{zb:0386.54001}}.\n\nCompare with {P38} and {P95}.\n\n----\n#### Meta-properties\n\n- $X$ satisfies this property iff its Kolmogorov quotient $\\text{Kol}(X)$ does.\n- This property is preserved by arbitrary products.\n- This property is preserved in any coarser topology.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Injectively path connected","aliases":["Arc connected"],"uid":"P000038","description":"Given any two distinct points $x,y\\in X$, there is an *injective path* in $X$ from $x$ to $y$;\nthat is, an injective continuous map $f:[0,1]\\to X$ with $f(0)=x$ and $f(1)=y$.\n\nCompare with {P37}, which does not require the path to be injective,\nand {P95}, which requires the path to be a homeomorphism onto its image.\n\nDefined on page 29 of {{zb:0386.54001}} with the name \"arc connected\".\nHere we reserve that name for the stronger property {P95},\nwhich is the more common usage in the literature.\n\n#### Meta-properties\n\n- This property is preserved by arbitrary products.\n- This property is preserved in any coarser topology.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Hyperconnected","aliases":["Anti-Hausdorff","Irreducible"],"uid":"P000039","description":"The space has no disjoint nonempty open sets.\n\nEquivalently, every nonempty open set is dense.\n\nDefined on page 29 of {{zb:0386.54001}}. Defined as \"anti-Hausdorff\" in {{doi:10.1016/0166-8641(93)90147-6}}.\n\nSee {{wikipedia:Hyperconnected_space}} for other equivalent characterizations.\n\n----\n#### Meta-properties\n\n- This property is hereditary with respect to open sets.\n- This property is preserved by arbitrary products.\n- This property is preserved by continuous images.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Hyperconnected space on wikipedia","wikipedia":"Hyperconnected_space"},{"name":"An anti-Hausdorff Fréchet space in which convergent sequences have unique limits","doi":"10.1016/0166-8641(93)90147-6"}]},{"name":"Ultraconnected","aliases":[],"uid":"P000040","description":"The space has no disjoint nonempty closed sets.\n\nEquivalently ({{mathse:5006424}}), the only neighborhood of the space's\ndiagonal $\\Delta=\\{\\langle x,x\\rangle:x\\in X\\}$ is $X^2$.\n\nDefined on page 29 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Are ultraconnected spaces divisible?","mathse":5006424}]},{"name":"Locally connected","aliases":[],"uid":"P000041","description":"The topology of $X$ has a base consisting of {P36} open sets.\n\nThis is equivalent to each of the following:\n\n* The connected components of every open set are open.\n\n* $X$ is *locally connected at $x$* for every $x\\in X$;\nthat is, each $x$ has a neighborhood base consisting of open connected sets.\n\n* $X$ is *connected im kleinen at $x$* for every $x\\in X$;\nthat is, each $x$ has a neighborhood base consisting of (not necessarily open) connected sets.\n\nNote: In general, local connectedness at a point $x$ is not equivalent to connectedness im kleinen at $x$.\nBut the corresponding global statements (where the condition holds at every point) are equivalent.\n\nFor terminology and the equivalence between the conditions above, see section 2 of {{zb:1342.30054}} and {{wikipedia:Locally_connected_space}}, plus the references therein.\n\nDefined on page 30 of {{zb:0386.54001}}.\n\n----\n#### Meta-properties\n\n- This property is hereditary with respect to open sets.","refs":[{"name":"The Mazurkiewicz distance and sets that are finitely connected at the boundary (Björn, Björn, Shanmugalingam)","zb":"1342.30054"},{"name":"Locally connected space on Wikipedia","wikipedia":"Locally_connected_space"},{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Locally path connected","aliases":["$LC^0$","Locally $0$-connected"],"uid":"P000042","description":"The topology of $X$ has a base consisting of {P37} open sets.\n\nThis is equivalent to each of the following:\n\n* The path components of every open set are open.\n\n* $X$ is *locally path connected at $x$* for every $x\\in X$;\nthat is, each $x$ has a neighborhood base consisting of open path connected sets.\n\n* $X$ is *path connected im kleinen at $x$* for every $x\\in X$;\nthat is, each $x$ has a neighborhood base consisting of (not necessarily open) path connected sets.\n\n* ($LC^0$ = *locally $0$-connected*) For each $x\\in X$, every neighborhood $U$ of $x$\ncontains a neighborhood $V$ of $x$ such that every continuous $\\phi:S^0\\to V$\nis null-homotopic in $U$ (where $S^0=\\{-1,1\\}$ with the discrete topology).\n\nNote: In general, local path connectedness at a point $x$ is not equivalent to path connectedness im kleinen at $x$.\nBut the corresponding global statements (where the condition holds at every point) are equivalent.\n\nFor terminology and the equivalence between the conditions above, see section 2 of {{zb:1342.30054}} and {{wikipedia:Locally_connected_space}}, plus the references therein,\nand also {{mathse:3002235}}.\n\nDefined on page 30 of {{zb:0386.54001}}.\n\nCompare with {P43} and {P96}.\n\n----\n### Meta-properties\n\n- This property is hereditary with respect to open sets.","refs":[{"name":"The Mazurkiewicz distance and sets that are finitely connected at the boundary (Björn, Björn, Shanmugalingam)","zb":"1342.30054"},{"name":"Locally connected space on Wikipedia","wikipedia":"Locally_connected_space"},{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Answer to \"Definition of locally pathwise connected\"","mathse":3002235}]},{"name":"Locally injectively path connected","aliases":["Locally arc connected"],"uid":"P000043","description":"The topology of $X$ has a base consisting of {P38} open sets.\n\nEquivalently, every point has a local base of {P38} open neighborhoods.\n\nCompare with {P42} and {P96}.\n\nDefined on page 30 of {{zb:0386.54001}} with the name \"locally arc connected\".\nHere we reserve that name for the stronger property {P96}.\n\n----\n#### Meta-properties\n\n- This property is preserved by arbitrary disjoint unions.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Biconnected","aliases":[],"uid":"P000044","description":"A {P36} space which cannot be partitioned into two {P36} subsets, each with at least two points.\n\nEquivalently, a {P36} space which does not contain two disjoint {P36} subsets, each with at least two points.\n\nSee for example {{mathse:4654942}} for a proof of the equivalence.\n\nDefined on page 33 of {{zb:0386.54001}}. See also for example {{doi:10.1016/0166-8641(95)00005-2}}.\n\n----\n#### Meta-properties\n\n- This property is hereditary with respect to connected sets.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Answer to \"Investigating a property related to the biconnected property\"","mathse":4654942},{"name":"A biconnected set in the plane (M.E. Rudin)","doi":"10.1016/0166-8641(95)00005-2"}]},{"name":"Has a dispersion point","aliases":[],"uid":"P000045","description":"A {P36} space $X$ with a \"dispersion point\" $p$, that is,\na point such that $X \\setminus \\{p\\}$ is {P47}.\n\nDefined on page 33 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Totally path disconnected","aliases":[],"uid":"P000046","description":"The only paths in the space are constant.\n\nDefined on page 31 of {{zb:0386.54001}}.\n\n----\n#### Meta-properties\n\n- This property is hereditary.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Totally disconnected","aliases":["Hereditarily disconnected"],"uid":"P000047","description":"All connected components of the space are singletons.\n\nDefined on page 32 of {{zb:0386.54001}}.\n\nSome authors (for example {{zb:0684.54001}}) use \"totally disconnected\" to mean {P48} and \"hereditarily disconnected\" to mean {P47}.\n\n----\n#### Meta-properties\n\n- This property is hereditary.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"name":"Totally separated","aliases":["Totally disconnected"],"uid":"P000048","description":"Given any two distinct points $x,y \\in X$, there exists a clopen subset $A\\subseteq X$ such that\n$x\\in A$ and $y\\not\\in A$.\n\nDefined on page 32 of {{zb:0386.54001}}.\n\nSome authors (for example {{zb:0684.54001}}) use \"totally disconnected\" to mean {P48} and \"hereditarily disconnected\" to mean {P47}.\n\n----\n#### Meta-properties\n\n- This property is hereditary.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"name":"Extremally disconnected","aliases":[],"uid":"P000049","description":"The closure of every open set in $X$ is open or, equivalently, clopen.\n\nEquivalently, any two disjoint open sets have disjoint closures.\n\nDefined in problem 15G of {{zb:1052.54001}} and problem 1H of {{zb:1380.46022}}.\n\n{{zb:0386.54001}} defines it on page 32 with the additional assumption of {P3},\nwhich we do not assume here.\n\n----\n#### Meta-properties\n\n- This property is hereditary with respect to open sets (see Problem 15G.2 in {{zb:1052.54001}}).\n- This property is hereditary with respect to dense sets (see {{mathse:3769214}}).\n- This property is hereditary with respect to locally dense sets (equivalent to previous two meta-properties; see also Proposition 1 of {{mathse:5025114}}).\nA set $A\\subseteq X$ is called *locally dense* (or *preopen*) if every $x\\in A$ has neighbourhood $U$ with $U\\cap A$ dense in $U$ (equivalently, $A\\subseteq\\operatorname{int}(\\overline A)$).\n- This property is preserved by arbitrary disjoint unions.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"},{"name":"Rings of Continuous Functions (Gillman & Jerison)","zb":"1380.46022"},{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Dense Subspace of Extremally Disconnected Space is Extremally Disconnected","mathse":3769214},{"name":"Answer to \"Which spaces admit only extremally disconnected compactifications?\"","mathse":5025114}]},{"name":"Zero dimensional","aliases":[],"uid":"P000050","description":"A space with a basis of clopen sets.\n\nDefined on page 33 of {{zb:0386.54001}}.\n\n----\n#### Meta-properties\n\n- This property is hereditary.\n- This property is preserved by arbitrary products.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Scattered","aliases":["Dispersed"],"uid":"P000051","description":"The space contains no nonempty dense-in-itself subset.\n\nEquivalently, every nonempty subspace $Y$ contains a point isolated in $Y$.\n\nDefined on page 33 of {{zb:0386.54001}}.\nDefined as \"dispersed\" in e.g. {{zb:0164.53101}}.\n\n----\n#### Meta-properties\n\n- This property is hereditary.\n- This property is preserved in any finer topology.\n- This property is preserved by arbitrary disjoint unions.\n- This property is preserved by finite products.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Scattered space on Wikipedia","wikipedia":"Scattered_space"},{"name":"Total paracompactness and paracompact dispersed spaces (Telgársky)","zb":"0164.53101"},{"name":"Countable products of scattered paracompact spaces. (Rudin and Watson)","zb":"0518.54021"}]},{"name":"Discrete","aliases":[],"uid":"P000052","description":"Every point is isolated, that is, $\\{x\\}$ is open for every point $x$.\n\nSee 3.2c of {{zb:1052.54001}}.\n\n----\n#### Meta-properties\n\n- This property is hereditary.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"name":"Metrizable","aliases":[],"uid":"P000053","description":"The topology on the space $X$ is induced by some metric $d$.\n\nA *metric* is a function $d:X \\times X \\to \\mathbb{R}$ such that for all $x,y,z \\in X$,\n(i) $d(x,y) \\geq 0$ and $d(x,y)=0$ if and only if $x=y$;\n(ii) $d(x,y) = d(y,x)$;\n(iii) $d(x,z) \\leq d(x,y) + d(y,z)$.\n\nGiven a metric $d$ and $x \\in X$, $\\epsilon>0$, define $B(x,\\epsilon) = \\{y \\in X: d(x,y)<\\epsilon\\}$. Then the sets $B(x,\\epsilon)$ form a base for a topology, called the topology induced by $d$.\n\nDefined on page 37 of {{zb:0386.54001}}.\n\n----\n#### Meta-properties\n\n- The Kolmogorov quotient $\\text{Kol}(X)$ is {P53} iff $X$ is {P121}.\n- This property is hereditary.\n- This property is preserved by arbitrary disjoint unions (see {{mathse:472003}}).\n- This property is preserved by countable products (see Theorem 4.2.2. in {{zb:0684.54001}}).","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"How to show that a disjoint sum of metric spaces is metrizable?","mathse":472003},{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"name":"Has a $\\sigma$-locally finite base","aliases":[],"uid":"P000054","description":"A space with a $\\sigma$-locally finite base for the topology.\n\nA collection of subsets of $X$ is *$\\sigma$-locally finite* if it is a countable union of locally finite collections.\n\nDefined on page 37 of {{zb:0386.54001}}.\n\n----\n#### Meta-properties\n\n- This property is hereditary.\n- This property is preserved by arbitrary disjoint unions.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Completely metrizable","aliases":["Topologically complete"],"uid":"P000055","description":"The topology on the space $X$ is induced by some complete metric $d$,\nthat is, a metric for which every Cauchy sequence converges.\n\nA sequence $(x_n)_n$ is *Cauchy* provided for each distance $\\epsilon>0$, there is\na natural number $N$ for which $d(x_i,x_j)<\\epsilon$ for all $i,j>N$.\n\nDefined in 24.2 of {{zb:1052.54001}}.\nCalled *topologically complete* on page 37 of {{zb:0386.54001}};\nthis last term has also been used for {P63} and {P221}.\n\n----\n#### Meta-properties\n\n- This property is hereditary with respect to $G_\\delta$ sets and,\nin particular, with respect to open sets and closed sets\n(see Theorem 24.12 in {{zb:1052.54001}}).\n- This property is preserved by countable products (see Theorem 24.11 in {{zb:1052.54001}}).","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"},{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Meager","aliases":["First category"],"uid":"P000056","description":"The space $X$ is a countable union of nowhere dense subsets of $X$.\n\nDefined on page 7 of {{zb:0386.54001}}.\n\n----\n#### Meta-properties\n\n- This property is preserved by arbitrary disjoint unions.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Meagre set on Wikipedia","wikipedia":"Meagre_set"}]},{"name":"Countable","aliases":["Cardinality $\\leq\\aleph_0$","Cardinality $\\leq\\beth_0$"],"uid":"P000057","description":"The cardinality of the space is less than or equal to the cardinality of $\\mathbb N$.\n\nSee sections 1.12-1.15 of {{zb:1052.54001}}.\n\n----\n#### Meta-properties\n\n- This property is hereditary.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"name":"Cardinality $\\lt\\mathfrak c$","aliases":["Smaller than the continuum","Cardinality $\\lt\\beth_1$"],"uid":"P000058","description":"The cardinality of the space is less than the cardinality of $\\mathbb R$.\n\nSee sections 1.12-1.15 of {{zb:1052.54001}}.\n\n----\n#### Meta-properties\n\n- This property is hereditary.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"name":"Cardinality $\\leq 2^{\\mathfrak c}$","aliases":["Smaller or same as the power set of the continuum","Cardinality $\\leq\\beth_2$"],"uid":"P000059","description":"The cardinality of the space is at most the cardinality of $\\mathcal{P}(\\mathbb R)$, the set of subsets of $\\mathbb R$.\n\nSee sections 1.12-1.15 of {{zb:1052.54001}}.\n\n----\n#### Meta-properties\n\n- This property is hereditary.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"name":"Strongly connected","aliases":["Ultrapseudocompact"],"uid":"P000060","description":"Every continuous function $f:X \\to \\mathbb R$ is constant.\n\nDefined on page 223 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"On ultrapseudocompact and related spaces (T. Nieminen)","doi":"10.5186/aasfm.1977.0321"}]},{"name":"Cozero complemented","aliases":[],"uid":"P000061","description":"Every cozero set $U$ in $X$ has a *cozero complement*, that is,\na cozero set $V$ disjoint from $U$ such that $U\\cup V$ is dense in $X$.\n\nA *cozero set* is a set of the form $\\text{coz}(f):=\\{x\\in X:f(x)\\ne 0\\}$\nfor some continuous function $f:X\\to\\mathbb R$; equivalently, the complement of a zero set.\n\nDefined in {{zb:1067.54015}}, with the additional usual assumption of {P6}.\nWe do not make that assumption here.\n\nNote: For a {P6} space, being cozero complemented is equivalent to various algebraic\nproperties of the ring $C(X)$ of continuous real-valued functions on $X$.\nSee Theorem 1.3 in {{zb:1067.54015}}.\n\n----\n#### Meta-properties\n\n- $X$ satisfies this property iff its Kolmogorov quotient $\\text{Kol}(X)$ does.\n- This property is hereditary with respect to cozero sets (see Lemma 1.5(b) in {{zb:1067.54015}}),\nand in particular, with respect to clopen sets.\n- This property is preserved by arbitrary disjoint unions.","refs":[{"name":"Cozero complemented spaces; when the space of minimal prime ideals of a $C(X)$ is compact (Henriksen & Woods)","zb":"1067.54015"}]},{"name":"Weakly Lindelöf","aliases":[],"uid":"P000062","description":"Every open cover of $X$ has a countable subcollection whose union is dense in $X$.\n\n----\n### Meta-properties\n\n- This property is hereditary with respect to clopen sets.\n- This property is preserved in any coarser topology.\n- If a nonempty product space satisfies the property, so does every factor.","refs":[{"name":"Products of weakly-א-compact spaces (M. Ulmer)","doi":"10.2307/1996310"}]},{"name":"Čech complete","aliases":["Topologically complete"],"uid":"P000063","description":"A {P6} space $X$ that is a $G_\\delta$ set in some compactification of $X$ (equivalently, in every compactification of $X$).\n\nEquivalently, there is a sequence $\\mathcal{U}_1, \\mathcal{U}_2, \\dots$ of open covers of $X$ such that whenever $\\mathcal{F}$ is a family of closed sets with the finite intersection property and such that for each $n$ there is some $F_n \\in \\mathcal{F}$ with $F_n \\subseteq U$ for some $U \\in \\mathcal{U}_n$, then $\\bigcap \\mathcal F \\neq \\emptyset$.\n\nSee Section 3.9 (\"Čech-complete spaces\") of {{zb:0684.54001}}, specifically Theorems 3.9.1 and 3.9.2 for the equivalences above.\n\nSuch spaces are called *topologically complete* in {{zb:0117.15903}};\nthis last term has also been used for {P55} and {P221}.\n\n----\n#### Meta-properties\n\n- This property is hereditary with respect to closed sets (see Theorem 3.9.6 in {{zb:0684.54001}}).\n- This property is hereditary with respect to $G_\\delta$ sets (see Theorem 3.9.6 in {{zb:0684.54001}}).","refs":[{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"},{"name":"A characterization of topologically complete spaces in the sense of E. Čech in terms of convergence of functions. (Frolik, 1963)","zb":"0117.15903"}]},{"name":"Baire","aliases":[],"uid":"P000064","description":"A space such that the countable union of closed nowhere dense subsets has empty interior; equivalently, such that the countable intersection of open dense subsets is still dense.\n\nSee {{wikipedia:Baire_space}} for several other equivalent characterizations.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"},{"name":"Baire space on Wikipedia","wikipedia":"Baire_space"}]},{"name":"Cardinality $=\\mathfrak c$","aliases":["Continuum-sized","Cardinality $=\\beth_1$"],"uid":"P000065","description":"The cardinality of the space is equal to the cardinality of $\\mathbb R$.\n\nSee sections 1.12-1.15 of {{zb:1052.54001}}.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"name":"Menger","aliases":[],"uid":"P000066","description":"A space that satisfies the selection principle\n\$S_{fin}(\\mathcal{O},\\mathcal{O})\$: for every sequence of open covers\n\$\\mathcal{U}_n\$ for \$n<\\omega\$, there exist finite subcollections\n$\\mathcal{F}_n\\subseteq\\mathcal{U}_n$ such that \$\\bigcup_{n<\\omega}\\mathcal{F}_n\$ is\nalso an open cover.\n\nSee section 5 of {{doi:10.1016/0166-8641(95)00067-4}}.\n\n----\n#### Meta-properties\n\n- This property is hereditary with respect to closed sets.","refs":[{"name":"Combinatorics of open covers I: Ramsey theory (M. Scheepers)","doi":"10.1016/0166-8641(95)00067-4"}]},{"name":"$T_6$","aliases":["Perfectly normal Hausdorff","Perfectly T_4","T6"],"uid":"P000067","description":"A space that is both {P2} and {P15}.\nEquivalently, a space that is both {P1} and {P15}.\n\nDefined on page 16 of {{zb:0386.54001}} as \"perfectly normal\".\n\n----\n\n#### Meta-properties\n\n- This property is hereditary since {P2} and {P15} are.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Normal space on Wikipedia","wikipedia":"Normal_space"}]},{"name":"Rothberger","aliases":["Property C''"],"uid":"P000068","description":"A space that satisfies the selection principle\n\$S_{1}(\\mathcal{O},\\mathcal{O})\$: for every sequence of open covers\n\$\\mathcal{U}_n\$ for \$n<\\omega\$, there exist choices\n\$V_n\\in\\mathcal{U}_n\$ such that \$\\{V_n:n<\\omega\\}\$ is\nalso an open cover.\n\nSee section 6 of {{doi:10.1016/0166-8641(95)00067-4}}.\n\n#### Meta-properties\n\n- This property is hereditary with respect to closed subspaces.","refs":[{"name":"Combinatorics of open covers I: Ramsey theory (M. Scheepers)","doi":"10.1016/0166-8641(95)00067-4"}]},{"name":"Strategic Menger","aliases":["$M^+$","Strategically Menger","$\\Omega M^+$","Strategically $\\Omega$-Menger"],"uid":"P000069","description":"Strategic Menger: The second player has a winning strategy in the Menger game. See Game 1.5 and Definition 1.6 of {{doi:10.14712/1213-7243.2015.201}} for more details.\n\nStrategically $\\Omega$-Menger: The second player has a winning strategy in the game $\\mathsf{G}_{\\mathrm{fin}}(\\Omega_X,\\Omega_X)$. See pages 2 and 3 of {{doi:10.1016/j.topol.2019.07.008}} for more details.\n\nThe equivalence of Strategic Menger and Strategically $\\Omega$-Menger is Theorem 35 of {{doi:10.1016/j.topol.2019.02.062}}.","refs":[{"name":"Applications of limited information strategies in Menger's game (S. Clontz)","doi":"10.14712/1213-7243.2015.201"},{"name":"Relating games of Menger, countable fan tightness, and selective separability (S. Clontz)","doi":"10.1016/j.topol.2019.02.062"},{"name":"Limited information strategies and discrete selectivity (Clontz & Holshouser)","doi":"10.1016/j.topol.2019.07.008"},{"name":"How are the Menger and Ω-Menger games related?","mathse":4730419}]},{"name":"Markov Menger","aliases":["Markov $\\Omega$-Menger"],"uid":"P000070","description":"Markov Menger: The second player has a Markov winning strategy in the Menger game (relying on only the round number and most recent move of the opponent). See Game 1.5 and Definition 2.2 of {{doi:10.14712/1213-7243.2015.201}} for more details.\n\nMarkov $\\Omega$-Menger: The second player has a Markov winning strategy in the game $\\mathsf{G}_{\\mathrm{fin}}(\\Omega_X,\\Omega_X)$ (relying on only the round number and most recent move of the opponent). See pages 2 and 3 of {{doi:10.1016/j.topol.2019.07.008}} for more details.\n\nThe equivalence of Markov Menger with Markov $\\Omega$-Menger is established in {{mathse:4730419}}.","refs":[{"name":"Applications of limited information strategies in Menger's game (S. Clontz)","doi":"10.14712/1213-7243.2015.201"},{"name":"Relating games of Menger, countable fan tightness, and selective separability (S. Clontz)","doi":"10.1016/j.topol.2019.02.062"},{"name":"Limited information strategies and discrete selectivity (Clontz & Holshouser)","doi":"10.1016/j.topol.2019.07.008"},{"name":"How are the Menger and Ω-Menger games related?","mathse":4730419}]},{"name":"$\\sigma$-relatively-compact","aliases":[],"uid":"P000071","description":"$X=\\bigcup_{n<\\omega}R_n$, where $R_n$ is relatively compact to $X$. Here, a set $A\\subseteq X$ is called *relatively compact to* $X$ if every open cover of $X$ has a finite subcollection which covers $A$.\n\nSee Definition 4.2 in {{doi:10.14712/1213-7243.2015.201}}.\n\n----\n#### Meta-properties\n\n- $X$ satisfies this property iff its Kolmogorov quotient $\\text{Kol}(X)$ does.","refs":[{"name":"Applications of limited information strategies in Menger's game (S. Clontz)","doi":"10.14712/1213-7243.2015.201"}]},{"name":"2-Markov Menger","aliases":[],"uid":"P000072","description":"The second player has a $2$-Markov winning strategy in the Menger game (relying on only the round number and two most recent moves of the opponent).\n\n----\n#### Meta-properties\n\n- $X$ satisfies this property iff its Kolmogorov quotient $\\text{Kol}(X)$ does.","refs":[{"name":"Applications of limited information strategies in Menger's game","doi":"10.14712/1213-7243.2015.201"}]},{"name":"Sober","aliases":[],"uid":"P000073","description":"Every nonempty irreducible (= {P39}) closed subset of $X$ is the closure of exactly one point of $X$;\nthat is, every nonempty irreducible closed subset has a unique *generic point*.\n(See {P201} for several equivalent formulations.)\n\nSpecial case:\n$X$ is sober and {P2} $\\iff$ every nonempty irreducible subset is a singleton.\n\nSee {{wikipedia:Sober_space}} and also \nthe section on [Irreducible components](https://stacks.math.columbia.edu/tag/004U) from the Stacks project.\n\nNote: Without the uniqueness of the generic points, the corresponding property is {P192}.\n\n----\n#### Meta-properties\n\n- The Kolmogorov quotient $\\text{Kol}(X)$ is {P73} iff $X$ is {P192}.\n- This property is hereditary with respect to closed sets.","refs":[{"name":"Sober space on Wikipedia","wikipedia":"Sober_space"},{"name":"Topology via logic (S. Vickers)","mr":1002193}]},{"name":"Cosmic","aliases":[],"uid":"P000074","description":"A space that is {P5} and {P182}.\n\nThe notion was originally defined in section 10 of {{mr:206907}}, which is available at .\n\n----\n#### Meta-properties\n\n- This property is hereditary.","refs":[{"name":"$\\aleph_0$-spaces (E. Michael)","mr":206907},{"name":"$\\mathfrak P$-spaces and related concepts (Gabriyelyan & Kakol)","doi":"10.1016/j.topol.2015.05.085"}]},{"name":"Spectral","aliases":["Coherent"],"uid":"P000075","description":"A {P16} {P73} space for which the collection of compact open subsets is\nclosed under finite intersections and forms a base for the topology.\n\nIn the theory of Stone duality, for every bounded distributive lattice there exists a unique spectral space whose compact open subsets are order-isomorphic to that lattice and whose points are the prime ideals of the lattice. Additionally, spectra of commutative rings endowed with the Zariski topology are spectral spaces and vice versa, although this association is not one-to-one. For a comprehensive overview of this and other information on spectral spaces, see {{doi:10.1017/9781316543870}}.\n\nSee example 21, section 2.6 of {{mr:1077251}} and the section on [Spectral spaces](https://stacks.math.columbia.edu/tag/08YF) from the Stacks project.","refs":[{"name":"Spectral spaces (Dickmann, Schwartz, Tressl)","doi":"10.1017/9781316543870"},{"name":"Stone spaces (Johnstone)","mr":861951},{"name":"Spectral space on Wikipedia","wikipedia":"Spectral_space"}]},{"name":"Proximal","aliases":[],"uid":"P000076","description":"A {P12} space such that there exists a uniformity\ngenerating the topology of the space such that the\nentourage-picker has a winning strategy in Bell's proximal game introduced\nin {{doi:10.1016/j.topol.2014.06.014}}.\n\nEquivalently, the entourage-picker has a winning strategy for\nthis variation defined as Game 1.11 of\n{{doi:10.1016/j.topol.2016.03.022}}: during round $0$,\nthe entourage-picker chooses an entourage $D_0$ of the diagonal from\nthe {P12} space's universal uniformity, and the point-picker\nchooses some point $x_0$. Then during round $n+1$, the entourage-picker chooses an\nentourage $D_{n+1}$, and the point-picker chooses some point $x_{n+1}\\in D_n[x_n]$.\nThe entourage-picker wins provided that either the sequence $x_n$ converges\nor the intersection $\\bigcap_{n<\\omega}D_n[x_n]$ is empty.\n\nNote that while many authors require {P76} spaces to be {P3}\n(equivalently, {P1}),\nwe do not require this separation axiom here.\nHowever, due to the first meta-property below, many results about {P3} {P76} spaces may\nbe generalized by considering the Kolmogorov quotient.\n\n----\n\n#### Meta-properties\n\n- $X$ satisfies this property iff its Kolmogorov quotient $\\text{Kol}(X)$ does. ({{mathse:5044579}})\n- This property is hereditary with respect to closed sets (see Lemma 7 in {{doi:10.1016/j.topol.2014.06.014}}).\n- This property is preserved by $\\Sigma$-products (see Theorem 8 in {{doi:10.1016/j.topol.2014.06.014}}).","refs":[{"name":"An infinite game with topological consequences (Bell)","doi":"10.1016/j.topol.2014.06.014"},{"name":"Tactic-proximal compact spaces are strong Eberlein compact (Clontz)","doi":"10.1016/j.topol.2016.03.022"},{"name":"Is a space proximal whenever its $T_0$ quotient is?","mathse":5044579}]},{"name":"Corson compact","aliases":[],"uid":"P000077","description":"$X$ is homeomorphic to a compact subspace of a $\\Sigma$-product of real lines.\nBy homogeneity of the real line, one can assume the $\\Sigma$-product is of the form\n\n$\\quad Y=\\{x=(x_\\alpha)\\in\\mathbb R^\\Gamma:\\{\\alpha:x_\\alpha\\ne 0\\}\\text{ is countable}\\}$\n\nfor some set $\\Gamma$.\nThe space $Y$ has the topology of pointwise convergence induced from\nthe product topology of $\\mathbb R^\\Gamma$.","refs":[{"name":"On function spaces of compact subspaces of Σ-products of the real line.","mr":584666}]},{"name":"Finite","aliases":["Cardinality $<\\aleph_0$","Cardinality $<\\beth_0$"],"uid":"P000078","description":"The cardinality of the space is finite.\n\nSee sections 1.12-1.15 of {{zb:1052.54001}}.\n\n----\n#### Meta-properties\n\n- This property is hereditary.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"name":"Sequential","aliases":[],"uid":"P000079","description":"A space in which every sequentially closed set is closed.\n\nEquivalently, for every set $A\\subseteq X$ that is *not* closed in $X$, there is some $x\\in\\overline A\\setminus A$ and a sequence in $A$ that converges to $x$.\n\n----\n### Meta-properties\n\n- This property is hereditary with respect to closed sets (see {{mathse:479808}}).\n- This property is hereditary with respect to open sets (see {{mathse:479808}}).\n- This property is preserved by arbitrary disjoint unions.\n- This property is preserved by quotient maps.","refs":[{"name":"Products of convergence properties (Gerlits, Nagy)","zb":"0521.54011"},{"name":"Sequential space on Wikipedia","wikipedia":"Sequential_space"},{"name":"Is a closed subspace (open subspace) of a sequential space sequential?","mathse":479808}]},{"name":"Fréchet Urysohn","aliases":[],"uid":"P000080","description":"A space such that for each $A\\subseteq X$ and each $p\\in \\overline A$ there is a sequence in $A$ converging to $p$.\n\nSee item (5) in the introduction to {{zb:0521.54011}}.\n{{wikipedia: Fréchet–Urysohn_space}} has various equivalent characterizations of the property.\n\nCompare with {P172}.\n\n----\n#### Meta-properties\n\n- This property is hereditary.","refs":[{"name":"Products of convergence properties","zb":"0521.54011"},{"name":"Fréchet Urysohn space on Wikipedia","wikipedia":"Fréchet–Urysohn_space"}]},{"name":"Countably tight","aliases":["Countably generated"],"uid":"P000081","description":"$X$ has *countable tightness*:\nFor each $A\\subseteq X$ and each $p\\in \\overline A$ there is a countable subset $D\\subseteq A$ such that $p\\in \\overline D$.\n\nEquivalently, $X$ is *countably generated*:\nFor each $A\\subseteq X$, $A$ is closed whenever $A\\cap C$ is closed in $C$ for every countable $C\\subseteq X$ (or the equivalent condition for open sets).\n\nFor a proof of the equivalence, see {{mathse:5086334}}.\n\n----\n#### Meta-properties\n\n- $X$ satisfies this property iff its Kolmogorov quotient $\\text{Kol}(X)$ does.\n- This property is hereditary.","refs":[{"name":"Products of convergence properties","mr":687569},{"name":"Countably generated space on Wikipedia","wikipedia":"Countably_generated_space"},{"name":"Countably tight and countably generated topological spaces","mathse":5086334}]},{"name":"Locally metrizable","aliases":[],"uid":"P000082","description":"Each point has a neighborhood which is {P53}.\n\nDefined in exercise 7 of section 34 in {{zb:0951.54001}}.\n\n----\n#### Meta-properties\n\n- This property is hereditary.\n- The Kolmogorov quotient $\\text{Kol}(X)$ is {P82} iff $X$ is {P144}.","refs":[{"name":"Topology (Munkres)","zb":"0951.54001"}]},{"name":"Meta-Lindelöf","aliases":[],"uid":"P000083","description":"Every open cover of $X$ has a point-countable open refinement.\n\n(A collection $\\mathscr V$ of subsets of $X$ is *point-countable* if each point $x\\in X$ belongs to at most countably many elements of $\\mathscr V$.)\n\nDefined on page 200 of {{zb:1059.54001}}.\n\nSee also .\n\n---\n#### Meta-properties\n\n- This property is hereditary with respect to closed sets.","refs":[{"name":"Encyclopedia of general topology","zb":"1059.54001"}]},{"name":"Locally Hausdorff","aliases":["Locally $T_2$"],"uid":"P000084","description":"Each point has a neighborhood that is {P3}.\n\nNote that, by Lemma 4.2 of {{doi:10.1090/S0002-9939-07-09100-9}},\neach point of such a space has a {P3} open neighborhood which is dense in the space.\n\n----\n#### Meta-properties\n\n- This property is hereditary.","refs":[{"name":"A note on the locally Hausdorff property (S. B. Niefield)","mr":702721},{"name":"Locally Hausdorff space on Wikipedia","wikipedia":"Locally_Hausdorff_space"},{"name":"Manifolds: Hausdorffness versus homogeneity (Baillif & Gabard)","doi":"10.1090/S0002-9939-07-09100-9"}]},{"name":"Basically disconnected","aliases":[],"uid":"P000085","description":"The closure of every cozero set in $X$ is open or, equivalently, clopen.\n\nA *cozero set* is a set of the form $\\{x\\in X:f(x)>0\\}$ for some continuous function $f:X\\to\\mathbb R$;\nequivalently, the complement of a zero set.\n\nEquivalently, any two disjoint open sets, at least one of which is a cozero set, have disjoint closures.\n\nDefined in problem 1H of {{zb:1380.46022}}.\n\nNo additional separation axiom is assumed here.\n\n----\n### Meta-properties\n\n- This property is hereditary with respect to clopen sets.","refs":[{"name":"Rings of Continuous Functions (Gillman & Jerison)","zb":"1380.46022"}]},{"name":"Homogeneous","aliases":[],"uid":"P000086","description":"A space $X$ such that for each $a,b\\in X$ there is a homeomorphism $\\phi : X \\to X$ such that $\\phi(a)=b$. In other words: the group of homeomorphisms of the space onto itself acts transitively.\n\n----\n#### Meta-properties\n\n- This property is preserved by arbitrary products.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"name":"Has a group topology","aliases":["Topological group"],"uid":"P000087","description":"$X$ is homeomorphic to a [topological group](https://en.wikipedia.org/wiki/Topological_group).\n\nEquivalently, there exists a continuous group operation $(x,y)\\mapsto x\\cdot y$ on $X$ such that\nthe inverse operation $x\\mapsto x^{-1}$ is also continuous.\n\nContrary to Munkres or Willard, we do not assume any separation axiom like {P3}, {P2} or {P1}.\n\n----\n#### Meta-properties\n\n- This property is preserved by arbitrary products.\n- This property is preserved by $\\Sigma$-products.","refs":[{"name":"Topology (Munkres)","zb":"0951.54001"}]},{"name":"Collectionwise normal","aliases":["CWN"],"uid":"P000088","description":"For every discrete family $(F_i)_{i \\in I}$ of closed subsets of $X$, there exists a pairwise disjoint family of open sets $(U_i)_{i \\in I}$ with $F_i \\subseteq U_i$ for all $i$.\n\nEquivalently, for every discrete family $(F_i)_{i \\in I}$ of closed subsets of $X$, there exists a discrete family of open sets $(U_i)_{i \\in I}$ with $F_i \\subseteq U_i$ for all $i$.\n\nTerminology:\n- A *discrete family* is a family of subsets such that each point of $X$ has a neighborhood meeting at most one of the subsets.\n- In the first definition, the family of open sets $(U_i)_i$ is sometimes called a *disjoint open expansion* of the family $(F_i)_i$.\n- In the second definition, the family of open sets $(U_i)_i$ is sometimes called a *discrete open expansion* of the family $(F_i)_i$. (See for example {{doi:10.1016/S0166-8641(96)00163-0}}.)\n\nThe equivalence between the two definitions is Theorem 5.1.17 of {{zb:0684.54001}}.\n\nDefined on page 168 of {{zb:0386.54001}};\nalso on page 305 of {{zb:0684.54001}} with the additional assumption of {P2}, which we do not assume here.\n\nOccasionally abbreviated as \"CWN\", as in {{zb:0529.54017}}.\n\n----\n#### Meta-properties\n\n- $X$ satisfies this property iff its Kolmogorov quotient $\\text{Kol}(X)$ does.\n- This property is preserved by arbitrary disjoint unions.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Collectionwise Normal Space on Wikipedia","wikipedia":"Collectionwise_normal_space"},{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"},{"name":"Collectionwise Hausdorff versus collectionwise normal with respect to compact sets (Reed)","zb":"0529.54017"},{"name":"Baireness of $C_k(X)$ for locally compact $X$ (Gruenhage & Ma)","doi":"10.1016/S0166-8641(96)00163-0"}]},{"name":"Fixed point property","aliases":[],"uid":"P000089","description":"Every continuous map from the space to itself has at least one fixed point.\n\n----\n#### Meta-properties\n\n- This property is hereditary with respect to retracts.\nA set $A\\subseteq X$ is called a [*retract*](https://en.wikipedia.org/wiki/Retraction_(topology)) if there exists a continuous $f:X\\to A$ fixing each point of $A$.","refs":[{"name":"Nonexpansive nonlinear operators in a Banach space.","mr":187120},{"name":"Fixed-Point Property on Wikipedia","wikipedia":"Fixed-point_property"}]},{"name":"Alexandrov","aliases":["Finitely generated","Alexandroff-discrete"],"uid":"P000090","description":"An arbitrary intersection of open sets is open.\n\nThis is equivalent to each of the following:\n\n* An arbitrary union of closed sets is closed.\n* Each $x \\in X$ has a smallest neighborhood.\n* For each $x \\in X$, the *neighborhood kernel of $x$*, i. e. $\\bigcap\\big\\{U\\text{ open in }X\\mid x\\in U\\big\\}$, is open.\n* $X$ is *finitely generated*:\nFor each $A \\subseteq X$, $A$ is open iff $A \\cap Y$ is open in the subspace $Y$ for each finite $Y \\subseteq X$.\n* There is a preorder (i.e., a reflexive and transitive relation) $\\preceq$ on $X$ such that the open sets are precisely the upward closed sets (i.e., sets $A\\subseteq X$ such that $x\\in A$ and $x\\preceq y$ imply $y\\in A$).\n\nSee {{wikipedia:Alexandrov_topology}} for a more extensive list.\nSee also {{zb:0944.54018}}.\n\n----\n#### Meta-properties\n\n- $X$ satisfies this property iff its Kolmogorov quotient $\\text{Kol}(X)$ does.\n- This property is hereditary.\n- This property is preserved by arbitrary disjoint unions.\n- This property is preserved by finite products.\n- This property is preserved by box products.","refs":[{"name":"Alexandroff spaces (F. G. Arenas)","zb":"0944.54018"},{"name":"Alexandrov Topology on Wikipedia","wikipedia":"Alexandrov_topology"}]},{"name":"Eberlein compact","aliases":[],"uid":"P000091","description":"Any compact subspace of a $\\Sigma^\\ast$-product of real lines.","refs":[{"name":"A note on Eberlein compacts","doi":"10.2140/pjm.1977.72.487"}]},{"name":"$k_{\\omega,3}$-space","aliases":["$k_\\omega$-space"],"uid":"P000092","description":"A space which is the union of an increasing sequence of {P16} {P3} subsets\n$K_1\\subseteq K_2 \\subseteq K_3 \\subseteq \\cdots$ such that a set is closed (resp. open) whenever\nits intersection with each $K_n$ is closed (resp. open) in $K_n$.\n\n(N.B. The requirement that this sequence be\nincreasing cannot be eliminated. For example,\n{S65} satisfies the requirement without the increasing condition,\nbut is not {P92}.)\n\nDefined as \"$k_\\omega$-space\" in {{zb:0416.54027}}\n(available [here](https://topology.nipissingu.ca/tp/reprints/v02/tp02105.pdf))\nand on page 209 of {{doi:10.1016/0016-660X(75)90021-5}}.\nThe corresponding notion without requiring that each $K_n$ be {P3} is\n{P98}. See also {P140}, {P141},\nand {P142}.","refs":[{"name":"A survey of $k_\\omega$-spaces (Franklin, Smith Thomas)","zb":"0416.54027"},{"name":"Free $k$-groups and free topological groups (Ordman)","doi":"10.1016/0016-660X(75)90021-5"}]},{"name":"Locally countable","aliases":[],"uid":"P000093","description":"The space has a base of open sets, each with countable cardinality.\n\n----\n#### Meta-properties\n\n- This property is hereditary.","refs":[{"name":"Countably compact, locally countable T2-spaces.","mr":574525}]},{"name":"Locally finite","aliases":[],"uid":"P000094","description":"Every point has a finite neighborhood. Equivalently, the space has a basis of finite sets.\n\n----\n#### Meta-properties\n\n- This property is hereditary.","refs":[{"name":"Some applications of minimal open sets (Nakaoka, Oda)","doi":"10.1155/S0161171201006482"},{"name":"Locally finite space on Wikipedia","wikipedia":"Locally_finite_space"}]},{"name":"Arc connected","aliases":[],"uid":"P000095","description":"Given any two distinct points $x,y\\in X$, there is an *arc* in $X$ from $x$ to $y$;\nthat is, a homeomorphic embedding $f:[0,1]\\to X$ with $f(0)=x$ and $f(1)=y$.\n\nSee Definition 27.1 in {{zb:1052.54001}}, which uses this usual definition of \"arc\" from the literature.\n\nCompare with {P37} and {P38}.\n\nNote: Page 29 of {{zb:0386.54001}} \nuses the terms \"arc\" for an injective path\nand \"arc connected\" for the weaker property {P38}.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"},{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Locally arc connected","aliases":[],"uid":"P000096","description":"The topology of $X$ has a base consisting of {P95} open sets.\n\nEquivalently, every point has a local base of {P95} open neighborhoods.\n\nCompare with {P42} and {P43}.\n\nNote: Page 30 of {{zb:0386.54001}} \nuses the term \"locally arc connected\" for the weaker property {P43}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Embeddable in $\\mathbb R$","aliases":[],"uid":"P000097","description":"Homeomorphic to a subspace of $\\mathbb R$ ({S25}).\n\n----\n#### Meta-properties\n\n- This property is hereditary.","refs":[{"name":"Real line on Wikipedia","wikipedia":"Real_line"}]},{"name":"$k_{\\omega,1}$-space","aliases":["Generated by countably-many compacts","$k_\\omega$-space"],"uid":"P000098","description":"A space which is the union of an increasing sequence of {P16} subsets\n$K_1\\subseteq K_2 \\subseteq K_3 \\subseteq \\cdots$ such that a set is closed (resp. open) whenever\nits intersection with each $K_n$ is closed (resp. open) in $K_n$.\n\nEquivalently, a space which is the union of countably-many\n{P16} subsets $K_n$ (without requiring\n$K_n\\subseteq K_{n+1}$)\nsuch that a set is closed (resp. open) whenever\nits intersection with each $K_n$ is closed (resp. open) in $K_n$.\n\nDefined as \"$k_\\omega$-space\" on page 13 of {{zb:0288.22006}}.\nThe corresponding notion requiring that each $K_n$ be {P3} is {P92}.\nSee also {P140}, {P141}, and {P142}.","refs":[{"name":"Quotients of k-semigroups. (Lawson, J.; Madison, B.)","zb":"0288.22006"}]},{"name":"US","aliases":["Sequentially Hausdorff","Unique sequential limits","Semi-Hausdorff"],"uid":"P000099","description":"Every convergent sequence in $X$ has exactly one limit.\n\nEquivalently, every {P20} set in $X$\nis sequentially closed (see {{mathse:5011974}}).\n\n----\n#### Meta-properties\n\n- This property is hereditary.","refs":[{"name":"Between T1 and T2 (Wilanski)","doi":"10.2307/2316017"},{"name":"Minimal properties between T1 and T2 (Alas & Wilson)","mr":2219327},{"name":"Semi-Hausdorff Spaces (Murdeshwar & Naimpally)","doi":"10.4153/CMB-1966-045-1"},{"name":"Answer to \"Can the US separation axiom be characterized by compact sets?\"","mathse":5011974}]},{"name":"KC","aliases":["Compacts are closed"],"uid":"P000100","description":"Every compact subset is closed.\n\n----\n#### Meta-properties\n\n- This property is hereditary.","refs":[{"name":"Between T1 and T2","doi":"10.2307/2316017"}]},{"name":"Has closed retracts","aliases":["Retracts are closed","RC"],"uid":"P000101","description":"Every retract subspace ($A\\subseteq X$ with a continuous $f:X\\to A$ fixing each point of $A$) is closed.","refs":[{"name":"\"All retracts are closed\" as separation axiom","mo":191016},{"name":"\"All retracts are closed\" and \"all compacts are closed\"","mo":434451}]},{"name":"Semimetrizable","aliases":[],"uid":"P000102","description":"For a set $X$, a function $d:X^2\\to[0,\\infty)$ is called a *symmetric*\nif it satisfies reflexivity ($d(x,y)=0\\Leftrightarrow x=y$) and symmetry\n($d(x,y)=d(y,x)$). A space $X$ is called *semimetrizable* if\nthere is a symmetric $d:X^2\\to[0,\\infty)$ such that the balls\n$B_d(x,\\varepsilon)=\\{y\\in X:d(x,y)<\\varepsilon\\}$ for $\\varepsilon>0$\nform a neighborhood basis at each point $x\\in X$. Such a symmetric is called a *semimetric*.\n(Note that {{zb:0116.14303}} demonstrates that\nthese neighborhood balls need not be open sets.)\n\nEquivalently:\n- There exists a symmetric $d$ such that $p$ lies in the closure of some $Y\\subseteq X$ iff $d(p,Y)=\\inf\\{d(p,y):y\\in Y\\}=0$.\nSuch a symmetric is automatically a semimetric.\n- (i) $X$ is {P2}; (ii) for each $x \\in X$, there is a non-increasing sequence of open sets $\\{G_n(x):n<\\omega\\}$ forming a local neighborhood basis for $x$; (iii) if for some sequence $x_n$ in $X$, $y\\in X$ belongs to $G_n(x_n)$ for all $n$, then $x_n\\to y$. This equivalence is shown in Theorem 3.2 of {{zb:0113.16501}}.\n\nDefined in e.g. {{zb:0113.16501}} and {{doi:10.2991/978-94-6239-216-8}}. Note that some sources, e.g., {{doi:10.2307/2370790}}, use the term semimetric to refer to a symmetric and the term semimetrizable to refer to {P104}.\n\n----\n#### Meta-properties\n\n- This property is hereditary. Compare with {P104}.","refs":[{"name":"Arc-wise connectedness in semi-metric spaces (Heath)","zb":"0113.16501"},{"name":"A regular semi-metric space for which there is no semimetric under which all spheres are open (Heath)","zb":"0116.14303"},{"name":"Generalized Metric Spaces and Mappings (Lin , Yun)","doi":"10.2991/978-94-6239-216-8"},{"name":"On semi-metric spaces (Wilson)","doi":"10.2307/2370790"}]},{"name":"Strongly KC","aliases":[],"uid":"P000103","description":"Every {P19} subset of $X$ is closed.\n\nIntroduced in {{zb:1224.54011}}.\n\n----\n#### Meta-properties\n\n- This property is hereditary.","refs":[{"name":"On minimal strongly KC-spaces","zb":"1224.54011"}]},{"name":"Symmetrizable","aliases":["Semimetrizable"],"uid":"P000104","description":"For a set $X$, a function $d:X^2\\to[0,\\infty)$ is called a *symmetric*\nif it satisfies reflexivity ($d(x,y)=0\\Leftrightarrow x=y$) and symmetry\n($d(x,y)=d(y,x)$). A space $X$ is called *symmetrizable* if\nthere is a symmetric such that a subset $U \\subseteq X$ is open iff for each $x \\in U$,\nthere is some $\\varepsilon > 0$ with $B_d(x,\\varepsilon)=\\{y\\in X:d(x,y)<\\varepsilon\\}$\nbeing contained in $U$.\n\nEquivalently, there exists a symmetric $d$ such that $Y \\subseteq X$ is closed iff $d(p,Y)=\\inf\\{d(p,y):y\\in Y\\} > 0$ for all $p \\in X \\setminus Y$.\n\nDefined in {{doi:10.2991/978-94-6239-216-8}}. Note that some sources, e.g., {{doi:10.2307/2370790}}, use the term semimetric\n(see {P102}) to refer to a symmetric and the term semimetrizable to refer to {P104}.\n\n----\n#### Meta-properties\n\n- This property is not hereditary. (Example: {S156|P104} and {S23|P104}.) Compare with {P102}.\n- This property is hereditary with respect to open sets.\n- This property is hereditary with respect to closed sets.","refs":[{"name":"Generalized Metric Spaces and Mappings (Lin , Yun)","doi":"10.2991/978-94-6239-216-8"},{"name":"On semi-metric spaces (Wilson)","doi":"10.2307/2370790"}]},{"name":"Para-Lindelöf","aliases":[],"uid":"P000105","description":"Every open cover of the space has a locally countable open refinement.\n\nDefined on page 200 of {{zb:1059.54001}}.\n\n----\n#### Meta-properties\n\n- This property is preserved by arbitrary disjoint unions.","refs":[{"name":"Encyclopedia of general topology (Hart et al)","zb":"1059.54001"}]},{"name":"Has a $G_\\delta$-diagonal","aliases":[],"uid":"P000106","description":"A space $X$ such that the diagonal $\\Delta=\\{(x,x):x\\in X\\}$ is a $G_\\delta$ set in $X\\times X$.\n\nEquivalently, $X$ has a *$G_\\delta$-diagonal sequence*, that is, a sequence of open covers\n$\\mathscr U_1,\\mathscr U_2, \\dots$ such that\n$\\{x\\}=\\bigcap_n\\operatorname{st}(x,\\mathscr U_n)$ for all $x\\in X$.\nHere, $\\operatorname{st}(x,\\mathscr U_n)=\\bigcup\\{U\\in\\mathscr U_n:x\\in U\\}$.\n\nFor the equivalence, see Theorem 2.2 in {{zb:0555.54015}} or Lemma 1 in\n.\n\nSee Definition 2.1 in {{zb:0555.54015}}.\n\n----\n#### Meta-properties\n\n- This property is hereditary.","refs":[{"name":"Generalized metric spaces (G. Gruenhage), Ch. 10 of Handbook of set-theoretic topology","zb":"0555.54015"}]},{"name":"Has a closed point","aliases":[],"uid":"P000107","description":"Some singleton $\\{x\\}\\subseteq X$ is a closed set.\n\nEquivalently, there exists a non-empty closed discrete subset\n(compare with {P21} and {P198}).\n\nCompare with {P139}.","refs":[]},{"name":"Hereditarily collectionwise normal","aliases":[],"uid":"P000108","description":"Every subspace of $X$ is {P88}.\n\nEquivalently, every open subspace of $X$ is {P88}.\n\nEquivalently, for every *separated family* $(F_i)_{i \\in I}$ of subsets of $X$,\nthere exists a pairwise disjoint family of open sets $(U_i)_{i \\in I}$ with $F_i \\subseteq U_i$ for all $i$.\n\n(The family $(F_i)_i$ is called a *separated family* if for each $i$, \n$F_i \\cap \\operatorname{cl}_X(\\bigcup_{j \\ne i}F_j) = \\emptyset$,\nthat is, if the family of the $F_i$ is discrete in its union.)\n\nFor the equivalence of the conditions above, see {{mathse:4737011}}.\n\n----\n#### Meta-properties\n\n- This property is hereditary.\n- This property is preserved by arbitrary disjoint unions.","refs":[{"name":"Collectionwise normal space on Wikipedia","wikipedia":"Collectionwise_normal_space"},{"name":"Definition of hereditarily collectionwise normal spaces using separated families","mathse":4737011}]},{"name":"Monotonically normal","aliases":[],"uid":"P000109","description":"There are several equivalent definitions.\nSee {{wikipedia:Monotonically_normal_space}} for a few more.\n\n**Definition (1)**: (see Definition 2.1 in {{zb:0269.54009}})\n\nThe space is {P2} and there is a function $G$ that assigns to each\nordered pair $(A,B)$ of disjoint closed sets in $X$ an open set $G(A,B)$ such that:\n- (i) $A\\subseteq G(A,B)\\subseteq \\overline{G(A,B)}\\subseteq X\\setminus B$;\n- (ii) $G(A,B)\\subseteq G(A',B')$ whenever $A\\subseteq A'$ and $B'\\subseteq B$.\n\nThe operator $G$ is called a *monotone normality operator*.\n\nOne can always choose $G$ to satisfy the property\n\n$\\quad G(A,B)\\cap G(B,A)=\\emptyset,$\n\nby replacing each $G(A,B)$ by $G(A,B)\\setminus\\overline{G(B,A)}$.\n\n----\n**Definition (2)**: (see Theorem 2.2(e) in {{zb:1327.54003}} and references there)\n\nThe space is {P2} and there is a function $\\mu$ that assigns to each pair $(x,U)$ with $U$ open in $X$ and $x\\in U$ an open set $\\mu(x,U)$ such that:\n- (i) $x\\in\\mu(x,U)$;\n- (ii) if $\\mu(x,U)\\cap\\mu(y,V)\\ne\\emptyset$, then $x\\in V$ or $y\\in U$.\n\nSuch a function $\\mu$ automatically satisfies\n\n$\\quad x\\in\\mu(x,U)\\subseteq\\overline{\\mu(x,U)}\\subseteq U$.\n\n*Note 1*: It is sufficient to assume the function $\\mu$ is defined for $U$ belonging to a base for the topology.\n\n*Note 2*: It is not necessarily the case that $\\mu(x,U)$ satisfies monotonicity in $U$,\nthat is, $\\mu(x,U)\\subseteq\\mu(x,U')$ if $x\\in U\\subseteq U'$.\nBut one can construct a modified function\n\n$\\quad \\nu(x,U)=\\bigcup\\{\\mu(x,W):W\\text{ open, }x\\in W\\subseteq U\\}$,\n\nwhich does satisfy (i), (ii), and monotonicity in $U$.\n\n----\n**Equivalence between the two definitions**: (see {{zb:0257.54018}})\n\nGiven $G$ from Definition (1) with the condition $G(A,B)\\cap G(B,A)=\\emptyset$,\nthe function $\\mu(x,U)=G(\\{x\\},X\\setminus U)$ satisfies Definition (2).\n\nGiven $\\mu$ from Definition (2) and the associated function $\\nu$,\nthe function $G(A,B)=\\bigcup\\{\\nu(x,X\\setminus B):x\\in A\\}$ satisfies Definition (1).\n\n----\n#### Meta-properties\n\n- This property is hereditary (easy consequence of Definition (2)).","refs":[{"name":"Monotonically normal spaces (Heath, Lutzer, Zenor)","zb":"0269.54009"},{"name":"Monotonically normal space on Wikipedia","wikipedia":"Monotonically_normal_space"},{"name":"Mary Ellen Rudin and monotone normality (Bennett & Lutzer)","zb":"1327.54003"},{"name":"A study of monotonically normal spaces (Borges)","zb":"0257.54018"}]},{"name":"Developable","aliases":[],"uid":"P000110","description":"The space has a development.\n\nA *development* for a space $X$ is a sequence $\\mathscr U_1,\\mathscr U_2,\\dots$\nof open covers of $X$ such that for each $x\\in X$\nthe collection $\\{\\operatorname{st}(x,\\mathscr U_n):n=1,2,\\dots\\}$ is a local base at $x$.\nHere, $\\operatorname{st}(x,\\mathscr U_n)=\\bigcup\\{U\\in\\mathscr U_n:x\\in U\\}$\nis the star of $x$ with respect to $\\mathscr U_n$.\n\n*Note*: If $X$ has a development, one can always choose a development such that\neach $\\mathscr U_{n+1}$ is a refinement of $\\mathscr U_n$.\n\nSee Definition 23.6 in {{zb:1052.54001}} and page 329 of {{zb:0684.54001}}.\n\n----\n#### Meta-properties\n\n- This property is hereditary.\n- $X$ satisfies this property iff its Kolmogorov quotient $\\text{Kol}(X)$ does.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"},{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"name":"Hemicompact","aliases":[],"uid":"P000111","description":"Defined in Section 8 of {{mr:0017525}}, see also {{zb:1052.54001}}. A space which is the union of countably-many\ncompact subspaces $K_n$, such that any compact subset is contained within some $K_n$.\n\n----\n#### Meta-properties\n\n- $X$ satisfies this property iff its Kolmogorov quotient $\\text{Kol}(X)$ does.","refs":[{"name":"A topology for spaces of transformations","mr":17525},{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"name":"Submetrizable","aliases":["Has a metrizable compression","Submetrisable"],"uid":"P000112","description":"The topology contains a coarser topology which is {P53}.\n\nThis is equivalent to each of the following:\n\n- There is a continuous injective map from $X$ into a metrizable space.\n- There is a continuous metric on $X$.\n\n----\n#### Meta-properties\n\n- This property is hereditary.\n- This property is preserved by arbitrary disjoint unions.","refs":[{"name":"Submetrizable spaces and almost $\\sigma$-compact function spaces (R. McCoy)","doi":"10.1090/S0002-9939-1978-0482639-9"}]},{"name":"Moore space","aliases":[],"uid":"P000113","description":"A space that is {P5} and {P110}.\n\nSee Definition 23.6 in {{zb:1052.54001}}.\n\n----\n#### Meta-properties\n\n- This property is hereditary.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"name":"Cardinality $=\\aleph_1$","aliases":["First-uncountable"],"uid":"P000114","description":"The cardinality of the space is equal to $\\aleph_1$, the cardinality of the first uncountable ordinal $\\omega_1$.\n\nSee section 1.19 of {{zb:1052.54001}}.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"name":"Subparacompact","aliases":["$\\sigma$-paracompact"],"uid":"P000115","description":"Every open cover of $X$ has a $\\sigma$-discrete closed refinement.\n\nThis is equivalent to each of the following:\n\n- Every open cover of $X$ has a $\\sigma$-locally finite closed refinement.\n\n- Every open cover of $X$ has a $\\sigma$-closure preserving closed refinement.\n\n- For every open cover $\\mathscr U$ of $X$, there is a sequence $\\left<\\mathscr V_n : n \\in \\omega\\right>$ of open covers where each $\\mathscr V_n$ is a refinement of $\\mathscr U$ and, for each $x \\in X$, there is some $n$ with $\\operatorname{st}(x,\\mathscr V_n)\\subseteq U$ for some $U\\in\\mathscr U$.\n\n- For every open cover $\\mathscr U$ of $X$, there is a sequence $\\left<\\mathscr V_n : n \\in \\omega\\right>$ of open covers where each $\\mathscr V_n$ is a refinement of $\\mathscr U$ and, for each $x \\in X$, there is some $n$ with $\\operatorname{ord}(x,\\mathscr V_n)=1$.\n\nSome of the terms used above: \nFor some topological property $\\mathcal P$, a collection of sets is called $\\sigma$-$\\mathcal P$ if it is a countable union of collections, each satisfying $\\mathcal P$.\nFor $\\mathscr W$ a collection of subsets of $X$, $\\mathscr W$ is *closure preserving*\nif for each $\\mathscr W'\\subseteq\\mathscr W$,\n$\\overline{\\bigcup\\mathscr W'}=\\bigcup\\{\\overline U:U\\in\\mathscr W'\\}$.\nAlso, $\\operatorname{st}(x,\\mathscr W)=\\bigcup\\{W\\in\\mathscr W:x\\in W\\}$\nand $\\operatorname{ord}(x,\\mathscr W)=|\\{W\\in\\mathscr W:x\\in W\\}|$.\n\nSee Theorem 3.1 in {{zb:0569.54022}}\n(the whole chapter assumes {P3}, but that property is not used to show the equivalences)\nand Theorem V.7 on p. 217 in {{zb:0598.54001}}.\nSee also {{zb:0569.54022}} for references and original names for the properties when they were first introduced.\n\n----\n#### Meta-properties\n\n- $X$ satisfies this property iff its Kolmogorov quotient $\\text{Kol}(X)$ does.\n- This property is hereditary with respect to closed sets.","refs":[{"name":"Covering properties (D. Burke), Ch. 9 of Handbook of set-theoretic topology","zb":"0569.54022"},{"name":"Modern General Topology (Nagata, 1985)","zb":"0598.54001"}]},{"name":"Polish","aliases":[],"uid":"P000116","description":"A space that is {P26} and {P55}.\n\nSee Definition 3.1 in {{doi:10.1007/978-1-4612-4190-4}}.","refs":[{"name":"Classical Descriptive Set Theory (Kechris)","doi":"10.1007/978-1-4612-4190-4"},{"name":"Polish space on Wikipedia","wikipedia":"Polish_space"}]},{"name":"Has a $\\sigma$-locally finite network","aliases":[],"uid":"P000117","description":"A space with a $\\sigma$-locally finite network.\n\nA family $\\mathcal N$ of subsets of $X$ is called a *network* if every open set is the union of a subfamily of $\\mathcal N$; that is, for every open set $U$ and point $x\\in U$ there is some $A\\in\\mathcal N$ with $x\\in A\\subseteq U$. The family $\\mathcal N$ is *$\\sigma$-locally finite* if it is a countable union of locally finite families.\n\nSee for example page 127 of {{zb:0684.54001}}.\n\n----\n#### Meta-properties\n\n- This property is hereditary.","refs":[{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"name":"Has a $\\sigma$-locally finite $k$-network","aliases":[],"uid":"P000118","description":"A space with a $\\sigma$-locally finite $k$-network.\n\nA family $\\mathcal N$ of subsets of $X$ is called a *$k$-network* if for every compact set $K$ and open set $U$ in $X$ with $K\\subseteq U$, there exists a finite $\\mathcal{N}^* \\subseteq \\mathcal{N}$ with $K \\subseteq \\bigcup\\mathcal{N}^* \\subseteq U$. The family $\\mathcal N$ is *$\\sigma$-locally finite* if it is a countable union of locally finite families.\n\nSee for example Definition 2.1 in {{doi:10.1016/j.topol.2015.05.085}}.\n\n----\n#### Meta-properties\n\n- This property is hereditary.\n- $X$ satisfies this property iff its Kolmogorov quotient $\\text{Kol}(X)$ does.","refs":[{"name":"$\\mathfrak P$-spaces and related concepts (Gabriyelyan & Kakol)","doi":"10.1016/j.topol.2015.05.085"}]},{"name":"Stonean","aliases":[],"uid":"P000119","description":"The space is {P16}, {P3}, and {P49}.\n\nSee and definition in {{wikipedia:Extremally_disconnected_space}}.\n\nNot to be confused with {P195}, which is a weaker property.","refs":[{"name":"Extremally disconnected space on Wikipedia","wikipedia":"Extremally_disconnected_space"}]},{"name":"Locally orderable","aliases":[],"uid":"P000120","description":"Each point has a neighbourhood that is orderable ({P133}).\n\nThis is equivalent to each of the following:\n- Each point has an open neighbourhood that is orderable.\n- Each point has a local base of orderable neighbourhoods.\n- The topology of $X$ has a base consisting of orderable open sets.\n\nThe equivalences can be shown similarly to the proof of Proposition 2.4 in {{zb:1458.54023}}.\n\nFirst introduced in the PhD dissertation of H. Herrlich {{mr:154246}}. \n\n----\n#### Meta-properties\n\n- This property is hereditary with respect to open sets.\n- This property is preserved by arbitrary disjoint unions.","refs":[{"name":"Locally ordered topological spaces (P. Pikul)","zb":"1458.54023"},{"name":"Ordnungsfähigkeit topologischer Räume (Herrlich)","mr":154246}]},{"name":"Pseudometrizable","aliases":[],"uid":"P000121","description":"A space $X$ whose topology is induced by some pseudometric.\n\nA *pseudometric* on $X$ is a map $d:X\\times X\\to\\Bbb{R}$ such that for all $x,y,z \\in X$\n(i) $d(x,y) \\geq 0$ and $d(x,x)=0$,\n(ii) $d(x,y) = d(y,x)$,\n(iii) $d(x,z) \\leq d(x,y) + d(y,z)$.\nThat is, a pseudometric satisfies the same properties as a metric, except that distinct points can have $d(x,y)=0$.\nThe topology induced by $d$ is the topology generated by all the open balls $B(x,\\epsilon) = \\{y \\in X:d(x,y)<\\epsilon\\}$ with $x\\in X$, $\\epsilon>0$.\n\nDefined in Example 3.2(a) of {{zb:1052.54001}}.\n\n----\n#### Meta-properties\n\n- $X$ is {P121} iff its Kolmogorov quotient $\\text{Kol}(X)$ is {P53}.\n- This property is hereditary.\n- This property is preserved by arbitrary disjoint unions.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"name":"Locally Euclidean","aliases":[],"uid":"P000122","description":"Each point has a neighborhood homeomorphic to an open subset of $\\mathbb R^n$\nwith its Euclidean topology for some integer $n\\ge 0$. Equivalently, each point has a neighborhood\nhomeomorphic to $\\mathbb R^n$ for some $n$. Note that the value of $n$ is allowed\nto differ between points; if it does not vary, then the space has the stronger\nproperty {P123}.\n\nIn the case $n=0$, a point with a neighborhood homeomorphic to\n$\\mathbb R^0=\\{0\\}$ is an isolated point.\n\n*Note*: In all the above, one can equivalently require the neighborhoods to be open.\n\nThe closely related {P123} property is\ndefined on page 316 of {{zb:0951.54001}} and on page 38 of {{mr:2766102}}.","refs":[{"name":"Topology (Munkres)","zb":"0951.54001"},{"name":"Introduction to topological manifolds (Lee)","mr":2766102}]},{"name":"Locally $n$-Euclidean","aliases":["Locally Euclidean of dimension $n$"],"uid":"P000123","description":"There exists some integer $n\\ge 0$ such that each point of the\nspace has a neighborhood homeomorphic to an open subset of $\\mathbb R^n$\nwith its Euclidean topology. Equivalently, each point has a neighborhood\nhomeomorphic to $\\mathbb R^n$. Note that the value of $n$ is not allowed\nto differ between points; if it can vary, then the space has the weaker\nproperty {P122}.\n\nIn the case that $n=0$, the point having a neighborhood homeomorphic to\n$\\mathbb R^0=\\{0\\}$ means it is an isolated point.\n\n*Note*: In all the above, one can equivalently require the neighborhoods to be open.\n\nDefined on page 316 of {{zb:0951.54001}} and on page 38 of {{mr:2766102}}.","refs":[{"name":"Topology (Munkres)","zb":"0951.54001"},{"name":"Introduction to topological manifolds (Lee)","mr":2766102}]},{"name":"Topological $n$-manifold","aliases":["$n$-dimensional topological manifold"],"uid":"P000124","description":"A {P123} space that is {P3} and {P27}.\n\nDefined on page 225 of {{zb:0951.54001}} and on page 39 of {{mr:2766102}}.","refs":[{"name":"Topology (Munkres)","zb":"0951.54001"},{"name":"Manifold on Wikipedia","wikipedia":"Manifold"},{"name":"Introduction to topological manifolds (Lee)","mr":2766102}]},{"name":"Has multiple points","aliases":["Has distinct points","Cardinality $\\geq 2$"],"uid":"P000125","description":"The cardinality of the space is at least $2$, that is, the space is not {S162} and not {S163}.","refs":[]},{"name":"Door","aliases":["Door space"],"uid":"P000126","description":"Every subset of the space is open or closed (or both).\n\nSee {{mr:0923909}}, available at .\n\n#### Meta-properties\n\n- This property is hereditary.","refs":[{"name":"Door spaces are identifiable (S. D. McCartan)","mr":923909},{"name":"Door space on Wikipedia","wikipedia":"Door_space"}]},{"name":"Dowker","aliases":[],"uid":"P000127","description":"A normal Hausdorff space $X$ such that the product $X\\times[0,1]$ is not normal.\n\nEquivalently, by Dowker's theorem, a {P7} space that is not {P32}.\n\nThe equivalence follows for example from Theorem 21.4 in {{zb:1052.54001}}.\nSee also {{mr:0776636}}.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"},{"name":"Dowker spaces (M.E. Rudin)","mr":776636}]},{"name":"$k$-Lindelöf","aliases":[],"uid":"P000128","description":"Every open $k$-cover of $X$ has a countable $k$-subcover. (A family $\\mathcal U$ of open subsets of $X$ is called a *$k$-cover* if each compact subset of $X$ is contained in an element of $\\mathcal U$ and $X \\notin \\mathcal U$.)\n\nDefined on page 3279 of {{doi:10.1016/j.topol.2005.07.015}}.","refs":[{"name":"Applications of k-covers II","doi":"10.1016/j.topol.2005.07.015"}]},{"name":"Indiscrete","aliases":["Trivial topology"],"uid":"P000129","description":"The only open sets are $\\emptyset$ and $X$.\n\nSee 3.2d of {{zb:1052.54001}}.\n\n----\n#### Meta-properties\n\n- This property is hereditary.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"name":"Locally compact","aliases":[],"uid":"P000130","description":"Each point has a local basis of compact neighborhoods.\n\nGiven as condition (3) in {{wikipedia:Locally_compact_space}}. See also the article \"What is locally compact?\" on p. 390 [here](http://www.pme-math.org/journal/issues/PMEJ.Vol.9.No.6.pdf), which discusses various kinds of local compactness.\n\n----\n#### Meta-properties\n\n- This property is hereditary with respect to open sets.\n- This property is hereditary with respect to closed sets.\n- This property is hereditary with respect to locally closed sets (equivalent to previous two meta-properties).\nA set $A \\subseteq X$ is called [*locally closed*](https://en.wikipedia.org/wiki/Locally_closed_subset) if every $x \\in A$ has neighbourhood $U$ with $U \\cap A$ closed in $U$ (equivalently, $A$ is the intersection of an open set and a closed set).\n- This property is preserved by arbitrary disjoint unions.","refs":[{"name":"Locally compact space on Wikipedia","wikipedia":"Locally_compact_space"}]},{"name":"Hereditarily Lindelöf","aliases":[],"uid":"P000131","description":"A space for which every subspace is {P18}. Equivalently, a space in which every open set is {P18}.\n\nThe equivalence between the two is shown for example in {{mathse:494918}}.\n\nDefined in 16E of {{zb:1052.54001}}.\n\n----\n#### Meta-properties\n\n- This property is hereditary.\n- $X$ satisfies this property iff its Kolmogorov quotient $\\text{Kol}(X)$ does.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"},{"name":"Lindelöf space","wikipedia":"Lindelöf_space"},{"name":"Showing that if every open subspace is Lindelöf, then the space is hereditarily Lindelöf.","mathse":494918}]},{"name":"$G_\\delta$ space","aliases":["Perfect"],"uid":"P000132","description":"A space in which every closed set is a $G_\\delta$ set (a countable intersection of open sets).\nEquivalently, a space in which every open set is an $F_\\sigma$ set (a countable union of closed sets).\n\nDefined on page 162 of {{zb:0386.54001}}.\n\nNote: A $G_\\delta$ space is sometimes called a \"perfect space\" (Exercise 1.5.H(a) in {{zb:0684.54001}}). Not to be confused with a space that is \"perfect\" in the sense of \"perfect set\" (= a set equal to its derived set = a closed set that is dense-in-itself), that is, a space without isolated point. See the discussion in .\n\n----\n#### Meta-properties\n\n- This property is hereditary.\n- $X$ satisfies this property iff its Kolmogorov quotient $\\text{Kol}(X)$ does.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"G-delta space","wikipedia":"Gδ_space"},{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"name":"LOTS","aliases":["Linearly ordered topological space","Orderable"],"uid":"P000133","description":"A space whose topology is the order topology for some total order on the underlying set.\n\nIf $X$ is a set and $<$ is a total order on $X$, the *order topology* on $(X,<)$ is generated by the subbase consisting of the open rays $(a,\\to) = \\{x\\in X: a < x\\}$ and $(\\leftarrow, a) = \\{x\\in X: x < a\\}$.\n\nDefined on page 84 of {{zb:0951.54001}}, in 6D of {{zb:1052.54001}}, 1.7.4 of {{zb:0684.54001}}, and example 39 of {{zb:0386.54001}}.\n\n----\n#### Meta-properties\n\n- This property is not hereditary with respect to open sets.\n(Example: $(0,1)\\cup\\{2\\}$ as a subspace of $[0,1]\\cup\\{2\\}$)\n- This property is not hereditary with respect to closed sets.\n(Example: $(0,1)\\cup\\{3\\}$ as a subspace of $(0,1)\\cup(2,4)$)","refs":[{"name":"Topology (Munkres)","zb":"0951.54001"},{"name":"General Topology (Willard)","zb":"1052.54001"},{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"},{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Order topology on Wikipedia","wikipedia":"Order_topology"}]},{"name":"$R_1$","aliases":["Preregular","R1"],"uid":"P000134","description":"A space in which any two topologically distinguishable points can be separated by disjoint neighbourhoods. (Two points are *topologically distinguishable* if there is an open set containing one of the points and not the other.)\n\nEquivalently, a space whose Kolmogorov quotient is Hausdorff.\n\nDefined in {{wikipedia:Hausdorff_space}}, and in definition 16.10, p. 439 of {{mr:1417259}}.\n\n----\n#### Meta-properties\n\n- $X$ is {P134} iff its Kolmogorov quotient $\\text{Kol}(X)$ is {P3}.\n- This property is hereditary.","refs":[{"name":"Handbook of Analysis and its Foundations (Schechter)","mr":1417259},{"name":"Hausdorff space on Wikipedia","wikipedia":"Hausdorff_space"}]},{"name":"$R_0$","aliases":["Symmetric","R0"],"uid":"P000135","description":"Given any two topologically distinguishable points, each has a neighborhood not including the other point. (Two points are *topologically distinguishable* if there is an open set containing one of the points and not the other.)\n\nEquivalently, any of the following equivalent properties holds:\n\n- The Kolmogorov quotient of the space is {P2}.\n- Every neighborhood of a point $x\\in X$ contains $\\operatorname{cl}\\{x\\}$.\n- Every open set is a union of closed sets.\n- For each $x\\in X$, the closure $\\operatorname{cl}\\{x\\}$ is precisely the set of points topologically indistinguishable from $x$.\n- The closures of any two points are either identical or disjoint (so $\\{\\operatorname{cl}\\{x\\}\\mid x\\in X\\}$ is a partition of the space $X$).\n\nSee {{wikipedia:T1_space}} for a more extensive list.\n\nAlso defined in definition 16.6, p. 438 of {{mr:1417259}} as \"symmetric space\". Not to be confused with symmetric spaces from Riemannian geometry.\n\n----\n#### Meta-properties\n\n- $X$ is {P135} iff its Kolmogorov quotient $\\text{Kol}(X)$ is {P2}.\n- This property is preserved by arbitrary products and box products.\n- This property is hereditary.","refs":[{"name":"T1 space on Wikipedia","wikipedia":"T1_space"},{"name":"Handbook of Analysis and its Foundations (Schechter)","mr":1417259}]},{"name":"Anticompact","aliases":["Compact subsets are finite"],"uid":"P000136","description":"Every compact subset of the space is finite.\n\n----\n#### Meta-properties\n\n- This property is hereditary.\n- This property is preserved by finite products.","refs":[{"name":"Can a hemicompact space fail to be weakly locally compact?","mathse":4566624},{"name":"The total negation of a topological property","doi":"10.1215/ijm/1256048236"}]},{"name":"Empty","aliases":["Cardinality $=0$"],"uid":"P000137","description":"A space with no element, necessarily equal to {S163}.\n\n----\n#### Meta-properties\n\n- This property is hereditary.","refs":[{"name":"Empty set on Wikipedia","wikipedia":"Empty_set"}]},{"name":"Countably-many continuous self-maps","aliases":[],"uid":"P000138","description":"A space with only countably-many continuous maps from the space to itself.","refs":[{"name":"Is there an infinite topological space with only countably many continuous functions to itself?","mo":418619}]},{"name":"Has an isolated point","aliases":["Not crowded","Not dense-in-itself"],"uid":"P000139","description":"Some singleton $\\{x\\}\\subseteq X$ is an open set.\n\n*Note*: A space that fails to satisfy {P139}, \ni.e., a space without any isolated points, is called \"dense-in-itself\" or \"crowded\". \nSee {{wikipedia:Dense-in-itself}} and {{mathse:3823020}}.\n\nCompare with {P107}.","refs":[{"name":"Reference request: crowded space","mathse":3823020},{"name":"Dense-in-itself on Wikipedia","wikipedia":"Dense-in-itself"}]},{"name":"$k_1$-space","aliases":["CG1","Generated by compact subspaces","Compactly generated","k-space"],"uid":"P000140","description":"A space whose topology coincides with the final topology with respect to the family of all inclusions from compact subspaces. A subset $A\\subseteq X$ is open (resp. closed) whenever $A\\cap K$ is open (resp. closed) in $K$ for every compact subspace $K$ of $X$.\n\nEquivalently, a space whose topology coincides with the final topology with respect to the family of all continuous maps from arbitrary compact spaces. A subset $A\\subseteq X$ is open (resp. closed) whenever $f^{-1}(A)$ is open (resp. closed) in $K$ for every compact space $K$ and every continuous map $f:K\\to X$.\n\n----\n#### Meta-properties\n\n- This property is hereditary with respect to closed sets.\n- This property is not hereditary with respect to open sets\n(e.g., {S23}, open in {S165}).","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"},{"name":"Unraveling the various definitions of k-space or compactly generated space","mathse":4646084},{"name":"Compactly generated space on Wikipedia","wikipedia":"Compactly_generated_space"}]},{"name":"$k_2$-space","aliases":["CG2","Generated by maps from compact Hausdorff spaces","Compactly generated","k-space"],"uid":"P000141","description":"A space whose topology coincides with the final topology with respect to the family of all continuous maps from arbitrary compact Hausdorff spaces. A subset $A\\subseteq X$ is open (resp. closed) whenever $f^{-1}(A)$ is open (resp. closed) in $K$ for every compact Hausdorff space $K$ and every continuous map $f:K\\to X$.","refs":[{"name":"Unraveling the various definitions of k-space or compactly generated space","mathse":4646084},{"name":"Compactly generated space on Wikipedia","wikipedia":"Compactly_generated_space"}]},{"name":"$k_3$-space","aliases":["CG3","Generated by compact Hausdorff subspaces","Compactly generated","k-space"],"uid":"P000142","description":"A space whose topology coincides with the final topology with respect to the family of all inclusions from compact Hausdorff subspaces. A subset $A\\subseteq X$ is open (resp. closed) whenever $A\\cap K$ is open (resp. closed) in $K$ for every compact Hausdorff subspace $K$ of $X$.","refs":[{"name":"Unraveling the various definitions of k-space or compactly generated space","mathse":4646084},{"name":"Compactly generated space on Wikipedia","wikipedia":"Compactly_generated_space"}]},{"name":"Weak Hausdorff","aliases":["Weakly Hausdorff"],"uid":"P000143","description":"A space $X$ such that for every compact Hausdorff space $K$ and every continuous map $f:K\\to X$, the image $f(K)$ is closed in $X$.\n\nOriginally defined on p. 275 of {{doi:10.1090/S0002-9947-1969-0251719-4}}. See also .\n\n----\n#### Meta-properties\n\n- This property is hereditary.","refs":[{"name":"Classifying spaces and infinite symmetric products (M. C. McCord)","doi":"10.1090/S0002-9947-1969-0251719-4"},{"name":"Weak Hausdorff space on Wikipedia","wikipedia":"Weak_Hausdorff_space"}]},{"name":"Locally pseudometrizable","aliases":[],"uid":"P000144","description":"Every point has a neighborhood that is {P121}.\n\n----\n#### Meta-properties\n\n- This property is hereditary.\n- $X$ is {P144} iff its Kolmogorov quotient $\\text{Kol}(X)$ is {P82}.","refs":[{"name":"How can the implication metrizable -> first countable be generalized?","mathse":4659902}]},{"name":"Strongly paracompact","aliases":["Hypocompact"],"uid":"P000145","description":"Every open cover has a star-finite open refinement.\n\nA family $\\mathscr A$ of subsets of $X$ is called *star-finite* if each element of $\\mathscr A$ meets at most finitely many elements of $\\mathscr A$.\n\nDefined on p. 326 of {{zb:0684.54001}} together with the condition that the space is {P3}. Here we do not assume any separation axiom as part of the definition.\nAlso called *hypocompact*, for example in Definition 2 of {{zb:0207.52704}}.\n\n----\n#### Meta-properties\n\n- $X$ satisfies this property iff its Kolmogorov quotient $\\text{Kol}(X)$ does.\n- This property is preserved by arbitrary disjoint unions.","refs":[{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"},{"name":"On some generalized compactness properties (J. Greever)","zb":"0207.52704"}]},{"name":"Ultraparacompact","aliases":[],"uid":"P000146","description":"Every open cover $\\mathscr U$ of $X$ has an open (hence clopen) refinement $\\mathscr V$ partitioning $X$.\n(We do not assume {P3}.)\n\n(Note: One can always choose the same index set for the two covers.\nGiven $\\mathscr U=\\{U_i:i\\in I\\}$, one can choose $\\mathscr V=\\{V_i:i\\in I\\}$ with $V_i\\subseteq U_i$ for each $i$.)\n\nEquivalently, every open cover of the space is refined by a locally finite clopen cover.\n\nThe equivalence between the two definitions is shown in Lemma 1.3 and Corollary 1.4 of {{zb:0182.25501}}.\n\n----\n#### Meta-properties\n\n- This property is preserved by arbitrary disjoint unions.","refs":[{"name":"Ultraparacompactness and Ultranormality (J. Van Name)","doi":"10.48550/arXiv.1306.6086"},{"name":"Extending continuous functions on zero-dimensional spaces (R. Ellis)","zb":"0182.25501"}]},{"name":"P-space","aliases":[],"uid":"P000147","description":"A space in which every countable intersection of open sets is open.\n\nEquivalently, a space in which every countable union of closed sets is closed.\n\nEquivalently, a space in which every point is a P-point, that is, a point for which the family of neighborhoods is closed under countable intersections.\n\nDefined in {{doi:10.1016/0016-660X(72)90026-8}} with the additional assumption of {P2}, which is not assumed here.\n\n----\n#### Meta-properties\n\n- This property is hereditary.\n- $X$ satisfies this property iff its Kolmogorov quotient $\\text{Kol}(X)$ does.","refs":[{"name":"A topological view of P-spaces (A. Misra)","doi":"10.1016/0016-660X(72)90026-8"}]},{"name":"CGWH","aliases":[],"uid":"P000148","description":"A space that is both {P141} and {P143}. Equivalently, a space that is both {P142} and {P143}.\n\nCGWH stands for \"compactly generated weak Hausdorff\". Mentioned for example in {{mo:251490}}.\n\nA good reference for these spaces is [N. Strickland, \"The category of CGWH spaces\"](https://ncatlab.org/nlab/files/StricklandCGHWSpaces.pdf).","refs":[{"name":"Answer to \"The 'right' topological spaces\"","mo":251490}]},{"name":"$\\omega$-Lindelöf","aliases":["$\\varepsilon$-space"],"uid":"P000149","description":"Every $\\omega$-cover of $X$ has a countable $\\omega$-subcover. (A family $\\mathcal U$ of open subsets of $X$ is called an *$\\omega$-cover* if each finite subset of $X$ is contained in an element of $\\mathcal U$ and $X \\notin \\mathcal U$.)\n\nEquivalently, every finite power of $X$ is {P18}.\n\nSee {{zb:0503.54020}}; the equivalence with every finite power being {P18} is on page 156.\n\nThe name $\\omega$-Lindelöf used in e.g.\n{{zb:1153.54009}} is motivated as the adaptation of\n{P18} for $\\omega$-covers. This property is\noften called $\\varepsilon$-space as it originally appeared\nas the fifth item \"$\\varepsilon$\" in a list from\n{{zb:0503.54020}}.\n\n---\n#### Meta-properties\n\n- This property is hereditary with respect to closed sets.","refs":[{"name":"Some properties of C(X), I (Gerlits & Nagy)","zb":"0503.54020"},{"name":"Selection principles related to α-properties (Kočinac)","zb":"1153.54009"}]},{"name":"$\\omega$-Rothberger","aliases":[],"uid":"P000150","description":"The space satisfies the selection principle $\\mathsf S_1(\\Omega,\\Omega)$: for every sequence $\\langle \\mathscr U_n : n \\in \\omega \\rangle$ of $\\omega$-covers of $X$, there exist choices $U_n \\in \\mathscr U_n$ so that $\\{ U_n :n \\in \\omega \\}$ is an $\\omega$-cover of $X$.\n(A family $\\mathcal U$ of open subsets of $X$ is called an *$\\omega$-cover* if each finite subset of $X$ is contained in an element of $\\mathcal U$ and $X \\notin \\mathcal U$.)\n\nEquivalently, every finite power of $X$ is {P68}.\n\nSee {{doi:10.1090/S0002-9939-1988-0964873-0}}; the equivalence with every finite power being {P68} is on page 918.","refs":[{"name":"Property $C''$ and Function Spaces (Sakai)","doi":"10.1090/S0002-9939-1988-0964873-0"}]},{"name":"Strategically Rothberger","aliases":["Strategically $\\Omega$-Rothberger"],"uid":"P000151","description":"Strategically Rothberger: The second player has a winning strategy in the Rothberger game $\\mathsf{G}_1(\\mathcal O_X,\\mathcal O_X)$. See pages 2 and 3 of {{doi:10.1016/j.topol.2019.07.008}} for more details.\n\nStrategically $\\Omega$-Rothberger: The second player has a winning strategy in the game $\\mathsf{G}_1(\\Omega_X,\\Omega_X)$. See pages 2 and 3 of {{doi:10.1016/j.topol.2019.07.008}} for more details.\n\nTheorem 15 of {{doi:10.1016/j.topol.2019.07.008}} states the equivalence for {P6} spaces, but the equivalence is shown to not require any separation axiom at {{mathse:4738368}}.\n\n----\n#### Meta-properties\n\n- $X$ satisfies this property iff its Kolmogorov quotient $\\text{Kol}(X)$ does.\n- $X$ satisfies the property if it is a countable union of subspaces with the property\n(idea: interleave strategies, one for each subspace).","refs":[{"name":"Limited information strategies and discrete selectivity (Clontz & Holshouser)","doi":"10.1016/j.topol.2019.07.008"},{"name":"Can P1 improve an open cover to an omega-cover in the finite-open game?","mathse":4738368}]},{"name":"Markov Rothberger","aliases":["Markov $\\Omega$-Rothberger","Topologically countable"],"uid":"P000152","description":"The second player has a Markov winning strategy in the Rothberger game $\\mathsf{G}_1(\\mathcal O_X,\\mathcal O_X)$ (relying on only the round number and most recent move of the opponent). See pages 2 and 3 of {{zb:1431.54009}} for more details.\n\nEquivalent conditions:\n\n- Markov $\\Omega$-Rothberger: The second player has a Markov winning strategy in the game $\\mathsf{G}_1(\\Omega_X,\\Omega_X)$ (relying on only the round number and most recent move of the opponent). See pages 2 and 3 of {{zb:1431.54009}} for more details.\n\n- Topologically countable: There is a set $\\{ x_n : n \\in \\omega \\} \\subseteq X$ so that, for every $x \\in X$, there is some $n \\in \\omega$ so that every neighborhood of $x_n$ contains $x$. See {{zb:1555.54009}} for more on this property.\n\nThe equivalence is shown in Theorem 4.17 of {{zb:1555.54009}} and also at {{mathse:4737285}}.","refs":[{"name":"Limited information strategies and discrete selectivity (Clontz & Holshouser)","zb":"1431.54009"},{"name":"Do uncountable spaces admit Markov strategies in Rothberger-style games?","mathse":4737285},{"name":"On traditional Menger and Rothberger variations (Caruvana, Clontz, Holshouser)","zb":"1555.54009"}]},{"name":"$\\omega$-Menger","aliases":[],"uid":"P000153","description":"The space satisfies the selection principle $\\mathsf S_{\\mathrm{fin}}(\\Omega,\\Omega)$: for every sequence $\\langle \\mathscr U_n : n \\in \\omega \\rangle$ of $\\omega$-covers of $X$, there exist choices $\\mathcal F_n$, a finite subset of $\\mathscr U_n$, so that $\\bigcup_{n\\in\\omega} \\mathcal F_n$ is an $\\omega$-cover of $X$.\n(A family $\\mathcal U$ of open subsets of $X$ is called an *$\\omega$-cover* if each finite subset of $X$ is contained in an element of $\\mathcal U$ and $X \\notin \\mathcal U$.)\n\nEquivalently, every finite power of $X$ is {P66}.\n\nSee {{doi:10.1016/S0166-8641(96)00075-2}}; the equivalence with every finite power being {P66} is Theorem 3.9.","refs":[{"name":"The combinatorics of open covers II","doi":"10.1016/S0166-8641(96)00075-2"}]},{"name":"GO-space","aliases":["Generalized ordered","Suborderable"],"uid":"P000154","description":"The space $X$ is {P2} and there is a linear order $<$ on $X$ such that every point\nhas a local base of order-convex neighborhoods.\n(A subset $A$ of an ordered set $(X,<)$ is *order-convex* if for each $a,b\\in A$ with $a)\nand {{mathse:4917398}}.\n\nSuch spaces are also called *suborderable* (as an example, in {{doi:10.1016/j.topol.2017.09.030}}).\n\n---\n### Meta-properties\n- This property is hereditary.","refs":[{"name":"On generalized ordered spaces (D. Lutzer)","zb":"0228.54026"},{"name":"Equivalent definitions for GO-spaces (generalized ordered spaces)","mathse":4917398},{"name":"The existence of continuous weak selections and orderability-type properties in products and filter spaces (Motooka et al.)","doi":"10.1016/j.topol.2017.09.030"}]},{"name":"Locally $1$-Euclidean","aliases":["Locally Euclidean of dimension $1$"],"uid":"P000155","description":"Each point has a neighborhood homeomorphic to {S25}.\n\nThis is the special case of {P123} with $n=1$.","refs":[{"name":"Introduction to topological manifolds (Lee)","mr":2766102}]},{"name":"$k$-Rothberger","aliases":[],"uid":"P000156","description":"The space satisfies the selection principle $\\mathsf S_1(\\mathcal K,\\mathcal K)$: for every sequence $\\langle \\mathscr U_n : n \\in \\omega \\rangle$ of $k$-covers of $X$, there exist choices $U_n \\in \\mathscr U_n$ so that $\\{ U_n :n \\in \\omega \\}$ is a $k$-cover of $X$.\n(A family $\\mathcal U$ of open subsets of $X$ is called a *$k$-cover* if each compact subset of $X$ is contained in an element of $\\mathcal U$ and $X \\notin \\mathcal U$.)","refs":[{"name":"Applications of k-covers II","doi":"10.1016/j.topol.2005.07.015"}]},{"name":"Strategically $k$-Rothberger","aliases":[],"uid":"P000157","description":"The second player has a winning strategy in the game $\\mathsf{G}_1(\\mathcal K_X,\\mathcal K_X)$. See Remark 2.4 and Definition 2.7 of {{doi:10.1016/j.topol.2021.107772}} for more details.","refs":[{"name":"Selection games and the Vietoris space","doi":"10.1016/j.topol.2021.107772"}]},{"name":"Markov $k$-Rothberger","aliases":[],"uid":"P000158","description":"The second player has a Markov winning strategy in the game $\\mathsf{G}_1(\\mathcal K_X,\\mathcal K_X)$ (relying on only the round number and most recent move of the opponent). See Remark 2.4 and Definition 2.7 of {{doi:10.1016/j.topol.2021.107772}} for more details.\n\n----\n#### Meta-properties\n\n- $X$ satisfies this property iff its Kolmogorov quotient $\\text{Kol}(X)$ does.","refs":[{"name":"Selection games and the Vietoris space","doi":"10.1016/j.topol.2021.107772"}]},{"name":"$k$-Menger","aliases":[],"uid":"P000159","description":"The space satisfies the selection principle $\\mathsf S_{\\mathrm{fin}}(\\mathcal K,\\mathcal K)$: for every sequence $\\langle \\mathscr U_n : n \\in \\omega \\rangle$ of $k$-covers of $X$, there exist choices $\\mathcal F_n$, a finite subset of $\\mathscr U_n$, so that $\\bigcup_{n\\in\\omega} \\mathcal F_n$ is a $k$-cover of $X$.\n(A family $\\mathcal U$ of open subsets of $X$ is called a *$k$-cover* if each compact subset of $X$ is contained in an element of $\\mathcal U$ and $X \\notin \\mathcal U$.)","refs":[{"name":"Applications of k-covers II","doi":"10.1016/j.topol.2005.07.015"}]},{"name":"Strategically $k$-Menger","aliases":[],"uid":"P000160","description":"The second player has a winning strategy in the game $\\mathsf{G}_{\\mathrm{fin}}(\\mathcal K_X,\\mathcal K_X)$. See Remark 2.4 and Definition 2.7 of {{doi:10.1016/j.topol.2021.107772}} for more details.","refs":[{"name":"Selection games and the Vietoris space","doi":"10.1016/j.topol.2021.107772"}]},{"name":"Markov $k$-Menger","aliases":[],"uid":"P000161","description":"The second player has a Markov winning strategy in the game $\\mathsf{G}_{\\mathrm{fin}}(\\mathcal K_X,\\mathcal K_X)$ (relying on only the round number and most recent move of the opponent). See Remark 2.4 and Definition 2.7 of {{doi:10.1016/j.topol.2021.107772}} for more details.","refs":[{"name":"Selection games and the Vietoris space","doi":"10.1016/j.topol.2021.107772"}]},{"name":"Realcompact","aliases":[],"uid":"P000162","description":"A space $X$ that is homeomorphic to a closed subset of $\\mathbb{R}^\\kappa$ for some cardinal $\\kappa$.\n\nEquivalently (see {{zb:1380.46022}}), $X$ is {P6} and every real $z$-ultrafilter $\\mathcal U$ on the space $X$ is fixed,\nthat is, $\\bigcap\\mathcal{U}\\neq\\emptyset$.\n\nA *$z$-ultrafilter* is an ultrafilter on the lattice of zero-sets of $X$ (see {{wikipedia:Ultrafilter}} for the general definition of an ultrafilter on a poset). A *real $z$-ultrafilter* is a $z$-ultrafilter with countable intersection property, that is, for any countable $\\mathcal{F}\\subseteq \\mathcal{U}$ we have $\\bigcap\\mathcal{F}\\neq \\emptyset$.\n\nSee also section 3.11 in {{zb:0684.54001}}.\n\nNote: If $X$ has {P164}, then $X$ is {P221} iff $\\text{Kol}(X)$ is {P162}.","refs":[{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"},{"name":"Rings of Continuous Functions (Gillman and Jerison)","zb":"1380.46022"},{"name":"Ultrafilter on Wikipedia","wikipedia":"Ultrafilter"}]},{"name":"Cardinality $\\leq\\mathfrak c$","aliases":[],"uid":"P000163","description":"The cardinality of the space is less than or equal to the cardinality of $\\mathbb R$.\n\n----\n#### Meta-properties\n\n- This property is hereditary.","refs":[]},{"name":"Cardinality less than every measurable cardinal","aliases":["Non-measurable cardinality"],"uid":"P000164","description":"The cardinality of the space is smaller than every measurable cardinal, if one exists.\n\nA cardinal $\\kappa$ is called *measurable* if $\\kappa$ is uncountable and there exists a measure $\\mu:2^\\kappa\\to \\{0, 1\\}$ such that \n\n1. $\\mu$ is *$\\kappa$-additive*: $\\mu(\\bigcup_{i\\in I} A_i) = \\sum_{i\\in I} \\mu(A_i)$ for any family $(A_i)_{i\\in I}\\subseteq 2^\\kappa$ of pairwise disjoint sets with $|I| < \\kappa$,\n\n2. $\\mu$ is non-trivial: $\\mu(\\kappa) = 1$ and $\\mu(\\{x\\}) = 0$ for all $x\\in \\kappa$. \n\nEquivalently, $\\kappa$ is uncountable and there exists a free ultrafilter $\\mathcal{U}$ on $\\kappa$ such that $\\mathcal{U}$ is *$\\kappa$-complete*, i.e., if $\\mathcal{F}\\subseteq \\mathcal{U}$ and $|\\mathcal{F}| < \\kappa$ then $\\bigcap\\mathcal{F}\\in \\mathcal{U}$. (See {{wikipedia:Measurable_cardinal}} for more details.)\n\nNote: Some authors, for example {{zb:1380.46022}}, refer to measurable cardinals as those cardinals $\\kappa$ for which there exists a $\\sigma$-additive measure $\\mu:2^\\kappa\\to \\{0, 1\\}$ which is non-trivial. If $\\kappa$ is the smallest such cardinal, then a non-trivial $\\sigma$-additive measure $\\mu:2^\\kappa\\to \\{0, 1\\}$ is $\\kappa$-additive (see lemma 10.2 of {{doi:10.1007/3-540-44761-X}} and comments preceding it), so $\\kappa$ is also measurable by the above definition.\n\n(The existence of a measurable cardinal cannot be proven in ZFC.\nSo spaces whose construction does not depend on set-theoretic axioms beyond ZFC should never have this property marked as false.)\n\n----\n#### Meta-properties\n\n- This property is hereditary.","refs":[{"name":"Measurable cardinal on Wikipedia","wikipedia":"Measurable_cardinal"},{"name":"Rings of Continuous Functions (Gillman and Jerison)","zb":"1380.46022"},{"name":"Set Theory (Jech)","doi":"10.1007/3-540-44761-X"}]},{"name":"Pseudonormal","aliases":["Weakly normal"],"uid":"P000165","description":"For every countable closed set $H$ and open set $U\\supseteq H$,\nthere exists an open set $V$ with $H\\subseteq V\\subseteq cl(V)\\subseteq U$.\nIn other words, countable closed sets have arbitrarily small closed neighborhoods.\n\nEquivalently, any two disjoint closed sets, at least one of which is countable, can be separated by open sets.\n\nDefined for example on p. 8 of {{zb:1411.54002}}.\n\nSee also .\n\n----\n#### Meta-properties\n\n- This property is hereditary with respect to closed sets.","refs":[{"name":"Pseudocompact topological spaces (Hrusak et al.)","zb":"1411.54002"},{"name":"Pseudonormal space on Wikipedia","wikipedia":"Pseudonormal_space"}]},{"name":"Has a coarser separable metrizable topology","aliases":["Has a separable metrizable compression"],"uid":"P000166","description":"The topology contains a coarser topology which is both {P26} and {P53}.\n\nN.B. if the space is also {P6}, by page 54 of {{doi:10.1007/BFb0098389}} this is equivalent to\nthe function space $C_p(X)$ being {P26}.\n\n----\n#### Meta-properties\n\n- This property is preserved by countable disjoint unions.\n- This property is hereditary (use the fact that [Metrizable + Separable ⇒ Hereditarily separable](https://topology.pi-base.org/spaces?q=metrizable+%2B+Separable+%2B+%7EHereditarily+separable)).","refs":[{"name":"Topological Properties of Spaces of Continuous Functions","doi":"10.1007/BFb0098389"}]},{"name":"Sequentially discrete","aliases":[],"uid":"P000167","description":"Every convergent sequence in the space is eventually constant.\n\nEquivalently, every set is sequentially closed. That is, the sequential coreflection of the space is discrete.\n\nSee {{mo:451796}}.\n\n----\n#### Meta-properties\n\n- This property is hereditary.","refs":[{"name":"Topological property of convergent sequences being eventually constant","mo":451796},{"name":"Sequential space on Wikipedia","wikipedia":"Sequential_space"}]},{"name":"Countable sets are discrete","aliases":["Countable sets are closed","Cid","$\\omega$C"],"uid":"P000168","description":"Every {P57} subspace is {P52}.\n\nEquivalently (see comments of {{mathse:4995501}}), every {P57}\nsubset is closed in the space.\n\nDefined as \"cid\" in {{zb:0665.54003}} (pg. 283 of \n[https://gdz.sub.uni-goettingen.de/id/PPN311571026_0039](https://gdz.sub.uni-goettingen.de/id/PPN311571026_0039?tify=%7B%22pages%22%3A%5B299%5D%2C%22view%22%3A%22%22%7D))\nwith the weaker meaning that \"countable infinite subspaces are discrete\". If $X$ is infinite, the two notions are equivalent.\n\n----\n#### Meta-properties\n\n- This property is hereditary.","refs":[{"name":"Answer to \"Does every door space with an infinite compact set have a single non-isolated point?\"","mathse":4995501},{"name":"On spaces whose denumerable subspaces are discrete (Ganster, Reilly, Vamanamurthy)","zb":"0665.54003"},{"name":"Answer to What separation is required to ensure extremally disconnected spaces are sequentially discrete?","mathse":4751818}]},{"name":"Semi-Hausdorff","aliases":["Semi $T_2$"],"uid":"P000169","description":"For any two distinct points $x,y$, there exists a regular open neighborhood\nof $x$ missing $y$, where an open set is *regular* provided it equals the interior\nof its closure.\n\nNot to be confused with other notions of \"semi-Hausdorff\" from the literature,\ne.g. separation by semi-open sets.","refs":[{"name":"Each regular paratopological group is completely regular","doi":"10.1090/proc/13318"},{"name":"Examples of strongly rigid countable (semi)Hausdorff spaces","mr":4614765}]},{"name":"$k_1$-Hausdorff","aliases":["k-Hausdorff","$k_1H$","$k_1T_2$"],"uid":"P000170","description":"A space $X$ such that the diagonal $\\Delta$ of $X\\times X$ is k-closed, where k-closed is in terms of the {P140} property (see Wikipedia). In other words, $\\Delta\\cap K$ is closed in $K$ for every compact subspace $K$ of $X\\times X$.\n\nEquivalently, every {P16} subspace is {P3}.\n\nEquivalently, every {P16} subspace is closed in $X$ and {P130}.\n\nThe property is defined in section 2 of {{doi:10.1007/BF02194829}}, where Theorem 2.1 shows the equivalences above.\n\nNote: this property is not to be confused with other variants of k-Hausdorff (e.g. {P171}), where k-closed is defined in terms of different notions of compactly generated space or k-space.\n\n----\n#### Meta-properties\n\n- This property is hereditary.","refs":[{"name":"Quotients of k-semigroups (Lawson & Madison)","doi":"10.1007/BF02194829"},{"name":"How are k-Hausdorff and weakly Hausdorff distinct?","mathse":4760309},{"name":"Compactly generated space on Wikipedia","wikipedia":"Compactly_generated_space"}]},{"name":"$k_2$-Hausdorff","aliases":["k-Hausdorff","$k_2H$","$k_2T_2$"],"uid":"P000171","description":"A space $X$ such that the diagonal $\\Delta$ of $X\\times X$ is k-closed, where k-closed is in terms of the {P141} property (see Wikipedia). In other words, the inverse image of $\\Delta$ is closed for every continuous map from a\n{P16} {P3} space to $X\\times X$.\n\nEquivalently, for every pair of continuous maps $f,g:K\\to X$ with {P16} {P3} domain $K$,\nthe set $\\{a\\in K:f(a)=g(a)\\}$ is closed in $K$.\n\nAnd alternatively, for every {P16} {P3} space $K$ and\ncontinuous map $f:K\\to X$ and points $k,l\\in K$ with $f(k)\\not=f(l)$,\nthere exist open neighborhoods $U,V$ of $k,l$ with $f[U],f[V]$\ndisjoint. See 4.2.4 of .\n\nSee also for an earlier\nreference to this weakening of {P3}.\n\nNote: this property is not to be confused with other variants of k-Hausdorff (e.g. {P170}),\nwhere k-closed is defined in terms of different notions of compactly generated space or k-space.\n\n----\n#### Meta-properties\n\n- This property is hereditary.","refs":[{"name":"Compactly generated space on Wikipedia","wikipedia":"Compactly_generated_space"}]},{"name":"Radial","aliases":["Fréchet chain-net"],"uid":"P000172","description":"A space such that for each $A\\subseteq X$ and each $p\\in \\overline A$ there is a transfinite sequence $(x_\\alpha)_{\\alpha<\\lambda}$ of points of $A$ converging to $p$. (Here $\\lambda$ is a limit ordinal and can always be taken to be a regular cardinal.)\n\nSee the chapter \"d-4 Pseudoradial spaces\" in {{zb:1059.54001}}.\n\nCompare with {P80}.\n\n----\n#### Meta-properties\n\n- This property is hereditary.\n- $X$ satisfies this property iff its Kolmogorov quotient $\\text{Kol}(X)$ does.\n- If each point has a neighborhood with the property, $X$ also has the property.\n- This property is not preserved by products\n(Example: {S78|P172}).","refs":[{"name":"Encyclopedia of general topology","zb":"1059.54001"}]},{"name":"Pseudoradial","aliases":["Chain-net"],"uid":"P000173","description":"A space in which every radially closed set is closed, where a set $A\\subseteq X$ is *radially closed* if it contains the limits of all convergent transfinite sequences consisting of points of $A$. In other words, for every non-closed set $A\\subseteq X$ there is a point $p\\in \\overline A\\setminus A$ and a transfinite sequence $(x_\\alpha)_{\\alpha<\\lambda}$ of points of $A$ that converges to $p$. (Here $\\lambda$ is a limit ordinal and can always be taken to be a regular cardinal.)\n\nSee the chapter \"d-4 Pseudoradial spaces\" in {{zb:1059.54001}}.","refs":[{"name":"Encyclopedia of general topology","zb":"1059.54001"}]},{"name":"Well-based","aliases":["lob-space"],"uid":"P000174","description":"Each point $x\\in X$ has a neighborhood base well-ordered by reverse inclusion.\n\nEquivalently, each point $x\\in X$ has a neighborhood base totally ordered by reverse inclusion. The two are equivalent since a linear order admits a well-ordered cofinal subset.\n\nSuch spaces are called *well-based* in {{doi:10.1016/j.topol.2014.08.002}} and *lob-spaces* (lob = \"linearly ordered (local) base\") in {{mr:0540476}}.\n\nOne says that $X$ is *well-based at the point $x$* if the conditions above hold at the specific point.\nSo $X$ is well-based if it is well-based at every point.\n\nSee also {{mo:202280}} and {{mo:322162}}.\n\n----\n#### Meta-properties\n\n- $X$ satisfies this property iff its Kolmogorov quotient $\\text{Kol}(X)$ does.\n- This property is hereditary.","refs":[{"name":"An internal characterisation of radiality (R. Leek)","doi":"10.1016/j.topol.2014.08.002"},{"name":"Spaces with linearly ordered local bases (S. W. Davis)","mr":540476},{"name":"Convergent filters generated by (not necessarily countable) chains","mo":202280},{"name":"Name for topological spaces where 'every point has a local base wellordered by reverse inclusion'","mo":322162}]},{"name":"Cardinality $\\geq 3$","aliases":[],"uid":"P000175","description":"The space contains at least three points.","refs":[]},{"name":"Cardinality $\\geq 4$","aliases":[],"uid":"P000176","description":"The space contains at least four points.","refs":[]},{"name":"$\\sigma$-space","aliases":[],"uid":"P000177","description":"A space that is {P5} and {P117}.\n\nThe name \"$\\sigma$-space\" was introduced by A. Okuyama (see {{doi:10.3792/pja/1195521153}}) to denote only the {P117} property. But most later papers, for example {{doi:10.1016/j.topol.2015.05.085}}, also include {P5} in this definition.\n\n----\n#### Meta-properties\n\n- This property is hereditary.","refs":[{"name":"$\\sigma$-spaces and closed mappings, I (A. Okuyama)","doi":"10.3792/pja/1195521153"},{"name":"$\\mathfrak P$-spaces and related concepts (Gabriyelyan & Kakol)","doi":"10.1016/j.topol.2015.05.085"}]},{"name":"$\\aleph$-space","aliases":[],"uid":"P000178","description":"A space that is {P5} and {P118}.\n\nThe notion was introduced in {{doi:10.1090/S0002-9939-1971-0276919-3}}.\n\n----\n#### Meta-properties\n\n- This property is hereditary.\n- The Kolmogorov quotient $\\text{Kol}(X)$ is {P178} iff $X$ is {P11} and {P118}.","refs":[{"name":"On paracompactness in function spaces with the compact-open topology (P. O'Meara)","doi":"10.1090/S0002-9939-1971-0276919-3"},{"name":"$\\mathfrak P$-spaces and related concepts (Gabriyelyan & Kakol)","doi":"10.1016/j.topol.2015.05.085"}]},{"name":"$\\aleph_0$-space","aliases":[],"uid":"P000179","description":"A space that is {P5} and {P183}.\n\nOriginally defined in {{mr:206907}}, available at .\n\n----\n#### Meta-properties\n\n- This property is hereditary.","refs":[{"name":"$\\aleph_0$-spaces (E. Michael)","mr":206907},{"name":"$\\mathfrak P$-spaces and related concepts (Gabriyelyan & Kakol)","doi":"10.1016/j.topol.2015.05.085"}]},{"name":"Hereditarily separable","aliases":[],"uid":"P000180","description":"A space for which every subspace is {P26}.\n\nDefined on page 182 of {{doi:10.1016/B978-0-444-50355-8.X5000-4}}.\n\n----\n#### Meta-properties\n\n- This property is hereditary.","refs":[{"name":"Encyclopedia of General Topology","doi":"10.1016/B978-0-444-50355-8.X5000-4"}]},{"name":"Countably infinite","aliases":["Cardinality $=\\aleph_0$","Cardinality $=\\beth_0$"],"uid":"P000181","description":"The cardinality of the space is equal to the cardinality of $\\mathbb N$.","refs":[]},{"name":"Has a countable network","aliases":[],"uid":"P000182","description":"A space with a (finite or infinite) countable network.\n\nA family $\\mathcal N$ of subsets of $X$ is called a *network* if every open set is the union of a subfamily of $\\mathcal N$. That is, for every open set $U$ and point $x\\in U$ there is some $A\\in\\mathcal N$ with $x\\in A\\subseteq U$.\n\nDefined on page 127 of {{zb:0684.54001}}.\n\n---\n### Meta-properties\n\n- This property is hereditary.\n- This property is preserved by countable products.","refs":[{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"name":"Has a countable $k$-network","aliases":[],"uid":"P000183","description":"A space with a (finite or infinite) countable $k$-network.\n\nEquivalently:\n- A space with countable pseudobase.\n- A space with countable $cs$-network.\n- A space with countable $cs^\\ast$-network.\n\nA family $\\mathcal N$ of subsets of $X$ is called a *$k$-network* if for every compact set $K$ and open set $U$ in $X$ with $K\\subseteq U$, there exists a finite $\\mathcal{N}^* \\subseteq \\mathcal{N}$ with $K \\subseteq \\bigcup\\mathcal{N}^* \\subseteq U$.\n\nA family $\\mathcal{N}$ of subsets of $X$ is called a *pseudobase* if for every compact set $K$ and open set $U$ in $X$ with $K\\subseteq U$, there exists some $A\\in\\mathcal{N}$ with $K \\subseteq A \\subseteq U$.\n\nA family $\\mathcal{N}$ of subsets of $X$ is called a *$cs$-network* if for every open set $U\\subseteq X$, point $x\\in U$ and sequence $(x_n)$ converging to $x$, there exists some $A\\in\\mathcal{N}$ with $x_n\\in A$ for all but finitely many $n$.\n\nA family $\\mathcal{N}$ of subsets of $X$ is called a *$cs^\\ast$-network* if for every open set $U\\subseteq X$, point $x\\in U$ and sequence $(x_n)$ converging to $x$, there exists some $A\\in\\mathcal{N}$ with $x_n\\in A$ for infinitely many $n$.\n\nTo see the equivalence, first note that a pseudobase is a $k$-network, a $k$-network is a $cs$-network and a $cs$-network is a $cs^\\ast$-network. In {{mo:506308}} it is shown that if $X$ has countable $cs^\\ast$-network, then it has a countable $k$-network. Then finite unions of elements of that $k$-network form a countable pseudobase.\n\nSee {{mr:206907}}, available at .\n\n---\n### Meta-properties\n\n- This property is hereditary.\n- This property is preserved by countable products (see Corollary in {{mo:506308}}).","refs":[{"name":"$\\aleph_0$-spaces (E. Michael)","mr":206907},{"name":"$\\mathfrak P$-spaces and related concepts (Gabriyelyan & Kakol)","doi":"10.1016/j.topol.2015.05.085"},{"name":"Answer to \"Product of spaces with countable $k$-networks\"","mo":506308}]},{"name":"Embeddable into Euclidean space","aliases":["Euclidean subspace"],"uid":"P000184","description":"Homeomorphic to a subspace of $\\mathbb R^n$ for some finite $n$.\n\nPer Theorems 1.1.2 and 1.11.4 of {{zb:0401.54029}}, this is equivalent to being {P27}, {P53}, and finite-dimensional, using any of the small inductive, large inductive, or covering dimensions by Theorem 1.7.7.\n\n----\n#### Meta-properties\n\n- This property is hereditary.","refs":[{"name":"Dimension Theory (Engelking)","zb":"0401.54029"}]},{"name":"Partition topology","aliases":[],"uid":"P000185","description":"There exists a basis for the topology that is also a partition into disjoint sets.\n\nThis is equivalent to each of the following:\n\n- Every open set is closed.\n- Every closed set is open.\n- The space's Kolmogorov quotient is {P52}.\n- The space is the disjoint union of a collection of {P129} spaces.\n\nFor proof of the equivalences and further characterizations, see section 13 of {{doi:10.5186/aasfm.1977.0321}}.\n\nDefined as example #5 (\"Partition Topology\")\nin {{zb:0386.54001}}.\n\n----\n#### Meta-properties\n\n- This property is hereditary.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"On ultrapseudocompact and related spaces (T. Nieminen)","doi":"10.5186/aasfm.1977.0321"}]},{"name":"Embeds in a topological $W$-group","aliases":[],"uid":"P000186","description":"Homeomorphic to a subspace of a space that is {P87} and {P187}.\n\n----\n#### Meta-properties\n\n- This property is hereditary.\n- This property is preserved by countable products.\n- This property is preserved by $\\Sigma$-products.","refs":[{"name":"A compact space $K$ is Corson compact if and only if $C_p(K)$ has a dense lc-scattered subspace (Tkachuk)","zb":"1535.54012"}]},{"name":"W-space","aliases":[],"uid":"P000187","description":"P1 has a winning strategy at each point $x\\in X$ \nfor the following game defined by Gruenhage in {{zb:0327.54019}}.\n\nDuring each round $n<\\omega$, P1 chooses some neighborhood $U_n$ of $x$,\nthen P2 chooses some point $x_n\\in U_n$. P1 wins provided the sequence\nof $x_n$ converges to $x$; P2 wins otherwise.\n\nSee also section 2 of {{zb:1141.54020}}.\n\n----\n#### Meta-properties\n\n- This property is hereditary.\n- This property is preserved by countable products (see Theorem 4.1 of {{zb:0327.54019}}).\n- This property is preserved by $\\Sigma$-products (see Theorem 4.6 of {{zb:0327.54019}}).\n- A locally $W$-space (every point has a neighborhood with the property) is a $W$-space.","refs":[{"name":"Infinite games and generalizations of first-countable spaces (Gruenhage)","zb":"0327.54019"},{"name":"The story of a topological game (Gruenhage)","zb":"1141.54020"}]},{"name":"Continuum","aliases":["Hausdorff continuum"],"uid":"P000188","description":"A {P3}, {P16}, and {P36} space, as defined in e.g.\n{{doi:10.1016/B978-0-444-50355-8.X5000-4}}.\n\nNote that many authors (e.g. {{doi:10.1201/9781315274089}}) use the\nterm \"continuum\" to refer to spaces that are also {P53}, which is\nnot assumed here.","refs":[{"name":"Encyclopedia of General Topology (Hart et al)","doi":"10.1016/B978-0-444-50355-8.X5000-4"},{"name":"Continuum Theory: An introduction (Nadler)","doi":"10.1201/9781315274089"}]},{"name":"$\\sigma$-connected","aliases":[],"uid":"P000189","description":"The space cannot be partitioned into more than one and at most countably infinitely many nonempty closed sets.\n\nEquivalently, $X$ is {P36} and cannot be written as a countably infinite union of pairwise disjoint nonempty closed sets.\n\nEquivalently, every continuous map from $X$ to {S15} is constant. (This is easy to check using the fact that the codomain is countable and $T_1$, and each of its proper closed subsets is a finite union of singletons.)\n\nDefined in Chapter 5, Section 4, Definition 5.15 of {{doi:10.1201/9781315274089}}.\n\n----\n#### Meta-properties\n\n- $X$ satisfies this property iff its Kolmogorov quotient $\\text{Kol}(X)$ does.\n- If a dense subspace of $X$ satisfies this property, so does $X$ (see Fact 3 in {{mathse:4975867}}).\n- The continuous image of a {P189} space is {P189} (see Fact 2 in {{mathse:4975867}}).","refs":[{"name":"Continuum Theory: An introduction (Nadler)","doi":"10.1201/9781315274089"},{"name":"Answer to \"$\\sigma$-connected spaces and path-connected property\"","mathse":4975867}]},{"name":"Ordinal space","aliases":[],"uid":"P000190","description":"The space is homeomorphic to an ordinal number $\\alpha$, viewed as the set of ordinals less than $\\alpha$, together with the corresponding order topology. For each point $\\beta<\\alpha$, the collection of intervals $(\\gamma,\\beta]$ with $\\gamma<\\beta$ forms a local base of open neighborhoods of $\\beta$.\n\nEquivalently, the topology is the order topology induced by a well-order on the underlying set.\n\nStudied as spaces #40-#43 in {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Ordinal number on Wikipedia","wikipedia":"Ordinal_number"}]},{"name":"Has points $G_\\delta$","aliases":["Countable pseudocharacter"],"uid":"P000191","description":"A space in which every singleton is a $G_\\delta$ set (a countable intersection of open sets).\n\nSee exercise 3.1.F(a) of {{zb:0684.54001}}.\n\n----\n#### Meta-properties\n\n- This property is hereditary.","refs":[{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"name":"Quasi-sober","aliases":[],"uid":"P000192","description":"Every nonempty irreducible (= {P39}) closed subset of $X$ is the closure of a point of $X$, not necessarily unique;\nthat is, every nonempty irreducible closed subset has at least one *generic point*.\n(See {P201} for several equivalent formulations.)\n\nEquivalently, the Kolmogorov quotient of the space is {P73}. (See {T512}.)\n\nSee for example Definition 5.8.6 in the section on [Irreducible components](https://stacks.math.columbia.edu/tag/004U) from the Stacks project,\nor the paragraph after Definition 1.2 in {{doi:10.35834/mjms/1316032836}}.\n\nNote: With an additional uniqueness condition on the generic points, the corresponding property is {P73}.\n\n----\n#### Meta-properties\n\n- $X$ is {P192} iff its Kolmogorov quotient $\\text{Kol}(X)$ is {P73}.","refs":[{"name":"Sober spaces and sober sets (Echi & Lazaar)","doi":"10.35834/mjms/1316032836"}]},{"name":"Shrinking","aliases":["Has the shrinking property"],"uid":"P000193","description":"A space in which every open cover admits a shrinking; that is, a space $X$ in which, given any open cover $\\{ U_\\alpha : \\alpha \\in A\\}$, there is an open cover $\\{ V_\\alpha : \\alpha \\in A\\}$ such that $\\overline{V_\\alpha} \\subseteq U_\\alpha$ for each $\\alpha \\in A$.\n\nSee also [Dan Ma's Topology Blog post on Spaces with shrinking properties](https://dantopology.wordpress.com/2017/01/05/spaces-with-shrinking-properties/).","refs":[{"name":"Encyclopedia of general topology","zb":"1059.54001"},{"name":"Generalized paracompactness (Y. Yasui)","zb":"0712.54016"},{"name":"Shrinking space","wikipedia":"Shrinking_space"}]},{"name":"Submetacompact","aliases":["$\\theta$-refinable"],"uid":"P000194","description":"Every open cover of $X$ has a $\\theta$-*sequence* of open refinements; that is, for every open cover $\\mathscr U$, there exists a sequence $\\{ \\mathscr V_n : n \\in \\omega\\}$ of open covers where each $\\mathscr V_n$ is a refinement of $\\mathscr U$ and, for each point $x \\in X$, there exists $n \\in \\omega$ such that $\\mathscr V_n$ is point-finite at $x$.\n\nThis property was introduced in {{zb:0132.18401}} under the name of *$\\theta$-refinable*, and later renamed to *submetacompact* in {{zb:0413.54027}} (full article available [here](http://topology.nipissingu.ca/tp/reprints/v03/tp03207s.pdf)).\n\n---\n#### Meta-properties\n- This property is hereditary with respect to closed sets.","refs":[{"name":"Characterizations of developable topological spaces (J. Worrell and H. Wicke)","zb":"0132.18401"},{"name":"On Submetacompactness (H. Junnila)","zb":"0413.54027"},{"name":"Generalized paracompactness (Y. Yasui)","zb":"0712.54016"}]},{"name":"Stone space","aliases":["Boolean","Profinite"],"uid":"P000195","description":"The space is {P16}, {P3}, and {P47}.\n\nThis is equivalent to each of the following:\n\n- The space is {P16} and {P48}.\n- The space is {P16}, {P1}, and {P50}.\n- The space is {P75} and {P2}.\n\nThe equivalence follows from various theorems in pi-base.\n\nAlso known as *profinite* spaces, referring to the fact that such spaces are precisely the cofiltered limits of finite discrete spaces.\nSee the section on [Profinite spaces](https://stacks.math.columbia.edu/tag/08ZW) at the Stacks project.\n\nNot to be confused with {P119}, which is a stronger property.","refs":[{"name":"Spectral spaces (Dickmann, Schwartz, Tressl)","doi":"10.1017/9781316543870"},{"name":"Stone spaces (Johnstone)","mr":861951},{"name":"Stone space on Wikipedia","wikipedia":"Stone_space"}]},{"name":"Hereditarily connected","aliases":["Totally ordered topology"],"uid":"P000196","description":"Every subspace is connected.\n\nThis is equivalent to each of the following:\n\n- The open sets are totally ordered by inclusion.\n- There is a basis of open sets that is totally ordered by inclusion.\n- The closed sets are totally ordered by inclusion.\n- The specialization preorder is total.\n\nFor proof of the equivalences and further characterizations, see section 12 of {{doi:10.5186/aasfm.1977.0321}}.\n\n----\n#### Meta-properties\n\n- This property is hereditary.\n- This property is preserved in any coarser topology.","refs":[{"name":"On ultrapseudocompact and related spaces (T. Nieminen)","doi":"10.5186/aasfm.1977.0321"}]},{"name":"Has countable spread","aliases":[],"uid":"P000197","description":"A space with countable spread, where the *spread* of $X$ is\n$s(X) = \\sup\\{|D| : D \\subseteq X, D \\text{ is discrete } \\} + \\omega$.\n\nThat is, every discrete subspace of $X$ is countable.\n\nEquivalently, every uncountable set $S\\subseteq X$ has a limit point in $S$.\n\nSee {{mr:0776620}} or section a-3 of {{doi:10.1016/B978-0-444-50355-8.X5000-4}}.\n\n----\n#### Meta-properties\n\n- $X$ satisfies this property iff its Kolmogorov quotient $\\text{Kol}(X)$ does.\n- This property is hereditary.","refs":[{"name":"Cardinal functions. I. Handbook of set-theoretic topology (R. Hodel)","mr":776620},{"name":"Encyclopedia of General Topology","doi":"10.1016/B978-0-444-50355-8.X5000-4"}]},{"name":"Has countable extent","aliases":[],"uid":"P000198","description":"A space with countable extent, where the *extent* of $X$ is\n$e(X) = \\sup\\{|D| : D \\subseteq X, D \\text{ is closed and discrete } \\} + \\omega$.\n\nThat is, every closed discrete subspace of $X$ is countable.\n\nEquivalently, every uncountable set $S\\subseteq X$ has a limit point in $X$.\n\nSee {{mr:0776620}} or section a-3 of {{doi:10.1016/B978-0-444-50355-8.X5000-4}}.\n\n----\n#### Meta-properties\n\n- If $X$ is covered by countably many subspaces, each having the property, then so does $X$.","refs":[{"name":"Cardinal functions. I. Handbook of set-theoretic topology (R. Hodel)","mr":776620},{"name":"Encyclopedia of General Topology","doi":"10.1016/B978-0-444-50355-8.X5000-4"}]},{"name":"Contractible","aliases":[],"uid":"P000199","description":"$X$ is [homotopy equivalent](https://en.wikipedia.org/wiki/Homotopy#Homotopy_equivalence) to {S162}.\n\nEquivalently: \n- $X \\neq \\emptyset$ and the identity map of $X$ is [homotopic](https://en.wikipedia.org/wiki/Homotopy#Formal_definition) to a constant map.\n- $X \\neq \\emptyset$, and for every nonempty space $Y$, any map $Y \\to X$ is homotopic to a constant map.\n\nDefined on page 4 of {{zb:1044.55001}}.\n\n----\n#### Meta-properties\n\n- This property is preserved by arbitrary products. (see {{mathse:4463999}})","refs":[{"name":"Contractible space on Wikipedia","wikipedia":"Contractible_space"},{"name":"Algebraic Topology (Hatcher)","zb":"1044.55001"},{"name":"The cartesian product of contractible spaces is contractible","mathse":4463999}]},{"name":"Simply connected","aliases":["1-connected"],"uid":"P000200","description":"Every pair of points in $X$ is connected by a unique homotopy class of paths. (See Proposition 1.6. of {{zb:1044.55001}}.)\n\nThis is equivalent to each of the following:\n- $X$ is {P37} and every continuous map $S^1 \\to X$ extends to a continuous map $D^2 \\to X$. \n- $X$ is {P37} and every continuous map $S^1 \\to X$ is homotopic to a constant map.\n\nIn case $X\\ne\\emptyset$, it is equivalent to the usual formulation:\n- $X$ is {P37} and the [fundamental group](https://en.wikipedia.org/wiki/Fundamental_group) $\\pi_1(X)$ is trivial.\n\n*Note*: The fundamental group is undefined for the empty space.\n\nAlso called $1$-*connected* if $X\\ne\\emptyset$. See [n-connected space in nLab](https://ncatlab.org/nlab/show/n-connected+space).\n\nDefined on page 28 of {{zb:1044.55001}}. See {{mathse:4044399}} for additional equivalent characterizations.\n\n----\n#### Meta-properties\n\n- $X$ satisfies this property iff its Kolmogorov quotient $\\text{Kol}(X)$ does.\n- This property is preserved by arbitrary products.","refs":[{"name":"Simply connected space on Wikipedia","wikipedia":"Simply_connected_space"},{"name":"Fundamental group on Wikipedia","wikipedia":"Fundamental_group"},{"name":"Algebraic Topology (Hatcher)","zb":"1044.55001"},{"name":"Characterizing simply connected spaces","mathse":4044399}]},{"name":"Has a generic point","aliases":[],"uid":"P000201","description":"There exists a generic point $p \\in X$.\n\nThe point $p$ is called a *generic point* of $X$ if any of the following equivalent conditions holds:\n\n- $\\{p\\}$ is dense in $X$.\n- $X$ is the only closed set containing $p$.\n- $p$ is an element of every nonempty open subset of $X$.\n- $p$ is a *generalization* of every point of $X$, according to the specialization preorder. See {{wikipedia:Specialization_(pre)order}}.\n- The topology is coarser than or equal to an included point/particular point topology (e.g. {S8}).\n\nCompare with {P202}.\n\nSee Definition 1.1.7 on page 3 of {{zb:1325.14001}}.\n\n----\n#### Meta-properties\n\n- This property is preserved in any coarser topology.","refs":[{"name":"Generic point on Wikipedia","wikipedia":"Generic_point"},{"name":"Specialization (pre)order on Wikipedia","wikipedia":"Specialization_(pre)order"},{"name":"Algebraic Geometry II (Mumford-Oda)","zb":"1325.14001"}]},{"name":"Has a point with a unique neighborhood","aliases":["Has a focal point","Has a point specializing every point"],"uid":"P000202","description":"There exists a point $p\\in X$ whose only neighborhood is the entire space.\n\nEquivalently:\n\n- $p$ is contained in all nonempty closed sets.\n- $p$ is a specialization of every point. See {{wikipedia:Specialization_(pre)order}}.\n- The topology is coarser than or equal to an excluded point topology (e.g. {S12}).\n\nSee Section C1.5 page 523 of {{zb:1071.18002}} using the terminology \"focal point\".\n\nCompare with {P201} and {P203}.\n\n----\n#### Meta-properties\n\n- This property is preserved in any coarser topology.","refs":[{"name":"Specialization (pre)order on Wikipedia","wikipedia":"Specialization_(pre)order"},{"name":"Sketches of an elephant. A topos theory compendium (Johnstone)","zb":"1071.18002"}]},{"name":"Almost discrete","aliases":[],"uid":"P000203","description":"All points except for one are isolated.\n\nEquivalently, the topology is not {P52} and is equal to or finer than an excluded point topology (e.g. {S12}).\n\nThere have been various slightly different definitions of \"almost discrete\" in the literature.\nMost definitions exclude discrete spaces; see for example {{zb:0791.54049}} and {{zb:1542.54005}}.\nSometimes {P3} is assumed, but we don't make that assumption here.\n\nCompare with {P202}.","refs":[{"name":"On the product of almost discrete Grothendieck spaces (Osipov)","zb":"1542.54005"},{"name":"Almost discrete SV-spaces (Henrikson, Wilson)","zb":"0791.54049"}]},{"name":"Has a cut point","aliases":[],"uid":"P000204","description":"There exists a cut point $p \\in X$.\nA point $p$ is called a *cut point* of $X$ if $X$ is {P36} but $X \\setminus \\{p\\}$ is not.\n\nSee also {P205}.","refs":[{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"},{"name":"Cut-point spaces (Honari & Bahrampour)","doi":"10.1090/S0002-9939-99-04839-X"},{"name":"Cut point on Wikipedia","wikipedia":"Cut_point"}]},{"name":"Cut point space","aliases":[],"uid":"P000205","description":"Every point in the space is a cut point, and the space is nonempty.\nA point $p$ is called a *cut point* of $X$ if $X$ is {P36} but $X \\setminus \\{p\\}$ is not.\n\nSee also {P204}.","refs":[{"name":"Cut-point spaces (Honari & Bahrampour)","doi":"10.1090/S0002-9939-99-04839-X"},{"name":"Cut point on Wikipedia","wikipedia":"Cut_point"}]},{"name":"Strongly Choquet","aliases":[],"uid":"P000206","description":"Player 2 has a winning strategy in the *strong Choquet game* played on a nonempty space.\n\nLet $V_{-1}=X$. During round $n<\\omega$\nof this game, Player 1 chooses a point $x_n$ with an open neighborhood $U_n\\subseteq V_{n-1}$,\nand Player 2 chooses some open neighborhood $V_n\\subseteq U_n$ of $x_n$.\n\nPlayer 2 wins this game provided\n$\\bigcap\\{U_n:n<\\omega\\}\\; (= \\bigcap\\{V_n:n<\\omega\\}) \\not=\\emptyset$.\n\nNote: We consider the {P137} space to be {P206} in the sense that\nPlayer 1 cannot make a legal move to start the game, and therefore loses immediately according\nto the {{wikipedia:Normal_play_convention}} of combinatorial games.\n\n----\n#### Meta-properties\n\n- $X$ satisfies this property iff its Kolmogorov quotient $\\text{Kol}(X)$ does.\n- This property is preserved by arbitrary products and box products.","refs":[{"name":"Choquet game on Wikipedia","wikipedia":"Choquet_game"},{"name":"Normal play convention on Wikipedia","wikipedia":"Normal_play_convention"},{"name":"Classical Descriptive Set Theory (Kechris)","zb":"0819.04002"}]},{"name":"Strongly collectionwise normal","aliases":["Divisible","Almost $2$-fully normal"],"uid":"P000207","description":"For each neighborhood $U$ of the diagonal $\\Delta=\\{(x,x)\\mid x\\in X\\}$\nin $X\\times X$, there is a neighborhood $V$ of the diagonal such that\n$V\\circ V\\subseteq U$.\n\nThis is equivalent to the family $\\mathcal{U}$ of neighborhoods of $\\Delta_X$ forming a uniformity, but the topology of $X$ and that of $(X, \\mathcal{U})$ do not need to agree. If $X$ is {P12} then both topologies coincide.\n\nIn Theorem 2.6 of {{doi:10.2307/1993026}} this property was shown to be equivalent to\n*almost $2$-fully normal*: each open cover $\\mathcal U$ has an open almost $2$-star\nrefinement $\\mathcal V$, that is $\\mathcal{V}$ is a refinement of $\\mathcal{U}$ and for any $x, y, z$ with $y, z\\in \\text{St}(x, \\mathcal{V})$ there exists $U\\in\\mathcal{U}$ with $y, z\\in U$.\n\nOriginally studied by Cohen in {{zb:0046.16403}}\n(),\nwhere it was shown to be\nstrictly stronger than {P88}.\nSee also {{doi:10.1090/S0002-9939-1981-0630058-4}}.\n\n----\n#### Meta-properties\n\n- This property is preserved by arbitrary disjoint unions.\n- This property is hereditary with respect to closed sets (corollary of Theorem 4.1 in {{zb:0078.14803}}).","refs":[{"name":"Sur une problème de M. Dieudonné (Cohen)","zb":"0046.16403"},{"name":"Strong collectionwise normality and M. E. Rudin’s Dowker space (Hart)","doi":"10.1090/S0002-9939-1981-0630058-4"},{"name":"Some generalizations of full normality (M. J. Mansfield)","zb":"0078.14803"}]},{"name":"Noetherian","aliases":["Hereditarily compact"],"uid":"P000208","description":"Every subset of $X$ is compact.\n\nEquivalently:\n- Every open set is compact.\n- The open sets satisfy the *ascending chain condition*: There is no infinite strictly increasing sequence $O_1 \\subsetneq O_2 \\subsetneq \\cdots$ of open sets.\n- The closed sets satisfy the *descending chain condition*: There is no infinite strictly decreasing sequence $Y_1 \\supsetneq Y_2 \\supsetneq \\cdots$ of closed sets.\n- Every nonempty collection of open sets has a maximal element.\n- Every nonempty collection of closed sets has a minimal element.\n\nDefined on page 5 of {{zb:1325.14001}}, and on page 248 of {{doi:10.1017/9781316543870}}. Also see the section on [Noetherian topological spaces](https://stacks.math.columbia.edu/tag/0050) from the Stacks project.\n\nCompare with {P226}.\n\n----\n#### Meta-properties\n\n- This property is hereditary.\n- This property is preserved by finite products (see Theorem 3 in {{mathse:5118758}}).\n- This property is preserved by finite disjoint unions.\n- This property is preserved in any coarser topology.","refs":[{"name":"Noetherian topological space on Wikipedia","wikipedia":"Noetherian_topological_space"},{"name":"Algebraic Geometry II (Mumford-Oda)","zb":"1325.14001"},{"name":"Spectral spaces (Dickmann, Schwartz, Tressl)","doi":"10.1017/9781316543870"},{"name":"Answer to \"Is the product of two Artinian topological spaces Artinian?\"","mathse":5118758}]},{"name":"Density $\\leq\\mathfrak c$","aliases":[],"uid":"P000209","description":"There exists a dense subset with cardinality $\\leq \\mathfrak c=2^{\\aleph_0}=|\\mathbb R|$.\n\n----\n#### Meta-properties\n- This property is preserved by countable products.","refs":[]},{"name":"$\\alpha_1$","aliases":["$\\alpha_1$-space"],"uid":"P000210","description":"For every $x\\in X$ and every sequence of countably infinite sets $S_1,S_2,\\dots\\subseteq X$ with\neach $S_n$ converging to $x$, there is a countably infinite set $S\\subseteq\\bigcup_n S_n$\nsuch that $S$ converges to $x$ and $S\\cap S_n$ is cofinite in $S_n$\n(i.e., $S_n\\setminus S$ is finite) for all $n$.\n\nEquivalently, the same condition holds with the additional requirement that the sets $S_n$\nare pairwise disjoint. (The equivalence is shown after Definition 1.1 in {{zb:1348.54033}}.)\n\nTerminology: A countably infinite set $A\\subseteq X$ *converges to $x$*\nif some (equivalently, all) bijective enumeration of $A$ converges to $x$;\nthat is, if every neighborhood of $x$ contains all but finitely many elements of $A$.\n\nThe definition above in terms of countably infinite sets follows {{zb:1348.54033}}.\nOne can also phrase the definition in terms of (injective) sequences of elements of $X$,\nas done in {{zb:0774.54019}}.\n\nThe $\\alpha_i$ properties for $i = 1, 2, 3, 4$ (sometimes called \"sheaf amalgamation properties\")\nare due to Arkhangel'skii ({{zb:0275.54004}}).\n\n----\n#### Meta-properties\n\n- This property is hereditary.\n- $X$ satisfies this property iff its Kolmogorov quotient $\\text{Kol}(X)$ does (see {{mathse:5123855}}).\n- This property is preserved by countable products (see {{mathse:5127008}}).\n- This property is preserved by arbitrary disjoint unions.\n- $X$ satisfies this property iff all its countable subspaces do.","refs":[{"name":"Arhangel'skii sheaf amalgamations in topological groups (Tsaban & Zdomskyy)","zb":"1348.54033"},{"name":"Subsets of ${}^\\omega\\omega$ and the Fréchet-Urysohn and $\\alpha_i$-properties (P. Nyikos)","zb":"0774.54019"},{"name":"The frequency spectrum of a topological space and the classification of spaces (A. V. Arkhangel'skii)","zb":"0275.54004"},{"name":"Answer to \"Are the Arkhangel'skii $\\alpha_i$ properties preserved by finite products?\"","mathse":5127008},{"name":"Does the Arkhangel'skii $\\alpha_1$ property hold for $X$ if it holds for its Kolmogorov quotient?","mathse":5123855}]},{"name":"$\\alpha_{1.5}$","aliases":["$\\alpha_{1.5}$-space"],"uid":"P000211","description":"For every $x\\in X$ and every sequence of pairwise disjoint countably infinite sets\n$S_1,S_2,\\dots\\subseteq X$ with\neach $S_n$ converging to $x$, there is a countably infinite set $S\\subseteq\\bigcup_n S_n$\nsuch that $S$ converges to $x$ and $S\\cap S_n$ is cofinite in $S_n$\n(i.e., $S_n\\setminus S$ is finite) for infinitely many $n$.\n\nThe requirement that the sequences be disjoint is important here.\nWithout it, one gets a property equivalent to {P210},\nas explained in the discussion after Definition 1.1 of {{zb:1348.54033}}.\n\nTerminology: A countably infinite set $A\\subseteq X$ *converges to $x$*\nif some (equivalently, all) bijective enumeration of $A$ converges to $x$;\nthat is, if every neighborhood of $x$ contains all but finitely many elements of $A$.\n\nThe definition above in terms of countably infinite sets follows {{zb:1348.54033}}.\nOne can also phrase the definition in terms of (injective) sequences of elements of $X$,\nas done in {{zb:0774.54019}}.\n\nThis property was introduced by Nyikos in {{zb:0774.54019}}.\n\n----\n#### Meta-properties\n\n- This property is hereditary.\n- $X$ satisfies this property iff its Kolmogorov quotient $\\text{Kol}(X)$ does (see {{mathse:5123855}}).\n- This property is preserved by countable products (see {{mathse:5127008}}).\n- This property is preserved by arbitrary disjoint unions.\n- $X$ satisfies this property iff all its countable subspaces do.","refs":[{"name":"Arhangel'skii sheaf amalgamations in topological groups (Tsaban & Zdomskyy)","zb":"1348.54033"},{"name":"Subsets of ${}^\\omega\\omega$ and the Fréchet-Urysohn and $\\alpha_i$-properties (P. Nyikos)","zb":"0774.54019"},{"name":"Answer to \"Are the Arkhangel'skii $\\alpha_i$ properties preserved by finite products?\"","mathse":5127008},{"name":"Does the Arkhangel'skii $\\alpha_1$ property hold for $X$ if it holds for its Kolmogorov quotient?","mathse":5123855}]},{"name":"$\\alpha_2$","aliases":["$\\alpha_2$-space"],"uid":"P000212","description":"For every $x\\in X$ and every sequence of countably infinite sets $S_1,S_2,\\dots\\subseteq X$ with\neach $S_n$ converging to $x$, there is a countably infinite set $S\\subseteq\\bigcup_n S_n$\nsuch that $S$ converges to $x$ and $S\\cap S_n$ is infinite for all $n$.\n\nEquivalently, the same condition holds with the additional requirement that the sets $S_n$\nare pairwise disjoint. (The equivalence follows from Lemma 1.2 in {{zb:0774.54019}}.)\n\nTerminology: A countably infinite set $A\\subseteq X$ *converges to $x$*\nif some (equivalently, all) bijective enumeration of $A$ converges to $x$;\nthat is, if every neighborhood of $x$ contains all but finitely many elements of $A$.\n\nThe definition above in terms of countably infinite sets follows {{zb:1348.54033}}.\nOne can also phrase the definition in terms of (injective) sequences of elements of $X$,\nas done in {{zb:0774.54019}}.\n\nThe $\\alpha_i$ properties for $i = 1, 2, 3, 4$ (sometimes called \"sheaf amalgamation properties\")\nare due to Arkhangel'skii ({{zb:0275.54004}}).\n\n----\n#### Meta-properties\n\n- This property is hereditary.\n- $X$ satisfies this property iff its Kolmogorov quotient $\\text{Kol}(X)$ does (see {{mathse:5123855}}).\n- This property is preserved by countable products (see {{mathse:5127008}}).\n- This property is preserved by arbitrary disjoint unions.\n- $X$ satisfies this property iff all its countable subspaces do.","refs":[{"name":"Arhangel'skii sheaf amalgamations in topological groups (Tsaban & Zdomskyy)","zb":"1348.54033"},{"name":"Subsets of ${}^\\omega\\omega$ and the Fréchet-Urysohn and $\\alpha_i$-properties (P. Nyikos)","zb":"0774.54019"},{"name":"The frequency spectrum of a topological space and the classification of spaces (A. V. Arkhangel'skii)","zb":"0275.54004"},{"name":"Answer to \"Are the Arkhangel'skii $\\alpha_i$ properties preserved by finite products?\"","mathse":5127008},{"name":"Does the Arkhangel'skii $\\alpha_1$ property hold for $X$ if it holds for its Kolmogorov quotient?","mathse":5123855}]},{"name":"$\\alpha_3$","aliases":["$\\alpha_3$-space"],"uid":"P000213","description":"For every $x\\in X$ and every sequence of countably infinite sets $S_1,S_2,\\dots\\subseteq X$ with\neach $S_n$ converging to $x$, there is a countably infinite set $S\\subseteq\\bigcup_n S_n$\nsuch that $S$ converges to $x$ and $S\\cap S_n$ is infinite for infinitely many $n$.\n\nEquivalently, the same condition holds with the additional requirement that the sets $S_n$\nare pairwise disjoint. (The equivalence follows from Lemma 1.2 in {{zb:0774.54019}}.)\n\nTerminology: A countably infinite set $A\\subseteq X$ *converges to $x$*\nif some (equivalently, all) bijective enumeration of $A$ converges to $x$;\nthat is, if every neighborhood of $x$ contains all but finitely many elements of $A$.\n\nThe definition above in terms of countably infinite sets follows {{zb:1348.54033}}.\nOne can also phrase the definition in terms of (injective) sequences of elements of $X$,\nas done in {{zb:0774.54019}}.\n\nThe $\\alpha_i$ properties for $i = 1, 2, 3, 4$ (sometimes called \"sheaf amalgamation properties\")\nare due to Arkhangel'skii ({{zb:0275.54004}}).\n\n----\n#### Meta-properties\n\n- This property is hereditary.\n- $X$ satisfies this property iff its Kolmogorov quotient $\\text{Kol}(X)$ does (see {{mathse:5123855}}).\n- This property is preserved by countable products (see {{mathse:5127008}}).\n- This property is preserved by arbitrary disjoint unions.\n- $X$ satisfies this property iff all its countable subspaces do.","refs":[{"name":"Arhangel'skii sheaf amalgamations in topological groups (Tsaban & Zdomskyy)","zb":"1348.54033"},{"name":"Subsets of ${}^\\omega\\omega$ and the Fréchet-Urysohn and $\\alpha_i$-properties (P. Nyikos)","zb":"0774.54019"},{"name":"The frequency spectrum of a topological space and the classification of spaces (A. V. Arkhangel'skii)","zb":"0275.54004"},{"name":"Answer to \"Are the Arkhangel'skii $\\alpha_i$ properties preserved by finite products?\"","mathse":5127008},{"name":"Does the Arkhangel'skii $\\alpha_1$ property hold for $X$ if it holds for its Kolmogorov quotient?","mathse":5123855}]},{"name":"$\\alpha_4$","aliases":["$\\alpha_4$-space"],"uid":"P000214","description":"For every $x\\in X$ and every sequence of countably infinite sets $S_1,S_2,\\dots\\subseteq X$ with\neach $S_n$ converging to $x$, there is a countably infinite set $S\\subseteq\\bigcup_n S_n$\nsuch that $S$ converges to $x$ and $S\\cap S_n\\ne\\emptyset$ for infinitely many $n$.\n\nEquivalently, the same condition holds with the additional requirement that the sets $S_n$\nare pairwise disjoint. (The equivalence follows from Lemma 1.2 in {{zb:0774.54019}}.)\n\nTerminology: A countably infinite set $A\\subseteq X$ *converges to $x$*\nif some (equivalently, all) bijective enumeration of $A$ converges to $x$;\nthat is, if every neighborhood of $x$ contains all but finitely many elements of $A$.\n\nThe definition above in terms of countably infinite sets follows {{zb:1348.54033}}.\nOne can also phrase the definition in terms of (injective) sequences of elements of $X$,\nas done in {{zb:0774.54019}}.\n\nThe $\\alpha_i$ properties for $i = 1, 2, 3, 4$ (sometimes called \"sheaf amalgamation properties\")\nare due to Arkhangel'skii ({{zb:0275.54004}}).\n\n----\n#### Meta-properties\n\n- This property is hereditary.\n- $X$ satisfies this property iff its Kolmogorov quotient $\\text{Kol}(X)$ does (see {{mathse:5123855}}).\n- This property is preserved by arbitrary disjoint unions.\n- $X$ satisfies this property iff all its countable subspaces do.","refs":[{"name":"Arhangel'skii sheaf amalgamations in topological groups (Tsaban & Zdomskyy)","zb":"1348.54033"},{"name":"Subsets of ${}^\\omega\\omega$ and the Fréchet-Urysohn and $\\alpha_i$-properties (P. Nyikos)","zb":"0774.54019"},{"name":"The frequency spectrum of a topological space and the classification of spaces (A. V. Arkhangel'skii)","zb":"0275.54004"},{"name":"Does the Arkhangel'skii $\\alpha_1$ property hold for $X$ if it holds for its Kolmogorov quotient?","mathse":5123855}]},{"name":"Hereditarily realcompact","aliases":[],"uid":"P000215","description":"Every subspace is {P162}. \n\nEquivalently, $X$ is {P6} and $X\\setminus \\{x\\}$ is realcompact for each $x\\in X$. (theorem 8.17 of {{zb:1380.46022}})\n\n----\n#### Meta-properties\n\n- This property is hereditary.","refs":[{"name":"Rings of Continuous Functions (Gillman & Jerison)","zb":"1380.46022"}]},{"name":"Hereditarily paracompact","aliases":[],"uid":"P000216","description":"Every subspace is {P30}. \n\nEquivalently, every open subspace is {P30}.\n\n----\n#### Meta-properties\n\n- This property is hereditary.\n- $X$ satisfies this property iff its Kolmogorov quotient $\\text{Kol}(X)$ does.\n- This property is preserved by arbitrary disjoint unions.","refs":[]},{"name":"Strongly zero-dimensional","aliases":[],"uid":"P000217","description":"Any two disjoint zero-sets are contained in disjoint clopen sets.\n\n(Note: The clopen sets can always be chosen to be complement of each other.)\n\nEquivalently:\n- Every finite cozero cover of $X$ has a finite clopen refinement which is a partition of $X$\n(see definition p. 360 and Theorem 6.2.4 in {{zb:0684.54001}}).\n\nIf $X$ is {P6}, this is also equivalent to:\n- The Stone–Čech compactification $\\beta X$ is {P50}\n(see Theorems 16.11 and 16.17 in {{zb:0093.30001}}).\n\nSome terminology used above:\nA *zero-set* is a set of the form $f^{-1}(0)$ for some continuous $f:X\\to\\mathbb{R}$.\nA *cozero set* is the complement of a zero-set.\n\n----\nDefined in section 6.2 of {{zb:0684.54001}} with the additional conditions that\n$X$ be Tychonoff ({P6}) and non-empty.\nHere we do not require these two conditions.\n\nSee also page 4 of {{doi:10.48550/arXiv.1306.6086}}.\n\nThe book {{zb:1471.54001}} uses \"strongly zero-dimensional\" with the meaning of non-empty {P218}.\nThe property {P217} is equivalent to $\\dim_0(X) \\leq 0$ where $\\dim_0$ is defined in chapter 11.\n\n----\n#### Meta-properties\n\n- $X$ satisfies this property iff its Kolmogorov quotient $\\text{Kol}(X)$ does.","refs":[{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"},{"name":"Rings of Continuous Functions (Gillman & Jerison)","zb":"0093.30001"},{"name":"Ultraparacompactness and Ultranormality (J. Van Name)","doi":"10.48550/arXiv.1306.6086"},{"name":"Dimension theory (M. Charalambous)","zb":"1471.54001"}]},{"name":"Ultranormal","aliases":["Strongly zero-dimensional"],"uid":"P000218","description":"Any two disjoint closed sets are contained in disjoint clopen sets.\n\n(Note: The clopen sets can always be chosen to be complement of each other.)\n\nEquivalently, every finite open cover of $X$ has a finite clopen refinement which is a partition of $X$.\n\n----\nDefined on page 11 of {{zb:0182.25501}}.\n\nSee also page 1 of {{doi:10.48550/arXiv.1306.6086}}.\n\nThe book {{zb:1471.54001}} uses \"strongly zero-dimensional\"\nfor a space that is {P218} and non-empty (see page 9); {P218} is equivalent to either one of $\\dim(X), \\text{Ind}(X)$ or $\\text{Ind}_0(X)$ being $\\leq 0$. See chapters 2 and 13 for definitions of $\\dim, \\text{Ind}, \\text{Ind}_0$.","refs":[{"name":"Extending continuous functions on zero-dimensional spaces (R. Ellis)","zb":"0182.25501"},{"name":"Ultraparacompactness and Ultranormality (J. Van Name)","doi":"10.48550/arXiv.1306.6086"},{"name":"Dimension theory (M. Charalambous)","zb":"1471.54001"}]},{"name":"Toronto","aliases":[],"uid":"P000219","description":"Every subspace $Y \\subseteq X$ with $|Y|=|X|$ is homeomorphic to $X$.\n\n*Note*: In {{zb:1286.54032}} it is shown that under GCH, every {P3} Toronto space is {P52}.","refs":[{"name":"Toronto space on Wikipedia","wikipedia":"Toronto_space"},{"name":"The Toronto Problem (W. R. Brian)","zb":"1286.54032"}]},{"name":"Ultrametrizable","aliases":[],"uid":"P000220","description":"There exists an ultrametric inducing the topology of $X$.\n\nA metric $d$ is called an *ultrametric* if it satisfies a property stronger than the triangle inequality, namely:\n\n$\\quad d(x, z) \\leq \\max(d(x, y), d(y, z)).$\n\n----\n#### Meta-properties\n\n- This property is hereditary.","refs":[{"name":"Ultrametric space on Wikipedia","wikipedia":"Ultrametric_space"}]},{"name":"Dieudonné complete","aliases":["Completely uniformizable","Topologically complete"],"uid":"P000221","description":"There exists at least one [complete uniformity](https://en.wikipedia.org/wiki/Uniform_space#Completeness)\nthat induces the topology of $X$.\n\nThis is equivalent to each of the following:\n- $X$ is homeomorphic to a closed subspace of a product of completely pseudometrizable spaces.\n- $X$ is homeomorphic to a closed subspace of a product of {P121} spaces.\n\nFor the equivalence, see Problem 8.5.13(z) in {{zb:0684.54001}} and page 285 of {{zb:0024.36301}}.\n\nTerminology:\nWe call a uniformity $\\mathcal{U}$ *complete* if every Cauchy filter $\\mathcal{F}$ on $(X, \\mathcal{U})$ converges. Equivalently, every Cauchy net $(x_i)_{i\\in I}$ on $(X, \\mathcal{U})$ converges. Here a *Cauchy filter* is a filter $\\mathcal{F}$ such that for every $U\\in\\mathcal{U}$ there exists $A\\in\\mathcal{F}$ such that $A\\times A\\subseteq U$. A *Cauchy net* is a net $(x_i)_{i\\in I}$ such that for every $U\\in\\mathcal{U}$ there exists $i_0$ such that $(x_j, x_k)\\in U$ for $j, k\\geq i_0$.\n(Compare with the definition of complete uniformity in 15.7 of {{zb:1380.46022}} where uniform structure is defined using pseudometrics.)\n\nSuch spaces are called *Dieudonné complete* (Problem 8.5.13 in {{zb:0684.54001}}) or *completely uniformizable* (Problem 39B in {{zb:1052.54001}}).\nThey are called *topologically complete* on page 208 of {{mr:370454}}; \nthis last term has also been used for {P55} and {P63}.\n\nNote: If $X$ has {P164}, then $X$ is {P221} iff $\\text{Kol}(X)$ is {P162}.\n\n----\n#### Meta-properties\n\n- $X$ satisfies this property iff its Kolmogorov quotient $\\text{Kol}(X)$ does.\n- This property is preserved by arbitrary products.\n- This property is hereditary with respect to closed sets.","refs":[{"name":"Completely uniformizable space","wikipedia":"Completely_uniformizable_space"},{"name":"Rings of Continuous Functions (Gillman & Jerison)","zb":"1380.46022"},{"name":"General Topology (Kelley)","mr":370454},{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"},{"name":"General Topology (Willard, 1970)","zb":"1052.54001"},{"name":"Sur les espaces uniformes complets (Dieudonné)","zb":"0024.36301"}]},{"name":"Has a cofinite topology","aliases":["Minimal $T_1$"],"uid":"P000222","description":"A set $U\\subseteq X$ is open iff $U$ is *cofinite* in $X$ (i.e., $X\\setminus U$ is finite) or $U = \\emptyset$.\n\nEquivalently, any of the following:\n\n- A set $A\\subseteq X$ is closed iff $A$ is finite or $A=X$.\n\n- $X$ is *minimal $T_1$*, i.e., {P2} with no strictly coarser {P2} topology.\n\n----\n#### Meta-properties\n\n- This property is hereditary.","refs":[]},{"name":"Locally contractible","aliases":[],"uid":"P000223","description":"$X$ admits a basis of open sets which are {P199}.\n\nEquivalently, for each $x \\in X$, every neighborhood of $x$ contains a {P199} open neighborhood of $x$.\n\n\"Locally contractible\" is used in {{zb:1044.55001}}, where it is defined informally with the meaning above as part of the statement of Proposition A.4 on page 522. (As explained on page 61, the book follows the convention of a space being \"locally P\" for a property P to mean that every point has arbitrarily small open neighborhoods with the property; this is very close to the [general convention used in pi-base](https://github.com/pi-base/data/wiki/Conventions-and-Style#Local-Properties).)","refs":[{"name":"Algebraic Topology (Hatcher)","zb":"1044.55001"}]},{"name":"Weakly locally contractible","aliases":[],"uid":"P000224","description":"Every point of $X$ has a neighborhood which is {P199}. \n\nThe name we have chosen for this property conforms to the [pi-base naming conventions](https://github.com/pi-base/data/wiki/Conventions-and-Style#Local-Properties).\nHowever, we have not seen this property mentioned with a specific name in the literature.\n\nThe terminology \"weakly locally contractible\" has been used for multiple concepts different from this one, but the name is not standardized.\nA relatively common usage among those is as a synonym for {P239}.","refs":[]},{"name":"$LC$","aliases":["Locally contractible"],"uid":"P000225","description":"$X$ is *locally contractible at the point* $x$ for all $x \\in X$,\nin the sense that every neighborhood $U$ of $x$ contains a neighborhood (equivalently, an open neighborhood) $V$ of $x$ that is contractible in $U$;\ni.e., such that the inclusion map $V \\hookrightarrow U$ is null-homotopic.\n\nEquivalently, every neighborhood $U$ of any point $x$ contains a neighborhood (or an open neighborhood) $V$ of $x$\nsuch that the inclusion map $V \\hookrightarrow U$ is homotopic to the constant map with value $x$.\n\nThis is the standard definition of \"locally contractible\" in the theory of ANRs, as originally introduced by Borsuk. Defined as *locally contractible* on page 28 of {{zb:0153.52905}}, page 347 of {{zb:1280.54001}}, and page 341 of {{zb:1059.54001}}. The abbreviation $LC$ is commonly used in this context.\n\n----\n#### Meta-properties\n- This property is preserved by retractions (Theorem 15.3 on p. 28 of {{zb:0153.52905}}).","refs":[{"name":"Theory of retracts (Borsuk)","zb":"0153.52905"},{"name":"Geometric aspects of general topology. (Sakai)","zb":"1280.54001"},{"name":"Encyclopedia of general topology","zb":"1059.54001"}]},{"name":"Artinian","aliases":[],"uid":"P000226","description":"Every nonempty collection of open sets has a minimal element.\n\nEquivalently:\n- Every nonempty collection of closed sets has a maximal element.\n- The open sets satisfy the *descending chain condition*: There is no infinite strictly decreasing sequence $O_1 \\supsetneq O_2 \\supsetneq \\cdots$ of open sets.\n- The closed sets satisfy the *ascending chain condition*: There is no infinite strictly increasing sequence $Y_1 \\subsetneq Y_2 \\subsetneq \\cdots$ of closed sets.\n- Every subspace has a finite dense subset.\n\nSee {{mathse:5117935}} for a proof of the equivalences.\n\nCompare with {P208}.\n\n----\n#### Meta-properties\n\n- This property is hereditary.\n- $X$ satisfies this property iff its Kolmogorov quotient $\\text{Kol}(X)$ does.\n- This property is preserved by finite disjoint unions.\n- This property is preserved by finite products (see {{mathse:5118732}}).","refs":[{"name":"Finite $T_0$-spaces and universal mappings. (Holsztyński, Pedersen)","zb":"0392.54005"},{"name":"Equivalent definitions of an Artinian space","mathse":5117935},{"name":"Is the product of two Artinian topological spaces Artinian?","mathse":5118732}]},{"name":"Has a discrete closed subset of size $\\mathfrak c$","aliases":[],"uid":"P000227","description":"$X$ has a discrete closed set of cardinality $\\mathfrak c=2^{\\aleph_0}$.\n\n*Note*: The discrete closed sets are exactly the subsets with no limit point in $X$.\n\n*Note*: This property implies $e(X)\\ge\\mathfrak c$,\nwhere the *extent* $e(X)$ is the supremum of the cardinality of discrete closed sets in $X$.\nBut, under certain set-theoretic assumptions (for example, if $\\mathfrak c=\\aleph_{\\omega_1}$),\nthere are spaces with $e(X)=\\mathfrak c$ and without discrete closed set of cardinality $\\mathfrak c$,\ni.e., where the supremum is not attained.\n\nCompare with these properties, where $D$ denotes a discrete closed set in $X$:\n- {P107} $(\\exists D: |D|=1)$\n- {P21} $(\\forall D: |D|<\\aleph_0)$\n- {P198} $(\\forall D: |D|\\le\\aleph_0)$\n\n----\n#### Meta-properties\n\n- This property is preserved in any finer topology.\n- If a closed subspace of $X$ satisfies this property, so does $X$.","refs":[]},{"name":"Weakly first countable","aliases":["g-first countable"],"uid":"P000228","description":"One can choose for each $x\\in X$ a decreasing sequence $\\mathcal T_x=(V_n(x))_{n\\in\\mathbb N}$ of subsets of $X$ containing $x$, such that the family $\\{\\mathcal T_x:x\\in X\\}$ satisfies:\n\n$\\quad$A subset $U\\subseteq X$ is open in $X$ iff for every $x\\in U$ there is some $n$ such that $V_n(x)\\subseteq U$.\n\n*Note*: The family $\\{\\mathcal T_x:x\\in X\\}$ is a special case of a *weak base* for $X$. See the references for details.\n\nSee Definition 2.3 in {{zb:0171.43603}}. This property is called *g-first countable* in {{zb:0285.54022}}.\n\n----\n#### Meta-properties\n\n- This property is hereditary with respect to open sets.\n- This property is hereditary with respect to closed sets.\n- $X$ satisfies this property if its Kolmogorov quotient $\\text{Kol}(X)$ does.\n- This property is preserved by arbitrary disjoint unions.","refs":[{"name":"Mappings and spaces (Arkhangel’skiĭ)","zb":"0171.43603"},{"name":"On defining a space by a weak base. (Siwiec)","zb":"0285.54022"}]},{"name":"Semilocally simply connected","aliases":["Semilocally $1$-connected"],"uid":"P000229","description":"Every point $x \\in X$ has a neighborhood $U$ such that the homomorphism $i_*:\\pi_1(U, x) \\to\\pi_1(X,x)$ of fundamental groups induced by the inclusion $i: U \\hookrightarrow X$ is trivial.\n\nEquivalently, every point $x \\in X$ has a neighborhood $U$ such that for every loop in $U$ based at $x$ there exists a basepoint-preserving homotopy in $X$ to the constant loop at $x$.\n\nDefined on page 494 of {{zb:0951.54001}} and on page 63 of {{zb:1044.55001}}.\n\n----\n#### Meta-properties\n\n- This property is hereditary with respect to clopen sets.\n- $X$ satisfies this property iff each of its path components does.\n- $X$ satisfies this property iff its Kolmogorov quotient $\\text{Kol}(X)$ does.\n- This property is preserved by arbitrary disjoint unions.\n- This property is preserved by finite products.\n- If each point has a neighborhood with the property, $X$ also has the property.","refs":[{"name":"Semilocally simply connected space on Wikipedia","wikipedia":"Semi-locally simply connected"},{"name":"Topology (Munkres)","zb":"0951.54001"},{"name":"Algebraic Topology (Hatcher)","zb":"1044.55001"}]},{"name":"Locally simply connected","aliases":[],"uid":"P000230","description":"$X$ admits a basis of open sets which are {P200}.\n\nEquivalently, for each $x \\in X$, every neighborhood of $x$ contains a simply connected open neighborhood of $x$.\n\nDefined on page 298 of {{zb:1209.57001}}, and listed as property $P_1$ in {{mo:487326}}.\n\nHas also been called \"strongly locally simply connected\", for example in {{zb:0209.54802}}.","refs":[{"name":"Introduction to topological manifolds (Lee)","zb":"1209.57001"},{"name":"Definition of locally simply connected space","mo":487326},{"name":"On the strong local simple connectivity of the decomposition spaces of toroidal decompositions (S. Armentrout)","zb":"0209.54802"}]},{"name":"Weakly locally simply connected","aliases":[],"uid":"P000231","description":"Every point of $X$ has a neighborhood which is {P200}. \n\nDefined as \"locally simply connected\" on page 54 of {{zb:0063.00842}} and listed as property $P_4$ in {{mo:487326}}.","refs":[{"name":"Theory of Lie groups. I (Chevalley)","zb":"0063.00842"},{"name":"Definition of locally simply connected space","mo":487326}]},{"name":"$LC^1$","aliases":["Locally simply connected","Locally $1$-connected"],"uid":"P000232","description":"$X$ is *locally $0$-connected* and *locally $1$-connected*.\n\nFollowing Borsuk's terminology (see page 30 of {{zb:0153.52905}}), this is the case $n=1$ in the hierarchy of $LC^n$ properties.\nA space $X$ is *locally $n$-connected* if for each $x\\in X$ every neighborhood $N$ of $x$ contains a neighborhood $U$ of $x$\nsuch that every map $S^n \\to N$ with values in $U$ is null-homotopic in $N$.\nAnd $X$ satisfies the $LC^n$ property if it is locally $k$-connected for $k=0,1,\\dots,n$.\n(Note: $LC^0$ is equivalent to {P42}.)\n\nEquivalently, for each $x \\in X$, every neighborhood $N$ of $x$ contains a {P37} neighborhood $U$ of $x$\nsuch that every loop $\\phi:S^1\\to U$ is null-homotopic in $N$. See {{mathse:5126526}}.\n\nSome authors, for example {{zb:0234.55001}} and {{zb:1280.54001}}, use \"locally 1-connected\" with the meaning of $LC^1$.\nHas also been called \"locally simply connected\", for example in {{zb:0198.56303}}.\nListed as property $P_{10}$ in {{mo:487326}}.\n\n----\n#### Meta-properties\n- This property is preserved by retractions (use Theorem 16.2 on p. 29 of {{zb:0153.52905}}).","refs":[{"name":"Theory of retracts (Borsuk)","zb":"0153.52905"},{"name":"Definition of locally simply connected space","mo":487326},{"name":"Is Borsuk's definition of $LC^1$ equivalent to this formulation of 'locally simply connected' ($P_{10}$)?","mathse":5126526},{"name":"Acyclicity in three-manifolds (McMillan)","zb":"0198.56303"},{"name":"Lectures on algebraic topology (Dold)","zb":"0234.55001"},{"name":"Geometric aspects of general topology. (Sakai)","zb":"1280.54001"}]},{"name":"Has open path components","aliases":[],"uid":"P000233","description":"Every path component of $X$ is open.\n\nThis is equivalent to each of the following:\n\n* Each point $x\\in X$ has a path connected neighborhood.\n\n* Each point $x\\in X$ has an open path connected neighborhood.\n\n* $X$ is a topological disjoint union of {P37} spaces.\n\nThe equivalence between the various conditions can be shown by arguments similar to those used in the definition of {P42}, in particular {{mathse:3002235}}.\n\nThis property is used in the literature, but usually without a separate name.\nOne exception is {{zb:0451.54017}} (accessible [here](https://topology.nipissingu.ca/tp/reprints/v04/tp04204.pdf)),\nwhere it is called \"PCGS\" (path connectedly generated space).\n\nCompare with {P42} and {P234}.\n\n----\n#### Meta-properties\n\n- $X$ satisfies this property iff its Kolmogorov quotient $\\text{Kol}(X)$ does.\n- This property is hereditary with respect to clopen sets.\n- This property is preserved by finite products.\n- This property is preserved by arbitrary disjoint unions.","refs":[{"name":"Topologies determined by paths (S. Franklin, B. Smith-Thomas)","zb":"0451.54017"},{"name":"Answer to \"Definition of locally pathwise connected\"","mathse":3002235}]},{"name":"Has open connected components","aliases":[],"uid":"P000234","description":"Every connected component of $X$ is open.\n\nThis is equivalent to each of the following:\n\n* Each point $x\\in X$ has a connected neighborhood.\n\n* Each point $x\\in X$ has an open connected neighborhood.\n\n* $X$ is a topological disjoint union of {P36} spaces.\n\nThe equivalence between the various conditions can be shown by arguments similar to those used in the definition of {P42}, in particular {{mathse:3002235}}.\n\nThis property is normally used without a separate name. But see {{zb:0451.54017}}\n(accessible [here](https://topology.nipissingu.ca/tp/reprints/v04/tp04204.pdf)),\nwhere it is called \"CGS\" (connectedly generated space).\n\nCompare with {P41} and {P233}.\n\n----\n#### Meta-properties\n\n- $X$ satisfies this property iff its Kolmogorov quotient $\\text{Kol}(X)$ does.\n- This property is hereditary with respect to clopen sets.\n- This property is preserved by finite products.\n- This property is preserved by arbitrary disjoint unions.","refs":[{"name":"Answer to \"Definition of locally pathwise connected\"","mathse":3002235},{"name":"Topologies determined by paths (S. Franklin, B. Smith-Thomas)","zb":"0451.54017"}]},{"name":"Locally a Euclidean half-space","aliases":[],"uid":"P000235","description":"Each point has a neighborhood homeomorphic to an open subset of the closed upper half-space $\\mathbb R^n_+$ for some integer $n\\ge 0$, where\n\n$\\quad\\quad\\mathbb R^n_+:=\\{(x_1,\\dots,x_n)\\in\\mathbb R^n : x_n\\ge 0\\}$\n\nhas the Euclidean subspace topology.\nEquivalently, each point has a neighborhood homeomorphic to $\\mathbb R^n_+$ for some $n$.\nNote that the value of $n$ is allowed to differ between points;\nif it does not vary, then the space has the stronger\nproperty {P236}.\n\nIn the case $n=0$, the half-space $\\mathbb R^0_+$ is considered the same as $\\mathbb R^0=\\{0\\}$;\nthus a point with a neighborhood homeomorphic to\n$\\mathbb R^0_+$ is an isolated point.\n\nAn equivalent condition for non-isolated points $p\\in X$ is that\none of the following holds:\n- (*\"manifold interior\" point*) $p$ has a neighborhood homeomorphic to $\\mathbb R^n;$\n- (*\"manifold boundary\" point*) $p$ has a neighborhood homeomorphic to $\\mathbb R^n_+$\nwith $p$ mapped to a point of $\\partial\\mathbb R^n_+:=\\{x\\in\\mathbb R^n: x_n=0\\}$.\n\n*Note*: In all the above, one can equivalently require the neighborhoods to be open.\n\nThe closely related {P236} property is the main ingredient used to define {P237}.","refs":[]},{"name":"Locally an $n$-Euclidean half-space","aliases":["Locally a Euclidean half-space of dimension $n$"],"uid":"P000236","description":"There exists some integer $n\\ge 0$ such that each point of $X$ has a neighborhood\nhomeomorphic to an open subset of the closed upper half-space $\\mathbb R^n_+$, where\n\n$\\quad\\quad\\mathbb R^n_+:=\\{(x_1,\\dots,x_n)\\in\\mathbb R^n : x_n\\ge 0\\}$\n\nhas the Euclidean subspace topology.\nEquivalently, each point has a neighborhood homeomorphic to $\\mathbb R^n_+$.\nNote that the value of $n$ is not allowed to differ between points;\nif it can vary, then the space has the weaker property {P235}.\n\nIn the case $n=0$, the half-space $\\mathbb R^0_+$ is considered the same as $\\mathbb R^0=\\{0\\}$;\nthus a point with a neighborhood homeomorphic to $\\mathbb R^0_+$ is an isolated point.\n\nEquivalently when $n>0$, each point $p\\in X$ satisfies one of the following:\n- (*\"manifold interior\" point*) $p$ has a neighborhood homeomorphic to $\\mathbb R^n;$\n- (*\"manifold boundary\" point*) $p$ has a neighborhood homeomorphic to $\\mathbb R^n_+$\nwith $p$ mapped to a point of $\\partial\\mathbb R^n_+:=\\{x\\in\\mathbb R^n: x_n=0\\}$.\n\n*Note*: In all the above, one can equivalently require the neighborhoods to be open.\n\nThis property is the main ingredient in the definition of {P237}.\n\nMentioned on page 42 of {{zb:1209.57001}}. See also {{wikipedia:Manifold_with_boundary}}.","refs":[{"name":"Introduction to topological manifolds (Lee, 2011)","zb":"1209.57001"},{"name":"Manifold with boundary on Wikipedia","wikipedia":"Manifold_with_boundary"}]},{"name":"Topological $n$-manifold with boundary","aliases":["$n$-dimensional topological manifold with boundary"],"uid":"P000237","description":"$X$ is {P236} and {P3} and {P27}.\n\nDefined on page 42 of {{zb:1209.57001}}. See also {{wikipedia:Manifold_with_boundary}}.","refs":[{"name":"Introduction to topological manifolds (Lee, 2011)","zb":"1209.57001"},{"name":"Manifold with boundary on Wikipedia","wikipedia":"Manifold_with_boundary"}]},{"name":"Has a real TVS topology","aliases":["Topological vector space","TVS"],"uid":"P000238","description":"$X$ is homeomorphic to a [topological vector space](https://en.wikipedia.org/wiki/Topological_vector_space) (TVS) over $\\mathbb R$.\n\nEquivalently, there exists a continuous commutative group operation $(x, y) \\mapsto x + y$ on $X$, and a continuous scalar multiplication operation $\\mathbb{R} \\times X \\to X$, $(\\lambda, x) \\mapsto \\lambda x$, where $\\mathbb{R}$ has the Euclidean topology, such that these operations together satisfy the axioms of a real vector space.\n\nSome others require separation axioms like {P3} or {P2}, though we do not.\n\n----\n#### Meta-properties\n\n- This property is preserved by arbitrary products.\n- This property is preserved by $\\Sigma$-products.","refs":[{"name":"Topological vector space on Wikipedia","wikipedia":"Topological_vector_space"},{"name":"Functional analysis (W. Rudin, 1991)","zb":"0867.46001"}]},{"name":"Semilocally contractible","aliases":["Weakly locally contractible"],"uid":"P000239","description":"Every point $x \\in X$ has a neighborhood (equivalently, an open neighborhood) $U$ that is\ncontractible in $X$; i.e., such that the inclusion map\n$U \\hookrightarrow X$ is a null-homotopy.\n\nEquivalently, every point $x \\in X$ has a neighborhood\n(equivalently, an open neighborhood) $U$ of\n$x$ such that the inclusion map $U \\hookrightarrow X$ is\nhomotopic to the constant map with value $x$.\n\nDefined as \"weakly locally contractible\" and given the\nacronym \"wlc\" in {{zb:0087.38203}}. Defined as \"semi-locally contractible\" in {{zb:0506.54015}}.\n\n----\n#### Meta-properties\n\n- This property is hereditary with respect to clopen sets.\n- $X$ satisfies this property iff its Kolmogorov quotient $\\text{Kol}(X)$ does.\n- This property is preserved by arbitrary disjoint unions.\n- This property is preserved by finite products.\n- If each point has a neighborhood with the property, $X$ also has the property.","refs":[{"name":"On fiber spaces (Fadell)","zb":"0087.38203"},{"name":"A characterization of local equi-connectedness (Sakai)","zb":"0506.54015"}]},{"name":"CW complex","aliases":[],"uid":"P000240","description":"$X$ has a chain of subspaces $\\empty = X_{-1}\\subseteq X_0\\subseteq X_1 \\subseteq \\dots$, such that:\n\n- $X_n$ is obtained from $X_{n-1}$ by attaching $n$-cells.\n- $X = \\bigcup_{n\\geq 0} X_n$ and a subspace $U \\subseteq X$ is open iff $U \\cap X_n$ open in $X_n$ for all $n \\geq -1$.\n\nHere *attaching $n$-cells* means that there is a discrete space $J$, and for every $j \\in J$ a continuous map $f_j:\\partial D^n \\to X_{n-1}$, such that $X_n$ is homeomorphic to the quotient $(X_{n-1} \\sqcup (J \\times D^n)) /{\\sim}$,\nwhere $\\sim$ is the equivalence relation generated by $(j,x) \\sim f_j(x)$ for all $(j,x)\\in J \\times \\partial D^n$.\nThe set $J$ is allowed to be empty, in which case $X_n=X_{n-1}$.\nThe $0$-skeleton $X_0$ has the discrete topology; if it is empty, $X$ itself is empty.\n\nHere $D^n$ is the closed unit disk in $\\mathbb R^n$ and $\\partial D^n=S^{n-1}$ is the unit sphere in $\\mathbb R^n$. We set $\\partial D^0=S^{-1}=\\emptyset$ by convention.\n\n*Note*: A *CW structure* on a topological space $X$ is a filtration $\\emptyset = X_{-1}\\subseteq X_0\\subseteq X_1 \\subseteq\\dots$ satisfying the conditions above.\nStrictly speaking, a *CW complex* is a space $X$ together with a compatible CW structure.\nFor simplicity and by a slight abuse of language, we call the space itself a CW complex if it admits a CW structure.\n\nDefined on page 5 of {{zb:1044.55001}}, also given in Theorem II.2.4 of {{zb:0207.21704}}.\nSee also .\n\n----\nAn equivalent definition can be given in terms of a decomposition of $X$ into cells and conditions on their associated characteristic maps.\nFor details and the meaning of this terminology, see Proposition A.2 on page 521 of {{zb:1044.55001}} or Definition II.1.1 in {{zb:0207.21704}}.\n\n----\n#### Meta-properties\n\n- This property is preserved by arbitrary disjoint unions.","refs":[{"name":"Algebraic Topology (Hatcher)","zb":"1044.55001"},{"name":"The Topology of CW Complexes (Lundell & Weingram)","zb":"0207.21704"},{"name":"CW complex on Wikipedia","wikipedia":"CW_complex"}]},{"name":"Locally a Euclidean half-line","aliases":["Locally a $1$-Euclidean half-space"],"uid":"P000241","description":"Each point has a (not necessarily open) neighborhood homeomorphic to the closed half-line $\\mathbb R_{\\ge 0}$,\nalso homeomorphic to {S210}.\n\nThis is the special case of {P236} with $n=1$.\n\nEquivalently, each point $p\\in X$ satisfies one of the following:\n- $p$ has an open neighborhood homeomorphic to {S25};\n- $p$ has an open neighborhood homeomorphic to {S210}\nwith $p$ mapped to the element $0\\in[0,1)$.\n\n----\n#### Meta-properties\n\n- This property is preserved by arbitrary disjoint unions.","refs":[]},{"name":"Weakly contractible","aliases":["Homotopically trivial","$C^\\infty$"],"uid":"P000242","description":"$X$ is nonempty and for each integer $n\\ge 0$ every continuous map $S^n\\to X$ is homotopic to a constant map.\n\nEquivalently, $X$ is nonempty, {P37} and all [homotopy groups](https://en.wikipedia.org/wiki/Homotopy_group) $\\pi_n(X)$ with $n\\ge 1$ are trivial.\nIn other words, $X$ is [weakly homotopy equivalent](https://en.wikipedia.org/wiki/Weak_equivalence_(homotopy_theory)) to {S162}.\n\nNotation above: $S^n$ is the unit sphere in $\\mathbb R^{n+1}$.\n\nDefined as *weakly contractible* for example on page 2 in {{zb:0979.55010}}\n(accessible [here](https://www.academia.edu/53860578/On_the_universal_space_for_group_actions_with_compact_isotropy)).\nDefined as *homotopically trivial* on page 281 of {{zb:1467.20042}}.\nBorsuk uses the terminology $C^\\infty$, defined as $X$ being *$n$-connected* for all $n\\ge 0$\n(see page 30 of {{zb:0153.52905}}).\n\n----\n#### Meta-properties\n\n- This property is preserved by arbitrary products (see Proposition 4.2 in {{zb:1044.55001}}).\n- This property is preserved by homotopy equivalences.","refs":[{"name":"Weakly contractible space on Wikipedia","wikipedia":"Weakly_contractible_space"},{"name":"Algebraic Topology (Hatcher)","zb":"1044.55001"},{"name":"On the universal space for group actions with compact isotropy (Lück, Meintrup)","zb":"0979.55010"},{"name":"A stronger reformulation of Webb’s conjecture in terms of finite topological spaces (Piterman)","zb":"1467.20042"},{"name":"Theory of retracts (Borsuk)","zb":"0153.52905"}]},{"name":"Has countable $\\pi$-weight","aliases":[],"uid":"P000243","description":"$X$ has a countable $\\pi$-base.\n\nA *$\\pi$-base* for $X$ is a collection $\\mathcal{V}$ of nonempty open sets in $X$ such that every nonempty open $U\\subseteq X$ contains some $V\\in\\mathcal V$.\n\nSuch a space has countable $\\pi$-weight, where the *$\\pi$-weight* is defined as\n\n$\\quad\\quad\\pi w(X):=\\min\\{|\\mathcal V|:\\mathcal V \\text{ is a }\\pi\\text{-base for }X\\}+\\omega.$\n\nDefined on page 14 of {{zb:0559.54003}}.\n\n----\n#### Meta-properties\n\n- This property is hereditary with respect to dense sets.\n- This property is hereditary with respect to open sets.\n- If some open dense subset of $X$ has the property, then so does $X$.\n- $X$ satisfies this property iff its Kolmogorov quotient $\\operatorname{Kol}(X)$ does.\n- This property is preserved by countable products.\n- This property is preserved by countable disjoint unions.","refs":[{"name":"Cardinal functions I (R. Hodel), Ch. 1 of Handbook of set-theoretic topology","zb":"0559.54003"}]},{"name":"Has countable $\\pi$-character","aliases":[],"uid":"P000244","description":"Every point in $X$ has a countable local $\\pi$-base.\n\nA *local $\\pi$-base* for $x \\in X$ is a collection $\\mathcal{V}_x$ of nonempty open sets in $X$ such that every neighbourhood $U$ of $x$ contains some $V \\in \\mathcal V_x$.\n(Note: We don't require that $x \\in V$, i.e., the elements of $\\mathcal{V}_x$ need not be neighbourhoods of $x$.)\n\nSuch a space has countable $\\pi$-character, where the *$\\pi$-character* is defined as\n\n$\\quad\\quad\\pi\\chi(X):=\\sup\\{\\pi\\chi(x,X):x\\in X\\}+\\omega$\n\nwith $\\pi\\chi(x,X)$ being the minimum size of a local $\\pi$-base for $x$.\n\nDefined on page 16 of {{zb:0559.54003}}.\n\n----\n#### Meta-properties\n\n- This property is hereditary with respect to dense sets.\n- This property is hereditary with respect to open sets.\n- $X$ satisfies this property iff its Kolmogorov quotient $\\operatorname{Kol}(X)$ does.\n- This property is preserved by countable products.","refs":[{"name":"Cardinal functions I (R. Hodel), Ch. 1 of Handbook of set-theoretic topology","zb":"0559.54003"}]},{"name":"Has finitely many open sets","aliases":[],"uid":"P000245","description":"$X$ has finitely many open sets.\n\nEquivalently, the Kolmogorov quotient of $X$ is finite.\n\n----\n#### Meta-properties\n\n- This property is hereditary.\n- This property is preserved in any coarser topology.\n- $X$ is {P245} iff its Kolmogorov quotient $\\text{Kol}(X)$ is {P78}.\n- This property is preserved by finite products.","refs":[]},{"name":"Collectionwise Hausdorff","aliases":["CWH"],"uid":"P000246","description":"Every closed {P52} subset $A\\subseteq X$ can be covered by a family of pairwise disjoint open subsets of $X$, each meeting $A$ in only one point.\n\nDefined for example on pages 382 and 640 of {{zb:0546.00022}}.\n\n----\n#### Meta-properties\n\n- $X$ satisfies this property if its Kolmogorov quotient $\\text{Kol}(X)$ does.\n- This property is preserved by arbitrary disjoint unions.","refs":[{"name":"Handbook of set-theoretic topology (Kunen & Vaughan, eds.)","zb":"0546.00022"},{"name":"Collectionwise Hausdorff space on Wikipedia","wikipedia":"Collectionwise_Hausdorff_space"},{"name":"Collectionwise Hausdorff versus collectionwise normal with respect to compact sets (Reed)","zb":"0529.54017"}]}],"spaces":[{"name":"Discrete topology on $\\{0,1\\}$","aliases":["Discrete topology on a two-point set","Finite discrete topology"],"uid":"S000001","counterexamples_id":1,"description":"Let $X=2=\\{0,1\\}$, a set with two elements, and\nlet all subsets of $X$ be open.\n\nDefined as counterexample #1 (\"Finite Discrete Topology\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Discrete space on Wikipedia","wikipedia":"Discrete_space"}]},{"name":"Discrete topology on $\\omega$","aliases":["Discrete topology on a countably infinite set","Countable discrete topology","Sierpinski's metric space","Natural numbers $\\mathbb N$","Integers $\\mathbb Z$"],"uid":"S000002","counterexamples_id":2,"description":"Let $X=\\omega=\\{0,1,2,\\dots\\}$, a set with a countably infinite\nnumber of points, and let all subsets of $X$ be open.\n\nDefined as counterexample #2 (\"Countable Discrete Topology\")\nin {{zb:0386.54001}}.\n\nThis space is metrizable, with the standard metric for a discrete space given by $d(x,y)=1$ for $x\\ne y$. A different metric that generates the same topology for $X$ is the Sierpiński metric ($d(x,y)=1+1/(x+y+2)$ for $x\\ne y$ in $\\omega$) given in counterexample #135 in {{zb:0386.54001}}. Since pi-base is meant to model topological properties and not properties of metrics, it only keeps track of topological spaces up to homeomorphism, and thus does not keep a separate entry for the Sierpiński metric.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Discrete space on Wikipedia","wikipedia":"Discrete_space"}]},{"name":"Discrete topology on $\\mathbb R$","aliases":["Discrete topology on the real numbers","Uncountable Discrete Topology"],"uid":"S000003","counterexamples_id":3,"description":"Let $X=\\mathbb R$ be the set of all real numbers,\nand let all subsets of $X$ be open.\n\nDefined as counterexample #3 (\"Uncountable Discrete Topology\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Discrete space on Wikipedia","wikipedia":"Discrete_space"}]},{"name":"Indiscrete topology on $\\{0,1\\}$","aliases":["Indiscrete topology on a two-point set"],"uid":"S000004","counterexamples_id":4,"description":"Let $X=2=\\{0,1\\}$, a set with two elements,\nand let $X$ and $\\emptyset$ be the only open sets.\n\nDefined as counterexample #4 (\"Indiscrete Topology\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Trivial topology on Wikipedia","wikipedia":"Trivial_topology"}]},{"name":"Odd-Even Topology","aliases":[],"uid":"S000005","counterexamples_id":6,"description":"Define a topology on $X = \\mathbb{N}$ by taking as a basis all pairs $\\{2k-1,2k\\}$ for $k \\in \\mathbb{N}$.\n(Here we assume $0\\notin\\mathbb{N}$.)\n\nHomeomorphic to the product of {S2} and {S4}.\n\nDefined as counterexample #6 (\"Odd-Even Topology\")\nin {{zb:0386.54001}}.\n\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Partition topology on Wikipedia","wikipedia":"Partition_topology"}]},{"name":"Deleted integer topology on $\\mathbb R\\setminus\\mathbb Z$","aliases":[],"uid":"S000006","counterexamples_id":7,"description":"Let $X=\\mathbb R\\setminus\\mathbb Z$, and\ntake as a basis $\\{(n,n+1)\\mid n \\in \\mathbb{Z}\\}$.\n\nEquivalently, the product of {S2} and {S194}.\n\nDefined as counterexample #7 (\"Deleted Integer Topology\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Partition topology on Wikipedia","wikipedia":"Partition_topology"}]},{"name":"Particular point topology on a three-point set","aliases":["Included point topology on a three-point set","Finite Particular Point Topology"],"uid":"S000007","counterexamples_id":8,"description":"Let $X=\\{0,1,2\\}$ be a finite set with three elements.\nA set is open in this topology if it contains the particular point $p=0$ or is empty.\n\nDefined as counterexample #8 (\"Finite Particular Point Topology\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Particular point topology on Wikipedia","wikipedia":"Particular_point_topology"}]},{"name":"Particular point topology on a countably infinite set","aliases":["Included point topology on $\\omega$","Countable Particular Point Topology"],"uid":"S000008","counterexamples_id":9,"description":"Let $X=\\omega=\\{0,1,2,\\dots\\}$.\nA set is open in this topology if it contains the particular point $p=0$ or is empty.\n\nDefined as counterexample #9 (\"Countable Particular Point Topology\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Particular point topology on Wikipedia","wikipedia":"Particular_point_topology"}]},{"name":"Particular point topology on $\\mathbb R$","aliases":["Included point topology on $\\mathbb R$","Uncountable Particular Point Topology"],"uid":"S000009","counterexamples_id":10,"description":"Let $X=\\mathbb R$ be the set of real numbers. A set is open in this\ntopology if it contains the particular point $p=0$ or is empty.\n\nDefined as counterexample #10 (\"Uncountable Particular Point Topology\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Particular point topology on Wikipedia","wikipedia":"Particular_point_topology"}]},{"name":"Sierpinski space","aliases":["Particular point topology on a two-point set","Excluded point topology on a two-point set","Right ray topology on a two-point set","Left ray topology on a two-point set"],"uid":"S000010","counterexamples_id":11,"description":"Let $X = \\{0,1\\}$ with open sets $\\{\\emptyset, \\{0\\}, X \\}$.\nThis space may be characterized as the particular point ($0$) topology on a\ntwo-point set, or the excluded point ($1$) topology on a two-point set.\n\nDefined as counterexample #11 (\"Sierpinski Space\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Sierpinski space on Wikipedia","wikipedia":"Sierpinski_space"}]},{"name":"Excluded Point Topology on a Three-Point Set","aliases":["Finite Excluded Point Topology"],"uid":"S000011","counterexamples_id":13,"description":"Let $X=\\{0,1,2\\}$ be a finite set with three elements.\nA set is closed in this topology if it contains the particular point $p=0$ or is empty. A set is open if it does not contain $p$ or is the whole space.\n\nDefined as counterexample #13 (\"Finite Excluded Point Topology\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Excluded point topology on Wikipedia","wikipedia":"Excluded_point_topology"}]},{"name":"Excluded Point Topology on a Countably Infinite Set","aliases":["Countable Excluded Point Topology"],"uid":"S000012","counterexamples_id":14,"description":"Let $X=\\omega=\\{0,1,2,\\dots\\}$.\nA set is closed in this topology if it contains $0$ or is empty.\n\nDefined as counterexample #14 (\"Countable Particular Point Topology\")\nin {{zb:0386.54001}}.\n\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Excluded point topology on Wikipedia","wikipedia":"Excluded_point_topology"}]},{"name":"Excluded point topology on $\\mathbb R$","aliases":["Uncountable excluded point topology"],"uid":"S000013","counterexamples_id":15,"description":"Let $X=\\mathbb R$ be the set of real numbers. A set is open in this\ntopology if it excludes a particular point $p=0$ or is the whole space.\n\nDefined as counterexample #15 (\"Uncountable Excluded Point Topology\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Excluded point topology on Wikipedia","wikipedia":"Excluded_point_topology"}]},{"name":"Either-Or Topology","aliases":[],"uid":"S000014","counterexamples_id":17,"description":"Let $X = [-1,1]$ as a set and declare $U \\subset X$ open if and only if $0 \\not\\in U$ or $(-1,1) \\subset U$.\n\nThe space is homeomorphic to the topological sum of\n{S13} and {S1}.\n(The sets $\\{-1\\}$, $\\{1\\}$, $(-1,1)$ form a partition of $X$ into clopen sets.\nAnd the subspace topology on the interval $(-1,1)$ is an \"excluded point topology\" on a set of cardinality $\\mathfrak c$.)\n\nDefined as counterexample #17 (\"Either-Or Topology\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Either-or topology on Wikipedia","wikipedia":"Either-or_topology"}]},{"name":"Cofinite topology on $\\omega$","aliases":["Finite complement topology on a countably infinite set","Finite Complement Topology on a Countable Space"],"uid":"S000015","counterexamples_id":18,"description":"The finite complement topology on a space $X$ is defined by taking\n$U \\subset X$ to be open if and only if $X \\setminus U$ is finite or\n$U = \\emptyset$. In this case we let \$X=\\omega=\\{0,1,2,\\dots\\}\$.\n\nDefined as counterexample #18 (\"Finite Complement Topology on a Countable Space\")\nin {{zb:0386.54001}}.\n\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Finite complement topology on Wikipedia","wikipedia":"Finite_complement_topology"}]},{"name":"Cofinite topology on $\\mathbb R$","aliases":["Finite complement topology on the real numbers","Finite Complement Topology on an Uncountable Space","Cofinite topology on the real numbers"],"uid":"S000016","counterexamples_id":19,"description":"The finite complement topology on a set $X$ is defined by taking\n$U \\subset X$ to be open if and only if $X \\setminus U$ is finite or\n$U = \\emptyset$. In this case we let \$X=\\mathbb R\$ so it has cardinality\n\$\\mathfrak c\$.\n\nDefined as counterexample #19 (\"Finite Complement Topology on an Uncountable Space\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Finite complement topology on Wikipedia","wikipedia":"Finite_complement_topology"}]},{"name":"Cocountable topology on $\\mathbb R$","aliases":["Cocountable topology on the real numbers","Countable complement topology"],"uid":"S000017","counterexamples_id":20,"description":"Let $X=\\mathbb R$ be the set of real numbers. Define the topology on $X$ by letting a set in $X$ be open iff it is empty or its complement is countable. \n\nDefined as counterexample #20 (\"Countable Complement Topology\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Cocountable topology on Wikipedia","wikipedia":"Cocountable_topology"}]},{"name":"Double pointed cocountable topology on $\\mathbb R$","aliases":["Double pointed countable complement topology on $\\mathbb R$"],"uid":"S000018","counterexamples_id":21,"description":"$X$ is the product of {S17} with {S4}.\n\nDefined as counterexample #21 (\"Double Pointed Countable Complement Topology\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Compact Complement Topology for the Euclidean Real Numbers","aliases":["Compact Complement Topology"],"uid":"S000019","counterexamples_id":22,"description":"Define a topology $\\tau$ on $\\mathbb{R}$ by letting $U \\subset \\mathbb{R}$ be open if and only if $\\mathbb{R} \\setminus U$ is compact in the usual Euclidean topology or $U=\\emptyset$.\n\nNote that this topology is coarser than {S25}.\n\nDefined as counterexample #22 (\"Compact Complement Topology\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Compact complement topology","wikipedia":"Compact_complement_topology"}]},{"name":"Fort space on a countably infinite set","aliases":["Countable Fort space","Converging sequence","Ordinal space $\\omega+1$"],"uid":"S000020","counterexamples_id":23,"description":"Let $X=\\omega\\cup\\{\\infty\\}=\\{0,1,2,\\dots\\}\\cup\\{\\infty\\}$.\nDefine $U \\subseteq X$ to be open if its complement either is finite or includes $\\infty$.\n\nThis space is the one-point compactification of a countably infinite discrete space.\nIt is homeomorphic to $\\{0\\}\\cup\\{2^{-n}:n<\\omega\\}$ as a subspace of {S25},\nand is also homeomorphic to the ordinal space $\\omega+1$.\n\nDefined as counterexample #23 (\"Countable Fort Space\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Fort space on Wikipedia","wikipedia":"Fort_space"}]},{"name":"Weak topology on separable Hilbert space","aliases":[],"uid":"S000021","counterexamples_id":null,"description":"Let $H$ be an infinite-dimensional separable real Hilbert space (such\nas the space $l^2$ of square-summable sequences) with inner product\n$\\langle \\cdot, \\cdot \\rangle$. (All such spaces are isometrically\nisomorphic.) The continuous dual $H^*$ is the set of all linear\nfunctionals $f : H \\to \\mathbb{R}$ that are continuous with respect to\nthe topology generated by the inner product. By the Riesz\nrepresentation theorem, every such functional $f$ is of the form $f(x)\n= \\langle x, y \\rangle$ for some $y \\in H$.\n\nThe weak topology on the set $H$ is the coarsest topology such that\nevery element of $H^*$ is continuous. That is, a basis for this\ntopology at $x \\in H$ is the collection of all sets of the form $\\{ y\n\\in H : |\\langle y -x , x_i\\rangle| < \\epsilon, i = 1, \\dots, n \\}$\nfor some $x_1, \\dots, x_n \\in H$ and some $\\epsilon > 0$.\n\nSee Definition 5.1.1 of {{doi:10.1007/978-93-86279-42-2}} or 28.16 of\n{{doi:10.1016/B978-0-12-622760-4.X5000-6}}. Note that the weak\ntopology on $H$ is homeomorphic to the weak-\\* topology on $H^*$.","refs":[{"name":"Handbook of Analysis and Its Foundations","doi":"10.1016/B978-0-12-622760-4.X5000-6"},{"name":"Functional Analysis","doi":"10.1007/978-93-86279-42-2"},{"name":"Weak topology on Wikipedia","wikipedia":"Weak_topology"}]},{"name":"Fortissimo space on $\\mathbb R$","aliases":["One-point Lindelöfication of uncountable discrete space"],"uid":"S000022","counterexamples_id":25,"description":"Let $X=\\mathbb R$ and let $p$ be a distinguished point of $X$.\nEvery point not equal to $p$ is isolated and the open neighborhoods of $p$ are the cocountable subsets of $X$ containing that point.\n\nThis space is the one-point Lindelöfication of an uncountable discrete space.\n\nCompare with {S155} and {S157}.\nWith a suitable assumption on $|\\mathbb R|$ so that the cardinalities match, $X$ will be homeomorphic to one of them.\n\nDefined as counterexample #25 (\"Fortissimo Space\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Fort space on Wikipedia","wikipedia":"Fort_space"}]},{"name":"Arens-Fort Space","aliases":[],"uid":"S000023","counterexamples_id":26,"description":"Let $X = \\{(n,m) : n,m \\in \\omega\\}$ with all singletons\nexcept $\\{(0,0)\\}$ open.\nDefine a set containing $(0,0)$ to be open if and only if it contains all but\na finite number of points in all but a finite number of columns.\n\nDefined as counterexample #26 (\"Arens-Fort Space\")\nin {{zb:0386.54001}}.\n\nThis space is homeomorphic to a subspace of {S156}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Arens-Fort space","wikipedia":"Arens-Fort_space"}]},{"name":"Modified Fort space on $\\mathbb R$","aliases":[],"uid":"S000024","counterexamples_id":27,"description":"Let $X=\\mathbb{R}\\cup\\{\\infty_1,\\infty_2\\}$.\nEvery point of $\\mathbb R$ is isolated and the open neighborhoods of each point $\\infty_i$ are the cofinite subsets of $X$ containing the point.\n\nThe subspaces $\\mathbb{R}\\cup\\{\\infty_1\\}$ and $\\mathbb{R}\\cup\\{\\infty_2\\}$ are each homeomorphic to {S154}.\n\nDefined as counterexample #27 (\"Modified Fort Space\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Fort space","wikipedia":"Fort_space"}]},{"name":"Euclidean real numbers $\\mathbb R$","aliases":["Real line","Euclidean topology"],"uid":"S000025","counterexamples_id":28,"description":"The usual topology on \$\\mathbb R\$ with a topology generated by the basis\nof open intervals \\(\\{(a,b):a0$ and $(a,1]$ with $a < 0$. Note that any $(a,b)$ with $a < 0 < b$ is open in $X$ and so this topology is coarser than the Euclidean topology on $[-1,1]$.\n\nDefined as counterexample #53 (\"Overlapping Interval Topology\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Overlapping interval topology on Wikipedia","wikipedia":"Overlapping_interval_topology"}]},{"name":"Interlocking interval topology","aliases":[],"uid":"S000046","counterexamples_id":54,"description":"Let $X = (0,\\infty)\\setminus\\mathbb Z$, the set of positive real numbers excluding the integers. Let $S_n = (0,1/n) \\cup (n,n+1)$ for $n \\geq 1$. Define a topology on $X$ by taking $\\{S_n\\}$ as a subbase.\n\nA base for the topology is given by the sets $S_n$ for $n\\ge 1$ together with the intervals $(0,1/n)$ for $n\\ge 2$.\n\nDefined as counterexample #54 (\"Interlocking Interval Topology\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Countable sum of Sierpinski spaces","aliases":["Hjalmar Ekdal topology"],"uid":"S000047","counterexamples_id":55,"description":"Let $X$ be the set of positive integers and define a topology by declaring $U \\subseteq X$ to be open if for every odd $p \\in U$, $p+1 \\in U$. Equivalently, $A \\subseteq X$ is closed iff for every even $p \\in A$, $p-1 \\in A$.\n\nThis space is a topological sum of countably-many copies of the {S10}.\n\nDefined as counterexample #55 (\"Hjalmar Ekdal Topology\")\nin {{zb:0386.54001}}.\n\nA piece of trivia: the origin of the name Hjalmar Ekdal is explained [here](https://proofwiki.org/wiki/Mathematician:Hjalmar_Ekdal).\n\n\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Cofinite topology on $\\omega$ extended by a non-open generic point","aliases":["$\\mathrm{Spec}(\\mathbb{Z})$","Prime ideal topology"],"uid":"S000048","counterexamples_id":56,"description":"Let $X = \\omega \\sqcup \\{p\\}$. A nonempty set is open if and only if it contains $p$ and has a finite complement. Hence the subspace $\\omega \\subseteq X$ equals {S15}. Note that $p$ is a generic point of $X$ and $\\{p\\}$ is not open; see {P201}.\n\n$X$ is homeomorphic to the set of prime ideals of the ring $\\mathbb{Z}$ with basis consisting of the sets $V_x = \\{P \\in X : x \\notin P\\}$ for all $x \\in \\mathbb{Z}_{\\ge 0}$. The zero ideal identifies with the generic point $p \\in X$. This is the Zariski topology on $\\mathrm{Spec}(\\mathbb{Z})$, the prime spectrum of the ring of integers. See {{wikipedia:Zariski_topology}} for the general construction of the Zariski topology on the prime spectrum of a commutative ring.\n\nAppears as counterexample #56 (\"Prime Ideal Topology\") in {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Zariski topology","wikipedia":"Zariski_topology"}]},{"name":"Divisor topology","aliases":[],"uid":"S000049","counterexamples_id":57,"description":"Let $X = \\{x \\in \\mathbb{N} : x \\geq 2\\}$ with topology generated by all $U_n = \\{x \\in X : x$ divides $n\\}$ for $n \\in X$.\n\nDefined as counterexample #57 (\"Divisor Topology\")\nin {{zb:0386.54001}}.\n\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Divisor topology on Wikipedia","wikipedia":"Divisor_topology"}]},{"name":"$\\mathbb Q$ extended by a focal point","aliases":["Rationals extended by a focal point","Non-Hausdorff cone over the rationals","Open extension of the rationals"],"uid":"S000050","description":"Let $X = \\mathbb{Q} \\cup \\{\\infty\\}$, where $\\mathbb{Q}$ has the usual topology\nof {S27} and is open in $X$,\nand the only neighborhood of $\\infty$ is $X$.\nSee {P202}.","refs":[]},{"name":"Khalimsky line","aliases":["Digital line"],"uid":"S000051","description":"The integers $\\mathbb Z$ with $\\{\\{2i-1,2i,2i+1\\}\\mid i\\in\\mathbb Z\\}\\cup\\{\\{2i+1\\}\\mid i\\in\\mathbb Z\\}$ as a basis for the topology.\n\nThis space is homeomorphic to:\n- The quotient of {S25} identifying the open interval $(i, i + 1)$ to a point for each $i \\in \\mathbb Z$.\n\nIt is the unique {P205} such that no proper subspace is also a cut point space.\nSee Theorem 4.5 of {{doi:10.1090/s0002-9939-99-04839-x}} for a proof.\n\nAlso named the *digital line* in the subject of digital topology, e.g. in {{doi:10.32219/isms.80.1_15}}.","refs":[{"name":"Cut-point spaces (Honari & Bahrampour)","doi":"10.1090/s0002-9939-99-04839-x"},{"name":"On Generalized Digital Lines (Nakaoka, Tamari, Maki)","doi":"10.32219/isms.80.1_15"}]},{"name":"Relatively prime integer topology","aliases":["Golomb Topology"],"uid":"S000052","counterexamples_id":60,"description":"Let $X=\\mathbb{Z}^+$, the set of positive integers. For $a,b \\in X$, let $U_a(b) = \\{b+na \\in X : n \\in \\mathbb{Z}\\} = (b+a\\mathbb{Z})\\cap X$. Then $\\{U_a(b) : a,b \\in X, \\gcd(a,b)=1\\}$ forms a basis for the relatively prime integer topology on $X$.\n\nAlternatively, a basis is given by $\\{U'_a(b) : a,b \\in X, \\gcd(a,b)=1\\}$ with $U'_a(b) = \\{b+na \\in X : n\\ge 0\\}$.\nSee {{mathse:4460847}} for a proof of the equivalence.\n\nAs a space, $X$ is often called the *Golomb space*, for example in {{zb:1474.54066}}.\nDefined as counterexample #60 (\"Relatively Prime Integer Topology\")\nin {{zb:0386.54001}}.\n\nCompare with {S53}, which has a coarser topology.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Arithmetic progression topologies on Wikipedia","wikipedia":"Arithmetic_progression_topologies"},{"name":"The Kirch topology is the same as the prime integer topology","mathse":4460847},{"name":"On continuous self-maps and homeomorphisms of the Golomb space (T. Banakh et al.)","zb":"1474.54066"}]},{"name":"Prime integer topology","aliases":["Kirch Topology"],"uid":"S000053","counterexamples_id":61,"description":"The prime integer topology on the set $X = \\mathbb{Z}^+$ of positive integers is generated by the subbasis of all sets of the form $U_p(b) = \\{b+np \\in X : n \\in \\mathbb{Z}\\}$ for all $p$ prime and $b$ not divisible by $p$. A base for the topology is given by the collection of all $U_a(b) = \\{b+na \\in X : n \\in \\mathbb{Z}\\}$ with $a,b>0$ and with $a$ squarefree and relatively prime to $b$.\n\nAlternatively, a base for the topology is given by $\\{U'_a(b) : a,b \\in X, \\gcd(a,b)=1, b\\text{ squarefree}\\}$ with $U'_a(b) = \\{b+na \\in X : n\\ge 0\\}$.\nSee {{mathse:4460847}} for a proof of the equivalence.\n\nAs a space, $X$ is often called the *Kirch space* (see references in {{wikipedia:Arithmetic_progression_topologies}}).\nDefined as counterexample #61 (\"Prime Integer Topology\")\nin {{zb:0386.54001}}.\n\nCompare with {S52}, which has a finer topology.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Arithmetic progression topologies on Wikipedia","wikipedia":"Arithmetic_progression_topologies"},{"name":"The Kirch topology is the same as the prime integer topology","mathse":4460847}]},{"name":"Double pointed reals","aliases":[],"uid":"S000054","counterexamples_id":62,"description":"The double pointed reals is the product $X$ of {S25} and {S4}.\n\nDefined as counterexample #62 (\"Double Pointed Reals\")\nin {{zb:0386.54001}}.\n\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Countable complement extension topology on the real numbers","aliases":[],"uid":"S000055","counterexamples_id":63,"description":"The countable complement extension topology on $\\mathbb{R}$ is the smallest topology generated by the union of the topologies from {S25} and {S17}.\n\nDefined as counterexample #63 (\"Countable Complement Extension Topology\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Smirnov's deleted sequence topology","aliases":["K-topology"],"uid":"S000056","counterexamples_id":64,"description":"Let $K = \\{\\frac{1}{n} : n \\in \\mathbb{N}\\}$. Smirnov's Deleted Sequence Topology is the topology on $\\mathbb{R}$ consisting of all sets of the form $U \\setminus B$ where $U \\subset \\mathbb{R}$ is open in the standard topology and $B \\subset K$.\n\nDefined as counterexample #64 (\"Smirnov's Deleted Sequence Topology\")\nin {{zb:0386.54001}}.\n\nCompare with {S206}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"K-topology on Wikipedia","wikipedia":"K-topology"}]},{"name":"Rational sequence topology","aliases":[],"uid":"S000057","counterexamples_id":65,"description":"For each irrational $x \\in \\mathbb{R}$ fix a sequence of rational numbers $x_i \\rightarrow x$. The rational sequence topology on $\\mathbb{R}$ is defined by declaring each rational point open and letting $U_n(x) = \\{x_i : i>n\\} \\cup \\{x\\}$ be a local basis at each irrational $x$.\n\nNote that in {{mo:447593}} it was shown there are $2^{\\mathfrak c}$ distinct topologies defined in this way, depending on the choices made for $x_i$. However, none of the properties currently tracked for this space rely on this information.\n\nDefined as counterexample #65 (\"Rational Sequence Topology\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Is every rational sequence topology homeomorphic?","mo":447593}]},{"name":"Indiscrete rational extension of $\\mathbb R$","aliases":[],"uid":"S000058","counterexamples_id":66,"description":"Let $D = \\mathbb{Q}$, the set of rational numbers.\nThe space $X$ is $\\mathbb R$ with the topology generated by the Euclidean open sets\ntogether with the sets of the form $D\\cap U$ with $U\\subseteq\\mathbb R$ Euclidean open.\n\nThe topology is finer than {S25}.\nAnd it is coarser than {S60}\nand coarser than {S62}.\n\nDefined as counterexample #66 (\"Indiscrete Rational Extension of $\\mathbb{R}$\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Indiscrete irrational extension of $\\mathbb R$","aliases":[],"uid":"S000059","counterexamples_id":67,"description":"Let $D=\\mathbb R\\setminus\\mathbb Q$, the set of irrational numbers.\nThe space $X$ is $\\mathbb R$ with the topology generated by the Euclidean open sets\ntogether with the sets of the form $D\\cap U$ with $U\\subseteq\\mathbb{R}$ Euclidean open.\n\nThe topology is finer than {S25}.\nAnd it is coarser than {S61}\nand coarser than {S63}.\n\nDefined as counterexample #67 (\"Indiscrete Irrational Extension of $\\mathbb{R}$\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Pointed rational extension of $\\mathbb{R}$","aliases":[],"uid":"S000060","counterexamples_id":68,"description":"Let $D = \\mathbb{Q}$, the set of rational numbers.\nThe space $X$ is $\\mathbb R$ with the topology generated by the sets of the form $\\{x\\}\\cup(D\\cap U)$\nwhere $U\\subset\\mathbb{R}$ is open in the Euclidean topology and $x\\in U$.\n\nThe topology is finer than {S58}.\n\nDefined as counterexample #68 (\"Pointed Rational Extension of $\\mathbb{R}$\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Pointed irrational extension of $\\mathbb R$","aliases":[],"uid":"S000061","counterexamples_id":69,"description":"Let $D=\\mathbb R\\setminus\\mathbb Q$, the set of irrational numbers.\nThe space $X$ is $\\mathbb R$ with the topology generated by the sets of the form $\\{x\\}\\cup(D\\cap U)$\nwhere $U\\subseteq\\mathbb R$ is open in the Euclidean topology and $x\\in U$.\n\nThe topology is finer than {S59}.\n\nDefined as counterexample #69 (\"Pointed Irrational Extension of $\\mathbb{R}$\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Discrete rational extension of $\\mathbb R$","aliases":[],"uid":"S000062","counterexamples_id":70,"description":"Let $D = \\mathbb{Q}$, the set of rational numbers.\nThe space $X$ is $\\mathbb R$ with the topology generated by the Euclidean open sets\ntogether with all singletons $\\{x\\}$ for $x\\in D$.\n(So any subset of $D$ is open in $X$.)\n\nThe topology is finer than {S58}.\n\nDefined as counterexample #70 (\"Discrete Rational Extension of $\\mathbb{R}$\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Michael line","aliases":["Discrete irrational extension of $\\mathbb R$"],"uid":"S000063","counterexamples_id":71,"description":"Let $D=\\mathbb R\\setminus\\mathbb Q$, the set of irrational numbers.\nThe space $X$ is $\\mathbb R$ with the topology generated by the Euclidean open sets\ntogether with all singletons $\\{x\\}$ for $x\\in D$.\n(So any subset of $D$ is open in $X$.)\n\nThe topology is finer than {S59}.\n\nThe use of this space by [Ernest Michael](https://en.wikipedia.org/wiki/Ernest_Michael) in {{doi:10.1090/S0002-9904-1963-10931-3}} resulted in it being known as the \"Michael line\".\nSee for example .\n\nDefined as counterexample #71 (\"Discrete Irrational Extension of $\\mathbb{R}$\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"The product of a normal space and a metric space need not be normal (E. Michael)","doi":"10.1090/S0002-9904-1963-10931-3"}]},{"name":"Rational extension of the plane","aliases":[],"uid":"S000064","counterexamples_id":72,"description":"If $(X,\\tau)$ is the {S176},\nwe define a new topology for $X$ by declaring open each point in the set\n$D = \\mathbb Q^2$, and each set of the form $\\{x\\} \\cup (D \\cap U)$\nwhere $x \\in U \\in \\tau$.\n\nThe topology is finer than that of {S176}.\n\nThe construction combines the ideas from\n{S60} and {S62}.\n\nDefined as counterexample #72 (\"Rational Extension in the Plane\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Telophase topology","aliases":["Interval with two right endpoints"],"uid":"S000065","counterexamples_id":73,"description":"Let $(X, \\tau)$ be the topological space formed by adding to the ordinary closed unit interval $[0,1]$ another right end point, say $1^{\\ast}$, with the sets $(a,1) \\cup \\{1^{\\ast}\\}$ as a local basis.\n\nDefined as counterexample #73 (\"Telophase Topology\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Double origin plane","aliases":[],"uid":"S000066","counterexamples_id":74,"description":"Let $X$ consist of the set of points of the plane $\\mathbb R^{2}$ together with an additional point $0^{\\ast}$.\nNeighborhoods of points other than the origin $0$ and the point $0^{\\ast}$ are the usual open sets of $\\mathbb{R}^{2}\\setminus \\{0\\}$;\nas a basis of neighborhoods of $0$ and $0^{\\ast}$, we take $V_{n}(0) = \\{(x,y):x^{2} + y^{2} < \\frac{1}{n^2},\\, y > 0\\} \\cup \\{0\\}$\nand $V_{n}(0^{\\ast}) = \\{(x,y):x^{2} + y^{2} < \\frac{1}{n^2},\\, y < 0\\} \\cup \\{0^{\\ast}\\}$.\n\nDefined as counterexample #74 (\"Double Origin Topology\")\nin {{zb:0386.54001}}. Note that this is *not* a copy\nof {S176} with the origin\ndoubled in the sense of {S83}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Double origin topology on Wikipedia","wikipedia":"Double_origin_topology"}]},{"name":"Irrational slope topology","aliases":["Bing's connected countable space"],"uid":"S000067","counterexamples_id":75,"description":"Let $X = \\{(x,y) : x,y \\in \\mathbb{Q}, y \\geq 0\\}$. For a fixed irrational $\\theta>0$, the irrational slope topology on $X$ is obtained by taking as a local base of open neighborhoods of each point $(x,y)\\in X$ the collection of sets $N_\\epsilon(x,y)$ with $\\epsilon>0$,\ndefined by $N_\\epsilon(x,y) = \\{(x,y)\\} \\cup B(x - y/\\theta, \\epsilon) \\cup B(x + y/\\theta, \\epsilon)$ where $B(a, \\epsilon):=((a-\\epsilon,a+\\epsilon)\\cap\\mathbb Q)\\times\\{0\\}$. That is, $N_\\epsilon(x,y)$ consists of the point $(x,y)$ together with the rational numbers in an $\\epsilon$-neighborhood around the two points where the lines passing through $(x,y)$ with slopes $\\theta$ and $-\\theta$ cross the real $x$-axis.\n\nDefined as counterexample #75 (\"Irrational Slope Topology\")\nin {{zb:0386.54001}}. Originally introduced in {{zb:0051.13902}} with $\\theta=\\sqrt 3$.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"A connected countable Hausdorff space (R.H. Bing)","zb":"0051.13902"}]},{"name":"Deleted diameter topology","aliases":[],"uid":"S000068","counterexamples_id":76,"description":"Let $X = \\mathbb{R}^2$. The deleted diameter topology on $X$ is generated by the basis consisting of all open discs with their horizontal diameters except the center deleted.\n\nDefined as counterexample #76 (\"Deleted Diameter Topology\")\nin {{zb:0386.54001}}.\n(That reference states that the deleted discs above form a subbasis for the topology, but they in fact form a basis.)","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Deleted radius topology","aliases":[],"uid":"S000069","counterexamples_id":77,"description":"Let $X = \\mathbb{R}^2$. The deleted radius topology on $X$ is generated by the subbasis consisting of all open discs with all of their right horizontal radii (excluding the center) deleted.\n\nDefined as counterexample #77 (\"Deleted Radius Topology\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Half-disc topology","aliases":[],"uid":"S000070","counterexamples_id":78,"description":"Let $H = \\{(x,y) \\in \\mathbb{R}^2 : y>0\\}$ be the upper half plane and $L$ the $x$-axis. The half disc topology on $X = H \\cup L$ is the topology generated by the following neighborhood bases: points $x\\in H$ have the usual Euclidean neighborhoods and points $x \\in L$ have neighborhoods of the form $\\{x\\} \\cup (H \\cap U)$ where $x \\in U$ and $U$ is open in the usual Euclidean topology.\n\nDefined as counterexample #78 (\"Half-Disc Topology\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Irregular lattice topology","aliases":[],"uid":"S000071","counterexamples_id":79,"description":"Let $A = \\{(i,k)\\in \\mathbb{N}^2: i>0\\}$, $B = \\{(i,0) : i \\in\\mathbb N \\}$ and $X = A \\cup B$.\nDeclare each point of $A$ to be open.\nLet the set of all $U_n(i) = \\{(i,k) : k=0$ or $k \\geq n\\}$ form a local basis at the point $(i,0)$ for $i > 0$\nand the set of all $V_n = \\{(i,k) : i=k=0$ or $i,k\\geq n\\}$ be a local basis at $(0,0)$.\n\nDefined as counterexample #79 (\"Irregular Lattice Topology\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Arens square","aliases":[],"uid":"S000072","counterexamples_id":80,"description":"Let $S=\\big((0,1)_{\\mathbb Q}\\setminus\\{\\frac{1}{2}\\}\\big)\\times(0,1)_{\\mathbb Q}$ and\n$X=S\\cup\\{\\langle 0,0\\rangle,\\langle 1,0\\rangle\\}\\cup\\{\\langle\\frac{1}{2},r\\sqrt{2}\\rangle:r\\in(0,\\frac{1}{\\sqrt 2})_{\\mathbb Q}\\}$\n(where $I_{\\mathbb Q}=I\\cap\\mathbb Q$).\n\nThis space is $X$ with the topology such that points of $S$ have their usual neighborhoods as a subspace of {S176}, and\n\n- $U_n(0,0) = \\{\\langle 0,0\\rangle\\} \\cup \\left((0,\\frac{1}{4})_{\\mathbb Q}\\times(0,\\frac{1}{n})_{\\mathbb Q}\\right)$,\n- $U_n(1,0) = \\{\\langle 1,0\\rangle\\} \\cup \\left((\\frac{3}{4},1)_{\\mathbb Q}\\times(0,\\frac{1}{n})_{\\mathbb Q}\\right)$, and\n- $U_n(\\frac{1}{2},r\\sqrt{2}) = \\left((\\frac{1}{4},\\frac{3}{4})\\times(r\\sqrt{2}-\\frac{1}{n},r\\sqrt{2}+\\frac{1}{n})\\right)\\cap X$\n for $r\\in(0,\\frac{1}{\\sqrt 2})_{\\mathbb Q}$.\n\nare local bases at $\\langle 0,0\\rangle$, $\\langle 1,0\\rangle$ and $\\langle\\frac{1}{2},r\\sqrt{2}\\rangle$ respectively.\n\nDefined as counterexample #80 (\"Arens Square\") in {{zb:0386.54001}}, where\nthis space was erroneously claimed to be {P4}.\nCompare with {S80} which modifies the construction to\nobtain a valid {P4} example.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Simplified arens square","aliases":[],"uid":"S000073","counterexamples_id":81,"description":"Let $S = (0,1)^2$ and $X = S \\cup \\{(0,0),(1,0)\\}$. Let $S$ have the subspace topology and let $U_n(0,0) = \\{(0,0)\\} \\cup \\{(x,y) : x < \\frac{1}{2}, y<\\frac{1}{n}\\}$ and $U_n(1,0) = \\{(1,0)\\} \\cup \\{(x,y) : \\frac{1}{2}0$.\n\nSee section 3 of {{doi:10.1090/S0002-9939-07-09100-9}}\nand the description of the general construction in {{mathse:3678440}}.\n\nThis provides an example of a {P123} space that is not {P3}, but is {P86}.","refs":[{"name":"Manifolds: Hausdorffness versus homogeneity (Baillif & Gabard)","doi":"10.1090/S0002-9939-07-09100-9"},{"name":"Answer to \"T1 spaces where the closure of a compact set is not compact\"","mathse":3678440}]},{"name":"Deleted Dieudonné plank","aliases":["Dieudonné plank"],"uid":"S000087","counterexamples_id":89,"description":"The subspace $X$ of {S191} obtained by removing the point $\\langle\\omega_1,\\omega\\rangle$.\nSo $X=(Y\\times Z)\\setminus\\{\\langle\\omega_1,\\omega\\rangle\\}$ \nwhere $Y=[0,\\omega_1]$ is given the topology with points of $[0,\\omega_1)$\nisolated and neighborhoods of $\\omega_1$ cocountable ({S155}),\nand $Z=[0,\\omega]$ is given the topology with points of $[0,\\omega)$\nisolated and neighborhoods of $\\omega$ cofinite ({S20}).\n\nDefined as counterexample #89 (\"Dieudonné Plank\")\nin {{zb:0386.54001}}. Note that the pi-Base qualifies this name with\n\"Deleted\" to align with e.g. {S78} and {S79}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Dieudonné plank on Wikipedia","wikipedia":"Dieudonné_plank"}]},{"name":"Tychonoff corkscrew","aliases":[],"uid":"S000088","counterexamples_id":90,"description":"The space $X=\\mathrm{DJ}(Y,H,K)$ is the result of applying the double-sided variation of the \"Jones machine\" to the space $Y=$ {S79} and its two closed subsets $H=\\omega_1\\times\\{\\omega\\}$ and $K=\\{\\omega_1\\}\\times\\omega$ (which have the property that they cannot be separated by open sets in $Y$). It is obtained by taking a two-sided sequence of copies of $Y$, plus two extra limit points $a^-, a^+$, and alternately gluing adjacent copies of $Y$ along either $H$ or $K$.\n\nFormally, define $Z = (Y\\times \\mathbb{Z}) \\cup \\{p^-, p^+\\}$, where $p^\\pm$ are two new points, to have topology such that $U\\subseteq Z$ is open iff $U\\cap (Y\\times \\mathbb{Z})$ is open in the product topology where $\\mathbb{Z}$ is discrete,\nand if $p^+\\in U$ (resp. $p^-\\in U$) then $Y\\times \\{n\\in \\mathbb{Z} : n\\geq N\\}\\subseteq U$\n(resp. $Y\\times \\{n\\in \\mathbb{Z} : n\\leq -N\\}\\subseteq U$) for some $N\\in\\mathbb{N}$.\nDefine $X=\\mathrm{DJ}(Y,H,K)$ as the quotient space $Z/{\\sim}$ where the equivalence relation $\\sim$ is given by $(x, 2n)\\sim (x, 2n+1)$ for $x\\in H$ and $(x, 2n-1)\\sim (x, 2n)$ for $x\\in K$, and no other relations between distinct points. If $q:Z\\to X$ is the quotient map, define $a^\\pm = q(p^\\pm)$.\n\nThe construction above is described in Problem 1Y in {{zb:0652.54016}}, based on the original idea in {{doi:10.1007/BFb0064021}}.\n\nThe space $X$ is defined in a more informal way as counterexample #90 (\"Tychonoff Corkscrew\") in {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Hereditarily separable, non-completely regular spaces","doi":"10.1007/BFb0064021"},{"name":"Extensions and absolutes of Hausdorff spaces","zb":"0652.54016"}]},{"name":"Deleted Tychonoff corkscrew","aliases":[],"uid":"S000089","counterexamples_id":91,"description":"The subspace $X := \\text{DJ}(Y, H, K)\\setminus \\{a^-\\}$ of the {S88} $\\text{DJ}(Y, H, K)$.\n\nDefined as counterexample #91 (\"Deleted Tychonoff Corkscrew\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Hewitt's condensed corkscrew","aliases":[],"uid":"S000090","counterexamples_id":92,"description":"If $T = \\text{DJ}(Y, H, K)$ is the {S88},\n*Hewitt's condensed corkscrew* is the set $X = (T\\setminus \\{a^+, a^-\\}) \\times \\omega_1$ with the subspace topology induced from\na certain topology $\\tau_A$ on $A = T \\times \\omega_1$ which we define below.\n\nLet $\\Gamma:X \\times X \\rightarrow \\omega_1$ be a fixed bijection and $\\pi_i:X\\times X\\to X$ be the coordinate projections. Define $\\psi: A \\setminus X \\rightarrow X$ by $\\psi(a^+, \\lambda) = \\pi_1 (\\Gamma^{-1}(\\lambda))$ and $\\psi(a^-, \\lambda) = \\pi_2 (\\Gamma^{-1}(\\lambda))$. Define $A_\\lambda = T\\times \\{\\lambda\\}$ for $\\lambda\\in \\omega_1$ and let $\\sigma_\\lambda$ be the product topology on $A_\\lambda$ (so that $(A_\\lambda, \\sigma_\\lambda)$ and $T$ are homeomorphic). Then define $$\\tau_A = \\{U\\subseteq A : \\psi^{-1}(U)\\subseteq U \\text{ and }U\\cap A_\\lambda\\in \\sigma_\\lambda\\text{ for all }\\lambda\\in\\omega_1\\}.$$\nNote that the topology $\\tau_A\\restriction A_\\lambda$ is coarser than $\\sigma_\\lambda$, and can be strictly coarser for a given $\\lambda$.\n\nDefined as counterexample #92 (\"Hewitt's Condensed Corkscrew\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Thomas plank","aliases":[],"uid":"S000091","counterexamples_id":93,"description":"Let $L_0 = \\{(x,0)\\ |\\ x \\in (0,1)\\}$ and for $i \\geq 1$, $L_i = \\{(x,\\frac{1}{i}) |\\ x \\in [0,1)\\}$, and $X = \\bigcup_{i=0}^\\infty L_i$. If $i \\geq 1$, each point of $L_i \\setminus \\{(0, \\frac{1}{i})\\}$ is open. Basis neighborhoods of $(0,\\frac{1}{i})$ are subsets $L_i$ with finite complements. The sets $U_i(x,0) = \\{(x,0)\\} \\cup \\{(x,\\frac{1}{n})\\ |\\ n > i\\}$ form a basis for the points in $L_0$.\n\nThis space is homeomorphic to $(Y_1\\times Y_2)\\setminus \\{(\\infty, \\infty)\\}$ where $Y_1 = ${S154} and $Y_2=${S20}.\n\nDefined as counterexample #93 (\"Thomas' Plank\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Thomas corkscrew","aliases":[],"uid":"S000092","counterexamples_id":94,"description":"The space $X=\\mathrm{DJ}(Y,H,K)$ is the result of applying the double-sided variation of the \"Jones machine\" to the space $Y=$ {S91} and its two closed subsets $H = L_0$ and $K = \\{(0, \\frac{1}{n}) : n\\in\\mathbb{N}\\}$ (which have the property that they cannot be separated by open sets in $Y$). It is obtained by taking a two-sided sequence of copies of $Y$, plus two extra limit points $a^-, a^+$, and alternately gluing adjacent copies of $Y$ along either $H$ or $K$.\n\nSee description of {S88} for detailed construction of $\\text{DJ}(Y, H, K)$.\n\nDefined as counterexample #94 (\"Thomas' Corkscrew\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Double arrow space","aliases":["Split interval","Weak parallel line topology"],"uid":"S000093","counterexamples_id":95,"description":"The set $X=([0,1]\\times\\{0,1\\})\\setminus\n\\{\\langle 0,0\\rangle,\\langle 1,1\\rangle \\}$ with the order topology given by the lexicographic order.\n$$\\langle x,t\\rangle < \\langle x',t'\\rangle\n\\iff xn\\}$ and $M_n^-(-a) = \\{-a\\}\\cup\\{(i,j)\\ |\\ i>\\omega,j>n\\}$.\n\nDefined as counterexample #100 (\"Minimal Hausdorff Topology\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Alexandroff square","aliases":[],"uid":"S000099","counterexamples_id":101,"description":"Let $X$ be the unit square $[0,1] \\times [0,1] \\subset \\mathbb{R}^2$. For a point $(x,y)$ not on the diagonal (for $x \\ne y$), basic neighborhoods are given by the vertical segments $\\{x\\} \\times (y - \\varepsilon,y + \\varepsilon)$ for $\\varepsilon > 0$. For a point $(x,x)$ on the diagonal, basic open neighborhoods are given by open horizontal strips minus a finite number of vertical lines, i.e., sets of the form $S \\times (x - \\varepsilon,x + \\varepsilon)$ for $\\varepsilon > 0$ and $S$ cofinite subset of $[0,1]$ containing $x$.\n\nEvery vertical slice is homeomorphic to {S158}. Every horizontal slice is homeomorphic to {S154}, with the homeomorphism sending the point on the diagonal to $\\infty$. The diagonal $\\{(x,x) | x \\in [0, 1]\\}$ is homeomorphic to {S158}.\n\nDefined as counterexample #101 (\"Alexandroff Square\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"David Gao's ultraconnected non-contractible space","aliases":[],"uid":"S000100","description":"David Gao's example of a nonempty {P40} space that is not {P199},\nas constructed in {{mathse:5001004}}.\n\nThe underlying set is the set $X$ of partial functions $f$ from $\\mathbb{Z}$ to $\\{0, 1\\}$ satisfying all of the following three conditions:\n\n1. There exists $n \\in \\mathbb{Z}$ such that the domain of $f$ is $\\mathbb{Z}_{\\leq n}$. We call $n$ the *length* of $f$, which shall be denoted by $l(f)$.\n2. There exists $n \\in \\mathbb{Z}$ such that $f(m) = 0$ for all $m \\leq n$.\n3. Either $\\text{range}(f) = \\{0\\}$ and $l(f) \\leq 0$; or there exists $n < 0$ such that $f(n) = 1$, and furthermore if $n_0 = \\min\\{n < 0: f(n) = 1\\}$, then $l(f) \\leq -n_0$.\n\nFor partial functions $f, g$ from $\\mathbb{Z}$ to $\\{0, 1\\}$, we shall say $g$ is a *truncation* of $f$, denoted by $g \\leq f$, if there exists $n \\in \\mathbb{Z}$ s.t. $g = f|_{\\text{dom}(f) \\cap \\mathbb{Z}_{\\leq n}}$. Truncations of elements of $X$ are also in $X$.\nFurthermore, $\\leq$ is a partial order on $X$.\n\nTo define the topology on $X$, for each $x \\in X$, let\n\n$$T_x = \\{y \\in X: y \\leq x\\}.$$\n\nAnd for each $n \\in \\mathbb{Z}$, let\n\n$$K_n = \\{y \\in X: l(y) \\leq n\\}.$$\n\nThe set $X$ is equipped with the topology generated by the closed subbasis\n\n$$\\{T_x: x \\in X\\} \\cup \\{K_n: n \\in \\mathbb{Z}\\};$$\n\nthat is, the topology on $X$ is generated by the subbasis of open sets consisting of the complements of the $T_x$ and the complements of the $K_n$.","refs":[{"name":"Answer to \"Are hereditarily connected spaces and ultraconnected spaces contractible?\"","mathse":5001004}]},{"name":"Continuum power of a countable discrete space","aliases":["Uncountable products of Z+"],"uid":"S000101","counterexamples_id":103,"description":"The product topology on \$\\omega^{\\mathfrak c}\$.\n\nDefined as counterexample #103 (\"Uncountable Products of \$\\mathbb{Z}^+\$\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Baire space of weight continuum $B(\\mathfrak c)$","aliases":["Countable product of continuum-sized discrete spaces","Baire metric on $\\mathbb R^\\omega$"],"uid":"S000102","counterexamples_id":104,"description":"$X$ is the product of countably many copies of {S3}.\nSo $X=\\mathbb R^\\omega$ with $\\mathbb R$ having the discrete topology.\n\nIn general, the *Baire space of weight $\\kappa$* for some infinite cardinal $\\kappa$\nis the countable power $D^\\omega$ for some discrete space $D$ with $|D|=\\kappa$,\nand is denoted $B(\\kappa)$. It has weight $\\kappa$.\nThe current space is $X=B(\\mathfrak c)=B(2^{\\aleph_0})$.\n\nThis space is metrizable using the Baire metric. For distinct elements\n$x=(x_i)_{i\\ge 0}, y=(y_i)_{i\\ge 0}$ in $X=D^\\omega$ the *Baire metric*\nis defined by $d(x,y)=\\frac{1}{i+1}$ where $i$ is the first coordinate with $x_i\\neq y_i$.\nAnd $d(x,x)=0$.\n\nSee Example 4.2.12 in {{zb:0684.54001}}.\n\nAlso defined as counterexample #104 (\"Baire Metric on \$\\mathbb{R}^\\omega\$\")\nin {{zb:0386.54001}}, where the index starts at $i=1$.\n\nThis space has a finer topology than {S30}.\n\nCompare with {S28}, which is the Baire space $B(\\aleph_0)$.","refs":[{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"},{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Continuum power of $[0,1]$","aliases":["Tychonoff cube of weight continuum","Continuum power of closed unit interval","$I^I$"],"uid":"S000103","counterexamples_id":105,"description":"The product topology on $X=I^I$, where $I$ is the {S158}.\nIn other words it is the set of all functions $f\\colon I\\to I$, endowed with the topology of pointwise convergence.\n\nIn general, the *Tychonoff cube of weight* $\\kappa$ for an infinite cardinal $\\kappa$ is the product space $[0,1]^\\kappa$, as defined on page 83 of {{zb:0684.54001}}.\nIt has topological weight $\\kappa$ (see Theorem 2.3.23 in the same reference).\nThe current space $X$ is the Tychonoff cube of weight $\\mathfrak c$.\n\nDefined as counterexample #105 (\"$I^I$\") in {{zb:0386.54001}}.\n\nCompare with {S32}, which is the Tychonoff cube of weight $\\aleph_0$.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"name":"Product of the first-uncountable ordinal with the continuum-power of unit intervals","aliases":["ω_1 times I^I"],"uid":"S000104","counterexamples_id":106,"description":"The product topology on $\\omega_1$ ({S35}) crossed with $I^I$ ({S103}).\n\nDefined as counterexample #106 (\"\$[0,\\Omega)\\times I^I\$\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Helly space","aliases":["Non-decreasing functions of I^I"],"uid":"S000105","counterexamples_id":107,"description":"Let $X \\subset I^I$ be the subspace\nof {S103} consisting of all non-decreasing functions.\n\nDefined as counterexample #107 (\"Helly Space\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Helly space on Wikipedia","wikipedia":"Helly_space"}]},{"name":"Countable box product of reals","aliases":["Countable boolean product of reals","Box product topology on R^ω","Box product topology on R^omega"],"uid":"S000107","counterexamples_id":109,"description":"The set $X = \\mathbb{R}^\\omega$ with the box product topology where basic open\nsets are of the form \$\\prod_{i<\\omega}U_i\$ for \$U_i\$ open in \$\\mathbb R\$.\nAlso known as the Boolean product.\n\nDefined as counterexample #109 (\"Box Product Topology on \$\\mathbb{R}^\\omega\$\")\nin {{zb:0386.54001}}.\n\n$X$ has a finer topology than {S30}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Box topology on Wikipedia","wikipedia":"Box_topology"}]},{"name":"Stone-Čech compactification $\\beta\\omega$ of the integers","aliases":["$\\beta\\mathbb{N}$"],"uid":"S000108","counterexamples_id":111,"description":"The Stone-Čech compactification of {S2}.\nOne way to describe $X=\\beta \\omega$ is as the set of all ultrafilters on $\\omega=\\{0,1,2,\\dots\\}$,\nwhere we identify $n \\in \\omega$ with the principal ultrafilter containing $n$.\nThe topology on $X$ is generated by all sets of the form\n$\\overline U = \\{F \\in \\beta\\omega : U \\in F\\}$ for $U \\subseteq \\omega$.\n\nDefined as counterexample #111 (\"Stone-Čech Compactification of the Integers\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Stone–Čech compactification on Wikipedia","wikipedia":"Stone–Čech_compactification"}]},{"name":"Novak space","aliases":[],"uid":"S000109","counterexamples_id":112,"description":"The space $X$ is the subspace $A_1 = \\mathbb N\\cup P$ of $\\beta\\mathbb N$\n({S108}) constructed in the proof\nof Theorem 3 in {{zb:0053.12404}}.\n\nUsing the alternative notation $\\beta\\omega$, the construction goes as follows.\nThere are $2^\\mathfrak{c}$ countably infinite subsets of $\\beta\\omega$.\nLet $\\{S_\\xi : \\xi<\\lambda\\}$ enumerate them all,\nwhere $\\lambda$ is the first ordinal of cardinality $2^\\mathfrak{c}$.\nBy transfinite recursion one can find elements $x_\\xi,y_\\xi\\in\\beta\\omega\\setminus\\omega$ for all $\\xi$\nsuch that:\n\n1. $x_\\xi, y_\\xi\\in \\overline{S_\\xi}\\setminus S_\\xi$;\n\n2. all the $x_\\xi$ are distinct and all the $y_\\xi$ are distinct;\n\n3. the sets $\\{x_\\xi:\\xi<\\lambda\\}$ and $\\{y_\\xi:\\xi<\\lambda\\}$\nform a partition of $\\beta\\omega\\setminus\\omega$.\n\nThen $X=\\omega\\cup\\{x_\\xi:\\xi<\\lambda\\}$ with the subspace topology induced from $\\beta\\omega$.\n\nDefined as counterexample #112 (\"Novak Space\") in {{zb:0386.54001}}.\nHowever, the construction in that book is incorrect as it assumes that\n$2^\\mathfrak{c}$ is a regular cardinal, while it is consistent with ZFC that it isn't.","refs":[{"name":"On the Cartesian product of two compact spaces (J. Novak)","zb":"0053.12404"},{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Strong ultrafilter topology","aliases":[],"uid":"S000110","counterexamples_id":113,"description":"Let $M$ be the set of all free ultrafilters on $\\omega$. The strong ultrafilter\ntopology is the set $X = \\omega \\cup M$ with topology generated by singletons\nof $\\omega$ along with all sets of the form $A \\cup \\{F\\}$ for $A \\in F \\in M$.\n\nDefined as counterexample #113 (\"Strong Ultrafilter Topology\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Single ultrafilter subspace of $\\beta\\omega$","aliases":["Single ultrafilter topology"],"uid":"S000111","counterexamples_id":114,"description":"Let $F$ be a non-principal ultrafilter on $\\omega$. The space $X$ is the set $\\omega\\cup\\{F\\}$ with points of $\\omega$ isolated\nand neighborhoods of $F$ all sets of the form $U \\cup \\{F\\}$ for $U \\in F$.\n\nThis is the subspace topology on $\\omega\\cup\\{F\\}$ induced as a subset of $\\beta\\omega$ ({S108}).\n\nDefined as counterexample #114 (\"Single Ultrafilter Topology\")\nin {{zb:0386.54001}}.\n\nNot to be confused with {S145}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Nested rectangles in the real plane","aliases":[],"uid":"S000112","counterexamples_id":115,"description":"Let $L_1$ be the line $x=1$, $L_2$ the line $x=-1$ and $R_n$ the boundary of the rectangle centered at $(0,0)$ with height $2n$ and width $\\frac{2n}{n+1}$. Let $X = L_1 \\cup L_2 \\cup \\bigcup_{n \\ge 1} R_n$ with the subspace topology inherited from $\\mathbb{R}^2$.\n\nDefined as counterexample #115 (\"Nested Rectangles\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Topologist's sine curve","aliases":[],"uid":"S000113","counterexamples_id":116,"description":"The set $X = \\{(x, \\sin\\frac{1}{x}):x \\in (0,1]\\} \\cup \\{(0,0)\\}$ with the subspace topology inherited from $\\mathbb{R}^2$.\n\nDefined as counterexample #116 (\"Topologist's Sine Curve\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Topologist's sine curve on Wikipedia","wikipedia":"Topologist's_sine_curve"}]},{"name":"Closed Topologist's Sine Curve","aliases":[],"uid":"S000114","counterexamples_id":117,"description":"The set $X=\\{(x, \\sin\\frac{1}{x}):x \\in (0,1]\\} \\cup \\{(0,y):y \\in [-1,1]\\}$ with the subspace topology inherited from $\\mathbb{R}^2$.\n\nDefined as counterexample #117 (\"Closed Topologist's Sine Curve\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Topologist's sine curve on Wikipedia","wikipedia":"Topologist's_sine_curve"}]},{"name":"Extended topologist's sine curve","aliases":[],"uid":"S000115","counterexamples_id":118,"description":"The set $X=\\{(x, \\sin\\frac{1}{x}):x \\in (0,1]\\} \\cup \\{(0,y):y \\in [-1,1]\\} \\cup \\{(x,1):x\\in[0,1]\\} $ with the subspace topology inherited from $\\mathbb{R}^2$.\n\nDefined as counterexample #118 (\"Extended Topologist's Sine Curve\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Topologist's sine curve on Wikipedia","wikipedia":"Topologist's_sine_curve"}]},{"name":"Infinite broom","aliases":[],"uid":"S000116","counterexamples_id":119,"description":"For each $n \\in \\omega$ let $L_n$ be the closed line segment from $(0,0)$ to $(1,\\frac{1}{n+1})$.\nThe Infinite Broom is the set $X=\\bigcup_{n \\in \\omega} L_n \\cup \\big((\\frac{1}{2},1] \\times \\{0\\}\\big) \\subset \\mathbb{R}^2$ with the subspace topology.\n\nThis space is a subspace of {S117}.\n\nDefined as counterexample #119 (\"The Infinite Broom\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Infinite broom on Wikipedia","wikipedia":"Infinite_broom"}]},{"name":"Closed infinite broom","aliases":[],"uid":"S000117","counterexamples_id":120,"description":"This space is the closure of {S116} in {S176}.\n\nSo $X$ is the union $\\bigcup_{n\\in\\omega}L_n\\cup L_\\infty\\subseteq\\mathbb R^2$\nwhere for each $n\\in\\omega$, $L_n$ is the closed line segment from $(0,0)$ to $(1,\\frac{1}{n+1})$,\nand $L_\\infty=[0,1]\\times\\{0\\}$ is the closed line segment from $(0,0)$ to $(1,0)$.\nIt has the subspace topology induced from {S176}.\n\nDefined as counterexample #120 (\"The Closed Infinite Broom\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Infinite broom on Wikipedia","wikipedia":"Infinite_broom"}]},{"name":"Integer broom","aliases":[],"uid":"S000118","counterexamples_id":121,"description":"Let $X \\subset \\mathbb{R}^2$ consist of points with polar coordinates $\\{(n,\\theta)\\ \\mid\\ n \\in \\omega, \\theta \\in \\{0\\} \\cup \\{1/n\\}_{n=1}^\\infty\\}$. Define a topology on $X$ by taking as a basis all sets of the form $U \\times V$ where $U$ is an open set in the right order topology on the positive integers and $V$ is open in $\\{0\\} \\cup \\{1/n\\}_{n=1}^\\infty \\subset \\mathbb{R}$. The only neighborhood of the origin is $X$ itself.\n\nThis space is the quotient of the product of {S200} with\n{S20} where all points whose first component is $0$ are identified.\n\nDefined as counterexample #121 (\"The Integer Broom\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Infinite broom topology on Wikipedia","wikipedia":"Integer_broom_topology"}]},{"name":"Nested angles in the real plane","aliases":[],"uid":"S000119","counterexamples_id":122,"description":"Let $X \\subset \\mathbb{R}^2$ be the set consisting of the line segments joining the points $(0,1)$ and $(n, \\frac{1}{n+1})$ for $n=1,2,\\dots$; the half-lines $y = \\frac{1}{n+1}$, $x \\leq n$ for $n=1,2,\\dots$; and the line $y=0$. Give $X$ the subspace topology induced from {S176}.\n\nDefined as counterexample #122 (\"Nested Angles\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Infinite cage","aliases":[],"uid":"S000120","counterexamples_id":123,"description":"For each integer $n\\geq 1$, let\n$A_n = \\{(\\frac{1}{n}, y, 0) \\in \\mathbb{R}^3\\ |\\ 0 \\leq y \\leq 3n\\}$,\n$B_n = \\{(0,y,0) \\in \\mathbb{R}^3\\ |\\ 2n - \\frac{1}{2} \\leq y \\leq 2n + \\frac{1}{2}\\}$ and\n$C_n = \\{(x,y,z) \\in \\mathbb{R}^3\\ |\\ 0 \\leq x \\leq \\frac{1}{n}, y=2n, z=x(\\frac{1}{n} -x)\\}$.\nThe Infinite Cage is the space $X = \\bigcup_{n \\geq 1} (A_n \\cup B_n \\cup C_n) \\subset \\mathbb{R}^3$ with the subspace topology.\n\nDefined as counterexample #123 (\"The Infinite Cage\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Bernstein's connected set","aliases":[],"uid":"S000121","counterexamples_id":124,"description":"Let $\\{C_\\alpha\\ : \\alpha<\\mathfrak c\\}$ be a well-ordering of the set of\nall closed, {P36} subsets of {S176} that contain at least two points\n(so in fact each $C_\\alpha$ has cardinality $\\mathfrak c$).\nNote that there are $\\mathfrak c$ such $C_\\alpha$.\nWe shall define sets $A_\\alpha$ and $B_\\alpha$ such that\n$A_\\alpha \\cap B_\\beta = \\emptyset$ for all $\\alpha, \\beta<\\mathfrak c$.\nFirst, let $A_0$, $B_0$ be distinct singletons in $\\mathbb{R}^2$.\nNow, assuming $A_\\beta,B_\\beta$ are defined for all $\\beta<\\alpha$\nand $|\\bigcup_{\\beta<\\alpha}(A_\\beta\\cup B_\\beta)|<\\mathfrak c$,\nwe may choose $A_\\alpha=\\{a_\\alpha\\}\\cup\\bigcup_{\\beta<\\alpha}A_\\beta$\nand $B_\\alpha=\\{b_\\alpha\\}\\cup\\bigcup_{\\beta<\\alpha}B_\\beta$ such that\n$a_\\alpha,b_\\alpha$ are distinct points in\n$C_\\alpha\\setminus\\left(\\bigcup_{\\beta<\\alpha}(A_\\beta\\cup B_\\beta)\\right)$.\n\nBernstein's connected set is $A=\\bigcup_{\\alpha<\\mathfrak c}A_\\alpha$\nwith its subspace topology induced from {S176}.\n\nDefined as counterexample #124 (\"Bernstein's Connected Sets\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Gustin's sequence space","aliases":[],"uid":"S000122","counterexamples_id":125,"description":"Let $\\mathbb{Z}^+$ denote the set of positive integers, $Y$ be the set of all finite sequences of positive integers of even length, and $W = \\{A \\subseteq Y\\ |\\ |A| = 2\\}$. Gustin's Sequence Space is the set $X = Y \\cup (\\mathbb{Z}^+ \\times W)$ with topology defined as follows:\n\nFor any finite sequences $\\alpha, \\beta$ of arbitrary length, let $\\alpha\\beta$ be the sequence formed by adjoining $\\beta$ to the end of $\\alpha$. Let $\\alpha \\geq i \\in \\mathbb{Z}$ abbreviate $a \\geq i$ for every $a \\in \\alpha$. Let $\\beta \\supseteq_i \\alpha$ abbreviate that there exists a sequence of finite length $\\gamma\\geq i$ with $\\beta = \\alpha\\gamma$ and $U_i(\\alpha) = \\{\\beta \\in Y\\ |\\ \\beta \\supseteq_i \\alpha\\}$.\n\nLet $q: \\mathbb{Z}^+ \\times W\\rightarrow \\mathbb{Z}^+$ be an injective function increasing on its first coordinate. (For example, fix a bijection $p$ from $W$ to the set of all positive primes, and define $q: \\mathbb{Z}^+ \\times W \\rightarrow \\mathbb{Z}^+$ by $q(n,w) = [p(w)]^n$.)\n\nDefine a topology on $X$ by letting neighborhoods of points $\\alpha \\in Y$ be sets of the form $U_i(\\alpha)$, and neighborhoods of points $(n,w) = (n, \\{\\alpha, \\beta\\})$ be sets of the form $V_i(n,w) = \\{(n,w)\\} \\cup U_i\\big(\\alpha q(n,w)\\big) \\cup U_i\\big(\\beta q(n,w)\\big) $.\n\nDefined as counterexample #125 (\"Gustin's Sequence Space\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Roy's lattice space","aliases":[],"uid":"S000123","counterexamples_id":126,"description":"Let $\\{C_i\\ |\\ i < \\omega\\}$ be a collection of disjoint dense subsets of $\\mathbb{Q}$. For specificity, let \$\\{p_i:i<\\omega\\}\$ enumerate the prime numbers,\nand let \$C_i\$ be the rational numbers of the form $\\frac{a}{p_i^n}$, with $a$ an integer not divisible by $p_i$.\n\nLet $X = \\{(r,i) \\in \\mathbb{Q} \\times \\omega\\ |\\ r \\in C_i\\} \\cup \\{\\omega\\}$ as a set and define a topology on $X$ as follows: neighborhoods of points of the form $(r,2n)$ are open intervals $U_\\epsilon(r,2n) = \\{(t,2n)\\ |\\ |r-t|<\\epsilon\\}$; neighborhoods of points of the form $(r,2n-1)$ have the form $V_\\epsilon(r,2n-1) = \\{(t,m)\\ |\\ |r-t|<\\epsilon, |m-n|\\leq 1\\}$; neighborhoods of $\\omega$ have the form $W_n(\\omega) = \\{(s,i)\\ |\\ i \\geq 2n\\}$.\n\nDefined as counterexample #126 (\"Roy's Lattice Space\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Roy's lattice subspace","aliases":[],"uid":"S000124","counterexamples_id":127,"description":"Let $X$ denote {S123}.\nRoy's lattice subspace is $X \\setminus \\{\\omega\\}$ with the subspace topology.\n\nDefined as counterexample #127 (\"Roy's Lattice Subspace\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Knaster-Kuratowski fan","aliases":["Cantor's leaky tent"],"uid":"S000125","counterexamples_id":128,"description":"For any $a \\in [0,1]$, let $L(a)$ be the line segment from $(a,0)$ to $p = (\\frac{1}{2}, \\frac{1}{2})$. Let $\\mathcal{C}$ be the middle-thirds Cantor set in the unit interval, $\\mathcal{E}$ the endpoints of the removed intervals and $\\mathcal{F} = \\mathcal{C} \\setminus \\mathcal{E}$. Define $A = \\{(x,y) \\in L(c)\\ |\\ c \\in \\mathcal{E}, y \\in \\mathbb{Q}\\}$ and $B = \\{(x,y) \\in L(c)\\ |\\ c \\in \\mathcal{F}, y \\not\\in \\mathbb{Q}\\}$. This space is $X = A \\cup B \\subset \\mathbb{R}^2$ with the subspace topology.\n\nThe subspace $X\\setminus\\{p\\}$ obtained by removing the apex point is {S126}.\n\nDefined as counterexample #128 (\"Cantor's Leaky Tent\")\nin {{zb:0386.54001}}.\n\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Knaster-Kuratowski fan","wikipedia":"Knaster-Kuratowski_fan"}]},{"name":"Punctured Knaster-Kuratowski fan","aliases":["Cantor's Teepee"],"uid":"S000126","counterexamples_id":129,"description":"{S000125} with the apex removed. Specifically: For any $a \\in [0,1]$, let $L(a)$ be the line segment from $(a,0)$ to $p = (\\frac{1}{2}, \\frac{1}{2})$. Let $\\mathcal{C}$ be the middle-thirds Cantor set in the unit interval, $\\mathcal{E}$ the endpoints of the removed intervals and $\\mathcal{F} = \\mathcal{C} \\setminus \\mathcal{E}$. Define $A = \\{(x,y) \\in L(c)\\ |\\ c \\in \\mathcal{E}, y \\in \\mathbb{Q}\\}$ and $B = \\{(x,y) \\in L(c)\\ |\\ c \\in \\mathcal{F}, y \\not\\in \\mathbb{Q}\\}$. This space is $X = (A \\cup B) \\setminus\\{p\\} \\subset \\mathbb{R}^2$ with the subspace topology.\n\nDefined as counterexample #129 (\"Cantor's Teepee\")\nin {{zb:0386.54001}}.\n\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Knaster-Kuratowski fan","wikipedia":"Knaster-Kuratowski_fan"}]},{"name":"Pseudo-arc","aliases":[],"uid":"S000127","counterexamples_id":130,"description":"The pseudo-arc is a non-degenerate, hereditarily indecomposable, chainable\n(i.e. arc-like) continuum. (All such continua are homeomorphic.)\n\nDefined as counterexample #130 (\"A Pseudo-Arc\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Pseudo-arc on Wikipedia","wikipedia":"Pseudo-arc"}]},{"name":"Miller's biconnected set","aliases":[],"ambiguous_construction":true,"uid":"S000128","counterexamples_id":131,"description":"Let $C \\subset [0,1]$ be the Cantor set. Let $W = C \\times [0,1] \\subset \\mathbb{R}^2$. Let $K$ be an indecomposable continuum such that $K \\cap [0,1]^2 = W$. Let $\\Delta$ be a countable dense subset of $K$ (which exists, as $K$ is compact and metrizable). Let $\\mathcal{C}$ be the set of composants of $K$, $\\mathcal{B}$ the set of continua which separate $K$ and $\\mathcal{D}$ the set of subsets of $\\Delta$ which are dense in some square $S \\subset [0,1]^2$ which meets $W$ and has edges parallel to the $x$- and $y$- axes. Choose well orderings $\\{C_\\alpha\\}$, $\\{B_\\alpha\\}$ and $\\{D_\\alpha\\}$ of $\\mathcal{C}$, $\\mathcal{B}$ and $\\mathcal{D}$ respectively for $\\alpha < \\kappa$ where $\\kappa$ is the least ordinal with $|\\kappa| = \\mathfrak{c}$.\n\nInductively define for each $\\alpha$ an $M_\\alpha \\subset K$ and simple closed curve $J_\\alpha$ such that:\n\n1. $M_\\alpha = p_\\alpha \\in B_\\alpha \\cap K$ if $B_\\alpha \\cap \\Delta = \\emptyset$\n2. $M_\\alpha = \\emptyset$ if $B_\\alpha \\cap \\Delta \\neq \\emptyset$\n3. For ordinals $\\mu \\neq \\lambda$ with $M_\\mu, M_\\lambda \\neq \\emptyset$, $M_\\mu$ and $M_\\lambda$ are in different composants of $K$\n4. $J_\\alpha$ separates $K$\n5. $J_\\alpha \\cap \\Delta \\setminus D_\\alpha = J_\\alpha \\cap M = \\emptyset$ where $M = \\bigcup_{\\alpha < \\kappa} M_\\alpha$\n\nLet $X = \\Delta \\cup M \\subset \\mathbb{R}^2$ with the subspace topology. \n\nDefined as counterexample #131 (\"Miller's Biconnected Set\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Wheel without its hub","aliases":[],"uid":"S000129","counterexamples_id":132,"description":"Let $X = \\{(x,y) \\in \\mathbb{R}^2\\ |\\ 0< x^2 + y^2 \\leq 1\\}$ with topology generated by the standard Euclidean topology, along with all open intervals on the radii contained in the open unit disc.\n\nDefined as counterexample #132 (\"Wheel without Its Hub\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Tangora's connected space","aliases":[],"uid":"S000130","counterexamples_id":133,"description":"Let $A,B,C$ partition the reals into three dense subspaces; for example, let\n$A = \\{\\frac{m}{2^n}:m\\in\\mathbb Z,n<\\omega\\} \\subset \\mathbb{Q}$ be the dyadic rationals,\n$B = \\mathbb{Q} \\setminus A$, and $C = \\mathbb{R}\\setminus\\mathbb{Q}$.\nLet $X$ extend the usual topology on $\\mathbb{R}$ by defining $A$ and $B$ open,\nalong with sets of the form $\\{c\\} \\cup \\big( (c-\\delta,c+\\delta) \\setminus C\\big)$ for\n$c\\in C$ and $\\delta > 0$.\n\nDefined as counterexample #133 (\"Tangora's Connected Space\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Sequential fan with $\\omega$-many spines","aliases":["$S_\\omega$"],"uid":"S000131","description":"The space $X=S_\\omega$ is obtained from the disjoint union of a countably infinite number of convergent sequences ({S20}) by identifying all the limit points to a single point $\\infty$.\n\nSo $X=(\\omega\\times\\omega)\\cup\\{\\infty\\}$ as sets, with all the points of $\\omega\\times\\omega$ isolated and neighborhoods of $\\infty$ being the sets omitting finitely many points from each of the columns of $\\omega\\times\\omega$.\nBasic open neighborhoods of $\\infty$ are given by the sets\n$U_f=\\{\\infty\\}\\cup\\{(m,n)\\in\\omega\\times\\omega:n\\ge f(m)\\}$\nfor some arbitrary function $f:\\omega\\to\\omega$.\n\nSee also for useful information.\n\nCompare with the coarser topology of {S202}.","refs":[{"name":"Sequential fans in topology (Eda, Gruenhage, et al.)","doi":"10.1016/0166-8641(95)00016-X"},{"name":"Answer to \"Can a Fréchet-Urysohn hemicompact Hausdorff space fail to be locally compact?\"","mathse":4844426}]},{"name":"Duncan's space","aliases":[],"uid":"S000132","counterexamples_id":136,"description":"Let $A$ be the set of all strictly increasing, infinite sequences of positive integers. For any $x\\in A$, define $N(n,x)$ as the number of elements of $x$ less than $n$. Let $\\delta (x) =\\lim_{n\\to\\infty} \\frac{N(n,x)}{n}$. Let $X$ be the subset of $A$ containing precisely the sequences $x$ for which $\\delta(x)$ exists. Now for any $x,y\\in X$ define $k(x,y)$ to be the smallest index $i$ such that $x_i\\neq y_i$. We define a metric $d(x,y)$ on $X$ as $$d(x,y)=\\frac{1}{k(x,y)}+|\\delta (x) - \\delta (y)| \\text{ with } d(x,x)=0.$$ Duncan's Space is the topology on $X$ induced by this metric.\n\nDefined as counterexample #136 (\"Duncan's Space\")\nin {{zb:0386.54001}}.\n\nIntroduced and studied by R. L. Duncan in \n{{zb:0085.03002}} ()\nand {{zb:0109.03302}} ().","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"A topology for sequences of integers I (R. L. Duncan)","zb":"0085.03002"},{"name":"A topology for sequences of integers I (R. L. Duncan)","zb":"0109.03302"}]},{"name":"Real line with post office metric","aliases":["Real plane with post office metric"],"uid":"S000133","counterexamples_id":139,"description":"A space $X$ with a particular point $p$ satisfying:\n\n- Each point $x\\ne p$ is isolated.\n- The point $p$ has a countable neighborhood base $\\{V_n:n<\\omega\\}$ such that $V_0=X$, $V_{n+1}\\subseteq V_n$ and $|V_n\\setminus V_{n+1}|=|\\mathbb R|$ for each n, and $\\bigcap_{n<\\omega}V_n=\\{p\\}$.\n\nThese conditions uniquely determine the space $X$ up to homeomorphism, as shown in {{mathse:4833751}}.\n\nHere are a few examples of spaces satisfying the conditions (see {{mathse:4834226}}).\n\n- The space $X=\\mathbb{R}$ with $p=0$. The basic open neighborhoods of $0$ are the Euclidean open intervals around $0$.\n\n- The post office metric space defined as\ncounterexample #139 (\"The Post Office Metric\")\nin {{zb:0386.54001}}, where $X=\\mathbb R^n$ with $n\\ge 1$ and $p$ is the origin.\nThe *post office metric* $d$ on $\\mathbb R^n$ is defined by:\n\n$$d(x,y) = \\begin{cases}\n 0 & \\text{if } x=y, \\\\\n \\|x\\| + \\|y\\| & \\text{otherwise.}\n\\end{cases}$$\n\n- The ordered space $(\\mathbb R\\times\\mathbb Z)\\cup\\{\\infty\\}$ with its order topology, where $\\mathbb R\\times\\mathbb Z$ has its lexicographical ordering and $\\infty$ is an additional maximum point (and the particular point).","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Name for topology generated by the standard open sets of reals and every point that isn't zero.","mathse":4833751}]},{"name":"Radial metric on the plane","aliases":["Paris metric on the plane"],"uid":"S000134","counterexamples_id":140,"description":"If $d_e$ is the usual Euclidean metric on $\\mathbb{R}^2$, the radial metric $r$ is defined by:\n\n$r(x,y) = \\begin{cases}\nd_e(x,y), & x \\text{ and } y \\text{ are collinear with } \\vec 0\\\\\nd_e(x,\\vec 0) + d_e(y,\\vec 0), & \\text{otherwise.}\n\\end{cases}$\n\nDefined as counterexample #140 (\"The Radial Metric\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Hedgehog space","wikipedia":"Hedgehog_space"}]},{"name":"Radial interval topology on $\\mathbb R^2$","aliases":[],"uid":"S000135","counterexamples_id":141,"description":"The radial interval topology on the plane $X=\\mathbb{R}^2$ is generated by the basis consisting of open intervals disjoint from the origin on lines passing through the origin, along with sets of the form $\\bigcup\\{I_\\theta : 0 \\leq \\theta < \\pi\\}$ where $I_\\theta$ is an open interval about the origin on the line with slope $\\tan\\theta$.\n\nThis space is homeomorphic to the quotient of the disjoint union of continuum many copies of {S25}\nwith the points $0$ of all of the copies identified to a single point.\n\nDefined as counterexample #141 (\"Radial Interval Topology\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Bing's Example G","aliases":["Bing's discrete extension space"],"uid":"S000136","counterexamples_id":142,"description":"Let $X = 2^{2^\\mathbb{R}}$. For $r \\in \\mathbb{R}$, let $x_r: 2^\\mathbb{R} \\rightarrow 2$ by $x_r(\\lambda) = 1 \\iff r \\in \\lambda$. Let $M = \\{x_r: r \\in \\mathbb{R}\\}$.\nFor the topology on $X$, points of $M$ have their usual neighborhoods from the product topology\nand points of $X\\setminus M$ are isolated.\n\nDefined as counterexample #142 (\"Bing's Discrete Extension Space\")\nin {{zb:0386.54001}}\nand Example G in {{doi:10.4153/CJM-1951-022-3}}.\n\nSee also .","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Metrization of topological spaces (R. H. Bing)","doi":"10.4153/CJM-1951-022-3"}]},{"name":"Michael's subspace of Bing's Example G","aliases":["Michael's closed subspace"],"uid":"S000137","counterexamples_id":143,"description":"This space is the subspace $Y = M \\cup F$ of $X=$ {S000136} where\n$F=\\{x\\in X: x(\\lambda)=1 \\text{ for only finitely many }\\lambda\\in 2^{\\Bbb{R}}\\}$.\nThe subspace $Y$ is closed in $X$.\n\nDefined as counterexample #143 (\"Michael's Closed Subspace\")\nin {{zb:0386.54001}};\noriginally published by E. Michael as Example 2 in {{doi:10.4153/CJM-1955-029-6}}.\n\nSee also .","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Point-finite and locally finite coverings (E. Michael)","doi":"10.4153/CJM-1955-029-6"}]},{"name":"Rudin's Dowker space","aliases":[],"uid":"S000138","description":"Rudin's Dowker space $X$ is the subset of the product $\\prod_{n<\\omega}(\\omega_{n+1}+1)$ with the box topology\nconsisting of all functions $f$ such that, for some $i<\\omega$, we have\n$\\omega< \\mathrm{cf}(f(n))<\\omega_i$ for all $n\\in\\omega$. By $\\mathrm{cf}(\\lambda)$ we denote the cofinality of $\\lambda$ (see {{wikipedia:Cofinality#Cofinality_of_ordinals_and_other_well-ordered_sets}}).\n\nSee {{zb:0224.54019}} and section 3.2(i) of {{zb:0554.54005}}.","refs":[{"name":"A normal space X for which X×I is not normal (M.E. Rudin)","zb":"0224.54019"},{"name":"Dowker spaces (M.E. Rudin)","zb":"0554.54005"},{"name":"Cofinality on Wikipedia","wikipedia":"Cofinality#Cofinality_of_ordinals_and_other_well-ordered_sets"}]},{"name":"Countable bouquet of circles","aliases":["$\\mathbb R$ with $\\mathbb Z$ collapsed to a point","Countable wedge sum of circles","Rose with countably many petals"],"uid":"S000139","description":"Let $X=\\mathbb R/\\mathbb Z$ to be the quotient of $\\mathbb R$ (with its Euclidean topology) obtained by identifying all the points of $\\mathbb Z$ to a single point $\\infty$.\n\nAlternatively, this space can be characterized as the *wedge sum*\n(see {{wikipedia:Wedge_sum}}) of countably-many circles,\nalso known as a *bouquet of countably many circles*.\n\nNot to be confused with {S201}, which has a coarser topology.","refs":[{"name":"Answer to \"Can a Fréchet-Urysohn hemicompact Hausdorff space fail to be locally compact?\"","mathse":4844916},{"name":"Wedge sum on Wikipedia","wikipedia":"Wedge_sum"},{"name":"Rose (topology) on Wikipedia","wikipedia":"Rose_(topology)"}]},{"name":"$\\mathbb{R}$ extended by a point with co-countable open neighborhoods","aliases":[],"uid":"S000140","description":"Let $X=\\mathbb R\\cup \\{\\infty\\}$. Let $\\mathbb R$ be open in $X$ and have the topology of {S25}, and let open neighborhoods of $\\infty$ be given by sets of the form $U\\cup\\{\\infty\\}$, where $U\\subseteq \\mathbb R$ is co-countable and open in the Euclidean topology. \n\nConstructed in {{mathse:4850979}} as an example of a space that is {P173} and {P81}, yet fails to be {P79}, yielding a counterexample to a natural analogue of {T211}. Elaborated on in {{mathse:4854178}}.","refs":[{"name":"Answer to 'Radial/pseudoradial implies Fréchet-Urysohn/sequential for locally countable spaces'","mathse":4850979},{"name":"What are the compactness properties of $\\mathbb R$, extended by a point with co-countable open neighborhoods?","mathse":4854178}]},{"name":"Ordered space $\\omega_1+1+\\omega^*$","aliases":[],"uid":"S000141","description":"Let $X$ be the ordered space $\\omega_1+1+\\omega^*$ with the corresponding order topology. Here $+$ is concatenation, $\\omega_1$ is {S35}, $1$ is the singleton $\\{p\\}$ for some point $p$, and $\\omega^*$ is $\\omega$ with the reverse order.\n\nThis provides an example of a {P133} space that is not {P174}.\n\nSee {{mathse:4884795}}.","refs":[{"name":"Are linearly ordered topological spaces well-based?","mathse":4884795}]},{"name":"Erdős space","aliases":[],"uid":"S000142","description":"The subspace $X$ of $\\ell^2$ consisting of those sequences $x=(x_i)_i$ with all $x_i\\in\\mathbb Q$.\n\nThe space $\\ell^2$ is the Banach space of sequences of real numbers $x=(x_i)_i$ with $\\sum_i x_i^2<\\infty$,\nequipped with the norm $\\|x\\|_2=(\\sum_i x_i^2)^{1/2}$ and corresponding distance and topology.\n\nThe space $\\ell^2$ is topologically homeomorphic to {S30};\nbut note the space $X$ is *not* homeomorphic to the subspace {S146} of $\\mathbb R^\\omega$.\n\nSee {{mathse:151954}} or Example 6.2.19 in {{zb:0684.54001}}.","refs":[{"name":"Answer to \"Totally disconnected implies base of closed sets?\"","mathse":151954},{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"name":"Butterfly space","aliases":["Bow-tie space"],"uid":"S000143","description":"The space $X=\\mathbb R\\times [0,\\infty)$ is the closed upper half-plane topologized as follows. Points above the $x$-axis have their usual Euclidean neighborhoods. A local base for a point $p=\\langle p_x,0\\rangle$ on the $x$-axis is formed by the \"butterfly neighborhoods\" $N_\\varepsilon(p)$ for $\\varepsilon>0$, where $N_\\varepsilon(p)$ consists of $p$ together with all points $q=\\langle q_x,q_y\\rangle$ at distance less than $\\varepsilon$ from $p$ and lying underneath the union of the two rays in $X$ that emanate from $p$ and have slopes $\\varepsilon$ and $-\\varepsilon$. In formula, $$N_\\varepsilon(p)=\\{p\\} \\cup \\{q\\in X:\\|q-p\\|<\\varepsilon\\text{ and }q_y<\\varepsilon|q_x-p_x|\\}.$$\n\nDefined as Example 12.1 in {{zb:0148.16701}} ().\n\nExercise 3.1.I in {{zb:0684.54001}} defines a very similar space of the same name, with a slightly different topology but essentially the same properties.","refs":[{"name":"$\\aleph_0$-spaces (E. Michael)","zb":"0148.16701"},{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"name":"Diamond poset $2\\times 2$ with Alexandrov topology","aliases":["Four-element Boolean algebra with Alexandrov topology"],"uid":"S000144","description":"Let $P=(\\{0,a,b,1\\},\\le)$ be the four-element poset with $0$ as minimum element, $1$ as maximum element, and the remaining elements $a$ and $b$ incomparable. This poset is order-isomorphic to the Boolean algebra with four elements. The space $X$ is $P$ with the corresponding Alexandrov topology, whose open sets are the upper sets for $\\le$.\n\nThe name \"diamond poset\" is used for example in {{doi:10.1016/j.jcta.2015.03.010}}.","refs":[{"name":"Alexandrov topology on Wikipedia","wikipedia":"Alexandrov_topology"},{"name":"Diamond-free families (Griggs, Li & Lu)","doi":"10.1016/j.jcta.2015.03.010"}]},{"name":"Free ultrafilter topology on $\\omega$","aliases":[],"uid":"S000145","description":"The space $X$ is the set $\\omega$ with topology $\\tau=\\mathscr U\\cup\\{\\emptyset\\}$,\nwhere $\\mathscr U$ is a free ultrafilter on $\\omega$.\n\nThe space is described for example in {{mathse:3088994}} and as item (3) in Theorem 1 of {{zb:1400.39025}}.\n\nNot to be confused with {S111}.\n\nNote: The topology on $X$ depends on the choice of ultrafilter. There are $2^{\\mathfrak c}$ homeomorphism classes of such spaces (see for example {{mathse:34841}}). However, the properties currently tracked in π-Base for this space do not depend on that choice.","refs":[{"name":"Examples of connected door spaces","mathse":3088994},{"name":"Connected door spaces and topological solutions of equations (Wu, Wang, Zhang)","zb":"1400.39025"},{"name":"Answer to \"Are there uncountably many non homeomorphic ways to topologize a countably infinite set?\"","mathse":34841}]},{"name":"Countable product of rationals $\\mathbb Q^\\omega$","aliases":[],"uid":"S000146","description":"The countable product of copies of {S27}.\n\nThis space is a subspace of {S30}.\nNote that it is not homeomorphic to {S142}.\n\nSee {{mathse:4416328}}, {{zb:0642.54033}}, or {{zb:0562.54054}}.","refs":[{"name":"What spaces are homeomorphic to $\\mathbb Q^\\omega = \\mathbb Q^\\mathbb N = \\mathbb Q^\\infty$?","mathse":4416328},{"name":"Characterizations of the countable infinite product of rationals and some related problems (van Engelen)","zb":"0642.54033"},{"name":"Countable products of zero-dimensional absolute $F_{\\sigma \\delta}$ spaces (van Engelen)","zb":"0562.54054"}]},{"name":"(CH) A Luzin subset of the reals (The special Lusin set $L$)","aliases":["Lusin set"],"ambiguous_construction":true,"uid":"S000147","description":"An uncountable subset of the reals such that its intersection with\nevery nowhere dense subset of the reals (equivalently,\nevery meager subset of the reals) is countable.\nFor background information, see the article {{zb:0389.54004}}, available at the [Topology Proceedings webpage](https://topology.nipissingu.ca/tp/reprints/v01/tp01021s.pdf).\n\nNote that the construction of such a set requires additional axioms beyond $ZFC$.\n\nHere we use the $CH$ construction from Lemma 2.6 of {{doi:10.1016/S0166-8641(96)00075-2}}\nin order to guarantee the failure of {P150}.","refs":[{"name":"Luzin spaces (K. Kunen)","zb":"0389.54004"},{"name":"The combinatorics of open covers II (Just et al)","doi":"10.1016/S0166-8641(96)00075-2"}]},{"name":"(CH) A Luzin subset of the reals (The generic Lusin set $H$)","aliases":["Lusin subset of the reals"],"ambiguous_construction":true,"uid":"S000148","description":"An uncountable subset of the reals such that its intersection with\nevery nowhere dense subset of the reals (equivalently,\nevery meager subset of the reals) is countable.\nFor background information, see the article {{zb:0389.54004}}, available at the [Topology Proceedings webpage](https://topology.nipissingu.ca/tp/reprints/v01/tp01021s.pdf).\n\nNote that the construction of such a set requires additional axioms beyond $ZFC$.\n\nHere we use the $CH$ construction from Theorem 2.13 of {{doi:10.1016/S0166-8641(96)00075-2}}\nin order to guarantee {P150}.","refs":[{"name":"Luzin spaces (K. Kunen)","zb":"0389.54004"},{"name":"The combinatorics of open covers II (Just et al)","doi":"10.1016/S0166-8641(96)00075-2"}]},{"name":"Two-sided long line","aliases":[],"uid":"S000149","description":"This space $X$ is constructed from {S38} with a mirror copy extending in the other direction.\n\nLet $L$ denote the {S38} $\\omega_1\\times[0,1)$ with the lexicographic order\nand let $L^\\ast$ be a copy of $L$ with the reverse order.\nAs an ordered set, $X$ is obtained from $L^\\ast$ followed by $L$ with $\\max(L^\\ast)$ and $\\min(L)$ identified.\nIt is given the corresponding order topology.","refs":[{"name":"Long line on Wikipedia","wikipedia":"Long_line_(topology)"}]},{"name":"Right \"closed-ray\" topology on $[0,1]\\cap\\mathbb Q$","aliases":[],"uid":"S000150","description":"The set $X=[0,1]\\cap\\mathbb Q$ with its order induced from $\\mathbb R$, and the topology generated by a base of open sets consisting of the intervals $[x,\\rightarrow)$ for $x\\in X$.\n\nThe open sets are exactly the [upward-closed sets](https://en.wikipedia.org/wiki/Upper_set) in the ordered set $(X,\\le)$. That includes \"closed rays\" $[x,\\rightarrow)$ and \"open rays\" $(x,\\rightarrow)$ for $x\\in X$, as well as sets of the form $(\\theta,1]\\cap\\mathbb Q$ for an irrational $\\theta\\in(0,1)$.\n\nThe closed sets are the downward-closed sets in $(X,\\le)$.\n\nCompare with the coarser topology of {S151}.","refs":[]},{"name":"Right \"open-ray\" topology on $[0,1]\\cap\\mathbb Q$","aliases":[],"uid":"S000151","description":"The set $X=[0,1]\\cap\\mathbb Q$ with its order induced from $\\mathbb R$, and the topology generated by a base of open sets consisting of $X$ and all the intervals $(x,\\rightarrow)$ for $x\\in X$.\n\nThe additional open sets are the sets of the form $(\\theta,1]\\cap\\mathbb Q$ for an irrational $\\theta\\in(0,1)$.\nNote that the \"closed rays\" $[x,\\rightarrow)$ for $x\\in X\\setminus\\{0\\}$ are not open sets.\n\nCompare with the finer topology of {S150}.","refs":[]},{"name":"Poset $\\{-1,0_a,0_b\\}\\cup\\{1/n\\}_{n=1}^\\infty$ with Alexandrov topology","aliases":[],"uid":"S000152","description":"Consider the set $\\{1/n\\}_{n=1}^\\infty$ with its usual order. Let\n\n$\\quad X=\\{-1,0_a,0_b\\}\\cup\\{1/n\\}_{n=1}^\\infty$\n\nwhere either of the zeros is smaller than all $1/n$, the two zeros are incomparable, and $-1$ is smaller than everything else. Then equip $X$ with the Alexandrov topology associated to the partial order defined above; that is, the open sets are the upper sets for $\\le$.\n\nThis space is described in part 2 of {{mathse:5007860}}.","refs":[{"name":"Alexandrov topology on Wikipedia","wikipedia":"Alexandrov_topology"},{"name":"Answer to \"For a compact sober \"highly non-$T_1$\" space, how much \"highly connectedness\" is needed to imply it's a spectral space?\"","mathse":5007860}]},{"name":"Open long ray","aliases":["Open long line"],"uid":"S000153","description":"The space obtained by taking $Y=\\omega_1 \\times [0,1)$ with its lexicographic order topology ({S38}) and removing the first element $\\langle 0,0 \\rangle$.\n\nThe resulting space $X$ is an open subspace of $Y$ with the subspace topology. This is the same as the order topology on $X$, since $X$ is order-convex in $Y$ (see Theorem 16.4 in {{zb:0951.54001}}).","refs":[{"name":"Long line on Wikipedia","wikipedia":"Long_line_(topology)"},{"name":"Topology (Munkres)","zb":"0951.54001"}]},{"name":"Fort space on the real numbers","aliases":["Uncountable Fort Space","One-point compactification of an uncountable discrete space"],"uid":"S000154","counterexamples_id":24,"description":"Let $X=\\mathbb R\\cup\\{\\infty\\}$. Every point of $\\mathbb R$ is isolated and the open neighborhoods of the point $\\infty$ are the cofinite subsets of $X$ containing that point.\n\nThis space is the one-point compactification of an uncountable discrete space (compare with {S22} and {S20}).\n\nDefined as counterexample #24 (\"Uncountable Fort Space\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Fort space","wikipedia":"Fort_space"}]},{"name":"Fortissimo space of size $\\aleph_1$","aliases":[],"uid":"S000155","description":"Let $X$ be a set of cardinality $\\aleph_1$ with a distinguished point $p\\in X$.\nEvery point not equal to $p$ is isolated and the open neighborhoods of $p$ are the cocountable subsets of $X$ containing that point.\n\nThis space is the one-point Lindelöfication of a discrete space of size $\\aleph_1$.\n\nIf (CH) $2^{\\aleph_0}=\\aleph_1$ holds, this space is homeomorphic to {S22}.\nCompare also with {S157}.\n\nFor a general $X$ with unspecified uncountable cardinality,\nsee counterexample #25 (\"Fortissimo Space\") in {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Fort space on Wikipedia","wikipedia":"Fort_space"}]},{"name":"Arens space","aliases":["$S_2$"],"uid":"S000156","description":"Let $X=\\omega\\cup\\{\\infty\\}$ be the one-point compactification of a\ncountable discrete space. Arens space $S_2$ is the minimal topology\non $(X\\times\\omega)\\cup\\{\\infty'\\}$ where\n$(\\{\\infty\\}\\times\\omega)\\cup\\{\\infty'\\}$ is homeomorphic to $X$.\n\nThis space was originally defined by Richard Arens in {{mr:0037500}}.\n\nThe subspace $(\\omega\\times\\omega)\\cup\\{\\infty'\\}$ of the Arens space is homeomorphic to {S23}, as mentioned in Examples 2.10 and 3.10 of {{mr:3269014}}.","refs":[{"name":"Note on convergence in topology (Arens)","mr":37500},{"name":"Mapping theorems on countable tightness and a question of F. Siwiec (Lin & Zhang)","mr":3269014}]},{"name":"Fortissimo space of size $\\aleph_2$","aliases":[],"uid":"S000157","description":"Let $X$ be a set of cardinality $\\aleph_2$ with a distinguished point $p\\in X$.\nEvery point not equal to $p$ is isolated and the open neighborhoods of $p$ are the cocountable subsets of $X$ containing that point.\n\nThis space is the one-point Lindelöfication of a discrete space of size $\\aleph_2$.\n\nIf $2^{\\aleph_0}=\\aleph_2$ holds, this space is homeomorphic to {S22}.\nCompare also with {S155}.\n\nFor a general $X$ with unspecified uncountable cardinality,\nsee counterexample #25 (\"Fortissimo Space\") in {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Fort space on Wikipedia","wikipedia":"Fort_space"}]},{"name":"Unit interval $[0,1]$","aliases":[],"uid":"S000158","description":"The subspace $I=[0,1]=\\{t\\in \\mathbb R : 0\\leq t \\leq 1\\}$ of {S25}.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"name":"Right \"open-ray\" topology on $[0,1]$","aliases":[],"uid":"S000159","description":"$X$ has underlying set $[0,1]$ and the open sets are exactly $X$ and $(a,1]$ for $0 \\leq a \\leq 1$.\n\nThis space is used in part 3 of {{mathse:5007860}}.","refs":[{"name":"Answer to \"For a compact sober \"highly non-$T_1$\" space, how much \"highly connectedness\" is needed to imply it's a spectral space?\"","mathse":5007860}]},{"name":"Right \"open-ray\" topology on $\\omega+1$","aliases":[],"uid":"S000160","description":"$X$ has underlying set $\\omega+1$. The open sets are exactly $X$ and $(a, \\omega]$ for $a \\in X$.","refs":[]},{"name":"Van Douwen's anti-Hausdorff Fréchet space","aliases":[],"uid":"S000161","description":"The topology on \$\\omega\$ constructed by Eric K. van Douwen in Section 5 of\n{{doi:10.1016/0166-8641(93)90147-6}}.","refs":[{"name":"An anti-Hausdorff Fréchet space in which convergent sequences have unique limits","doi":"10.1016/0166-8641(93)90147-6"}]},{"name":"The Singleton","aliases":["The Single-Point Space"],"uid":"S000162","description":"\$X=\\{x\\}\$ with its only valid topology \$\\{\\emptyset,X\\}\$.","refs":[{"name":"Finite topological space","wikipedia":"Finite_topological_space"}]},{"name":"The Empty Space","aliases":[],"uid":"S000163","description":"\$X=\\emptyset\$ with its only valid topology \$\\{\\emptyset\\}\$.","refs":[{"name":"Initial and terminal objects","wikipedia":"Initial_and_terminal_objects"}]},{"name":"Sum of singleton and two-point indiscrete space","aliases":[],"uid":"S000164","description":"The space \$X=\\{a,b,c\\}\$ with the topology \$\\{\\emptyset,\\{a,b\\},\\{c\\},X\\}\$, that is,\nthe topological sum of {S162} and {S4}.","refs":[]},{"name":"One-point compactification of the Arens-Fort space","aliases":[],"uid":"S000165","description":"The one-point compactification $X$ of {S23}.\n\nSee {{mo:435257}}.","refs":[{"name":"Answer to \"\"All retracts are closed\" and \"all compacts are closed\"\"","mo":435257}]},{"name":"Left ray topology on $\\omega+1$","aliases":[],"uid":"S000166","description":"$X$ has underlying set $\\omega+1$. The open sets are exactly $X$ and $[0,a)$ for $a \\in X$. Equivalently, the open sets are exactly downwards closed subsets of $X$. This is the Alexandrov topology for the reverse ordering on $\\omega+1$.","refs":[{"name":"Alexandrov topology on Wikipedia","wikipedia":"Alexandrov_topology"}]},{"name":"Right \"open-ray\" topology on $\\omega+1+\\omega^\\ast$","aliases":[],"uid":"S000167","description":"$X$ has underlying totally ordered set $(\\omega+1)+\\omega^\\ast$, where the second $+$ is concatenation, $\\omega+1$ is equipped with its usual order as an ordinal, and $\\omega^\\ast$ is $\\omega$ with the reverse order. The open sets are exactly $X$ and $(a,\\to)$ for $a \\in X$.","refs":[]},{"name":"Real projective plane $\\mathbb RP^2$","aliases":["RP2"],"uid":"S000168","description":"The quotient of {S169} which identifies its antipodal points.\n\nThis space is homeomorphic to:\n- $\\left( \\mathbb R^3 \\setminus \\{0\\} \\right) / \\sim$, where $x \\sim y$ if and only if $x = c y$ for some nonzero real number $c$.\n- The space of $1$-dimensional subspaces of the real vector space $\\mathbb{R}^3$, $P(\\mathbb{R}^3)$, with the quotient topology induced by the map $\\mathbb R^3 \\setminus \\{0\\} \\to P(\\mathbb{R}^3)$ which sends a vector to its span.\n- The quotient of $\\left[ -1, 1 \\right]^2$ identifying $(t, 1)$ with $(-t, -1)$ and identifying $(1, t)$ with $(-1, -t)$, for each $t$.\n\nDefined in Example 3.51 and Proposition 6.2 of {{mr:2766102}}.","refs":[{"name":"Real projective plane on Wikipedia","wikipedia":"Real_projective_plane"},{"name":"Introduction to topological manifolds (Lee)","mr":2766102}]},{"name":"Sphere $S^2$","aliases":["Two-dimensional sphere"],"uid":"S000169","description":"The subspace $\\{x \\in \\mathbb{R}^3: \\|x\\| = 1\\}$ of the Euclidean space $\\mathbb{R}^3$\nconsisting of points with distance 1 from the origin.\n\nThis space is homeomorphic to:\n- The one-point compactification of {S176},\n- The suspension of {S170}, i.e., the quotient of $S^1 \\times [0,1]$ where $S^1 \\times \\{0\\}$ is identified to a\nsingle point and $S^1 \\times \\{1\\}$ is identified to a single point.","refs":[{"name":"Sphere on Wikipedia","wikipedia":"Sphere"}]},{"name":"Circle $S^1$","aliases":["One-dimensional sphere"],"uid":"S000170","description":"The subspace $\\{x \\in \\mathbb{R}^2: \\|x\\| = 1\\}$ of\n{S176} consisting of points with distance $1$ from the origin.\n\nThis space is homeomorphic to:\n- The one-point compactification of {S25},\n- The quotient of the interval $[0,1]$ identifying its endpoints.","refs":[{"name":"Circle on Wikipedia","wikipedia":"Circle"}]},{"name":"Brian's stack of Bernstein sets","aliases":[],"uid":"S000171","description":"Let $X = \\mathbb{R}$ and let $(X_i)_{i\\in \\omega}$ be a countably infinite partition of $X$ into\n[Bernstein sets](https://en.wikipedia.org/wiki/Bernstein_set) (for the Euclidean topology).\nGiven $x \\in X$, let $n \\in \\omega$ be such that $x \\in X_n$.\nThen basic open neighbourhoods of $x$ are given by the sets $\\{x\\} \\cup (U \\cap \\bigcup_{i0$ has a local base consisting of the sets\n$(\\beta,\\alpha]\\times(\\gamma,\\omega_1]$ for $\\beta<\\alpha$ and $\\gamma<\\omega_1$.\nThe point $\\langle 0,\\omega_1\\rangle$ has a local base consisting of the sets\n$\\{0\\}\\times(\\gamma,\\omega_1]$ for $\\gamma<\\omega_1$.\nThe resulting space $X$ has a finer topology than {S218}.\n\nEach vertical slice $\\{\\alpha\\}\\times[0,\\omega_1]$ for $\\alpha<\\omega_1$\nis homeomorphic to {S155}.\nNote however that $X$ is *not* homeomorphic to the product of\n{S35} and {S155}.\n\nH. J. Cohen provides this example, communicated by Bing, in {{zb:0046.16403}}\n()\nto show that {P207}\nis strictly stronger than {P88}.\nSee {{mathse:5004796}} for details in English.","refs":[{"name":"Sur une problème de M. Dieudonné (H. J. Cohen)","zb":"0046.16403"},{"name":"Answer to \"Is every $T_4$ topological space divisible?\"","mathse":5004796}]},{"name":"Hewitt realcompactification of Rudin's Dowker space","aliases":[],"uid":"S000208","description":"$X$ is the subspace of the product $\\prod_{n\\in\\omega}(\\omega_{n+1}+1)$ with the box topology consisting of all $f\\in \\prod_{n\\in \\omega}(\\omega_{n+1}+1)$ such that $\\omega< \\text{cf}(f(n))$ for all $n$ (see {{wikipedia:Cofinality#Cofinality_of_ordinals_and_other_well-ordered_sets}}).\n\nDefined (as the space called $X'$) and shown to be the Hewitt realcompactification of {S138} in section IV.4 of {{zb:0224.54019}} \n(see remark 8.8 of {{zb:1380.46022}} for the definition of Hewitt realcompactification).","refs":[{"name":"A normal space X for which X×I is not normal (M.E. Rudin)","zb":"0224.54019"},{"name":"Rings of Continuous Functions (Gillman & Jerison)","zb":"1380.46022"},{"name":"Cofinality on Wikipedia","wikipedia":"Cofinality#Cofinality_of_ordinals_and_other_well-ordered_sets"}]},{"name":"Circle with two origins","aliases":[],"uid":"S000209","description":"Choose a point $0 \\in S^1$ to be called the \"origin\", and replace $0$ with two origins $0_1$ and $0_2$. Basic open neighborhoods of a point $x \\neq 0$ are Euclidean open neighborhoods of $x$ not containing $0$. Basic open neighborhoods of each origin $0_i$ are of the form $(U\\setminus\\{0\\})\\cup\\{0_i\\}$ with $U$ a Euclidean open neighborhood of $0$.\n\nLet $\\{1, 2\\}$ have the discrete topology. $X$ is homeomorphic to the quotient space of $S^1 \\times \\{1, 2\\}$ obtained by identifying $\\langle \\theta, 1 \\rangle$ and $\\langle \\theta, 2 \\rangle$ exactly when $\\theta$ is not the \"origin\" point.\n\n$X$ is homeomorphic to the Alexandroff one-point compactification of {S83}.","refs":[{"name":"Alexandroff extension on Wikipedia","wikipedia":"Alexandroff_extension"}]},{"name":"Interval $[0,1)$","aliases":[],"uid":"S000210","description":"The subspace $[0,1)=\\{t\\in \\mathbb R : 0\\leq t < 1\\}$ of {S25}.","refs":[]},{"name":"Join of cofinite and modified right \"closed-ray\" topologies on $\\omega_1+1$","aliases":[],"uid":"S000211","description":"The set $X=\\omega_1+1$ with the topology generated by the base\n$\\{[\\alpha,\\omega_1]\\setminus F:\\alpha<\\omega_1,F \\text{ finite subset of }X\\}$.\nThe topology is the join in the [lattice of topologies](https://en.wikipedia.org/wiki/Lattice_of_topologies) on $X$ of\n(1) the cofinite topology and\n(2) the topology with base consisting of the rays $[\\alpha,\\omega_1]$ for all $\\alpha<\\omega_1$ (but not $\\alpha=\\omega_1$).\n\nSee {{mathse:5017415}}.\n\nA related example was constructed by Isbell in section 2 of\n{{doi:10.1090/S0002-9939-1987-0891170-3}}\nas an example of a {P130} space which is not {P141}. We make it {P2} by applying section 3 of the same article.\nSee also {{mathse:3904987}}.","refs":[{"name":"A distinguishing example in k-spaces (Isbell)","doi":"10.1090/S0002-9939-1987-0891170-3"},{"name":"Answer to \"Is there a locally compact space which is not a k-space\"","mathse":3904987}]},{"name":"Lexicographically ordered Hilbert cube $[0,1]^\\omega$","aliases":["Lexicographic Hilbert cube $[0,1]^\\omega$"],"uid":"S000212","description":"The space $X=[0,1]^\\omega$ ordered lexicographically and with the corresponding order topology.","refs":[{"name":"Answer to \"Is there a nontrivial LOTS that is connected and totally path disconnected?\"","mathse":4453440}]},{"name":"Pseudocircle","aliases":[],"uid":"S000213","description":"Let $X$ be the *complete bipartite poset* $K_{2,2}$, with two incomparable elements in the bottom layer, two incomparable elements in the top layer, and every bottom element less than every top element.\n\nSpecifically, take $X=\\{0,1,2,3\\}$ with bottom elements $0,2$ and top elements $1,3$.\n(The ordered pairs $(a,b)$ with $a.","refs":[{"name":"Pseudocircle on Wikipedia","wikipedia":"Pseudocircle"},{"name":"Answer to \"Is there a relationship between a quotient group of the fundamental group of X and the fundamental group of a quotient topology of X?\"","mo":293177}]},{"name":"Lexicographically ordered $\\mathbb Z^{\\omega_1}$","aliases":[],"uid":"S000214","description":"The space $X = \\mathbb Z ^ {\\omega_1}$ with the order topology induced by lexicographic order. \nThat is, for $f,g \\in X$ we have $f < g$ if and only if there exists $\\alpha \\in \\omega_1$ such that $f(\\alpha) < g(\\alpha)$ and $f(\\beta) = g(\\beta)$ for all $\\beta < \\alpha$.\n\nSee {{mo:500191}}.","refs":[{"name":"Answer to: \"Find a topological group which is T_5 but not T_6\"","mo":500191}]},{"name":"Katětov's non-normal subspace of $\\beta\\mathbb{N}$","aliases":[],"uid":"S000216","description":"Fix a bijection $\\varphi:\\mathbb{N}\\to\\mathbb{Q}$. For each irrational $r$ fix a sequence of rational numbers $s_n\\to r$, and let $E_r = \\{\\varphi^{-1}(s_n) : n\\in\\mathbb{N}\\}$. Let $\\mathcal{E} = \\{E_r : r\\in\\mathbb{R}\\setminus\\mathbb{Q}\\}$. Let $E'$ be the set of limit points for a subset $E$ of {S108}. Then $E'\\neq \\emptyset$ for $E \\in\\mathcal{E}$. For each $E\\in\\mathcal{E}$ pick some $p_E\\in E'$.\n\nKatětov's non-normal subspace of $\\beta\\mathbb{N}$ is the space $X=\\mathbb{N}\\cup D$ where $D = \\{p_E : E\\in\\mathcal{E}\\}$.\n\nConstructed in exercise 6Q of {{zb:1380.46022}}.","refs":[{"name":"Rings of Continuous Functions (Gillman & Jerison)","zb":"1380.46022"}]},{"name":"Left ray topology on $\\omega_1$","aliases":[],"uid":"S000217","description":"Let $X=\\omega_1$, with a base for the topology consisting of the left rays\n$[0,\\alpha]=[0,\\alpha+1)$ for $\\alpha\\in\\omega_1$.\n\nThe open sets are the downward closed subsets of $X$, which are $X$, $\\emptyset$,\nthe basic open sets above, and the rays $[0,\\alpha)$ with $\\alpha$ limit ordinal.\nThis is the Alexandrov topology for the reverse ordering on $\\omega_1$.","refs":[{"name":"Alexandrov topology on Wikipedia","wikipedia":"Alexandrov_topology"}]},{"name":"Product $\\omega_1\\times(\\omega_1+1)$","aliases":[],"uid":"S000218","description":"The product of {S35} and {S36}.","refs":[]},{"name":"Discrete space of size $2^{\\mathfrak c}$","aliases":[],"uid":"S000219","description":"Let $X$ be a set of cardinality $2^{\\mathfrak c}$, for example the power set of $\\mathbb R$,\nand let all subsets of $X$ be open.","refs":[{"name":"Discrete space on Wikipedia","wikipedia":"Discrete_space"}]},{"name":"Right \"closed-ray\" topology on $\\omega_1$","aliases":[],"uid":"S000220","description":"As a set, $X=\\omega_1=[0,\\omega_1)$ with its usual order.\nThe open sets are the [upward-closed sets](https://en.wikipedia.org/wiki/Upper_set) in the ordered set $(X,\\le)$,\nconsisting of $\\emptyset$ and the \"closed rays\" $[\\alpha,\\rightarrow)$ for $\\alpha\\in X$.\n\n- The open sets include the \"open rays\" since $(\\beta,\\rightarrow)=[\\beta+1,\\rightarrow)$.\n- The topology is the {P90} topology corresponding to $(X,\\le)$.\n\nThe closed sets are the downward-closed sets in $(X,\\le)$.\n\nCompare with the coarser topology of {S221}.","refs":[]},{"name":"Right \"open-ray\" topology on $\\omega_1$","aliases":[],"uid":"S000221","description":"As a set, $X=\\omega_1=[0,\\omega_1)$ with its usual order.\nThe open sets for the topology are $X,\\emptyset$\nand all \"open rays\" $(\\alpha,\\rightarrow)=[\\alpha+1,\\rightarrow)$ for $\\alpha\\in X$.\n\nNote that the \"closed rays\" $[\\beta,\\rightarrow)$ with $\\beta$ a limit ordinal are not open sets.\n\nThe nonempty proper closed sets are the intervals $[0,\\alpha]$ for $\\alpha\\in X.$\n\nCompare with the finer topology of {S220}.","refs":[]},{"name":"Product topology on $\\omega^{2^\\mathfrak{c}}$","aliases":["Uncountable product of $\\mathbb Z^+$","Supercontinuum power of $\\omega$"],"uid":"S000222","description":"The product topology on $\\omega^{2^{\\mathfrak c}}=\\omega^{\\beth_2}$,\nwhere each factor is {S2}.\n\nDefined more generally as counterexample #103\n(\"Uncountable Products of $\\mathbb{Z}^+$\")\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"name":"Ellentuck topology on $[\\omega]^\\omega$","aliases":[],"uid":"S000223","description":"Let $X = [\\omega]^\\omega$, the set of infinite sets of nonnegative integers, and give it the topology with basis consisting of all sets\n\n$$[s, A] = \\{B \\in X: s \\subseteq B \\subseteq s \\cup A \\;\\text{ and }\\; \\max(s) < \\min(B \\setminus s)\\}$$\n\nfor a finite $s\\subseteq\\omega$ and an infinite $A\\subseteq\\omega$.\nIf $s = \\emptyset$ we let $\\max(s) = -1$.\n\n*Note*: Omitting the condition $\\max(s) < \\min(B \\setminus s)$ would give another base for the same topology, but the form chosen above is usually more convenient.\n\nIntroduced by Ellentuck in {{zb:0292.02054}} (). See also Definition 26.25 in {{zb:1007.03002}} and the section \"The Ellentuck Topology\" on p. 248 of {{zb:1400.03002}}","refs":[{"name":"A new proof that analytic sets are Ramsey (E. Ellentuck)","zb":"0292.02054"},{"name":"Set theory (T. Jech, 2003)","zb":"1007.03002"},{"name":"Combinatorial set theory (L. Halbeisen, 2017)","zb":"1400.03002"}]},{"name":"Closed upper half-plane $\\mathbb R^2_+$","aliases":[],"uid":"S000225","description":"The subspace $\\mathbb R^2_+=\\{(x,y)\\in\\mathbb R^2 : y\\ge 0\\}$ of {S176}.\n\nThis space is homemorphic to the product of {S25} and {S210}.","refs":[]},{"name":"Fortissimo space of size $\\aleph_1$ and Sierpinski space glued at their limit points","aliases":[],"uid":"S000226","description":"Let $X$ be a set of cardinality $\\aleph_1$ with two distinguished points $p,q\\in X$.\nEvery point $x\\ne p$ is isolated and the open neighborhoods of $p$ are the cocountable subsets of $X$ containing both $p$ and $q$.\n\nTwo subspaces are worth noting:\n- $Y=X\\setminus\\{q\\}$ is homeomorphic to {S155} with $p$ as limit point;\n- $Z=\\{p,q\\}$ is homeomorphic to {S10} with $p$ as limit point.\n\n$X$ could also be described as the result of \"gluing\"\n{S155} and {S10} at their limit points;\nthat is, $X$ is homeomorphic to the quotient of the topological sum of these two spaces with their limit points identified.","refs":[]}],"theorems":[{"when":{"kind":"atom","property":"P000016","value":true},"then":{"kind":"atom","property":"P000019","value":true},"uid":"T000001","description":"Immediate from the definitions.","refs":[]},{"when":{"kind":"atom","property":"P000019","value":true},"then":{"kind":"atom","property":"P000021","value":true},"uid":"T000002","description":"Let $A \\subset X$ be countably infinite. If $A \\subset X$ has no limit point then it must be closed. Moreover, for every $a \\in A$ there is some open $U_a \\subset X$ such that $U_a \\cap A = \\{a\\}$. Then the set of all $U_a$ together with $X \\setminus A$ forms a countable cover of $X$ and must have some finite subcover. Thus $A$ may be covered by finitely many $U_a$, contradicting the fact that $A$ is infinite. Thus $A$ must have a limit point.\n\nAsserted on page 21 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"atom","property":"P000020","value":true},"then":{"kind":"atom","property":"P000019","value":true},"uid":"T000003","description":"If $X$ is not countably compact then there is some countable open cover $\\{U_n\\}_{n \\in \\omega}$ of $X$ with no finite subcover. Let $x_n \\in X \\setminus \\cup_{i < n} U_i \\neq \\emptyset$ for all $n \\in \\omega$. Then for any $p \\in X$, $p \\in U_n$ for some least $n$ and so no infinite subsequence of $\\{x_n\\}$ converges to $p$.\n\nAsserted on page 21 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"atom","property":"P000019","value":true},"then":{"kind":"atom","property":"P000022","value":true},"uid":"T000004","description":"If $X$ is countably compact and $f:X \\rightarrow \\mathbb{R}$, the collection of all sets $f^{-1}((-n,n))$ for $n \\in \\omega$ forms a countable cover of $X$ and so has a finite subcover. Thus there is some maximal $r \\in \\omega$ so that $X \\subset f^{-1}((-r,r))$ and so $f$ is bounded.\n\nAsserted on page 21 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"atom","property":"P000025","value":true},"then":{"kind":"atom","property":"P000111","value":true},"uid":"T000005","description":"See the proof of the theorem in {{mathse:4569500}}.","refs":[{"name":"Answer to \"Does locally compact and σ-compact non-Hausdorff space imply hemicompact?\"","mathse":4569500}]},{"when":{"kind":"atom","property":"P000016","value":true},"then":{"kind":"atom","property":"P000024","value":true},"uid":"T000006","description":"If a space is compact, the entire space is a compact open neighborhood of any of its points.\n\nAsserted on page 22 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"atom","property":"P000024","value":true},"then":{"kind":"atom","property":"P000023","value":true},"uid":"T000007","description":"Trivially, the closure of an open neighborhood of a point with compact closure is a compact neighborhood of that point.\n\nAsserted on page 22 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"atom","property":"P000025","value":true},"then":{"kind":"atom","property":"P000023","value":true},"uid":"T000008","description":"By definition.\n\nAsserted on page 22 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"atom","property":"P000016","value":true},"then":{"kind":"atom","property":"P000025","value":true},"uid":"T000009","description":"Since compact spaces are $\\sigma$-compact and locally compact.\n\nAsserted on page 22 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000049","value":true},{"kind":"atom","property":"P000084","value":true}]},"then":{"kind":"atom","property":"P000167","value":true},"uid":"T000010","description":"Shown in the answer by Ulli to {{mathse:4751804}}. Stated for Hausdorff spaces, but the proof generalizes by restricting to an open Hausdorff neighborhood of the limit point.","refs":[{"name":"Answer to \"What separation is required to ensure extremally disconnected spaces are sequentially discrete?\"","mathse":4751804}]},{"when":{"kind":"atom","property":"P000183","value":true},"then":{"kind":"atom","property":"P000182","value":true},"uid":"T000011","description":"Evident from the definitions, as a $k$-network is a network.","refs":[]},{"when":{"kind":"atom","property":"P000034","value":true},"then":{"kind":"atom","property":"P000030","value":true},"uid":"T000012","description":"Proved in {{mathse:4862626}}.","refs":[{"name":"Answer to \"Fully normal implies paracompact without a $T_1$ assumption?\"","mathse":4862626},{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"when":{"kind":"atom","property":"P000016","value":true},"then":{"kind":"atom","property":"P000145","value":true},"uid":"T000013","description":"This holds as any finite cover is star-finite.","refs":[]},{"when":{"kind":"atom","property":"P000030","value":true},"then":{"kind":"atom","property":"P000031","value":true},"uid":"T000014","description":"This holds as any locally finite refinement is point finite.\n\nAsserted on page 24 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"atom","property":"P000030","value":true},"then":{"kind":"atom","property":"P000032","value":true},"uid":"T000015","description":"This holds as any countable open cover is an open cover and thus has a locally finite refinement.\n\nAsserted on page 24 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"atom","property":"P000194","value":true},"then":{"kind":"atom","property":"P000033","value":true},"uid":"T000016","description":"See Corollary 2.3 of {{zb:0268.54018}}.","refs":[{"name":"Some results on weak covering conditions (Gittings, Raymond F.)","zb":"0268.54018"}]},{"when":{"kind":"atom","property":"P000019","value":true},"then":{"kind":"atom","property":"P000032","value":true},"uid":"T000017","description":"This holds as any finite cover is locally finite.\n\nAsserted on page 24 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"atom","property":"P000032","value":true},"then":{"kind":"atom","property":"P000033","value":true},"uid":"T000018","description":"This holds as any locally finite refinement is point finite.\n\nAsserted on page 24 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000002","value":true},{"kind":"atom","property":"P000021","value":true}]},"then":{"kind":"atom","property":"P000019","value":true},"uid":"T000019","description":"Proven on pages 19-20 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"atom","property":"P000121","value":true},"then":{"kind":"atom","property":"P000054","value":true},"uid":"T000020","description":"A portion of Theorem 40.3 (Nagata-Smirnov metrization theorem) of {{zb:0951.54001}}.\nSee {{mathse:3777706}} for the weakening of {P53} to {P121}.","refs":[{"name":"Topology (Munkres)","zb":"0951.54001"},{"name":"Nagata-Smirnov Metrization Theorem for Pseudometric Spaces","mathse":3777706}]},{"when":{"kind":"atom","property":"P000026","value":true},"then":{"kind":"atom","property":"P000029","value":true},"uid":"T000021","description":"If $\\{x_n\\}$ is a countable dense subset and $U_\\alpha$ a pairwise disjoint collection of open sets, then choose some $x_\\alpha \\in U_\\alpha$ for each $\\alpha$. The map $U_\\alpha \\mapsto x_\\alpha$ is injective and so $|U_\\alpha| \\leq |\\{x_n\\}|$.\n\nAsserted on page 22 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000136","value":true},{"kind":"atom","property":"P000057","value":true}]},"then":{"kind":"atom","property":"P000183","value":true},"uid":"T000022","description":"In an {P136} space the collection of all singletons is a $k$-network, which is countable if the space is {P57}.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000011","value":true},{"kind":"atom","property":"P000054","value":true}]},"then":{"kind":"atom","property":"P000121","value":true},"uid":"T000023","description":"Theorem 40.3 (Nagata-Smirnov metrization theorem) of {{zb:0951.54001}},\ngeneralized to not assume {P1} in {{mathse:3777706}}.","refs":[{"name":"Topology (Munkres)","zb":"0951.54001"},{"name":"Nagata-Smirnov Metrization Theorem for Pseudometric Spaces","mathse":3777706}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000083","value":true},{"kind":"atom","property":"P000026","value":true}]},"then":{"kind":"atom","property":"P000018","value":true},"uid":"T000024","description":"See {{mathse:1095781}} or Theorem 1 in .","refs":[{"name":"Answer to \"Does separability imply the Lindelöf property?\"","mathse":1095781}]},{"when":{"kind":"atom","property":"P000124","value":true},"then":{"kind":"atom","property":"P000123","value":true},"uid":"T000025","description":"By definition: see page 316 of {{zb:0951.54001}}.","refs":[{"name":"Topology (Munkres)","zb":"0951.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000003","value":true},{"kind":"atom","property":"P000025","value":true}]},"then":{"kind":"atom","property":"P000007","value":true},"uid":"T000026","description":"Asserted in Figure 7 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000023","value":true},{"kind":"atom","property":"P000134","value":true}]},"then":{"kind":"atom","property":"P000012","value":true},"uid":"T000027","description":"By Theorem 19.3 in {{zb:1052.54001}} locally compact Hausdorff spaces are completely regular. The general case follows by passing to the Kolmogorov quotient, as explained for example in {{mathse:4503299}}.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"},{"name":"Locally compact preregular spaces are completely regular","mathse":4503299}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000003","value":true},{"kind":"atom","property":"P000019","value":true},{"kind":"atom","property":"P000028","value":true}]},"then":{"kind":"atom","property":"P000005","value":true},"uid":"T000028","description":"Suppose we have a closed set $A$ and a point $x\\notin A$.\nLet $(U_n)_{n\\in \\mathbb{N}}$ be a countable neighbourhood basis for $x$.\nSince $X$ is {P3}, the (countably many) open sets $X\\setminus\\overline{U_n}$ cover $A$.\nSince $A$ is closed in $X$, it is also {P19};\nconsequently $A\\subseteq\\bigcup_{n\\in I}(X\\setminus\\overline{U_n})$ for some finite $I\\subseteq\\mathbb N$.\nTaking complements, one sees that $\\bigcap_{n\\in I}\\overline{U_n}$, which is a closed neighborhood of $x$, is disjoint from $A$.","refs":[]},{"when":{"kind":"atom","property":"P000182","value":true},"then":{"kind":"atom","property":"P000117","value":true},"uid":"T000029","description":"Evident from the definitions, as a countable family of sets in $X$ is $\\sigma$-locally finite.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000011","value":true},{"kind":"atom","property":"P000018","value":true}]},"then":{"kind":"atom","property":"P000145","value":true},"uid":"T000030","description":"Corollary 5.3.11 of {{zb:0684.54001}} (which uses \"Lindelöf\" to mean {P18} and {P5}) shows that every {P5} {P18} space is {P145}.\nThe result follows by passing to the Kolmogorov quotient.","refs":[{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000123","value":true},{"kind":"atom","property":"P000003","value":true},{"kind":"atom","property":"P000027","value":true}]},"then":{"kind":"atom","property":"P000124","value":true},"uid":"T000031","description":"By definition: see page 316 of {{zb:0951.54001}}.","refs":[{"name":"Topology (Munkres)","zb":"0951.54001"}]},{"when":{"kind":"atom","property":"P000004","value":true},"then":{"kind":"atom","property":"P000003","value":true},"uid":"T000032","description":"Asserted on page 14 of {{zb:0386.54001}}\n(where Comp. Haus means $T_{2.5}$).","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"atom","property":"P000005","value":true},"then":{"kind":"atom","property":"P000004","value":true},"uid":"T000033","description":"If $X$ is $T_3$, it is $T_2$. So given distinct points $a,b\\in X$, there are disjoint open $O_a$ and $O_b$ containing $a$ and $b$ respectively. Then $a \\notin \\operatorname{cl}(O_b)$ and by regularity there is an open $U$ containing $a$ with $\\operatorname{cl}(U) \\cap \\operatorname{cl}(O_b) = \\emptyset$.\n\nAsserted in Figure 1 of {{zb:0386.54001}}\n(where T_3 means regular).","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"atom","property":"P000118","value":true},"then":{"kind":"atom","property":"P000117","value":true},"uid":"T000034","description":"Evident from the definitions, as a $k$-network is a network.","refs":[]},{"when":{"kind":"atom","property":"P000012","value":true},"then":{"kind":"atom","property":"P000011","value":true},"uid":"T000035","description":"If $f:X \\rightarrow [0,1]$ is a continuous function with $f(a)=0$ and $f(B)=\\{1\\}$, then $f^{-1}([0,\\frac{1}{3}))$ and $f^{-1}((\\frac{2}{3},1])$ are disjoint open sets with disjoint closures containing $a$ and $B$ respectively.\n\nAsserted in Figure 1 of {{zb:0386.54001}}\n(where T_3 means regular and T_{3.5} means completely regular).","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"atom","property":"P000014","value":true},"then":{"kind":"atom","property":"P000013","value":true},"uid":"T000036","description":"Asserted in Figure 1 of {{zb:0386.54001}}\n(where T_4 means normal and T_5 means completely normal).","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000013","value":true},{"kind":"atom","property":"P000135","value":true}]},"then":{"kind":"atom","property":"P000012","value":true},"uid":"T000037","description":"See {{mathse:3990826}}.","refs":[{"name":"$R_0$ and normal implies completely regular","mathse":3990826}]},{"when":{"kind":"atom","property":"P000040","value":true},"then":{"kind":"atom","property":"P000037","value":true},"uid":"T000038","description":"If $X$ is ultraconnected and $a,b \\in X$ then $cl(\\{a\\}) \\cap cl(\\{b\\}) \\neq \\emptyset$ so let $p \\in cl(\\{a\\}) \\cap cl(\\{b\\})$. The map $f:[0,1] \\rightarrow X$ by $f([0,\\frac{1}{2})) = \\{a\\}$, $f(\\frac{1}{2})=p$ and $f((\\frac{1}{2},1])=\\{b\\}$ is continuous.\n\nAsserted on page 29 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"atom","property":"P000038","value":true},"then":{"kind":"atom","property":"P000037","value":true},"uid":"T000039","description":"Evident from the definitions.","refs":[]},{"when":{"kind":"atom","property":"P000037","value":true},"then":{"kind":"atom","property":"P000036","value":true},"uid":"T000040","description":"If $X$ is path connected and $a,b \\in X$ then there is a path from $a$ to $b$ in $X$ and so $a$ and $b$ are in the same connected component. Thus $X$ has one connected component.\n\nAsserted on page 29 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"atom","property":"P000045","value":true},"then":{"kind":"atom","property":"P000137","value":false},"uid":"T000041","description":"Immediate from the definitions.","refs":[]},{"when":{"kind":"atom","property":"P000052","value":true},"then":{"kind":"atom","property":"P000002","value":true},"uid":"T000042","description":"Asserted on Figure 9 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000002","value":true},{"kind":"atom","property":"P000051","value":true}]},"then":{"kind":"atom","property":"P000047","value":true},"uid":"T000043","description":"If $X$ is $T_1$, any nontrivial connected subset is dense-in-itself. Thus if $X$ is scattered and $T_1$, it has no nontrivial connected subsets.\n\nAsserted on Figure 9 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"atom","property":"P000185","value":true},"then":{"kind":"atom","property":"P000049","value":true},"uid":"T000044","description":"In a {P185} space, every open set is closed.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000049","value":true},{"kind":"atom","property":"P000003","value":true}]},"then":{"kind":"atom","property":"P000048","value":true},"uid":"T000045","description":"Given distinct points $x,y \\in X$, take disjoint neighborhoods $U_x$ and $U_y$ of $x$ and $y$ respectively, by {P3}. Using {P49}, $\\operatorname{cl}(U_x)$ is a clopen set containing $x$ and not $y$, which shows the space is {P48}.\n\nAsserted on Figure 9 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"atom","property":"P000048","value":true},"then":{"kind":"atom","property":"P000047","value":true},"uid":"T000046","description":"If $X$ is totally separated and $x,y \\in C \\subset X$ then there is a separation $X = U \\cup V$ of $X$ with $x \\in U$ and $y \\in V$. Thus $C = (C \\cap U) \\cup (C \\cap V)$ is disconnected.\n\nAsserted on Figure 9 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"atom","property":"P000047","value":true},"then":{"kind":"atom","property":"P000046","value":true},"uid":"T000047","description":"Holds as path components are contained in components.\n\nAsserted on Figure 9 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"atom","property":"P000048","value":true},"then":{"kind":"atom","property":"P000009","value":true},"uid":"T000048","description":"If $X$ is totally separated and $a,b \\in X$, let $a \\in U$ and $b \\in V$ with $U,V$ open and disjoint and $X=U \\cup V$. Define $f:X \\rightarrow [0,1]$ by $f(U)=\\{0\\}$ and $f(V)=\\{1\\}$. Then $f$ is a continuous function with $f(a)=0$ and $f(b)=1$.\n\nAsserted on Figure 9 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"atom","property":"P000046","value":true},"then":{"kind":"atom","property":"P000002","value":true},"uid":"T000049","description":"See {{mathse:4580119}}.","refs":[{"name":"Totally path disconnected spaces are $T_1$","mathse":4580119}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000003","value":true},{"kind":"atom","property":"P000026","value":true}]},"then":{"kind":"atom","property":"P000059","value":true},"uid":"T000050","description":"Let $D$ be a countable dense subset of $X$. Then the map $\\Phi : X \\rightarrow 2^{P(D)}$ by $\\Phi(x)(A)=1$ if and only if $A=D \\cap U_x$ for some neighborhood $U_x$ of $x$ is injective if $X$ is Hausdorff.\n\nShown on page 26 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"atom","property":"P000039","value":true},"then":{"kind":"atom","property":"P000041","value":true},"uid":"T000051","description":"Any open subset of a hyperconnected space is connected.\n\nAsserted on page 26 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000047","value":true},{"kind":"atom","property":"P000125","value":true}]},"then":{"kind":"atom","property":"P000036","value":false},"uid":"T000052","description":"Asserted on page 32 of {{zb:0386.54001}}; note that\nthe singleton is ruled out.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000018","value":true},{"kind":"atom","property":"P000023","value":true}]},"then":{"kind":"atom","property":"P000025","value":true},"uid":"T000053","description":"See {{mathse:4568032}}.\n\nAsserted on page 22 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Answer to \"A question about local compactness and σ-compactness\"","mathse":4568032}]},{"when":{"kind":"atom","property":"P000050","value":true},"then":{"kind":"atom","property":"P000012","value":true},"uid":"T000054","description":"If $X$ is zero dimensional, then given any open $U \\subset X$ and $x \\in U$ there is a clopen $V$ with $x \\in V \\subset U$. The function equal to $0$ on $V$ and $1$ outside of $V$ is continuous and separates $x$ from the complement of $U$.\n\nAsserted on page 33 of {{zb:0386.54001}} for the regularity property. Completely regularity is the simple modification above.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"atom","property":"P000074","value":true},"then":{"kind":"atom","property":"P000182","value":true},"uid":"T000055","description":"Follows from the definition.","refs":[]},{"when":{"kind":"atom","property":"P000121","value":true},"then":{"kind":"atom","property":"P000034","value":true},"uid":"T000056","description":"Asserted on page 34 of {{zb:0386.54001}} for metrizable spaces. The result extends to pseudometrizable spaces, since the Kolmogorov quotient of a pseudometrizable space is metrizable, and the quotient induces a bijection between topologies, which preserves the qualities of being an open cover and of being a star refinement.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"atom","property":"P000144","value":true},"then":{"kind":"atom","property":"P000028","value":true},"uid":"T000057","description":"Take a pseudometrizable neighborhood. The balls of rational radius within this neighborhood form a local basis.","refs":[{"name":"How can the implication metrizable -> first countable be generalized?","mathse":4659902}]},{"when":{"kind":"atom","property":"P000023","value":true},"then":{"kind":"atom","property":"P000140","value":true},"uid":"T000058","description":"See {{mathse:919892}}.","refs":[{"name":"Every locally compact space is compactly generated","mathse":919892}]},{"when":{"kind":"atom","property":"P000079","value":true},"then":{"kind":"atom","property":"P000141","value":true},"uid":"T000059","description":"As shown in {{mathse:2026072}}, {P79} spaces are {P140}. The stronger conclusion of {P141} is Proposition 1.6 in [N. Strickland: The category of CGWH spaces](https://ncatlab.org/nlab/files/StricklandCGHWSpaces.pdf).","refs":[{"name":"A first countable Hausdorff space is compactly generated","mathse":2026072}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000140","value":true},{"kind":"atom","property":"P000170","value":true}]},"then":{"kind":"atom","property":"P000142","value":true},"uid":"T000060","description":"Immediate from the definitions, since {P170} means all\n{P16} subspaces are {P3}.","refs":[{"name":"Compactly generated space on Wikipedia","wikipedia":"Compactly_generated_space"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000182","value":true},{"kind":"atom","property":"P000005","value":true}]},"then":{"kind":"atom","property":"P000074","value":true},"uid":"T000061","description":"Follows from the definition.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000054","value":true},{"kind":"atom","property":"P000062","value":true}]},"then":{"kind":"atom","property":"P000027","value":true},"uid":"T000062","description":"Every locally finite collection of nonempty open sets in a {P62} space is countable.\n(Proof: Let $\\mathcal V$ be such a collection. Each $x \\in X$ has an open neighborhood \n$N_x$ meeting only finitely many members of $\\mathcal{V}$. As $X$ is {P62}, there are \n$x_n$ for $n<\\omega$ such that $\\{N_{x_n}:n<\\omega\\}$ covers a dense subset of $X$. Since\neach member of $ \\mathcal{V}$ meets some $N_{x_n}$, $\\mathcal{V}$ is necessarily countable.)\n\nTherefore any $\\sigma$-locally finite base in a {P62} space is countable, and a {P62}\nspace with a $\\sigma$-locally finite base is {P27}.","refs":[{"name":"Is a σ-locally finite collection of open sets locally countable?","mathse":4878507}]},{"when":{"kind":"atom","property":"P000043","value":true},"then":{"kind":"atom","property":"P000042","value":true},"uid":"T000063","description":"Evident from the definitions.","refs":[]},{"when":{"kind":"atom","property":"P000042","value":true},"then":{"kind":"atom","property":"P000041","value":true},"uid":"T000064","description":"Since any path connected set is connected.\n\nAsserted on page 30 of {{zb:0386.54001}}","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000023","value":true},{"kind":"atom","property":"P000053","value":true}]},"then":{"kind":"atom","property":"P000055","value":true},"uid":"T000065","description":"Proven as Theorem 2.3.30 of {{doi:10.1007/b98956}}.","refs":[{"name":"A Course on Borel Sets","doi":"10.1007/b98956"}]},{"when":{"kind":"atom","property":"P000133","value":true},"then":{"kind":"atom","property":"P000154","value":true},"uid":"T000066","description":"By definition.","refs":[]},{"when":{"kind":"atom","property":"P000057","value":true},"then":{"kind":"atom","property":"P000058","value":true},"uid":"T000067","description":"Since $\\aleph_0 < \\mathfrak{c} = 2^{\\aleph_0}$.\n\n----\nThe converse is independent of ZFC. It is true iff CH holds.","refs":[{"name":"Continuum hypothesis","wikipedia":"Continuum_hypothesis"}]},{"when":{"kind":"atom","property":"P000058","value":true},"then":{"kind":"atom","property":"P000163","value":true},"uid":"T000068","description":"Immediate from the definitions.","refs":[]},{"when":{"kind":"atom","property":"P000148","value":true},"then":{"kind":"atom","property":"P000142","value":true},"uid":"T000069","description":"See Lemma 1.4(c) in [N. Strickland: The category of CGWH spaces](https://ncatlab.org/nlab/files/StricklandCGHWSpaces.pdf). See also {{mathse:4303611}}.","refs":[{"name":"Equivalent criteria for being a compactly closed set","mathse":4303611}]},{"when":{"kind":"atom","property":"P000142","value":true},"then":{"kind":"atom","property":"P000002","value":true},"uid":"T000070","description":"Consider a singleton $\\{p\\}\\subseteq X$. Its intersection with every compact Hausdorff (hence $T_1$) subspace $K$ of $X$ is empty or a single point, which is closed in $K$. Hence the singleton is closed in $X$.","refs":[{"name":"Answer to Unraveling the various definitions of k-space or compactly generated space","mathse":4647078}]},{"when":{"kind":"atom","property":"P000039","value":true},"then":{"kind":"atom","property":"P000060","value":true},"uid":"T000071","description":"As defined on page 223 of {{zb:0386.54001}}\nall continuous functions from a hyperconnected space to the\nreals are constant.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"atom","property":"P000040","value":true},"then":{"kind":"atom","property":"P000088","value":true},"uid":"T000072","description":"In an {P40} space, since any two nonempty closed sets intersect, any discrete family of closed sets can only contain one nonempty set (which is contained in the open set $X$).","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000050","value":true},{"kind":"atom","property":"P000001","value":true}]},"then":{"kind":"atom","property":"P000048","value":true},"uid":"T000073","description":"Given two distinct points $a,b\\in X$, by {P1} there is an open set $U$\ncontaining one of the points and not the other, say $a\\in U$ and $b\\notin U$.\nBy {P50} there is a clopen set $V$ with $a\\in V\\subseteq U$.\nThen $V$ is a clopen set containing $a$ and not $b$;\nand $X\\setminus V$ is a clopen set containing $b$ and not $a$.\n\nAsserted in Figure 9 (page 32) of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"atom","property":"P000057","value":true},"then":{"kind":"atom","property":"P000017","value":true},"uid":"T000074","description":"A countable set is a countable union of finite subsets and any finite set is compact.\n\nAsserted in item 4 of space 26 in {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000038","value":true},{"kind":"atom","property":"P000125","value":true}]},"then":{"kind":"atom","property":"P000058","value":false},"uid":"T000075","description":"There is an injective path $f:[0,1]\\to X$ joining two distinct points.\nTherefore $|X| \\geq |[0,1]| = \\mathfrak{c}$.","refs":[]},{"when":{"kind":"atom","property":"P000060","value":true},"then":{"kind":"atom","property":"P000022","value":true},"uid":"T000076","description":"As defined on page 223 of {{zb:0386.54001}}\nall continuous functions from a strongly connected space to the\nreals are constant and thus have compact images.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"atom","property":"P000055","value":true},"then":{"kind":"atom","property":"P000053","value":true},"uid":"T000077","description":"Defined as such on page 37 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"atom","property":"P000044","value":true},"then":{"kind":"atom","property":"P000036","value":true},"uid":"T000078","description":"Defined as such on page 33 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"atom","property":"P000060","value":true},"then":{"kind":"atom","property":"P000036","value":true},"uid":"T000079","description":"If $X$ is disconnected, there are disjoint nonempty open sets $U$ and $V$ with $X = U \\cup V$. Then $f:X \\rightarrow \\mathbb{R}$ by $f(U) = \\{0\\}$ and $f(V) = \\{1\\}$ is continuous and nonconstant.\n\nAsserted in Figure 31 on page 223 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000009","value":true},{"kind":"atom","property":"P000125","value":true}]},"then":{"kind":"atom","property":"P000060","value":false},"uid":"T000080","description":"Choose distinct $x,y \\in X$, and\nby Complete Hausdorff there is a continuous\n\$f:X \\rightarrow \\mathbb{R}\$ with \$f(x)=0 \\neq 1=f(y)\$.\n\nAsserted on page 223 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000023","value":true},{"kind":"atom","property":"P000100","value":true}]},"then":{"kind":"atom","property":"P000024","value":true},"uid":"T000081","description":"Evident from the definitions.","refs":[]},{"when":{"kind":"atom","property":"P000235","value":true},"then":{"kind":"atom","property":"P000096","value":true},"uid":"T000082","description":"Every point $x\\in X$ has an open neighborhood $U$ homeomorphic to some convex open subset of $\\mathbb R^n_+$.\nThe open balls centered at $x$ within the metric space $U$\n(identified with a subset of $\\mathbb R^n_+$)\nform a local base of neighborhoods open in $X$ and {P95}.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000043","value":true},{"kind":"atom","property":"P000052","value":false}]},"then":{"kind":"atom","property":"P000058","value":false},"uid":"T000083","description":"By non-discreteness choose a non-isolated point of $X$.\nIt has a {P38} neighborhood with at least two distinct points,\nso by {T75} this neighborhood has cardinality at least $\\mathfrak c$.","refs":[]},{"when":{"kind":"atom","property":"P000177","value":true},"then":{"kind":"atom","property":"P000005","value":true},"uid":"T000084","description":"Follows from the definition.","refs":[]},{"when":{"kind":"atom","property":"P000052","value":true},"then":{"kind":"atom","property":"P000055","value":true},"uid":"T000085","description":"The topology on a discrete space may be generated by the discrete metric $d(x,y)=1$ for all $x \\neq y$. Then any Cauchy sequence is eventually constant and thus converges.\n\nSee item 6 of spaces 1-3 on page 41 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"atom","property":"P000009","value":true},"then":{"kind":"atom","property":"P000004","value":true},"uid":"T000086","description":"If $X$ is Urysohn and $x,y \\in X$ then there is a continuous $f:X \\rightarrow [0,1]$ with $f(x)=0$ and $f(y)=1$. Then $f^{-1}([0,\\frac{1}{3}))$ and $f^{-1}((\\frac{2}{3},1])$ are open sets about $x$ and $y$ with disjoint closures.\n\nAsserted on page 17 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"atom","property":"P000040","value":true},"then":{"kind":"atom","property":"P000218","value":true},"uid":"T000087","description":"By definition.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000037","value":true},{"kind":"atom","property":"P000125","value":true}]},"then":{"kind":"atom","property":"P000046","value":false},"uid":"T000088","description":"Follows from the definition on page 31 of {{zb:0386.54001}}\nas the path connecting two distinct points would yield a nonsingleton path\ncomponent.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000233","value":true},{"kind":"atom","property":"P000046","value":true}]},"then":{"kind":"atom","property":"P000052","value":true},"uid":"T000089","description":"By {P46} the path components of $X$ are singletons, \nso by {P233} every singleton is open.","refs":[]},{"when":{"kind":"atom","property":"P000027","value":true},"then":{"kind":"atom","property":"P000054","value":true},"uid":"T000090","description":"By the definition of a \$\\sigma\$-locally finite base, see e.g.\n23.9 of {{zb:1052.54001}}. If \$\\{B_n:n<\\omega\\}\$ is a countable base,\nthen \$\\mathcal F_n=\\{B_n\\}\$ is locally finite and\n\$\\bigcup_{n<\\omega}\\mathcal F_n\$ is a \$\\sigma\$-locally finite base.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000039","value":true},{"kind":"atom","property":"P000013","value":true}]},"then":{"kind":"atom","property":"P000040","value":true},"uid":"T000091","description":"If $X$ is not {P40}, there are two disjoint nonempty closed sets $C$ and\n$D$. If $X$ is additionally {P39}, there are no disjoint nonempty open sets (containing $C$\nand $D$), so $X$ is not {P13}.","refs":[]},{"when":{"kind":"atom","property":"P000045","value":true},"then":{"kind":"atom","property":"P000044","value":true},"uid":"T000092","description":"The space $X$ is {P36} by definition of {P45}. If $X=A\\cup B$ with $A$ and $B$ disjoint and connected of size at least $2$ and if the dispersion point $p$ of $X$ is in $A$ for example, then $B$ would be a connected subset of $X\\setminus\\{p\\}$ of size at least $2$. That is not possible since $X\\setminus\\{p\\}$ is {P47}.\n\nAsserted on page 33 of {{zb:0386.54001}}. See also {{mathse:4812062}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Relationship between dispersion point and biconnected property","mathse":4812062}]},{"when":{"kind":"atom","property":"P000182","value":true},"then":{"kind":"atom","property":"P000180","value":true},"uid":"T000093","description":"Having a countable network is a hereditary property; and a space with a countable network is {P26}, since choosing one point from each element of the network provides a dense set in $X$.\n\nSee page 127 and the diagram on page 225 in {{zb:0684.54001}}.","refs":[{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000038","value":true},{"kind":"atom","property":"P000125","value":true}]},"then":{"kind":"atom","property":"P000044","value":false},"uid":"T000094","description":"There is an injective path $f:[0,1]\\to X$ joining two distinct points.\nThen, $f([0,1/3])$ and $f([2/3,1])$ are disjoint connected sets, each of size at least $2$.\nThus $X$ is not {P44}.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000036","value":true},{"kind":"atom","property":"P000233","value":true}]},"then":{"kind":"atom","property":"P000037","value":true},"uid":"T000095","description":"All path components of $X$ are open, hence also clopen.","refs":[]},{"when":{"kind":"atom","property":"P000039","value":true},"then":{"kind":"atom","property":"P000049","value":true},"uid":"T000096","description":"Any nonempty open set $U\\subseteq X$ is dense in $X$, so its closure is the whole space, which is open.\n\nSee {{mathse:4742443}}.","refs":[{"name":"Extremally disconnected without Hausdorff","mathse":4742443}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000049","value":true},{"kind":"atom","property":"P000036","value":true}]},"then":{"kind":"atom","property":"P000039","value":true},"uid":"T000097","description":"A nonempty open set $U\\subseteq X$ has a nonempty clopen closure by {P49}. Since the space is {P36}, that closure must be the whole space, that is, $U$ is dense in $X$.\n\nSee {{mathse:4742443}}.","refs":[{"name":"Extremally disconnected without Hausdorff","mathse":4742443}]},{"when":{"kind":"atom","property":"P000007","value":true},"then":{"kind":"atom","property":"P000002","value":true},"uid":"T000098","description":"By definition as in 15.1 of {{zb:1052.54001}}.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000002","value":true},{"kind":"atom","property":"P000013","value":true}]},"then":{"kind":"atom","property":"P000007","value":true},"uid":"T000099","description":"By definition as in 15.1 of {{zb:1052.54001}}.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"when":{"kind":"atom","property":"P000008","value":true},"then":{"kind":"atom","property":"P000002","value":true},"uid":"T000100","description":"By definition as on page 12 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000002","value":true},{"kind":"atom","property":"P000014","value":true}]},"then":{"kind":"atom","property":"P000008","value":true},"uid":"T000101","description":"By definition as on page 12 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"atom","property":"P000028","value":true},"then":{"kind":"atom","property":"P000174","value":true},"uid":"T000102","description":"Given a countable local base $\\{A_n:n\\in\\omega\\}$ at a point $x$, the collection $\\{B_n:n\\in\\omega\\}$ with $B_n=\\bigcap_{k\\le n}A_k$ is a totally ordered local base at $x$.","refs":[{"name":"Spaces with linearly ordered local bases (S. W. Davis)","mr":540476}]},{"when":{"kind":"atom","property":"P000174","value":true},"then":{"kind":"atom","property":"P000172","value":true},"uid":"T000103","description":"Mentioned in the Introduction section of {{doi:10.1016/j.topol.2014.08.002}}.\n\nGiven a set $A\\subseteq X$ and a point $x\\in\\overline A$, let $(V_\\alpha)_{\\alpha<\\lambda}$ be a local base at $x$ well-ordered by reverse inclusion. For each $\\alpha$, take $x_\\alpha\\in V_\\alpha\\cap A$. Then $(x_\\alpha)_{\\alpha<\\lambda}$ is a transfinite sequence in $A$ that converges to $x$.","refs":[{"name":"An internal characterisation of radiality (R. Leek)","doi":"10.1016/j.topol.2014.08.002"}]},{"when":{"kind":"atom","property":"P000035","value":true},"then":{"kind":"atom","property":"P000002","value":true},"uid":"T000104","description":"By definition as on page 23 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000002","value":true},{"kind":"atom","property":"P000034","value":true}]},"then":{"kind":"atom","property":"P000035","value":true},"uid":"T000105","description":"By definition as on page 23 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000018","value":true},{"kind":"atom","property":"P000019","value":true}]},"then":{"kind":"atom","property":"P000016","value":true},"uid":"T000106","description":"If $X$ is Lindelöf, any open cover has a countable subcover, and if $X$ is countably compact, this subcover has a finite subcover.\n\nAsserted under 17.1 of {{zb:1052.54001}}.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000019","value":true},{"kind":"atom","property":"P000083","value":true}]},"then":{"kind":"atom","property":"P000016","value":true},"uid":"T000107","description":"Theorem 1 and the following Remark of {{doi:10.3792/pja/1195526497}} show that\nin any topological space admitting a point-countable open cover without a countable subcover,\none can construct a sequence without an\n[accumulation point](https://en.wikipedia.org/wiki/Accumulation_point_(sequence));\nand thus such a space cannot be {P19}.\n\nNow let $\\mathcal U$ be an open cover of $X$.\nIf $X$ is {P83}, $\\mathcal U$ has a point-countable open refinement $\\mathcal V$.\nIf $X$ is also {P19}, $\\mathcal V$ has a countable subcover by the previous paragraph;\nand by {P19} again, $\\mathcal V$ has a finite subcover,\nwhich is a finite open refinement of $\\mathcal U$. This proves $X$ is {P16}.","refs":[{"name":"A note on compactness and metacompactness (Boyte)","doi":"10.3792/pja/1195526497"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000234","value":true},{"kind":"atom","property":"P000047","value":true}]},"then":{"kind":"atom","property":"P000052","value":true},"uid":"T000108","description":"By {P47} the connected components of $X$ are singletons,\nso by {P234} every singleton is open.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000040","value":true},{"kind":"atom","property":"P000135","value":true}]},"then":{"kind":"atom","property":"P000129","value":true},"uid":"T000109","description":"If $X$ is {P135} and $x,y\\in X$ are topologically distinct, then $\\operatorname{cl}\\{x\\}$ and $\\operatorname{cl}\\{y\\}$ are disjoint closed subsets of $X$.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000013","value":true},{"kind":"atom","property":"P000022","value":true}]},"then":{"kind":"atom","property":"P000021","value":true},"uid":"T000110","description":"Suppose $X$ is a normal space which is not weakly countably compact. There is a countably infinite closed discrete subset $A=\\{x_1,x_2,\\ldots\\}$ of $X$. By the Tietze Extension Theorem, the continuous function $f$ on $A$ defined by $f(x_n)=n$ can be extended to an (unbounded) real-valued continuous function on all of $X$. So $X$ is not pseudocompact.\n\nEssentially shown on page 20 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000044","value":true},{"kind":"atom","property":"P000176","value":true}]},"then":{"kind":"atom","property":"P000001","value":true},"uid":"T000111","description":"See {{mathse:4814029}}.","refs":[{"name":"Are biconnected spaces T_0?","mathse":4814029}]},{"when":{"kind":"atom","property":"P000008","value":true},"then":{"kind":"atom","property":"P000007","value":true},"uid":"T000112","description":"If $A, B \\subset X$ are disjoint closed sets, $cl(A) \\cap B = A \\cap cl(B) = \\emptyset$.\n\nSee page 17 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"atom","property":"P000007","value":true},"then":{"kind":"atom","property":"P000006","value":true},"uid":"T000113","description":"In a $T_1$ space points are closed. By Urysohn's Lemma, for any point $a$ and closed set $B$ disjoint from $a$, there is a continuous function $f:X \\to [0,1]$ sending $a$ to 0 and $B$ to 1.\n\nSee page 17 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"atom","property":"P000006","value":true},"then":{"kind":"atom","property":"P000009","value":true},"uid":"T000114","description":"Since $X$ is {P2} points of $X$ are closed, and since $X$ is {P12} there is a function separating any point from any closed set.\n\nSee page 17 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"atom","property":"P000006","value":true},"then":{"kind":"atom","property":"P000005","value":true},"uid":"T000115","description":"If $f:X \\rightarrow [0,1]$ with $f(a)=0$ and $f(B)=\\{1\\}$ for some point $a$ and closed set $B$, then $O_a = f^{-1}([0,\\frac{1}{2}))$ and $O_B = f^{-1}((\\frac{1}{2},1])$ are disjoint open sets containing $a$ and $B$ respectively.\n\n\nSee page 17 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"atom","property":"P000179","value":true},"then":{"kind":"atom","property":"P000183","value":true},"uid":"T000116","description":"Follows from the definition.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000183","value":true},{"kind":"atom","property":"P000005","value":true}]},"then":{"kind":"atom","property":"P000179","value":true},"uid":"T000117","description":"Follows from the definition.","refs":[]},{"when":{"kind":"atom","property":"P000003","value":true},"then":{"kind":"atom","property":"P000002","value":true},"uid":"T000118","description":"By definition, see page 11 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"atom","property":"P000002","value":true},"then":{"kind":"atom","property":"P000001","value":true},"uid":"T000119","description":"By definition, see page 11 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"atom","property":"P000097","value":true},"then":{"kind":"atom","property":"P000154","value":true},"uid":"T000120","description":"By definition, since $\\mathbb R$ is a {P133}.","refs":[]},{"when":{"kind":"atom","property":"P000016","value":true},"then":{"kind":"atom","property":"P000017","value":true},"uid":"T000121","description":"By definition; see Figure 3 on page 21 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"atom","property":"P000017","value":true},"then":{"kind":"atom","property":"P000018","value":true},"uid":"T000122","description":"If $X = \\bigcup_{n \\in \\omega} C_n$ with each $C_n$ compact and $\\mathcal{U}$ is any open cover, then some finite subcollection $\\mathcal{U}_n$ covers each $C_n$ and $\\bigcup_n \\mathcal{U}_n$ is a countable subcover of $\\mathcal{U}$.\n\n\nSee Figure 3 on page 21 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000018","value":true},{"kind":"atom","property":"P000032","value":true}]},"then":{"kind":"atom","property":"P000030","value":true},"uid":"T000123","description":"If $X$ is Lindelöf, any open cover has a countable subcover, and if $X$ is countably paracompact, this subcover has a locally finite subcover.\n\nAsserted on page 24 of {{zb:0386.54001}}","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000018","value":true},{"kind":"atom","property":"P000033","value":true}]},"then":{"kind":"atom","property":"P000031","value":true},"uid":"T000124","description":"If $X$ is Lindelöf, any open cover has a countable subcover, and if $X$ is countably metacompact, this subcover has a point finite subcover.\n\nAsserted on page 24 of {{zb:0386.54001}}","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000154","value":true},{"kind":"atom","property":"P000016","value":true}]},"then":{"kind":"atom","property":"P000133","value":true},"uid":"T000125","description":"Let $\\tau$ be the topology on $X$ and let $<$ be the order on $X$ in the definition of {P154}, with corresponding order topology $\\sigma$. Then $\\sigma\\subseteq\\tau$ and the identity map $(X,\\tau)\\to(X,\\sigma)$ is a continuous bijection from a compact space to a Hausdorff space, hence a homeomorphism. So $\\tau=\\sigma$.\n\nSee Lemma 6.1 in {{zb:0228.54026}} ().","refs":[{"name":"On generalized ordered spaces (D. Lutzer)","zb":"0228.54026"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000016","value":true},{"kind":"atom","property":"P000003","value":true},{"kind":"atom","property":"P000049","value":true}]},"then":{"kind":"atom","property":"P000119","value":true},"uid":"T000126","description":"By definition.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000086","value":true},{"kind":"atom","property":"P000018","value":true},{"kind":"atom","property":"P000084","value":true},{"kind":"atom","property":"P000064","value":true}]},"then":{"kind":"atom","property":"P000003","value":true},"uid":"T000127","description":"See Theorem 4.1 in {{doi:10.1090/S0002-9939-07-09100-9}}.","refs":[{"name":"Manifolds: Hausdorffness versus homogeneity (Baillif & Gabard)","doi":"10.1090/S0002-9939-07-09100-9"}]},{"when":{"kind":"atom","property":"P000018","value":true},"then":{"kind":"atom","property":"P000062","value":true},"uid":"T000128","description":"By the definition given in e.g. {{doi:10.2307/1996310}}.","refs":[{"name":"Products of weakly-א-compact spaces","doi":"10.2307/1996310"}]},{"when":{"kind":"atom","property":"P000029","value":true},"then":{"kind":"atom","property":"P000062","value":true},"uid":"T000129","description":"$wc(X)\\leq c(X)$ where $wc(X)$ is the weak covering number of $X$ and $c(X)$ is the cellularity of $X$.\n$X$ has CCC iff $c(X)\\leq\\aleph_0$ and is weakly Lindelöf iff $wc(X)\\leq \\aleph_0$.\nThus if $X$ has CCC, then it's weakly Lindelöf.\n\nSee chapter 1, section 3 of {{doi:10.1016/c2009-0-12309-7}}.\n\nOr see {{mathse:4743458}} for a direct proof.","refs":[{"name":"Handbook of Set-Theoretic Topology","doi":"10.1016/c2009-0-12309-7"},{"name":"Is every ccc space weakly Lindelöf?","mathse":4743458}]},{"when":{"kind":"atom","property":"P000054","value":true},"then":{"kind":"atom","property":"P000028","value":true},"uid":"T000130","description":"Suppose $\\mathcal{U} = \\bigcup_{n \\in \\mathbb{N}} U_n$ is a $\\sigma$-locally finite base. For any $x \\in X$, the set $B_n = \\{B \\in U_n | x \\in B \\}$ is finite, and so $\\bigcup_{n \\in \\mathbb{N}} B_n$ is a countable base at $x$.\n\nFollows from the definition given in e.g. 23.9 of {{zb:1052.54001}}.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000154","value":true},{"kind":"atom","property":"P000036","value":true}]},"then":{"kind":"atom","property":"P000133","value":true},"uid":"T000131","description":"The given topology on $X$ and the order topology induced by the order in the definition of {P154} coincide in this case.\n \nSee Lemma 6.1 in {{zb:0228.54026}} ().","refs":[{"name":"On generalized ordered spaces (D. Lutzer)","zb":"0228.54026"},{"name":"Equivalent definitions for GO-spaces (generalized ordered spaces)","mathse":4917398}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000053","value":true},{"kind":"atom","property":"P000206","value":true}]},"then":{"kind":"atom","property":"P000055","value":true},"uid":"T000132","description":"Shown in {{zb:0819.04002}}. See also {{wikipedia:Choquet_game}}.","refs":[{"name":"Choquet game","wikipedia":"Choquet_game"},{"name":"Classical Descriptive Set Theory (Kechris)","zb":"0819.04002"}]},{"when":{"kind":"atom","property":"P000055","value":true},"then":{"kind":"atom","property":"P000063","value":true},"uid":"T000133","description":"Theorem 4.3.26 of {{zb:0684.54001}}.","refs":[{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000064","value":true},{"kind":"atom","property":"P000137","value":false}]},"then":{"kind":"atom","property":"P000056","value":false},"uid":"T000134","description":"By their definitions, see 25.1 and 25.2 of {{zb:1052.54001}}.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"when":{"kind":"atom","property":"P000206","value":true},"then":{"kind":"atom","property":"P000064","value":true},"uid":"T000135","description":"A consequence of Exercise 8.15 and the statement before Exercise 8.13 in \n{{zb:0819.04002}}.","refs":[{"name":"Classical Descriptive Set Theory (Kechris)","zb":"0819.04002"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000023","value":true},{"kind":"atom","property":"P000011","value":true}]},"then":{"kind":"atom","property":"P000064","value":true},"uid":"T000136","description":"See Theorem 34, p. 200 of {{mr:0370454}}, or Proposition 20.18, p. 538 of {{mr:1417259}}.","refs":[{"name":"General Topology (Kelley)","mr":370454},{"name":"Handbook of Analysis and its Foundations (Schechter)","mr":1417259}]},{"when":{"kind":"atom","property":"P000054","value":true},"then":{"kind":"atom","property":"P000118","value":true},"uid":"T000137","description":"Evident from the definitions, as a base for the topology is a $k$-network.","refs":[]},{"when":{"kind":"atom","property":"P000065","value":true},"then":{"kind":"atom","property":"P000058","value":false},"uid":"T000138","description":"See {{zb:1052.54001}} for a discussion on cardinalities.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"when":{"kind":"atom","property":"P000065","value":true},"then":{"kind":"atom","property":"P000163","value":true},"uid":"T000139","description":"Immediate from the definitions.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"when":{"kind":"atom","property":"P000066","value":true},"then":{"kind":"atom","property":"P000018","value":true},"uid":"T000140","description":"For any open cover $\\mathcal U$, apply Menger to $\\langle\\mathcal U,\\mathcal U,\\dots\\rangle$ to produce $\\langle\\mathcal F_0,\\mathcal F_1,\\dots\\rangle$ with each $\\mathcal F_n$ a finite subset of $\\mathcal U$ such that $\\bigcup_{n<\\omega} \\mathcal F_n$ is a countable subcover of $\\mathcal U$.\n\nSee proposition 1.2 of {{doi:10.14712/1213-7243.2015.201}}.","refs":[{"name":"Applications of limited information strategies in Menger's game (S. Clontz)","doi":"10.14712/1213-7243.2015.201"}]},{"when":{"kind":"atom","property":"P000017","value":true},"then":{"kind":"atom","property":"P000070","value":true},"uid":"T000141","description":"See the proof of Theorem 2.2 in {{doi:10.1016/S0166-8641(96)00075-2}}, which we summarize here.\n\nAn $\\omega$-cover of a compact space admits a finite subcover that covers all $k$-sized subsets for any positive integer $k$. Write the space $X$ as an ascending union of $\\{K_n:n\\in\\omega\\}$ where each $K_n$ is compact. Given an $\\omega$-cover $\\mathscr U_n$ of $X$, let $\\mathscr V_n$ be a finite subset of $\\mathscr U_n$ that covers all $n$-sized subsets of $K_n$. Note that this is a winning Markov strategy in the $\\omega$-Menger game (equivalent to {P70}).","refs":[{"name":"The combinatorics of open covers II","doi":"10.1016/S0166-8641(96)00075-2"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000029","value":true},{"kind":"atom","property":"P000012","value":true}]},"then":{"kind":"atom","property":"P000061","value":true},"uid":"T000142","description":"Given a cozero set $U$ in $X$, the open set $X\\setminus\\overline U$\ncontains a maximal collection $\\mathscr V$ of pairwise disjoint nonempty cozero sets of $X$.\nBy {P29} $\\mathscr V$ is countable,\nhence $V=\\bigcup\\mathscr V$ is a cozero set.\nAnd since $X$ is {P12}, the maximality of $\\mathscr V$\nimplies that $V$ is dense in $X\\setminus\\overline U$.\nSo $V$ is a cozero complement of $U$.\n\nSee Example 1.6(d) in {{zb:1067.54015}}.","refs":[{"name":"Cozero complemented spaces; when the space of minimal prime ideals of a $C(X)$ is compact (Henriksen & Woods)","zb":"1067.54015"}]},{"when":{"kind":"atom","property":"P000126","value":true},"then":{"kind":"atom","property":"P000001","value":true},"uid":"T000143","description":"Given two distinct points $x$ and $y$, by the {P126} property either $\\{x\\}$ is open (and is a neighborhood of $x$ not containing $y$) or $X\\setminus\\{x\\}$ is open (and is a neighborhood of $y$ not containing $x$). Therefore the two points are topologically distinguishable.","refs":[{"name":"How do finite door spaces work?","mo":405781}]},{"when":{"kind":"atom","property":"P000052","value":true},"then":{"kind":"atom","property":"P000126","value":true},"uid":"T000144","description":"Evident from the definitions.","refs":[{"name":"Door space on Wikipedia","wikipedia":"Door_space"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000126","value":true},{"kind":"atom","property":"P000003","value":true}]},"then":{"kind":"atom","property":"P000051","value":true},"uid":"T000145","description":"See {{mathse:3789612}}.","refs":[{"name":"In a Hausdorff door space, at most one point is a limit point","mathse":3789612}]},{"when":{"kind":"atom","property":"P000005","value":true},"then":{"kind":"atom","property":"P000011","value":true},"uid":"T000146","description":"See 14.1 of {{zb:1052.54001}}.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"when":{"kind":"atom","property":"P000177","value":true},"then":{"kind":"atom","property":"P000117","value":true},"uid":"T000147","description":"Follows from the definition.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000011","value":true},{"kind":"atom","property":"P000001","value":true}]},"then":{"kind":"atom","property":"P000005","value":true},"uid":"T000148","description":"\$T_3\$ is often defined as regular + \$T_1\$, but on page 12 of\n{{zb:0386.54001}} the authors note that regular + \$T_0\$\nimplies \$T_2\$, and therefore \$T_3\$ (using their usual reversed\nlanguage surrounding \$T_x\$ axioms).","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"atom","property":"P000006","value":true},"then":{"kind":"atom","property":"P000012","value":true},"uid":"T000149","description":"See e.g. {{zb:1052.54001}}","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000117","value":true},{"kind":"atom","property":"P000005","value":true}]},"then":{"kind":"atom","property":"P000177","value":true},"uid":"T000150","description":"Follows from the definition.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000012","value":true},{"kind":"atom","property":"P000001","value":true}]},"then":{"kind":"atom","property":"P000006","value":true},"uid":"T000151","description":"\$T_{3.5}\$ is often defined as regular + \$T_1\$, but on page 14 of\n{{zb:0386.54001}} the authors note that completely regular +\n\$T_0\$ implies \$T_2\$, and therefore \$T_{3.5}\$ (using their usual reversed\nlanguage surrounding \$T_x\$ axioms).","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"atom","property":"P000067","value":true},"then":{"kind":"atom","property":"P000002","value":true},"uid":"T000152","description":"By definition, see {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000002","value":true},{"kind":"atom","property":"P000015","value":true}]},"then":{"kind":"atom","property":"P000067","value":true},"uid":"T000153","description":"By definition, see {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"atom","property":"P000067","value":true},"then":{"kind":"atom","property":"P000008","value":true},"uid":"T000154","description":"See Figure 2 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"atom","property":"P000011","value":true},"then":{"kind":"atom","property":"P000010","value":true},"uid":"T000155","description":"Let $(X,\\tau)$ be a regular topological space. Let also $x \\in X$ and $V \\in \\tau$ be arbitrary. To show that $(X,\\tau)$ has a base of regular open sets, we show there is $U$ satisfying $x \\in U \\subset V$ and $\\operatorname{int}(\\operatorname{cl} (U)) = U$.\n\nSince $x \\notin V^c$, it follows from regularity that $x$ and $V^c$ can be separated by open sets. Let $U_0$ be the open set containing $x$ in this separation. Define $U$ to be $\\operatorname{int}(\\operatorname{cl} (U_0))$.\n\nSince $U_0$ is open, we know $U_0 \\subset U$, and so $x \\in U$.\n\nSince $U_0$ is a separation of $x$ from $V^c$, we know $\\operatorname{cl}(U_0) \\subset V$, and so $U \\subset V$.\n\nFinally, $\\operatorname{int}(\\operatorname{cl} (U)) = \\operatorname{int}(\\operatorname{cl} (\\operatorname{int}(\\operatorname{cl} (U_0)))) = \\operatorname{int}(\\operatorname{cl} (U_0)) = U$.\n\nAsserted in 14E of {{zb:1052.54001}}.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"when":{"kind":"atom","property":"P000015","value":true},"then":{"kind":"atom","property":"P000014","value":true},"uid":"T000156","description":"Separation by a continuous function implies separation by open sets.\n\nSee exercise 6b of section 33 in {{zb:0951.54001}}.","refs":[{"name":"Topology (Munkres)","zb":"0951.54001"}]},{"when":{"kind":"atom","property":"P000127","value":true},"then":{"kind":"atom","property":"P000007","value":true},"uid":"T000157","description":"By definition.","refs":[{"name":"Dowker space on Wikipedia","wikipedia":"Dowker_space"}]},{"when":{"kind":"atom","property":"P000127","value":true},"then":{"kind":"atom","property":"P000032","value":false},"uid":"T000158","description":"By definition.","refs":[{"name":"Dowker space on Wikipedia","wikipedia":"Dowker_space"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000007","value":true},{"kind":"atom","property":"P000032","value":false}]},"then":{"kind":"atom","property":"P000127","value":true},"uid":"T000159","description":"By definition.","refs":[{"name":"Dowker space on Wikipedia","wikipedia":"Dowker_space"}]},{"when":{"kind":"atom","property":"P000068","value":true},"then":{"kind":"atom","property":"P000066","value":true},"uid":"T000160","description":"By definition, see e.g. {{doi:10.14712/1213-7243.2015.201}}.","refs":[{"name":"A note on the Menger (Rothberger) property and topological games (Peng & Shen)","doi":"10.1016/j.topol.2015.12.059"}]},{"when":{"kind":"atom","property":"P000069","value":true},"then":{"kind":"atom","property":"P000066","value":true},"uid":"T000161","description":"See Figure 3 of {{doi:10.14712/1213-7243.2015.201}}.","refs":[{"name":"Applications of limited information strategies in Menger's game (S. Clontz)","doi":"10.14712/1213-7243.2015.201"}]},{"when":{"kind":"atom","property":"P000070","value":true},"then":{"kind":"atom","property":"P000072","value":true},"uid":"T000162","description":"See Figure 3 of {{doi:10.14712/1213-7243.2015.201}}.","refs":[{"name":"Applications of limited information strategies in Menger's game (S. Clontz)","doi":"10.14712/1213-7243.2015.201"}]},{"when":{"kind":"atom","property":"P000017","value":true},"then":{"kind":"atom","property":"P000071","value":true},"uid":"T000163","description":"All compact subsets are relatively compact. See {{mathse:4702452}}.","refs":[{"name":"Applications of limited information strategies in Menger's game (S. Clontz)","doi":"10.14712/1213-7243.2015.201"},{"name":"Definitions of relatively compact","mathse":4702452}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000011","value":true},{"kind":"atom","property":"P000071","value":true}]},"then":{"kind":"atom","property":"P000017","value":true},"uid":"T000164","description":"The closure of a relatively-compact set in a {P11} space is compact. See Proposition 4.4 in {{doi:10.14712/1213-7243.2015.201}}.","refs":[{"name":"Applications of limited information strategies in Menger's game (S. Clontz)","doi":"10.14712/1213-7243.2015.201"}]},{"when":{"kind":"atom","property":"P000070","value":true},"then":{"kind":"atom","property":"P000071","value":true},"uid":"T000165","description":"Let $\\sigma(\\mathcal{U}, n)$ be a winning Markov strategy for $F$ in the Menger\ngame, and let $\\mathfrak{C}$ be the collection of all open covers of $X$. Then for\n\n$R_n = \\bigcap_{\\mathcal{U}\\in\\mathfrak{C}} \\bigcup\\sigma(\\mathcal{U},n)$\n\nit follows that $R_n$ is relatively compact to $X$, and\n$\\bigcup_{n<\\omega} R_n = X$.\n\nSee Corollary 4.7 of {{doi:10.14712/1213-7243.2015.201}}.","refs":[{"name":"Applications of limited information strategies in Menger's game (S. Clontz)","doi":"10.14712/1213-7243.2015.201"}]},{"when":{"kind":"atom","property":"P000071","value":true},"then":{"kind":"atom","property":"P000070","value":true},"uid":"T000166","description":"The second player may cover the nth relatively compact subset during the nth round of the game.\n\nSee Corollary 4.7 of {{doi:10.14712/1213-7243.2015.201}}.","refs":[{"name":"Applications of limited information strategies in Menger's game","doi":"10.14712/1213-7243.2015.201"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000027","value":true},{"kind":"atom","property":"P000069","value":true}]},"then":{"kind":"atom","property":"P000070","value":true},"uid":"T000167","description":"Let $\\sigma(\\mathcal{U}_0,\\dots,\\mathcal{U}_{n-1})$ be a winning strategy for $F$, and observe that since $X$ is second-countable, we may assume all covers are countable. Let $\\mathfrak{C}$ be the collection of all countable covers of $X$. We define $R_s,\\mathcal{U}_s$ for $s\\in\\omega^{<\\omega}$ as follows:\n\n* $R_\\emptyset = \\bigcap_{\\mathcal{U}\\in\\mathfrak{C}} \\left(\\bigcup \\sigma(\\mathcal{U})\\right)$\n* Note that there are only countably many distinct finite subsets $\\sigma(\\mathcal{U})$ of the countable collection $\\mathcal U$. Thus define each $\\mathcal U_{\\langle n\\rangle}$ so that $R_\\emptyset =\\bigcap_{n\\lt\\omega}\\left(\\bigcup\\sigma(\\mathcal{U}_{\\langle n\\rangle})\\right)$\n* $R_s = \\bigcap_{\\mathcal{U}\\in\\mathfrak{C}} \\left(\\bigcup \\sigma(\\mathcal{U}_{s\\restriction 1},\\mathcal{U}_{s\\restriction 2},\\dots,\\mathcal{U}_s,\\mathcal{U})\\right)$\n* Again, note that there are only countably many distinct finite subsets $\\sigma(\\mathcal{U}_{s\\restriction 1},\\mathcal{U}_{s\\restriction 2},\\dots,\\mathcal{U}_s,\\mathcal{U})$ of the countable collection $\\mathcal U$. Thus define each $\\mathcal U_{s{^\\frown}\\langle n\\rangle}$ so that\n$R_s = \\bigcap_{n<\\omega} \\left(\\bigcup \\sigma(\\mathcal{U}_{s\\restriction 1}, \\mathcal{U}_{s\\restriction 2}, \\dots, \\mathcal{U}_s, \\mathcal{U}_{s{^\\frown}\\langle n\\rangle})\\right)$\n\nSee Lemma 4.10 of {{doi:10.14712/1213-7243.2015.201}}.","refs":[{"name":"Applications of limited information strategies in Menger's game (S. Clontz)","doi":"10.14712/1213-7243.2015.201"}]},{"when":{"kind":"atom","property":"P000072","value":true},"then":{"kind":"atom","property":"P000069","value":true},"uid":"T000168","description":"See Figure 3 of {{doi:10.14712/1213-7243.2015.201}}.","refs":[{"name":"Applications of limited information strategies in Menger's game (S. Clontz)","doi":"10.14712/1213-7243.2015.201"}]},{"when":{"kind":"atom","property":"P000051","value":true},"then":{"kind":"atom","property":"P000001","value":true},"uid":"T000169","description":"By definition on page 33 of {{zb:0386.54001}},\ngiven two distinct points \$x,y\$, the subspace \$\\{x,y\\}\$ has\nan isolated point, say, \$x\$. Thus \$\\{x\\}\$ witnesses \$T_0\$.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000134","value":true},{"kind":"atom","property":"P000030","value":true}]},"then":{"kind":"atom","property":"P000034","value":true},"uid":"T000170","description":"Follows from {{mathse:4969398}} ({P134} and {P30} imply {P11}), {{zb:0684.54001}} Theorem 5.1.5 ({P30} and {P11} imply {P13}) and {{mathse:4862626}} ({P13} and {P30} imply {P34}).\n\nNote that while {{zb:0684.54001}} defines paracompact spaces to be {P3}, neither {P3} nor even {P2} are required to upgrade from {P11} to {P13} in the second half of the proof of Theorem 5.1.5.","refs":[{"name":"Answer to \"Fully normal implies paracompact without a $T_1$ assumption?\"","mathse":4862626},{"name":"Answer to \"Showing that $R_1$ paracompact spaces are regular.\"","mathse":4969398},{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000235","value":true},{"kind":"atom","property":"P000125","value":true}]},"then":{"kind":"atom","property":"P000039","value":false},"uid":"T000171","description":"If some point $x\\in X$ has an open neighborhood $U$ homeomorphic to\n$\\mathbb R^n$ or $\\mathbb R^n_+$ with $n\\ge 1$, the open set $U$ contains two disjoint\nnonempty open sets, themselves open in $X$. Otherwise, every point is isolated;\nso two distinct points are contained in two disjoint nonempty open sets.","refs":[]},{"when":{"kind":"atom","property":"P000235","value":true},"then":{"kind":"atom","property":"P000206","value":true},"uid":"T000172","description":"After Player 1 chooses an open set $U_0$ and a point $x_0\\in U_0$,\nPlayer 2 can choose a open neighborhood $V_0\\subseteq U_0$ of $x_0$\nhomeomorphic to some $\\mathbb R^n$ or $\\mathbb R^n_+$.\nThe rest of the game is played on $V_0$ and Player 2 has a winning strategy\nsince $\\mathbb R^n$ and $\\mathbb R^n_+$ are both {P206}\n[(Explore)](https://topology.pi-base.org/spaces?q=Completely+metrizable+%2B+%7EStrongly+Choquet).","refs":[]},{"when":{"kind":"atom","property":"P000084","value":true},"then":{"kind":"atom","property":"P000073","value":true},"uid":"T000173","description":"See Proposition 3.5 of {{mr:0702721}}.","refs":[{"name":"A note on the locally Hausdorff property (S. B. Niefield)","mr":702721}]},{"when":{"kind":"atom","property":"P000073","value":true},"then":{"kind":"atom","property":"P000001","value":true},"uid":"T000174","description":"See page 124 of {{mr:1002193}}.","refs":[{"name":"Topology via logic","mr":1002193}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000235","value":true},{"kind":"atom","property":"P000018","value":true}]},"then":{"kind":"atom","property":"P000027","value":true},"uid":"T000175","description":"Apply the argument in {{mathse:4416020}},\nusing the fact that both $\\mathbb R^n$ and $\\mathbb R^n_+$ are {P27}.","refs":[{"name":"Lindelof and locally Euclidean implies second countable","mathse":4416020}]},{"when":{"kind":"atom","property":"P000075","value":true},"then":{"kind":"atom","property":"P000073","value":true},"uid":"T000176","description":"By definition.\nSee example 21, section 2.6 of {{mr:1077251}}.","refs":[{"name":"General topology I (Arkhangelʹskiĭ, Pontryagin)","mr":1077251}]},{"when":{"kind":"atom","property":"P000121","value":true},"then":{"kind":"atom","property":"P000076","value":true},"uid":"T000177","description":"The entourage-picker in the proximal game may choose the metric entourage of radius $2^{-n}$ during round $n$ of the game.\n\nSee Lemma 4 of {{doi:10.1016/j.topol.2014.06.014}}. The non-{P1} case follows via the Kolmogorov quotient.","refs":[{"name":"An infinite game with topological consequences (Bell)","doi":"10.1016/j.topol.2014.06.014"}]},{"when":{"kind":"atom","property":"P000077","value":true},"then":{"kind":"atom","property":"P000076","value":true},"uid":"T000178","description":"Real lines are {P53} and therefore {P76},\nand the $\\Sigma$-products and closed subsets of {P76} are {P76}.\n\nSee the note preceding Problem 10 in {{mr:3288115}}.","refs":[{"name":"Proximal and semi-proximal spaces.","mr":3288115}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000003","value":true},{"kind":"atom","property":"P000016","value":true},{"kind":"atom","property":"P000076","value":true}]},"then":{"kind":"atom","property":"P000077","value":true},"uid":"T000179","description":"Main result of {{doi:10.1016/j.topol.2014.05.010}}.","refs":[{"name":"Proximal compact spaces are Corson compact","doi":"10.1016/j.topol.2014.05.010"}]},{"when":{"kind":"atom","property":"P000015","value":true},"then":{"kind":"atom","property":"P000061","value":true},"uid":"T000180","description":"In a {P15} space, every open set is a cozero set.\nSo given a cozero set $U\\subseteq X$, \nthe open set $X\\setminus\\overline U$ is a cozero complement of $U$.","refs":[]},{"when":{"kind":"atom","property":"P000053","value":true},"then":{"kind":"atom","property":"P000082","value":true},"uid":"T000181","description":"$X$ is a metrizable neighborhood for each point of $X$.","refs":[]},{"when":{"kind":"atom","property":"P000178","value":true},"then":{"kind":"atom","property":"P000118","value":true},"uid":"T000182","description":"Follows from the definition.","refs":[]},{"when":{"kind":"atom","property":"P000028","value":true},"then":{"kind":"atom","property":"P000080","value":true},"uid":"T000183","description":"Fix a sequence of neighborhoods and pick a point from each of these neighborhoods.\n\nAsserted in the introduction to {{mr:0687569}}.","refs":[{"name":"Products of convergence properties","mr":687569}]},{"when":{"kind":"atom","property":"P000080","value":true},"then":{"kind":"atom","property":"P000079","value":true},"uid":"T000184","description":"Asserted in the introduction to {{mr:0687569}}.","refs":[{"name":"Products of convergence properties","mr":687569}]},{"when":{"kind":"atom","property":"P000079","value":true},"then":{"kind":"atom","property":"P000081","value":true},"uid":"T000185","description":"Asserted in the introduction to {{mr:0687569}}.","refs":[{"name":"Products of convergence properties","mr":687569}]},{"when":{"kind":"atom","property":"P000093","value":true},"then":{"kind":"atom","property":"P000081","value":true},"uid":"T000186","description":"As shown in {{mathse:3809662}}.","refs":[{"name":"Locally countable implies countably tight?","mathse":3809662}]},{"when":{"kind":"atom","property":"P000078","value":true},"then":{"kind":"atom","property":"P000057","value":true},"uid":"T000187","description":"See {{zb:1052.54001}} for a discussion on cardinalities.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000020","value":true},{"kind":"atom","property":"P000167","value":true}]},"then":{"kind":"atom","property":"P000078","value":true},"uid":"T000188","description":"In an infinite {P167} space, a sequence with distinct terms has no convergent subsequence.","refs":[]},{"when":{"kind":"atom","property":"P000078","value":true},"then":{"kind":"atom","property":"P000245","value":true},"uid":"T000189","description":"Immediate from the definition.","refs":[]},{"when":{"kind":"atom","property":"P000114","value":true},"then":{"kind":"atom","property":"P000163","value":true},"uid":"T000190","description":"Evident from the definitions.\n\nSee section 1.19 of {{zb:1052.54001}}.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"when":{"kind":"atom","property":"P000114","value":true},"then":{"kind":"atom","property":"P000057","value":false},"uid":"T000191","description":"Evident from the definitions.\n\nSee section 1.19 of {{zb:1052.54001}}.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000141","value":true},{"kind":"atom","property":"P000171","value":true}]},"then":{"kind":"atom","property":"P000148","value":true},"uid":"T000192","description":"See Proposition 11.4 in [C. Rezk, \"Compactly generated spaces\"](https://ncatlab.org/nlab/files/Rezk_CompactlyGeneratedSpaces.pdf)\nor Proposition 2.14 in [N. Strickland, \"The category of CGWH spaces\"](https://ncatlab.org/nlab/files/StricklandCGHWSpaces.pdf).\n\n(also mentioned implicitly as an equivalent definition of CGWH in {{mo:455457}})","refs":[{"name":"Do CGWH spaces form an exponential ideal in Condensed Sets?","mo":455457}]},{"when":{"kind":"atom","property":"P000003","value":true},"then":{"kind":"atom","property":"P000084","value":true},"uid":"T000193","description":"Follows directly from their definitions.","refs":[{"name":"The equivalence relations of local homeomorphisms and Fell algebras","mr":3084709}]},{"when":{"kind":"atom","property":"P000148","value":true},"then":{"kind":"atom","property":"P000100","value":true},"uid":"T000194","description":"See {{mathse:1072014}}.","refs":[{"name":"Answer to \"Is every compact space compactly generated?\"","mathse":1072014}]},{"when":{"kind":"atom","property":"P000084","value":true},"then":{"kind":"atom","property":"P000002","value":true},"uid":"T000195","description":"Given $a,b\\in X$ with $a\\neq b$. We want to show that $a$ has a neighborhood that does not contain $b$. Since $X$ is locally $T_2$, there is an open neighborhood $U\\subseteq X$ of $a$ which is Hausdorff. If $b\\notin U$, the claim is proved. Assume now $b\\in U$. Then $a$ and $b$ have disjoint open neighborhoods in $U$ which are also open in $X$.\n\nSee also {{mathse:3142975}}.","refs":[{"name":"The equivalence relations of local homeomorphisms and Fell algebras","mr":3084709},{"name":"Locally Euclidean space implies T1 space","mathse":3142975}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000023","value":true},{"kind":"atom","property":"P000003","value":true}]},"then":{"kind":"atom","property":"P000063","value":true},"uid":"T000196","description":"Theorem 3.3.9 of {{zb:0684.54001}} asserts that any locally compact Hausdorff\nspace is an open subset of any Hausdorff compactification.","refs":[{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000118","value":true},{"kind":"atom","property":"P000005","value":true}]},"then":{"kind":"atom","property":"P000178","value":true},"uid":"T000197","description":"Follows from the definition.","refs":[]},{"when":{"kind":"atom","property":"P000245","value":true},"then":{"kind":"atom","property":"P000208","value":true},"uid":"T000198","description":"Immediate from the definitions.","refs":[]},{"when":{"kind":"atom","property":"P000116","value":true},"then":{"kind":"atom","property":"P000026","value":true},"uid":"T000199","description":"By definition.","refs":[{"name":"Polish space on Wikipedia","wikipedia":"Polish_space"}]},{"when":{"kind":"atom","property":"P000116","value":true},"then":{"kind":"atom","property":"P000055","value":true},"uid":"T000200","description":"By definition.","refs":[{"name":"Polish space on Wikipedia","wikipedia":"Polish_space"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000026","value":true},{"kind":"atom","property":"P000055","value":true}]},"then":{"kind":"atom","property":"P000116","value":true},"uid":"T000201","description":"By definition.","refs":[{"name":"Polish space on Wikipedia","wikipedia":"Polish_space"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000144","value":true},{"kind":"atom","property":"P000134","value":true},{"kind":"atom","property":"P000030","value":true}]},"then":{"kind":"atom","property":"P000121","value":true},"uid":"T000202","description":"If $X$ is {P144}, {P134} and {P30}, its Kolmogorov quotient is {P82}, {P3} and {P30}.\nBy the \"Smirnov metrization theorem\" (Theorem 42.1 in {{zb:0951.54001}}), a space with the last three properties is {P53}.\nHence $X$ is {P121}.","refs":[{"name":"Topology (Munkres)","zb":"0951.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000154","value":true},{"kind":"atom","property":"P000041","value":true}]},"then":{"kind":"atom","property":"P000023","value":true},"uid":"T000203","description":"Given a {P41} {P154} $X$, there is a base for the topology consisting of {P36} open sets. Each of these open sets $U$ is itself a {P154} (as a subspace of a {P154}), hence is a {P133} by {T131}. {{mathse:4968515}} shows that a {P36} {P133} is {P23}. So $U$ is {P23} and the same holds for $X$. See also {{mathse:4968678}}.","refs":[{"name":"Cardinality of connected LOTS","mathse":4968515},{"name":"Connected sets in GO-spaces","mathse":4968678}]},{"when":{"kind":"atom","property":"P000052","value":true},"then":{"kind":"atom","property":"P000086","value":true},"uid":"T000204","description":"Evident from the definitions as all self-bijections are homeomorphisms.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"when":{"kind":"atom","property":"P000172","value":true},"then":{"kind":"atom","property":"P000173","value":true},"uid":"T000205","description":"Follows directly from the definitions: given a radially closed set $A$ and $a\\in cl(A)$,\nby {P172} there exists a transfinite sequence of points in $A$ converging to $a$. It follows\nthat $a\\in A$ as $A$ is radially closed; therefore $A=cl(A)$ is closed.","refs":[{"name":"Encyclopedia of general topology","zb":"1059.54001"}]},{"when":{"kind":"atom","property":"P000080","value":true},"then":{"kind":"atom","property":"P000172","value":true},"uid":"T000206","description":"Evident from the definitions, as ordinary sequences are transfinite sequences.\n\nSee the chapter \"d-4 Pseudoradial spaces\" in {{zb:1059.54001}}.","refs":[{"name":"Encyclopedia of general topology","zb":"1059.54001"}]},{"when":{"kind":"atom","property":"P000079","value":true},"then":{"kind":"atom","property":"P000173","value":true},"uid":"T000207","description":"Evident from the definitions, as ordinary sequences are transfinite sequences.\n\nSee the chapter \"d-4 Pseudoradial spaces\" in {{zb:1059.54001}}.","refs":[{"name":"Encyclopedia of general topology","zb":"1059.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000129","value":true},{"kind":"atom","property":"P000125","value":true}]},"then":{"kind":"atom","property":"P000139","value":false},"uid":"T000208","description":"Evident from the definitions.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000139","value":true},{"kind":"atom","property":"P000086","value":true}]},"then":{"kind":"atom","property":"P000052","value":true},"uid":"T000209","description":"Let $a$ be an isolated point of $X$. Let $x\\in X$; there is a homeomorphism of $X$ onto itself taking $a$ to $x$, showing that every point is isolated.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000093","value":true},{"kind":"atom","property":"P000173","value":true}]},"then":{"kind":"atom","property":"P000079","value":true},"uid":"T000210","description":"Given a transfinite sequence with values in a set $A$ and converging to a point $p\\in X$, first replace it by a tail end belonging to a countable neighborhood of $p$ by {P93}. So we can assume without loss of generality that $A$ is countable. Then, as explained in {{mathse:4785088}}, there exists a countable sequence of points of $A$ converging to $p$.","refs":[{"name":"Radial/pseudoradial implies Fréchet-Urysohn/sequential for locally countable spaces","mathse":4785088}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000081","value":true},{"kind":"atom","property":"P000172","value":true}]},"then":{"kind":"atom","property":"P000080","value":true},"uid":"T000211","description":"Established in {{mathse:4850979}}.","refs":[{"name":"Answer to \"Radial/pseudoradial implies Fréchet-Urysohn/sequential for locally countable spaces\"","mathse":4850979}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000057","value":true},{"kind":"atom","property":"P000028","value":true}]},"then":{"kind":"atom","property":"P000027","value":true},"uid":"T000212","description":"Evident from the definitions: the countable union of the countable\npoint-bases is a countable basis.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"atom","property":"P000088","value":true},"then":{"kind":"atom","property":"P000013","value":true},"uid":"T000213","description":"Evident from the definitions as\ntwo closed disjoint subsets form a discrete family.\n\nAsserted in Figure 17 on page 169 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"atom","property":"P000034","value":true},"then":{"kind":"atom","property":"P000207","value":true},"uid":"T000214","description":"Shown in {{mathse:5005987}} using techniques from {{zb:0078.14803}}.","refs":[{"name":"Answer to \"Fully normal implies collectionwise normal?\"","mathse":5005987},{"name":"Some generalizations of full normality (Mansfield)","zb":"0078.14803"}]},{"when":{"kind":"atom","property":"P000077","value":true},"then":{"kind":"atom","property":"P000080","value":true},"uid":"T000215","description":"Proven in U.120 of {{doi:10.1007/978-3-319-16092-4}}.","refs":[{"name":"A Cp-Theory Problem Book","doi":"10.1007/978-3-319-16092-4"}]},{"when":{"kind":"atom","property":"P000154","value":true},"then":{"kind":"atom","property":"P000172","value":true},"uid":"T000216","description":"The result for {P133} is stated as evident in the first sentence of the proof of Satz 1 in {{doi:10.4064/fm-61-1-79-81}}.\n\nGiven a subset $A$ of a {P133} space $X$, a point $p\\in\\overline A\\setminus A$ is in the closure of the part of $A$ to one side of $p$, say the left side. So the set $B=\\{a\\in A:a.","refs":[{"name":"How are k-Hausdorff and weakly Hausdorff distinct?","mathse":4760309}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000028","value":true},{"kind":"atom","property":"P000099","value":true}]},"then":{"kind":"atom","property":"P000003","value":true},"uid":"T000230","description":"See {{mathse:1369459}}.","refs":[{"name":"If every convergent sequence has a unique limit point in space X, then is X Hausdorff?","mathse":1369459}]},{"when":{"kind":"atom","property":"P000097","value":true},"then":{"kind":"atom","property":"P000184","value":true},"uid":"T000231","description":"By definition.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000046","value":true},{"kind":"atom","property":"P000097","value":true}]},"then":{"kind":"atom","property":"P000050","value":true},"uid":"T000232","description":"See {{mathse:3824535}}.","refs":[{"name":"Answer to \"A subspace of $\\mathbb R$ is zero-dimensional iff it is totally disconnected\"","mathse":3824535}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000036","value":true},{"kind":"atom","property":"P000097","value":true}]},"then":{"kind":"atom","property":"P000042","value":true},"uid":"T000233","description":"The connected subsets of $\\mathbb R$ are the order-convex sets (that is, sets $A\\subseteq\\mathbb R$ such that $a\\le x\\le b$ with $a,b\\in A$ implies $x\\in A$), which are essentially intervals and rays of various types (see {{mathse:3617167}}).\nWith the subspace topology, they are {P42}.","refs":[{"name":"Answer to \"The convex subsets of $\\mathbb R$ are intervals or singletons\"","mathse":3617167}]},{"when":{"kind":"atom","property":"P000103","value":true},"then":{"kind":"atom","property":"P000100","value":true},"uid":"T000234","description":"Follows from the definitions.\n\nNoted on page 308 of {{doi:10.1007/s10587-009-0022-6}}.","refs":[{"name":"On minimal strongly KC-spaces","doi":"10.1007/s10587-009-0022-6"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000079","value":true},{"kind":"atom","property":"P000099","value":true}]},"then":{"kind":"atom","property":"P000103","value":true},"uid":"T000235","description":"Shown in Lemma 3.10 of {{doi:10.1007/s10587-009-0022-6}}.","refs":[{"name":"On minimal strongly KC-spaces","doi":"10.1007/s10587-009-0022-6"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000154","value":true},{"kind":"atom","property":"P000081","value":true}]},"then":{"kind":"atom","property":"P000028","value":true},"uid":"T000236","description":"Shown by KP Hart at {{mo:312803}}. To see this,\nsuppose $X$ is a {P154} for some linear order $<$ and let $a\\in X$.\n\nIf $a$ is not isolated to the left (that is, in $(\\leftarrow,a]$), it is in the closure of $(\\leftarrow,a)$ and by {P81} there is a countably infinite set $C\\subseteq(\\leftarrow,a)$ with $a=\\sup C$. We can then extract a strictly increasing sequence $(x_n)$ in $C$ with $a=\\sup x_n$. Similarly, if $a$ is not isolated to the right, there is a strictly decreasing sequence $(y_n)$ with $a=\\inf y_n$.\n\nSo, if $a$ is not isolated on either side, any neighborhood of $a$ contains an order-convex neighborhood, which necessarily contains points on either side of $a$, and hence contains an interval $(x_n,y_n)$. Thus the collection of intervals $(x_n,y_n)$ forms a countable local base at $a$. And if $a$ is isolated on one side or both, one can take the collection of intervals $(x_n,a]$ or $[a,y_n)$ or the singleton $\\{a\\}$.","refs":[{"name":"Is there a generalized order space $X$ with countable tightness which is not first countable?","mo":312803}]},{"when":{"kind":"atom","property":"P000111","value":true},"then":{"kind":"atom","property":"P000017","value":true},"uid":"T000237","description":"17I.1 of {{zb:1052.54001}}","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"when":{"kind":"atom","property":"P000057","value":true},"then":{"kind":"atom","property":"P000093","value":true},"uid":"T000238","description":"Every open set is countable in a countable space.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000016","value":true},{"kind":"atom","property":"P000036","value":true},{"kind":"atom","property":"P000041","value":true},{"kind":"atom","property":"P000053","value":true}]},"then":{"kind":"atom","property":"P000038","value":true},"uid":"T000239","description":"This is Theorem 31.2 in {{zb:1052.54001}}.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000037","value":true},{"kind":"atom","property":"P000143","value":true}]},"then":{"kind":"atom","property":"P000095","value":true},"uid":"T000240","description":"It is a classical result that {P37} {P3} spaces are {P95}.\nThis is shown for example in Jeremy Brazas's\n[expository note](https://www.wcupa.edu/sciences-mathematics/mathematics/jBrazas/documents/Constructing_arcs_from_paths_9-9-21.pdf)\nreferenced from {{mathse:4247643}}.\nSee also Corollary 31.6 in {{zb:1052.54001}}.\n\nNow suppose $X$ is {P37} and {P143}.\nGiven two distinct points $x,y\\in X$ and a path $f:[0,1]\\to X$,\nthe image $f([0,1])$ is Hausdorff (for example from Lemma 1 in {{mathse:964546}}).\nBy the classical case above, there is an arc (homeomorphic embedding) from $x$ to $y$\nwithin $f([0,1])$, which is also an arc in $X$.","refs":[{"name":"Answer to \"Does path-connected imply simple path-connected?\"","mathse":4247643},{"name":"General Topology (Willard)","zb":"1052.54001"},{"name":"Answer to \"Weak Hausdorff space not KC\"","mathse":964546}]},{"when":{"kind":"atom","property":"P000179","value":true},"then":{"kind":"atom","property":"P000074","value":true},"uid":"T000241","description":"Evident from the definitions, as a $k$-network is a network.","refs":[]},{"when":{"kind":"atom","property":"P000074","value":true},"then":{"kind":"atom","property":"P000177","value":true},"uid":"T000242","description":"Evident from the definitions, as a countable family of sets is $\\sigma$-locally finite.","refs":[]},{"when":{"kind":"atom","property":"P000179","value":true},"then":{"kind":"atom","property":"P000178","value":true},"uid":"T000243","description":"Evident from the definitions, as a countable family of sets is $\\sigma$-locally finite.","refs":[]},{"when":{"kind":"atom","property":"P000178","value":true},"then":{"kind":"atom","property":"P000177","value":true},"uid":"T000244","description":"Evident from the definitions, as a $k$-network is a network.","refs":[]},{"when":{"kind":"atom","property":"P000130","value":true},"then":{"kind":"atom","property":"P000023","value":true},"uid":"T000245","description":"Having a local base of compact neighborhoods implies at least one compact neighborhood. See {{wikipedia:Locally_compact_space}}.","refs":[{"name":"Locally compact space on Wikipedia","wikipedia":"Locally_compact_space"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000023","value":true},{"kind":"atom","property":"P000011","value":true}]},"then":{"kind":"atom","property":"P000130","value":true},"uid":"T000246","description":"By Theorem 17 in chapter 5, p. 146, of {{mr:0370454}} every point in a weakly locally compact regular space has a local base of closed compact neighborhoods.\n\nSee also {{mathse:3812324}} and {{wikipedia:Locally_compact_space}}.","refs":[{"name":"General Topology (Kelley)","mr":370454},{"name":"Weakly locally compact vs. locally compact","mathse":3812324},{"name":"Locally compact space on Wikipedia","wikipedia":"Locally_compact_space"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000052","value":true},{"kind":"atom","property":"P000129","value":true}]},"then":{"kind":"atom","property":"P000125","value":false},"uid":"T000247","description":"The only spaces that are both discrete and indiscrete are the\nempty and singleton spaces.","refs":[{"name":"Finite Topological space","wikipedia":"Finite_topological_space"}]},{"when":{"kind":"atom","property":"P000125","value":false},"then":{"kind":"atom","property":"P000052","value":true},"uid":"T000248","description":"The spaces with less than two points are the empty space and the\nsingleton space, which are both discrete.","refs":[{"name":"Finite Topological space","wikipedia":"Finite_topological_space"}]},{"when":{"kind":"atom","property":"P000125","value":false},"then":{"kind":"atom","property":"P000129","value":true},"uid":"T000249","description":"The spaces with less than two points are the empty space and the\nsingleton space, which are both indiscrete.","refs":[{"name":"Finite Topological space","wikipedia":"Finite_topological_space"}]},{"when":{"kind":"atom","property":"P000078","value":false},"then":{"kind":"atom","property":"P000125","value":true},"uid":"T000250","description":"Trivially from the definitions.","refs":[{"name":"Finite set","wikipedia":"Finite_set"}]},{"when":{"kind":"atom","property":"P000129","value":true},"then":{"kind":"atom","property":"P000245","value":true},"uid":"T000251","description":"By definition.","refs":[]},{"when":{"kind":"atom","property":"P000185","value":true},"then":{"kind":"atom","property":"P000121","value":true},"uid":"T000252","description":"The pseudometric where $d(x,y)=0$ if the points $x$ and $y$ lie in the same element of the partition and $d(x,y)=1$ otherwise results in the topology.\n\nSee item #6 for space #5 in {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"What topological properties are trivially/vacuously satisfied by any indiscrete space?","mathse":3844039}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000125","value":true},{"kind":"atom","property":"P000001","value":true}]},"then":{"kind":"atom","property":"P000129","value":false},"uid":"T000253","description":"Take two distinct points in $X$. Since $X$ is $T_0$, there is an open set containing one of the points and not the other. That's a nonempty proper subset, showing that the topology is not indiscrete.","refs":[{"name":"Kolmogorov space","wikipedia":"Kolmogorov_space"}]},{"when":{"kind":"atom","property":"P000131","value":true},"then":{"kind":"atom","property":"P000018","value":true},"uid":"T000254","description":"Evident from the definitions.","refs":[{"name":"Lindelöf space","wikipedia":"Lindelöf_space"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000018","value":true},{"kind":"atom","property":"P000132","value":true}]},"then":{"kind":"atom","property":"P000131","value":true},"uid":"T000255","description":"If $X$ is Lindelöf and a $G_\\delta$ space, every open set in $X$ is an $F_\\sigma$ and hence is also Lindelöf.\nThis uses the facts that closed sets in a Lindelöf space are Lindelöf, and a countable union of Lindelöf subspaces is Lindelöf.\n\nSee Theorem 16.6 in {{zb:1052.54001}}.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"when":{"kind":"atom","property":"P000015","value":true},"then":{"kind":"atom","property":"P000132","value":true},"uid":"T000256","description":"Follows from the definition of {P15}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000013","value":true},{"kind":"atom","property":"P000132","value":true}]},"then":{"kind":"atom","property":"P000015","value":true},"uid":"T000257","description":"Follows from the definition of {P15}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000011","value":true},{"kind":"atom","property":"P000131","value":true}]},"then":{"kind":"atom","property":"P000015","value":true},"uid":"T000258","description":"Shown in {{mathse:322506}}.","refs":[{"name":"Answer to \"Another question on hereditarily Lindelöf space\"","mathse":322506}]},{"when":{"kind":"atom","property":"P000057","value":true},"then":{"kind":"atom","property":"P000182","value":true},"uid":"T000259","description":"Evident, as the collection of all singletons in $X$ is a network.\n\nSee page 127 and the diagram on page 225 in {{zb:0684.54001}}.","refs":[{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"when":{"kind":"atom","property":"P000182","value":true},"then":{"kind":"atom","property":"P000131","value":true},"uid":"T000260","description":"Having a countable network is a hereditary property; and a space with a countable network is {P18}, as shown for example in Theorem 3.8.12 of {{zb:0684.54001}}.\n\nSee also the diagram on page 225 in {{zb:0684.54001}}.","refs":[{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000057","value":true},{"kind":"atom","property":"P000135","value":true}]},"then":{"kind":"atom","property":"P000132","value":true},"uid":"T000261","description":"Every open subset of \$X\$ is an \$F_\\sigma\$, as it is the countable union of the closures of its singletons.","refs":[{"name":"T1 space","wikipedia":"T1_space"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000134","value":true},{"kind":"atom","property":"P000039","value":true}]},"then":{"kind":"atom","property":"P000129","value":true},"uid":"T000262","description":"If two points in a {P134} space are topologically distinguishable, they have disjoint open neighborhoods. However, in a {P39} space, no two nonempty open sets are disjoint. Therefore all points are topologically indistinguishable from each other, so the space is {P129}.","refs":[{"name":"Hyperconnected space","wikipedia":"Hyperconnected_space"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000131","value":true},{"kind":"atom","property":"P000051","value":true}]},"then":{"kind":"atom","property":"P000057","value":true},"uid":"T000263","description":"See {{mathse:4955301}}.","refs":[{"name":"Every hereditarily Lindelöf scattered space is countable","mathse":4955301}]},{"when":{"kind":"atom","property":"P000053","value":true},"then":{"kind":"atom","property":"P000121","value":true},"uid":"T000264","description":"Evident from the definitions.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000121","value":true},{"kind":"atom","property":"P000001","value":true}]},"then":{"kind":"atom","property":"P000053","value":true},"uid":"T000265","description":"If two distinct points $x$ and $y$ satisfy $d(x,y)=0$, they are not topologically distinguishable.\n\nSee also {{mathse:1644528}}.","refs":[{"name":"Pseudometric space","wikipedia":"Pseudometric_space"}]},{"when":{"kind":"atom","property":"P000078","value":true},"then":{"kind":"atom","property":"P000094","value":true},"uid":"T000266","description":"All open sets in a finite space are finite by definition.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000090","value":true},{"kind":"atom","property":"P000002","value":true}]},"then":{"kind":"atom","property":"P000052","value":true},"uid":"T000267","description":"Stated on p. 18 of {{mr:1711071}}.\n\nEvery subset of an Alexandrov $T_1$ space is closed, being a union of\nsingletons, which are closed.","refs":[{"name":"Alexandroff spaces","mr":1711071}]},{"when":{"kind":"atom","property":"P000121","value":true},"then":{"kind":"atom","property":"P000015","value":true},"uid":"T000268","description":"Every closed subset $A$ of $X$ is the zero-set of a real-valued continuous function, namely, $f(x)=d(x,A)$.\n\nThis is shown for example in {{mathse:73609}} for metric spaces, but the same identical proof works for pseudometrizable spaces.\n\nAsserted on page 134, Exercise K, in {{mr:0370454}}.","refs":[{"name":"General Topology (Kelley)","mr":370454},{"name":"Is a metric space perfectly normal?","mathse":73609}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000023","value":true},{"kind":"atom","property":"P000003","value":true},{"kind":"atom","property":"P000047","value":true}]},"then":{"kind":"atom","property":"P000050","value":true},"uid":"T000269","description":"See {{mathse:11423}} or Theorem 6.2.9 in {{zb:0684.54001}} (where {P47} is called \"hereditarily disconnected\"). See also Theorem 29.7 in {{zb:1052.54001}}.","refs":[{"name":"Answer to \"Any two points in a Stone space can be disconnected by clopen sets\"","mathse":11423},{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"},{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"when":{"kind":"atom","property":"P000027","value":true},"then":{"kind":"atom","property":"P000028","value":true},"uid":"T000270","description":"Let $\\mathcal{B}=\\{U_n\\}_{n \\in \\omega}$ be a countable basis for $X$. Then for any $x \\in X$, $\\{U \\in \\mathcal{B}\\mid x \\in U\\}$ is a countable local basis at $x$.\n\nSee 16.1 of {{zb:1052.54001}}.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"when":{"kind":"atom","property":"P000027","value":true},"then":{"kind":"atom","property":"P000183","value":true},"uid":"T000271","description":"Follows from the definitions, as a base for the topology is a $k$-network.","refs":[]},{"when":{"kind":"atom","property":"P000027","value":true},"then":{"kind":"atom","property":"P000128","value":true},"uid":"T000272","description":"See bof's answer at {{mathse:4727903}}.","refs":[{"name":"Is every second countable space k-Lindelof?","mathse":4727903}]},{"when":{"kind":"atom","property":"P000154","value":true},"then":{"kind":"atom","property":"P000008","value":true},"uid":"T000273","description":"For a {P133} space, see items #3-6 of example #39 in {{zb:0386.54001}} or {{mathse:322683}}.\n\nSince {P8} is a hereditary property, the result also holds for {P154}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Why are ordered spaces normal?","mathse":322683}]},{"when":{"kind":"atom","property":"P000052","value":true},"then":{"kind":"atom","property":"P000133","value":true},"uid":"T000274","description":"Since discrete spaces are homeomorphic precisely when they have the same cardinality, it suffices to show that for every cardinality $k$ there is a set of cardinality $k$ with a total order that induces the discrete topology.\n\nLet $\\kappa$ be a cardinal number. If $\\kappa$ is finite, then any total order on $k$ induces the discrete topology. If $\\kappa$ is infinite, let $X=\\alpha \\times \\mathbb{Z}$ where $\\alpha$ is an ordinal of cardinality $\\kappa$, together with the lexicographical ordering. Then we see that $X$ with the order topology is discrete. \n\nSee also {{mathse:3429751}}.","refs":[{"name":"Verification of basic properties of order topology","mathse":3429751}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000133","value":true},{"kind":"atom","property":"P000036","value":true},{"kind":"atom","property":"P000026","value":true}]},"then":{"kind":"atom","property":"P000027","value":true},"uid":"T000275","description":"Let $X$ be a connected linearly ordered topological space with countable dense subset $Q$.\nThen the collection of all open intervals and rays of the form $(p, q)$ with $p,q\\in Q\\cup \\{-\\infty, \\infty\\}$ forms a countable base for $X$.\nTo see this, let $x\\in(a,b)$ - as $X$ is connected if either of $(a,x)$ or $(x,b)$ is empty then $x$ is the minimum (resp maximum) value of $X$, so choose $p\\in(a,x)\\cap Q$ and $q\\in(x,b)\\cap Q$ if they are nonempty, otherwise choose $p=-\\infty$ or $q=\\infty$ as necessary.\n\nSee also {{mathse:3534793}}.","refs":[{"name":"Is a totally ordered, separable and connected topological space metrizable (in the order topology)?","mathse":3534793}]},{"when":{"kind":"atom","property":"P000131","value":true},"then":{"kind":"atom","property":"P000197","value":true},"uid":"T000276","description":"Follows as Lindelöf discrete spaces are countable.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000154","value":true},{"kind":"atom","property":"P000029","value":true}]},"then":{"kind":"atom","property":"P000131","value":true},"uid":"T000277","description":"For the result with the stronger hypothesis of {P133} in place of {P154}, see Theorem 2.2 in {{doi:10.1090/S0002-9939-1969-0248762-3}}.\n\nThe general result with {P154} is Proposition 2.10(b) in {{zb:0228.54026}} () and is proved as follows. Suppose $X$ is a GO-space satisfying the CCC condition. It can be topologically embedded as a dense subspace of a LOTS $Y$ (see for example {{mathse:4917398}}). Since $X$ is dense in $Y$, the space $Y$ also satisfies CCC, hence is {P131} by the first result. The conclusion then follows since {P131} is a hereditary property.","refs":[{"name":"Separability, the countable chain condition and the Lindelöf property in linearly orderable spaces (Lutzer & Bennet)","doi":"10.1090/S0002-9939-1969-0248762-3"},{"name":"On generalized ordered spaces (D. Lutzer)","zb":"0228.54026"},{"name":"Equivalent definitions for GO-spaces (generalized ordered spaces)","mathse":4917398}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000154","value":true},{"kind":"atom","property":"P000029","value":true}]},"then":{"kind":"atom","property":"P000028","value":true},"uid":"T000278","description":"For the result with the stronger hypothesis of {P133} in place of {P154}, Exercise 3.12.4(a) in {{zb:0684.54001}} states that the character of a linearly ordered topological space does not exceed its cellularity. So if a LOTS satisfies CCC, it is first countable. See Lemma 5 in , noting that the compactness assumption is not used.\n\nFor the general result with {P154}, suppose $X$ is a GO-space satisfying the CCC condition. It can be topologically embedded as a dense subspace of a LOTS $Y$ (see for example {{mathse:4917398}}). Since $X$ is dense in $Y$, the space $Y$ also satisfies CCC, hence is {P28} by the first result. The conclusion then follows since {P28} is a hereditary property.","refs":[{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"},{"name":"Equivalent definitions for GO-spaces (generalized ordered spaces)","mathse":4917398}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000111","value":true},{"kind":"atom","property":"P000028","value":true}]},"then":{"kind":"atom","property":"P000023","value":true},"uid":"T000279","description":"See {{mathse:2919068}}.","refs":[{"name":"A first countable hemicompact space is locally compact","mathse":2919068}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000093","value":true},{"kind":"atom","property":"P000002","value":true}]},"then":{"kind":"atom","property":"P000046","value":true},"uid":"T000280","description":"Let $I$ be the interval $[0,1]$ and consider a path $f:I\\to X$.\nSuppose first that $X$ is {P57} and {P2}. Each $f^{-1}(x)$ for $x\\in X$ is a (possibly empty) closed subset of $I$. The nonempty $f^{-1}(x)$ partition $I$ into countably many closed sets. The number of these nonempty closed sets cannot be finite $>1$, since $I$ is connected. And as shown in {{mo:48970}}, it cannot be countably infinite either (theorem of Sierpiński (1918)). So there is only one such closed set and $f$ is constant.\n\nSuppose now that $X$ is {P93} and {P2}. For each $t\\in I$ the function $f$ maps some closed interval around $t$ to some countable subspace of $X$. By the countable case above, $f$ is constant on that interval; so $f$ is locally constant. A standard argument using the connectedness of $I$ then implies that $f$ is constant.","refs":[{"name":"Why are the integers with the cofinite topology not path-connected?","mo":48970}]},{"when":{"kind":"atom","property":"P000003","value":true},"then":{"kind":"atom","property":"P000134","value":true},"uid":"T000281","description":"Follows directly from the definitions, and stated in {{wikipedia:Hausdorff_space}}.","refs":[{"name":"Hausdorff space on Wikipedia","wikipedia":"Hausdorff_space"}]},{"when":{"kind":"atom","property":"P000011","value":true},"then":{"kind":"atom","property":"P000134","value":true},"uid":"T000282","description":"Follows directly from the definitions, and stated in {{wikipedia:Hausdorff_space}}.","refs":[{"name":"Hausdorff space on Wikipedia","wikipedia":"Hausdorff_space"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000134","value":true},{"kind":"atom","property":"P000001","value":true}]},"then":{"kind":"atom","property":"P000003","value":true},"uid":"T000283","description":"Follows directly from the definitions, and stated in {{wikipedia:Hausdorff_space}}.","refs":[{"name":"Hausdorff space on Wikipedia","wikipedia":"Hausdorff_space"}]},{"when":{"kind":"atom","property":"P000090","value":true},"then":{"kind":"atom","property":"P000130","value":true},"uid":"T000284","description":"See Theorem 5 in . The smallest (open) neighborhood $U$ of a point $x$ in an Alexandrov space is always compact. Indeed, in $U$ itself with the subspace topology any open cover of $U$ contains a neighborhood of $x$ included in $U$. Such a neighborhood is necessarily equal to $U$, so the open cover admits $\\{U\\}$ as a finite subcover. \n\nSee also {{wikipedia:Alexandrov_topology}}.","refs":[{"name":"Alexandrov topology on Wikipedia","wikipedia":"Alexandrov_topology"}]},{"when":{"kind":"atom","property":"P000090","value":true},"then":{"kind":"atom","property":"P000028","value":true},"uid":"T000285","description":"The smallest neighborhood of a point in an Alexandrov space forms a local base with a single element at that point.","refs":[{"name":"Alexandrov topology on Wikipedia","wikipedia":"Alexandrov_topology"}]},{"when":{"kind":"atom","property":"P000134","value":true},"then":{"kind":"atom","property":"P000135","value":true},"uid":"T000286","description":"Follows directly from the definitions, and stated in {{wikipedia:Separation_axiom}}.","refs":[{"name":"Separation axiom on Wikipedia","wikipedia":"Separation_axiom"}]},{"when":{"kind":"atom","property":"P000002","value":true},"then":{"kind":"atom","property":"P000135","value":true},"uid":"T000287","description":"Follows directly from the definitions, and stated in {{wikipedia:T1_space}}.","refs":[{"name":"$T_1$ space on Wikipedia","wikipedia":"T1_space"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000135","value":true},{"kind":"atom","property":"P000001","value":true}]},"then":{"kind":"atom","property":"P000002","value":true},"uid":"T000288","description":"Follows directly from the definitions. See {{wikipedia:T1_space}}.","refs":[{"name":"$T_1$ space on Wikipedia","wikipedia":"T1_space"}]},{"when":{"kind":"atom","property":"P000132","value":true},"then":{"kind":"atom","property":"P000135","value":true},"uid":"T000289","description":"See {{mathse:4547643}}.","refs":[{"name":"$G_\\delta$ spaces are $R_0$","mathse":4547643}]},{"when":{"kind":"atom","property":"P000237","value":true},"then":{"kind":"atom","property":"P000236","value":true},"uid":"T000290","description":"By definition.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000136","value":true},{"kind":"atom","property":"P000057","value":true}]},"then":{"kind":"atom","property":"P000111","value":true},"uid":"T000291","description":"Let $X=\\omega$ be a countable space such that every compact subset is finite. Then $X$ is hemicompact: let $K_n=\\{0,\\dots,n\\}$, then every compact set is finite and thus contained in some $K_n$. See {{mathse:4566624}}.","refs":[{"name":"Can a hemicompact space fail to be weakly locally compact?","mathse":4566624}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000136","value":true},{"kind":"atom","property":"P000140","value":true}]},"then":{"kind":"atom","property":"P000094","value":true},"uid":"T000292","description":"An {P136} {P140} has a topology generated by its finite subsets, i.e. it is {P90}.\nThen by {T284} and anticompactness every point has a finite neighborhood.","refs":[]},{"when":{"kind":"atom","property":"P000094","value":true},"then":{"kind":"atom","property":"P000136","value":true},"uid":"T000293","description":"Any compact subset is covered by finitely many finite sets and is therefore finite.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000154","value":true},{"kind":"atom","property":"P000026","value":true}]},"then":{"kind":"atom","property":"P000180","value":true},"uid":"T000294","description":"For the result with the stronger hypothesis of {P133} in place of {P154}, see Theorem 3.3 in {{doi:10.1090/S0002-9939-1969-0248762-3}}.\n\nThe general result with {P154} is Proposition 2.10(a) in {{zb:0228.54026}} () and is proved as follows. Suppose $X$ is a GO-space containing a countable topologically dense subset $D$. $X$ can be topologically embedded as a dense subspace of a LOTS $Y$ (see for example {{mathse:4917398}}). Since $X$ is dense in $Y$, the set $D$ is also topologically dense in $Y$, hence $Y$ is {P180} by the first result. The conclusion then follows since {P180} is a hereditary property.","refs":[{"name":"Separability, the countable chain condition and the Lindelöf property in linearly orderable spaces (Lutzer & Bennet)","doi":"10.1090/S0002-9939-1969-0248762-3"},{"name":"On generalized ordered spaces (D. Lutzer)","zb":"0228.54026"},{"name":"Equivalent definitions for GO-spaces (generalized ordered spaces)","mathse":4917398}]},{"when":{"kind":"atom","property":"P000125","value":true},"then":{"kind":"atom","property":"P000137","value":false},"uid":"T000295","description":"A space with at least two points is not empty.","refs":[{"name":"Empty set on Wikipedia","wikipedia":"Empty_set"}]},{"when":{"kind":"atom","property":"P000129","value":true},"then":{"kind":"atom","property":"P000196","value":true},"uid":"T000296","description":"The two open sets $\\emptyset$ and $X$ form a chain under inclusion.","refs":[]},{"when":{"kind":"atom","property":"P000138","value":true},"then":{"kind":"atom","property":"P000057","value":true},"uid":"T000297","description":"Each constant map is a continuous map.","refs":[{"name":"Is there an infinite topological space with only countably many continuous functions to itself?","mo":418619}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000138","value":true},{"kind":"atom","property":"P000011","value":true}]},"then":{"kind":"atom","property":"P000078","value":true},"uid":"T000298","description":"See [an anonymous comment](https://mathoverflow.net/questions/418619/is-there-an-infinite-topological-space-with-only-countably-many-continuous-funct#comment1075675_418619) in {{mo:418619}}.\nThis extends the result of {{mathse:4231158}} that {P138} and {P53} spaces are {P78}.","refs":[{"name":"Is there an infinite topological space with only countably many continuous maps to itself?","mo":418619},{"name":"Is there an infinite topological space with only countably many continuous maps to itself?","mathse":4231158}]},{"when":{"kind":"atom","property":"P000078","value":true},"then":{"kind":"atom","property":"P000138","value":true},"uid":"T000299","description":"A {P78} space only has finitely-many self-maps, continuous or not.","refs":[{"name":"Is there an infinite topological space with only countably many continuous functions to itself?","mo":418619}]},{"when":{"kind":"atom","property":"P000058","value":true},"then":{"kind":"atom","property":"P000217","value":true},"uid":"T000300","description":"If $Z_1, Z_2$ are disjoint zero-sets, there exists a continuous $f:X\\to [0, 1]$ such that $f^{-1}(0) = Z_1$ and $f^{-1}(1) = Z_2$ (see theorem 1.5.14 in {{zb:0684.54001}}). Since $|X| < \\mathfrak{c}$ there exists $r\\in (0, 1)$ such that $f^{-1}(r) = \\emptyset$. Then $V = f^{-1}([0, r)) = f^{-1}([0, r])$ is a clopen set such that $Z_1\\subseteq V$ and $Z_2\\cap V = \\emptyset$, so $X$ is {P217}.","refs":[{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000138","value":true},{"kind":"atom","property":"P000090","value":true}]},"then":{"kind":"atom","property":"P000078","value":true},"uid":"T000301","description":"See {{mathse:4231158}} and {{mathse:4575225}}.","refs":[{"name":"Is there an infinite topological space with only countably many continuous maps to itself?","mathse":4231158},{"name":"Does a countably infinite preorder admit continuum many endomorphisms?","mathse":4575225}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000057","value":true},{"kind":"atom","property":"P000016","value":true},{"kind":"atom","property":"P000134","value":true}]},"then":{"kind":"atom","property":"P000121","value":true},"uid":"T000302","description":"It is shown in {{mathse:3705764}} that countable compact Hausdorff spaces are metrizable. The case of countable compact $R_1$ spaces being pseudometrizable follows by passing to the Kolmogorov quotient.","refs":[{"name":"Every countable compact Hausdorff space is metrizable?","mathse":3705764}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000136","value":true},{"kind":"atom","property":"P000016","value":true}]},"then":{"kind":"atom","property":"P000078","value":true},"uid":"T000303","description":"Follows directly from the definitions.","refs":[{"name":"The total negation of a topological property (P. Bankston)","doi":"10.1215/ijm/1256048236"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000136","value":true},{"kind":"atom","property":"P000017","value":true}]},"then":{"kind":"atom","property":"P000057","value":true},"uid":"T000304","description":"Follows directly from the definitions.","refs":[{"name":"The total negation of a topological property (P. Bankston)","doi":"10.1215/ijm/1256048236"}]},{"when":{"kind":"atom","property":"P000167","value":true},"then":{"kind":"atom","property":"P000046","value":true},"uid":"T000305","description":"See {{mathse:4882280}}.","refs":[{"name":"Every sequentially discrete space is totally path disconnected","mathse":4882280}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000051","value":true},{"kind":"atom","property":"P000137","value":false}]},"then":{"kind":"atom","property":"P000139","value":true},"uid":"T000306","description":"The space itself is non-empty and thus contains an isolated point.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000094","value":true},{"kind":"atom","property":"P000001","value":true}]},"then":{"kind":"atom","property":"P000051","value":true},"uid":"T000307","description":"First assume $X$ is nonempty and show it has an isolated point. Take a nonempty finite open set $U\\subseteq X$ that is minimal under inclusion. It cannot contain more than one point, otherwise intersecting with another open set that contains one point and not another in $U$ would give a strictly smaller open set, which is not possible. So $U$ consists of one isolated point.\n\nNow given any nonempty subspace $A\\subseteq X$, it is itself locally finite and $T_0$ and hence contains a point isolated in $A$. This shows that $X$ is scattered.","refs":[{"name":"Answer to \"How many T0 topologies are possible for a set of 3 elements?\"","mathse":4583879}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000135","value":true},{"kind":"atom","property":"P000139","value":true},{"kind":"atom","property":"P000125","value":true}]},"then":{"kind":"atom","property":"P000036","value":false},"uid":"T000308","description":"By {P139} there is a singleton $\\{x\\}$ that is open.\nBy {P135} the open set $\\{x\\}$ contains $\\overline{\\{x\\}}$, hence the singleton is clopen.\nAnd since it is a nonempty proper subset of $X$, the space is not {P36}.","refs":[]},{"when":{"kind":"atom","property":"P000028","value":true},"then":{"kind":"atom","property":"P000228","value":true},"uid":"T000309","description":"For every $x \\in X$, let $(C_n(x))_{n \\in \\mathbb{N}}$ be a countable neighborhood basis induced by {P28}.\nThen $V_n(x) := \\bigcap_{k \\leq n}C_k(x)$ works.","refs":[]},{"when":{"kind":"atom","property":"P000003","value":true},"then":{"kind":"atom","property":"P000101","value":true},"uid":"T000310","description":"See {{mathse:805274}}.","refs":[{"name":"is retract of a hausdorff space closed in that space?","mathse":805274}]},{"when":{"kind":"atom","property":"P000101","value":true},"then":{"kind":"atom","property":"P000002","value":true},"uid":"T000311","description":"See {{mo:191016}}.","refs":[{"name":"\"All retracts are closed\" as separation axiom","mo":191016}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000140","value":true},{"kind":"atom","property":"P000100","value":true}]},"then":{"kind":"atom","property":"P000101","value":true},"uid":"T000312","description":"If $A$ is a retract, $r:X\\to A$ is a retraction and $K\\subseteq X$ is compact, let $L = r^{-1}(K)\\cap K$. \nSince $X$ is {P100}, it follows that $L$ is compact. Then $r(L) = K\\cap A$ so that $K\\cap A$ is compact, and so closed in $K$.\nBecause $X$ is {P140} it follows that $A$ is closed.","refs":[{"name":"\"All retracts are closed\" and \"all compacts are closed\"","mo":434451}]},{"when":{"kind":"atom","property":"P000039","value":true},"then":{"kind":"atom","property":"P000029","value":true},"uid":"T000313","description":"In a hyperconnected space, no two nonempty open sets can be disjoint. So the space satisfies {P29}.","refs":[{"name":"Hyperconnected space on Wikipedia","wikipedia":"Hyperconnected_space"}]},{"when":{"kind":"atom","property":"P000139","value":true},"then":{"kind":"atom","property":"P000056","value":false},"uid":"T000314","description":"A space containing an isolated point cannot be meager, because no set containing the isolated point can be nowhere dense.","refs":[{"name":"Meagre set on Wikipedia","wikipedia":"Meagre_set"}]},{"when":{"kind":"atom","property":"P000137","value":true},"then":{"kind":"atom","property":"P000056","value":true},"uid":"T000315","description":"The empty space is nowhere dense in itself, hence is a meager space.","refs":[{"name":"Meagre set on Wikipedia","wikipedia":"Meagre_set"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000236","value":true},{"kind":"atom","property":"P000003","value":true},{"kind":"atom","property":"P000027","value":true}]},"then":{"kind":"atom","property":"P000237","value":true},"uid":"T000316","description":"By definition.","refs":[]},{"when":{"kind":"atom","property":"P000051","value":true},"then":{"kind":"atom","property":"P000206","value":true},"uid":"T000317","description":"See two paragraphs before Lemma 5.6 in {{zb:1348.91079}}.","refs":[{"name":"Coding strategies, the Choquet game, and domain representability (L. Yengulalp)","zb":"1348.91079"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000136","value":true},{"kind":"atom","property":"P000002","value":true}]},"then":{"kind":"atom","property":"P000170","value":true},"uid":"T000318","description":"Every {P16} subset is {P78} and {P2}, and thus\n{P52} and {P3}.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000057","value":true},{"kind":"atom","property":"P000139","value":false},{"kind":"atom","property":"P000002","value":true}]},"then":{"kind":"atom","property":"P000056","value":true},"uid":"T000319","description":"As $X$ is $T_1$ and without isolated point, every singleton is closed and with empty interior, and thus nowhere dense in $X$. So $X$ is meager as a countable union of the singletons.","refs":[{"name":"Meagre set on Wikipedia","wikipedia":"Meagre_set"}]},{"when":{"kind":"atom","property":"P000040","value":true},"then":{"kind":"atom","property":"P000189","value":true},"uid":"T000320","description":"By definition, an ultraconnected space cannot be partitioned into disjoint closed sets.","refs":[]},{"when":{"kind":"atom","property":"P000098","value":true},"then":{"kind":"atom","property":"P000111","value":true},"uid":"T000321","description":"Shown in {{mathse:4585825}}.","refs":[{"name":"Without Hausdorff, what implications can we prove about $k_\\omega$ related to other covering properties?","mathse":4585825}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000098","value":true},{"kind":"atom","property":"P000002","value":true},{"kind":"atom","property":"P000028","value":true}]},"then":{"kind":"atom","property":"P000023","value":true},"uid":"T000322","description":"See {{mathse:4585825}}.","refs":[{"name":"Without Hausdorff, what implications can we prove about kω related to other covering properties?","mathse":4585825}]},{"when":{"kind":"atom","property":"P000098","value":true},"then":{"kind":"atom","property":"P000140","value":true},"uid":"T000323","description":"Let $K_n$ witness {P98}, and let $C$ have closed intersection with every \n{P16} subspace.\nThen $C$ has closed intersection with every $K_n$, and is thus closed in the space.\n\nSee also {{mathse:4633318}}.","refs":[{"name":"Is every $k_\\omega$ space a $k$ space?","mathse":4633318}]},{"when":{"kind":"atom","property":"P000142","value":true},"then":{"kind":"atom","property":"P000141","value":true},"uid":"T000324","description":"See {{mathse:4646084}}.","refs":[{"name":"Unraveling the various definitions of k-space or compactly generated space","mathse":4646084}]},{"when":{"kind":"atom","property":"P000141","value":true},"then":{"kind":"atom","property":"P000140","value":true},"uid":"T000325","description":"See {{mathse:4646084}}.","refs":[{"name":"Unraveling the various definitions of k-space or compactly generated space","mathse":4646084}]},{"when":{"kind":"atom","property":"P000121","value":true},"then":{"kind":"atom","property":"P000144","value":true},"uid":"T000326","description":"The entire space is the desired neighborhood.","refs":[{"name":"How can the implication metrizable -> first countable be generalized?","mathse":4659902}]},{"when":{"kind":"atom","property":"P000082","value":true},"then":{"kind":"atom","property":"P000144","value":true},"uid":"T000327","description":"Every metric is a pseudometric.","refs":[{"name":"How can the implication metrizable -> first countable be generalized?","mathse":4659902}]},{"when":{"kind":"atom","property":"P000082","value":true},"then":{"kind":"atom","property":"P000084","value":true},"uid":"T000328","description":"Around every point there is a neighborhood that is {P53}, hence {P3}.","refs":[{"name":"Metric space on Wikipedia","wikipedia":"Metric_space"}]},{"when":{"kind":"atom","property":"P000235","value":true},"then":{"kind":"atom","property":"P000082","value":true},"uid":"T000329","description":"Every point has a neighborhood homeomorphic to some $\\mathbb R^n_+$, hence {P53}.","refs":[]},{"when":{"kind":"atom","property":"P000144","value":true},"then":{"kind":"atom","property":"P000135","value":true},"uid":"T000330","description":"Around every point there is a neighborhood that is {P121}, hence $R_0$. And a space that is \"locally $R_0$\" is $R_0$, as shown in {{mathse:3142975}}.","refs":[{"name":"Locally Euclidean space implies T1 space","mathse":3142975}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000144","value":true},{"kind":"atom","property":"P000001","value":true}]},"then":{"kind":"atom","property":"P000082","value":true},"uid":"T000331","description":"Every point has a neighborhood that is {P121} and {P1}, hence {P53} via {T265}.","refs":[{"name":"Pseudometric space on Wikipedia","wikipedia":"Pseudometric_space"}]},{"when":{"kind":"atom","property":"P000235","value":true},"then":{"kind":"atom","property":"P000130","value":true},"uid":"T000332","description":"Every point $p$ has a neighborhood $U$ homeomorphic to some $\\mathbb R^n_+$.\nSince $\\mathbb R^n_+$ is {P130}, $p$ has a local base of compact neighborhoods in $U$,\nwhich also form a local base in $X$.","refs":[{"name":"Every manifold is locally compact?","mathse":103774}]},{"when":{"kind":"atom","property":"P000237","value":true},"then":{"kind":"atom","property":"P000003","value":true},"uid":"T000333","description":"By definition.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000182","value":true},{"kind":"atom","property":"P000001","value":true}]},"then":{"kind":"atom","property":"P000163","value":true},"uid":"T000334","description":"See page 127 in {{zb:0684.54001}}.","refs":[{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"when":{"kind":"atom","property":"P000007","value":true},"then":{"kind":"atom","property":"P000013","value":true},"uid":"T000335","description":"By definition as in 15.1 of {{zb:1052.54001}}.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"when":{"kind":"atom","property":"P000008","value":true},"then":{"kind":"atom","property":"P000014","value":true},"uid":"T000336","description":"By definition as on page 12 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"atom","property":"P000035","value":true},"then":{"kind":"atom","property":"P000034","value":true},"uid":"T000337","description":"By definition as on page 23 of {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"atom","property":"P000067","value":true},"then":{"kind":"atom","property":"P000015","value":true},"uid":"T000338","description":"By definition, see {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"atom","property":"P000075","value":true},"then":{"kind":"atom","property":"P000016","value":true},"uid":"T000339","description":"By definition.\nSee example 21, section 2.6 of {{mr:1077251}}.","refs":[{"name":"General topology I (Arkhangelʹskiĭ, Pontryagin)","mr":1077251}]},{"when":{"kind":"atom","property":"P000237","value":true},"then":{"kind":"atom","property":"P000027","value":true},"uid":"T000340","description":"By definition.","refs":[]},{"when":{"kind":"atom","property":"P000057","value":true},"then":{"kind":"atom","property":"P000152","value":true},"uid":"T000341","description":"Suppose $X$ is countable and let $\\{ F_n : n \\in \\omega \\}$ be an enumeration of the non-empty finite subsets of $X$. In the $n^{\\mathrm{th}}$ inning of the game, given an $\\omega$-cover $\\mathscr U_n$ of $X$, let $U_n \\in \\mathscr U_n$, be so that $F_n \\subseteq U_n$. Note that this defines a winning Markov strategy for Two in the $\\omega$-Rothberger game.","refs":[{"name":"Rothberger space","wikipedia":"Rothberger_space"},{"name":"Limited information strategies and discrete selectivity (Clontz & Holshouser)","doi":"10.1016/j.topol.2019.07.008"}]},{"when":{"kind":"atom","property":"P000146","value":true},"then":{"kind":"atom","property":"P000145","value":true},"uid":"T000342","description":"Any partition is star-finite.","refs":[{"name":"Answer to What separation properties are satisfied by the Arens space?","mathse":4673639}]},{"when":{"kind":"atom","property":"P000145","value":true},"then":{"kind":"atom","property":"P000030","value":true},"uid":"T000343","description":"Any star-finite collection of open sets is locally finite.\n\nStated on p. 326 of {{zb:0684.54001}}.","refs":[{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000018","value":true},{"kind":"atom","property":"P000050","value":true}]},"then":{"kind":"atom","property":"P000146","value":true},"uid":"T000344","description":"See Proposition 4 of {{doi:10.48550/arXiv.1306.6086}}.","refs":[{"name":"Ultraparacompactness and Ultranormality (J. Van Name)","doi":"10.48550/arXiv.1306.6086"}]},{"when":{"kind":"atom","property":"P000087","value":true},"then":{"kind":"atom","property":"P000012","value":true},"uid":"T000345","description":"For each open neighborhood $U$ of the identity, $D_U=\\{(x,y):xy^{-1}\\in U\\}$\nis a basic entourage forming a uniformity inducing the topology.\nSee {{mathse:427510}}.","refs":[{"name":"Topological groups are completely regular","mathse":427510}]},{"when":{"kind":"atom","property":"P000087","value":true},"then":{"kind":"atom","property":"P000137","value":false},"uid":"T000346","description":"A group cannot be empty.","refs":[{"name":"Topological group on Wikipedia","wikipedia":"Topological_group"}]},{"when":{"kind":"atom","property":"P000087","value":true},"then":{"kind":"atom","property":"P000086","value":true},"uid":"T000347","description":"Every element can be sent to every other element by some left-translation, which is a homeomorphism.","refs":[{"name":"Topological group on Wikipedia","wikipedia":"Topological_group"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000087","value":true},{"kind":"atom","property":"P000028","value":true}]},"then":{"kind":"atom","property":"P000121","value":true},"uid":"T000348","description":"This is essentially the Birkhoff-Kakutani theorem (), which says that a topological group is metrizable if and only if it is Hausdorff and first countable. The current version without Hausdorff is obtained by passing to the Kolmogorov quotient.\n\nOne way to prove it is to show that a uniformity on a set is induced by a pseudometric if it has a countable base. And if a topological group is first countable, the usual left (or right) uniformity on the group has a countable base.\n\nSee Theorem 38.3 and Problem 38C(1) in {{zb:1052.54001}}, or Theorem 8.1.21 and Exercise 8.1.G in {{zb:0684.54001}}.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"},{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"when":{"kind":"atom","property":"P000129","value":true},"then":{"kind":"atom","property":"P000086","value":true},"uid":"T000349","description":"All bijections of the space onto itself are continuous.","refs":[]},{"when":{"kind":"atom","property":"P000090","value":true},"then":{"kind":"atom","property":"P000147","value":true},"uid":"T000350","description":"Follows immediately from definitions.","refs":[{"name":"A topological view of P-spaces (A. Misra)","doi":"10.1016/0016-660X(72)90026-8"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000011","value":true},{"kind":"atom","property":"P000147","value":true}]},"then":{"kind":"atom","property":"P000050","value":true},"uid":"T000351","description":"Shown in the proof of Proposition 3.2 in {{doi:10.1016/0016-660X(72)90026-8}}.","refs":[{"name":"A topological view of P-spaces (A. Misra)","doi":"10.1016/0016-660X(72)90026-8"}]},{"when":{"kind":"atom","property":"P000183","value":true},"then":{"kind":"atom","property":"P000118","value":true},"uid":"T000352","description":"Evident from the definitions, as a countable family of sets in $X$ is $\\sigma$-locally finite.","refs":[]},{"when":{"kind":"atom","property":"P000069","value":true},"then":{"kind":"atom","property":"P000153","value":true},"uid":"T000353","description":"Evident from the definitions.","refs":[{"name":"Limited information strategies and discrete selectivity (Clontz & Holshouser)","doi":"10.1016/j.topol.2019.07.008"}]},{"when":{"kind":"atom","property":"P000152","value":true},"then":{"kind":"atom","property":"P000151","value":true},"uid":"T000354","description":"Evident from the definitions.","refs":[{"name":"Limited information strategies and discrete selectivity (Clontz & Holshouser)","doi":"10.1016/j.topol.2019.07.008"}]},{"when":{"kind":"atom","property":"P000151","value":true},"then":{"kind":"atom","property":"P000150","value":true},"uid":"T000355","description":"Evident from the definitions.","refs":[{"name":"Limited information strategies and discrete selectivity (Clontz & Holshouser)","doi":"10.1016/j.topol.2019.07.008"}]},{"when":{"kind":"atom","property":"P000153","value":true},"then":{"kind":"atom","property":"P000149","value":true},"uid":"T000356","description":"Evident from the definitions and similar to the proof of {T140}.","refs":[{"name":"Selection games on hyperspaces (Caruvana & Holshouser)","doi":"10.1016/j.topol.2021.107771"}]},{"when":{"kind":"atom","property":"P000152","value":true},"then":{"kind":"atom","property":"P000070","value":true},"uid":"T000357","description":"Evident from the definitions and similar to the comment that every {P68} space is {P66} in {{doi:10.1016/0166-8641(95)00067-4}}.","refs":[{"name":"Combinatorics of open covers I Ramsey theory (Scheepers)","doi":"10.1016/0166-8641(95)00067-4"}]},{"when":{"kind":"atom","property":"P000151","value":true},"then":{"kind":"atom","property":"P000069","value":true},"uid":"T000358","description":"Evident from the definitions and similar to the comment that every {P68} space is {P66} in {{doi:10.1016/0166-8641(95)00067-4}}.","refs":[{"name":"Combinatorics of open covers I Ramsey theory (Scheepers)","doi":"10.1016/0166-8641(95)00067-4"}]},{"when":{"kind":"atom","property":"P000150","value":true},"then":{"kind":"atom","property":"P000153","value":true},"uid":"T000359","description":"Evident from the definitions and similar to the comment that every {P68} space is {P66} in {{doi:10.1016/0166-8641(95)00067-4}}.","refs":[{"name":"Combinatorics of open covers I Ramsey theory (Scheepers)","doi":"10.1016/0166-8641(95)00067-4"}]},{"when":{"kind":"atom","property":"P000149","value":true},"then":{"kind":"atom","property":"P000018","value":true},"uid":"T000360","description":"Evident from the definitions.","refs":[{"name":"Some properties of C(X), I (Gerlits & Nagy)","doi":"10.1016/0166-8641(82)90065-7"}]},{"when":{"kind":"atom","property":"P000150","value":true},"then":{"kind":"atom","property":"P000068","value":true},"uid":"T000361","description":"Evident from the definitions.","refs":[{"name":"Property $C''$ and Function Spaces (Sakai)","doi":"10.1090/S0002-9939-1988-0964873-0"}]},{"when":{"kind":"atom","property":"P000153","value":true},"then":{"kind":"atom","property":"P000066","value":true},"uid":"T000362","description":"Evident from the definitions.","refs":[{"name":"The combinatorics of open covers II","doi":"10.1016/S0166-8641(96)00075-2"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000002","value":true},{"kind":"atom","property":"P000152","value":true}]},"then":{"kind":"atom","property":"P000057","value":true},"uid":"T000363","description":"Proved for {P6} spaces as one of the implications in Theorem 17 of {{doi:10.1016/j.topol.2019.07.008}}. Shown to only require {P2} in {{mathse:4737285}}.","refs":[{"name":"Limited information strategies and discrete selectivity (Clontz & Holshouser)","doi":"10.1016/j.topol.2019.07.008"},{"name":"Do uncountable spaces admit Markov strategies in Rothberger-style games?","mathse":4737285}]},{"when":{"kind":"atom","property":"P000128","value":true},"then":{"kind":"atom","property":"P000149","value":true},"uid":"T000364","description":"See the proof provided at {{mathse:4717687}}.","refs":[{"name":"Is every k-Lindelof space an epsilon-space?","mathse":4717687},{"name":"Applications of k-covers","doi":"10.1007/s10114-005-0753-8"}]},{"when":{"kind":"atom","property":"P000158","value":true},"then":{"kind":"atom","property":"P000157","value":true},"uid":"T000365","description":"Evident from the definitions.","refs":[{"name":"Selection games and the Vietoris space","doi":"10.1016/j.topol.2021.107772"}]},{"when":{"kind":"atom","property":"P000157","value":true},"then":{"kind":"atom","property":"P000156","value":true},"uid":"T000366","description":"Evident from the definitions.","refs":[{"name":"Selection games and the Vietoris space","doi":"10.1016/j.topol.2021.107772"}]},{"when":{"kind":"atom","property":"P000161","value":true},"then":{"kind":"atom","property":"P000160","value":true},"uid":"T000367","description":"Evident from the definitions.","refs":[{"name":"Selection games and the Vietoris space","doi":"10.1016/j.topol.2021.107772"}]},{"when":{"kind":"atom","property":"P000160","value":true},"then":{"kind":"atom","property":"P000159","value":true},"uid":"T000368","description":"Evident from the definitions.","refs":[{"name":"Selection games and the Vietoris space","doi":"10.1016/j.topol.2021.107772"}]},{"when":{"kind":"atom","property":"P000159","value":true},"then":{"kind":"atom","property":"P000128","value":true},"uid":"T000369","description":"Evident from the definitions and similar to the proof of {T140}.","refs":[{"name":"Selection games and the Vietoris space","doi":"10.1016/j.topol.2021.107772"}]},{"when":{"kind":"atom","property":"P000156","value":true},"then":{"kind":"atom","property":"P000159","value":true},"uid":"T000370","description":"Evident from the definitions and similar to the comment that every {P68} space is {P66} in {{doi:10.1016/0166-8641(95)00067-4}}.","refs":[{"name":"Applications of k-covers II","doi":"10.1016/j.topol.2005.07.015"}]},{"when":{"kind":"atom","property":"P000157","value":true},"then":{"kind":"atom","property":"P000160","value":true},"uid":"T000371","description":"Evident from the definitions and similar to the comment that every {P68} space is {P66} in {{doi:10.1016/0166-8641(95)00067-4}}.","refs":[{"name":"Selection games and the Vietoris space","doi":"10.1016/j.topol.2021.107772"}]},{"when":{"kind":"atom","property":"P000158","value":true},"then":{"kind":"atom","property":"P000161","value":true},"uid":"T000372","description":"Evident from the definitions and similar to the comment that every {P68} space is {P66} in {{doi:10.1016/0166-8641(95)00067-4}}.","refs":[{"name":"Selection games and the Vietoris space","doi":"10.1016/j.topol.2021.107772"}]},{"when":{"kind":"atom","property":"P000111","value":true},"then":{"kind":"atom","property":"P000158","value":true},"uid":"T000373","description":"See Theorem 3.22 of {{mr:3991109}}.\n\nLet $\\langle K_n : n \\in \\omega \\rangle$ be a sequence of compact sets so that every compact $K \\subseteq X$ is a subset of $K_n$ for some $n$.\n\nIn the $n^{\\mathrm{th}}$ inning of the game, Two need only pick $U_n$ so that $K_n \\subseteq U_n$. This is a Markov strategy.","refs":[{"name":"Closed discrete selection in the compact open topology","mr":3991109}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000002","value":true},{"kind":"atom","property":"P000028","value":true},{"kind":"atom","property":"P000159","value":true}]},"then":{"kind":"atom","property":"P000111","value":true},"uid":"T000374","description":"See Proposition 5 of {{doi:10.1016/j.topol.2005.07.015}}. In that proof, one can take $O_n(K)$ to be $X \\setminus \\{ x_n(K) \\}$, which is open by {P2}.","refs":[{"name":"Applications of k-covers II","doi":"10.1016/j.topol.2005.07.015"}]},{"when":{"kind":"atom","property":"P000159","value":true},"then":{"kind":"atom","property":"P000153","value":true},"uid":"T000375","description":"Theorem 6 of {{doi:10.1007/s10114-005-0753-8}} states that a space $X$ is $k$-Menger if and only if $X^m$ is $k$-Menger for every positive integer $m$. To see that a $k$-Menger space is Menger in each of its finite powers, it suffices to see that a space being $k$-Menger is Menger. Indeed, given any sequence $\\langle \\mathscr U_n : n \\in \\omega \\rangle$ of open covers of a $k$-Menger space $X$, we can form $\\mathscr V_n$ to be the finite unions of elements of $\\mathscr U_n$. Then $\\mathscr V_n$ is a $k$-cover of $X$. We can apply the $k$-Menger selection principle to produce a $k$-cover of $X$ and note that such a selection corresponds to finite selections from each $\\mathscr U_n$ which result in a cover of $X$.","refs":[{"name":"Applications of k-covers","doi":"10.1007/s10114-005-0753-8"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000117","value":true},{"kind":"atom","property":"P000018","value":true}]},"then":{"kind":"atom","property":"P000182","value":true},"uid":"T000376","description":"Every $\\sigma$-locally finite collection of sets in a {P18} space is countable.\n\nSee Lemma 5.1.24 in {{zb:0684.54001}} for locally finite collections, and take a countable union of those.","refs":[{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000136","value":true},{"kind":"atom","property":"P000149","value":true}]},"then":{"kind":"atom","property":"P000128","value":true},"uid":"T000377","description":"Evident from the definitions since all compact sets are finite.","refs":[{"name":"Limited information strategies and discrete selectivity (Clontz & Holshouser)","doi":"10.1016/j.topol.2019.07.008"},{"name":"Selection games and the Vietoris space","doi":"10.1016/j.topol.2021.107772"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000136","value":true},{"kind":"atom","property":"P000153","value":true}]},"then":{"kind":"atom","property":"P000159","value":true},"uid":"T000378","description":"Evident from the definitions since all compact sets are finite.","refs":[{"name":"Limited information strategies and discrete selectivity (Clontz & Holshouser)","doi":"10.1016/j.topol.2019.07.008"},{"name":"Selection games and the Vietoris space","doi":"10.1016/j.topol.2021.107772"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000136","value":true},{"kind":"atom","property":"P000151","value":true}]},"then":{"kind":"atom","property":"P000157","value":true},"uid":"T000379","description":"Evident from the definitions since all compact sets are finite.","refs":[{"name":"Limited information strategies and discrete selectivity (Clontz & Holshouser)","doi":"10.1016/j.topol.2019.07.008"},{"name":"Selection games and the Vietoris space","doi":"10.1016/j.topol.2021.107772"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000136","value":true},{"kind":"atom","property":"P000150","value":true}]},"then":{"kind":"atom","property":"P000156","value":true},"uid":"T000380","description":"Evident from the definitions since all compact sets are finite.","refs":[{"name":"Limited information strategies and discrete selectivity (Clontz & Holshouser)","doi":"10.1016/j.topol.2019.07.008"},{"name":"Selection games and the Vietoris space","doi":"10.1016/j.topol.2021.107772"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000135","value":true},{"kind":"atom","property":"P000158","value":true}]},"then":{"kind":"atom","property":"P000111","value":true},"uid":"T000381","description":"Proved for {P6} spaces as one of the implications of Theorem 3.22 in {{zb:1446.54012}}.\nShown in {{mo:450531}} to only require {P2}. \nBy taking Kolmogorov quotient we can further replace the {P2} assumption by {P135}.","refs":[{"name":"Closed discrete selection in the compact open topology (Caruvana & Holshouser)","zb":"1446.54012"},{"name":"Does there exist a non-hemicompact regular space for which the 2nd player in the K-Rothberger game has a winning Markov strategy?","mo":450531}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000007","value":true},{"kind":"atom","property":"P000194","value":true},{"kind":"atom","property":"P000164","value":true}]},"then":{"kind":"atom","property":"P000162","value":true},"uid":"T000382","description":"By the main theorem of the article {{doi:10.1090/S0002-9939-1973-0322812-9}},\nif $X$ is {P7}, embeds as a closed subspace of a product of {P194} spaces (e.g. it is itself {P194}),\nand every closed discrete subspace of $X$ is {P164} (e.g. it is itself {P164}), then $X$ is {P162}.","refs":[{"name":"Certain Subsets of Products of θ-refinable Spaces are Realcompact (P. Zenor)","doi":"10.1090/S0002-9939-1973-0322812-9"},{"name":"Rings of Continuous Functions (Gillman & Jerison)","zb":"1380.46022"}]},{"when":{"kind":"atom","property":"P000059","value":true},"then":{"kind":"atom","property":"P000164","value":true},"uid":"T000383","description":"See Theorem 12.5 in {{zb:1380.46022}}: in ZFC a measurable cardinal must be strongly inaccessible.\n\nAlso {{wikipedia:Measurable_cardinal}}.","refs":[{"name":"Rings of Continuous Functions (Gillman and Jerison)","zb":"1380.46022"},{"name":"Measurable cardinal on Wikipedia","wikipedia":"Measurable_cardinal"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000005","value":true},{"kind":"atom","property":"P000018","value":true}]},"then":{"kind":"atom","property":"P000162","value":true},"uid":"T000384","description":"See Theorem 3.8.2 of {{zb:0684.54001}} (where the {P18} property assumes {P5}).\n\nAlso Theorem 8.2 of {{zb:1380.46022}} (where all spaces are assumed {P6}).","refs":[{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"},{"name":"Rings of Continuous Functions (Gillman and Jerison)","zb":"1380.46022"}]},{"when":{"kind":"atom","property":"P000162","value":true},"then":{"kind":"atom","property":"P000006","value":true},"uid":"T000385","description":"{P6} is hereditary, and {P162} spaces are [closed] subsets of {P6} spaces.","refs":[{"name":"Realcompact space on Wikipedia","wikipedia":"Realcompact_space"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000022","value":true},{"kind":"atom","property":"P000221","value":true}]},"then":{"kind":"atom","property":"P000016","value":true},"uid":"T000386","description":"By taking Kolmogorov quotient we can assume $X$ is $T_0$. If $X\\subseteq \\prod_\\alpha X_\\alpha$ is closed where $X_\\alpha$ are metric spaces, and $\\pi_\\alpha:X\\to X_\\alpha$ are projections, then $\\pi_\\alpha(X)\\subseteq X_\\alpha$ is {P22} and {P53}, and so {P16} [(Explore)](https://topology.pi-base.org/spaces?q=pseudocompact+%2B+metrizable+%2B+%7Ecompact). It follows that $X$ is a closed subspace of the {P16} space $\\prod_\\alpha \\pi_\\alpha(X)$, and so {P16}.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000118","value":true},{"kind":"atom","property":"P000018","value":true}]},"then":{"kind":"atom","property":"P000183","value":true},"uid":"T000387","description":"Every $\\sigma$-locally finite collection of sets in a {P18} space is countable.\n\nSee Lemma 5.1.24 in {{zb:0684.54001}} for locally finite collections, and take a countable union of those.","refs":[{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"when":{"kind":"atom","property":"P000132","value":true},"then":{"kind":"atom","property":"P000033","value":true},"uid":"T000388","description":"We will use Ishikawa's characterization of {P33}.\n\nSuppose $X$ is {P132}.\nLet $F_0 \\supseteq F_1 \\supseteq \\cdots$ be a nested sequence of closed sets with empty intersection.\nSince each $F_n$ is a $G_\\delta$ set, $F_n = \\bigcap_{m \\in \\omega} U_{n, m}$ for some open sets $U_{n, m}$.\n\nDefine $U_k := \\bigcap_{n \\leq k, m \\leq k} U_{n, m}$.\nClearly each $U_k$ is an open set and $F_k = \\bigcap_{n \\leq k} F_n = \\bigcap_{n \\leq k, m \\in \\omega} U_{n, m} \\subseteq U_k$.\nFinally, $\\bigcap_{k \\in \\omega} U_k = \\bigcap_{n \\in \\omega, m \\in \\omega} U_{n, m} = \\bigcap_{n \\in \\omega} F_n = \\varnothing$.\n\nTherefore, $X$ is {P33}.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000033","value":true},{"kind":"atom","property":"P000013","value":true}]},"then":{"kind":"atom","property":"P000032","value":true},"uid":"T000389","description":"See theorem 2 of {{doi:10.4153/CJM-1951-026-2}}.","refs":[{"name":"On Countably Paracompact Spaces","doi":"10.4153/CJM-1951-026-2"}]},{"when":{"kind":"atom","property":"P000163","value":true},"then":{"kind":"atom","property":"P000059","value":true},"uid":"T000390","description":"Since for any cardinal $\\kappa$, $\\kappa < 2^{\\kappa}$.\n\nSee 1.15 of {{zb:1052.54001}} for a discussion of Cantor's theorem.","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000163","value":true},{"kind":"atom","property":"P000058","value":false}]},"then":{"kind":"atom","property":"P000065","value":true},"uid":"T000391","description":"Immediate from the definitions.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000011","value":true},{"kind":"atom","property":"P000161","value":true}]},"then":{"kind":"atom","property":"P000111","value":true},"uid":"T000392","description":"See {{mo:450531}}.","refs":[{"name":"Does there exist a non-hemicompact regular space for which the 2nd player in the 𝐾-Rothberger game has a winning Markov strategy?","mo":450531}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000049","value":true},{"kind":"atom","property":"P000010","value":true}]},"then":{"kind":"atom","property":"P000050","value":true},"uid":"T000393","description":"If a space is {P10}, it has a basis $\\mathcal B$ of sets $O\\in\\mathcal B$ with $\\operatorname{int}\\operatorname{cl}O=O$. In a {P49} space, $\\operatorname{int}\\operatorname{cl}U=\\operatorname{cl}U$ for any open set $U$. So we have a basis of sets $O\\in\\mathcal B$ where $\\operatorname{cl}O=O$, i.e. a basis of clopen sets, and the space is {P50}.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000053","value":true},{"kind":"atom","property":"P000162","value":true}]},"then":{"kind":"atom","property":"P000164","value":true},"uid":"T000394","description":"See {{mo:469593}}. In particular [this answer by K. P. Hart](https://mathoverflow.net/a/469610/150060).","refs":[{"name":"If $X$ is metrizable, then $X$ is realcompact iff $|X|$ is non-measurable","mo":469593}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000147","value":true},{"kind":"atom","property":"P000018","value":true},{"kind":"atom","property":"P000003","value":true}]},"then":{"kind":"atom","property":"P000013","value":true},"uid":"T000395","description":"See {{mathse:4744652}}.\n\nStated as Proposition 4.2 b) in {{doi:10.1016/0016-660x(72)90026-8}}.","refs":[{"name":"Every Hausdorff Lindelöf P-space is normal","mathse":4744652},{"name":"A topological view of P-spaces (A. Misra)","doi":"10.1016/0016-660x(72)90026-8"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000147","value":true},{"kind":"atom","property":"P000009","value":true}]},"then":{"kind":"atom","property":"P000048","value":true},"uid":"T000396","description":"See Corollary 5.2 in {{doi:10.1016/0016-660x(72)90026-8}} (where \"totally disconnected\" is used with the meaning of {P48}).","refs":[{"name":"A topological view of P-spaces (A. Misra)","doi":"10.1016/0016-660x(72)90026-8"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000168","value":true},{"kind":"atom","property":"P000019","value":true}]},"then":{"kind":"atom","property":"P000078","value":true},"uid":"T000397","description":"Let $A$ be a countable subset of $X$.\nSince $X$ is {P168},\n$A$ is closed in $X$ and hence {P19}.\nFurthermore, the subspace topology on $A$ is discrete.\nSo $A$ is necessarily finite.\nThis proves that $X$ itself is finite.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000147","value":true},{"kind":"atom","property":"P000005","value":true},{"kind":"atom","property":"P000022","value":true}]},"then":{"kind":"atom","property":"P000078","value":true},"uid":"T000398","description":"See proposition 4.1 e) in {{doi:10.1016/0016-660x(72)90026-8}} and {{mathse:4744697}}.","refs":[{"name":"Is every T_3 pseudocompact P-space finite?","mathse":4744697},{"name":"A topological view of P-spaces (A. Misra)","doi":"10.1016/0016-660x(72)90026-8"}]},{"when":{"kind":"atom","property":"P000190","value":true},"then":{"kind":"atom","property":"P000061","value":true},"uid":"T000399","description":"See Example 1.6(f) in {{zb:1067.54015}}.","refs":[{"name":"Cozero complemented spaces; when the space of minimal prime ideals of a $C(X)$ is compact (Henriksen & Woods)","zb":"1067.54015"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000036","value":true},{"kind":"atom","property":"P000217","value":true}]},"then":{"kind":"atom","property":"P000060","value":true},"uid":"T000400","description":"Suppose $X$ is {P217} and suppose that $f:X\\to \\mathbb{R}$ is a non-constant continuous function, say $f(x)\\neq f(y)$. Then $Z_1 = f^{-1}(f(x))$ and $Z_2 = f^{-1}(f(y))$ are disjoint non-empty zero-sets, and so there exists a clopen set $U$ with $Z_1\\subseteq U$ and $Z_2\\cap U=\\emptyset$. Since $U$ is a non-empty proper subset of $X$, $X$ is not {P36}.","refs":[]},{"when":{"kind":"atom","property":"P000013","value":true},"then":{"kind":"atom","property":"P000165","value":true},"uid":"T000401","description":"Immediate from the definitions.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000002","value":true},{"kind":"atom","property":"P000165","value":true}]},"then":{"kind":"atom","property":"P000011","value":true},"uid":"T000402","description":"Let $H$ be a closed set and $x$ be a point not in $H$. The singleton $\\{x\\}$ is countable and closed by {P2}. So by {P165} $H$ and $x$ can be separated by open sets.","refs":[{"name":"Why do pseudonormal spaces have property D?","mathse":4745000}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000011","value":true},{"kind":"atom","property":"P000147","value":true}]},"then":{"kind":"atom","property":"P000165","value":true},"uid":"T000403","description":"Shown in {{mathse:4744732}}.","refs":[{"name":"Answer to \"Is every T3 pseudocompact P-space finite?\"","mathse":4744732}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000079","value":true},{"kind":"atom","property":"P000099","value":true},{"kind":"atom","property":"P000209","value":true}]},"then":{"kind":"atom","property":"P000163","value":true},"uid":"T000404","description":"Proved in {{mathse:4850951}} with a generalization of the ideas in Theorem 4.4 of {{mr:0776620}}.","refs":[{"name":"Cardinal functions. I. Handbook of set-theoretic topology (R. Hodel)","mr":776620},{"name":"Answer to \"Does a separable, $US$, sequential space have cardinality at most the continuum?\"","mathse":4850951}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000003","value":true},{"kind":"atom","property":"P000018","value":true},{"kind":"atom","property":"P000028","value":true}]},"then":{"kind":"atom","property":"P000163","value":true},"uid":"T000405","description":"See Theorem 4.5 of {{mr:0776620}}.","refs":[{"name":"Cardinal functions. I. Handbook of set-theoretic topology (R. Hodel)","mr":776620}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000165","value":true},{"kind":"atom","property":"P000057","value":true}]},"then":{"kind":"atom","property":"P000013","value":true},"uid":"T000406","description":"Immediate from the definitions.","refs":[]},{"when":{"kind":"atom","property":"P000053","value":true},"then":{"kind":"atom","property":"P000112","value":true},"uid":"T000407","description":"Immediate from the definition.","refs":[]},{"when":{"kind":"atom","property":"P000112","value":true},"then":{"kind":"atom","property":"P000009","value":true},"uid":"T000408","description":"Given two points $x,y$, there exists a function $f:X\\to[0,1]$ continuous\nin the coarser {P53} topology with $f(x)=0$ and $f(y)=1$. \nThis function is then continuous\nin the topology for the {P112} space as well.","refs":[{"name":"How separated is a submetrizable space?","mathse":4749941}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000026","value":true},{"kind":"atom","property":"P000112","value":true}]},"then":{"kind":"atom","property":"P000166","value":true},"uid":"T000409","description":"Suppose $D$ is a countable dense set in $X$. If $\\tau$ is a coarser metrizable topology on $X$, the set $D$ is still dense for the coarser topology. So $\\tau$ is separable metrizable.","refs":[]},{"when":{"kind":"atom","property":"P000166","value":true},"then":{"kind":"atom","property":"P000112","value":true},"uid":"T000410","description":"Immediate from the definition.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000052","value":true},{"kind":"atom","property":"P000163","value":true}]},"then":{"kind":"atom","property":"P000166","value":true},"uid":"T000411","description":"Embed the space as a subset of $\\mathbb R$, then\n{S25} witnesses {P166}.","refs":[]},{"when":{"kind":"atom","property":"P000166","value":true},"then":{"kind":"atom","property":"P000163","value":true},"uid":"T000412","description":"Suppose $X$ has a coarser topology that is {P26} and {P53}, hence also {P3} and {P28}. By {T404}, the space has at most continuum cardinality.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000136","value":true},{"kind":"atom","property":"P000002","value":true}]},"then":{"kind":"atom","property":"P000167","value":true},"uid":"T000413","description":"Suppose the sequence $(x_n)$ converges to the point $a\\in X$. The set $A=\\{x_n:n\\in\\mathbb N\\}\\cup\\{a\\}$ is compact, hence finite by {P136}. Since $A$ is finite and $T_1$, it is discrete and the singleton $\\{a\\}$ is open in $A$. As the sequence converges to $a$ in the subspace $A$ as well, the sequence is eventually constant with value $a$.","refs":[{"name":"Answer to What separation is required to ensure extremally disconnected spaces are sequentially discrete?","mathse":4751818}]},{"when":{"kind":"atom","property":"P000167","value":true},"then":{"kind":"atom","property":"P000099","value":true},"uid":"T000414","description":"Suppose the sequence $(x_n)$ converges to a point $x$. Then the sequence $(x_1,x,x_2,x,\\dots)$ that alternates between $x_n$ and $x$ also converges to $x$. If only eventually constant sequences converge, the $x_n$ are eventually all equal to $x$, and $x$ must be the tail value of the sequence.","refs":[{"name":"Answer to What separation is required to ensure extremally disconnected spaces are sequentially discrete?","mathse":4751818}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000147","value":true},{"kind":"atom","property":"P000002","value":true}]},"then":{"kind":"atom","property":"P000168","value":true},"uid":"T000415","description":"If singletons are closed and countable unions of closed sets are closed, then every countable set is closed.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000079","value":true},{"kind":"atom","property":"P000167","value":true}]},"then":{"kind":"atom","property":"P000052","value":true},"uid":"T000416","description":"If $X$ is sequential, its topology is equal to its sequential coreflection. So if the sequential coreflection is discrete, the space itself is discrete.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000026","value":true},{"kind":"atom","property":"P000168","value":true}]},"then":{"kind":"atom","property":"P000057","value":true},"uid":"T000417","description":"The space $X$ contains a countable dense subset, which is closed by {P168}, hence equal to $X$. So the space is countable.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000081","value":true},{"kind":"atom","property":"P000168","value":true}]},"then":{"kind":"atom","property":"P000052","value":true},"uid":"T000418","description":"Let $A$ be an arbitrary subset of $X$. By {P81}, every point $p\\in\\overline A$ is in the closure of a countable subset $D\\subseteq A$. By {P168}, $p\\in D$, which shows that $A$ is closed.","refs":[]},{"when":{"kind":"atom","property":"P000169","value":true},"then":{"kind":"atom","property":"P000002","value":true},"uid":"T000419","description":"Immediate from the definitions.","refs":[]},{"when":{"kind":"atom","property":"P000003","value":true},"then":{"kind":"atom","property":"P000169","value":true},"uid":"T000420","description":"Assume the points $x,y$ are separated by disjoint open sets $U$ and $V$.\nThen $\\operatorname{cl}(U)\\cap V=\\emptyset$.\nThe set $U'=\\operatorname{int}(\\operatorname{cl}(U))$ is regular open\nand satisfies $x\\in U\\subseteq U'$ and $U'\\cap V = \\emptyset$, so that $y\\notin U'$.","refs":[{"name":"Each regular paratopological group is completely regular (Banakh and Ravsky)","doi":"10.1090/proc/13318"},{"name":"Reference search: separation by regular open sets","mathse":4840550}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000002","value":true},{"kind":"atom","property":"P000010","value":true}]},"then":{"kind":"atom","property":"P000169","value":true},"uid":"T000421","description":"Given distinct points $x,y$, let $U$ be an open neighborhood of $x$ missing $y$ by {P2}.\nBy {P10}, $U$ contains a regular open neighborhood of $x$ missing $y$.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000169","value":true},{"kind":"atom","property":"P000125","value":true}]},"then":{"kind":"atom","property":"P000039","value":false},"uid":"T000422","description":"Let $x,y$ be distinct points with a regular open set $U$ containing $x$\nand missing $y$. Then $U$ is not dense, since the interior of its\nclosure is not the whole space.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000023","value":true},{"kind":"atom","property":"P000170","value":true}]},"then":{"kind":"atom","property":"P000003","value":true},"uid":"T000423","description":"Given distinct points $x , y \\in X$, each has a compact neighborhood. \nThe union of these two neighborhoods is a compact set $K$ containing $x$ and $y$ in its interior. \nSince $X$ is {P170}, $K$ is Hausdorff, and so is $\\text{int}(K)$, which is open in $X$. \nThe points $x$ and $y$ can be separated by open sets in $\\text{int}(K)$, which are necessarily open sets in $X$. See also {{mathse:5113190}}.","refs":[{"name":"Answer to \"Unions of two Hausdorff spaces\"","mathse":5113190}]},{"when":{"kind":"atom","property":"P000170","value":true},"then":{"kind":"atom","property":"P000100","value":true},"uid":"T000424","description":"A characterization of {P170} in {{doi:10.1007/BF02194829}}\nis that all {P16} subsets are closed and {P130}.\n\nTo see that {P16} subsets being {P3} implies\n{P16} subsets are closed, let $K$ be {P16}, and $x\\in X\\setminus K$.\nThen $K\\cup\\{x\\}$ is {P16} and thus {P3}, and contains a {P16} and thus\nclosed subspace $K$. It follows that $\\{x\\}$ is open in $K\\cup\\{x\\}$ and thus\n$X$ has an open neighborhood of $x$ missing $K$.","refs":[{"name":"Quotients of k-semigroups","doi":"10.1007/BF02194829"}]},{"when":{"kind":"atom","property":"P000171","value":true},"then":{"kind":"atom","property":"P000099","value":true},"uid":"T000425","description":"See {{mathse:4760309}}.","refs":[{"name":"How are k-Hausdorff and weakly Hausdorff distinct?","mathse":4760309}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000002","value":true},{"kind":"atom","property":"P000045","value":true}]},"then":{"kind":"atom","property":"P000046","value":true},"uid":"T000426","description":"See {{mathse:4807248}}.","refs":[{"name":"Why does the existence of a dispersion point imply total path disconnectedness?","mathse":4807248}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000025","value":true},{"kind":"atom","property":"P000100","value":true}]},"then":{"kind":"atom","property":"P000030","value":true},"uid":"T000427","description":"See {{mathse:4810248}} for a proof.","refs":[{"name":"Answer to \"Is a locally compact Hausdorff space, that is also σ-compact, normal?\"","mathse":4810248}]},{"when":{"kind":"atom","property":"P000175","value":true},"then":{"kind":"atom","property":"P000125","value":true},"uid":"T000428","description":"By definition.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000036","value":true},{"kind":"atom","property":"P000175","value":false},{"kind":"atom","property":"P000137","value":false}]},"then":{"kind":"atom","property":"P000045","value":true},"uid":"T000429","description":"By definition, since nonempty connected spaces where {P175} fails\n(exactly {S162}, {S10}, and {S4})\nbecome {S163} or {S162} and thus {P47} upon the removal of any point.","refs":[]},{"when":{"kind":"atom","property":"P000176","value":true},"then":{"kind":"atom","property":"P000175","value":true},"uid":"T000430","description":"By definition.","refs":[]},{"when":{"kind":"atom","property":"P000078","value":false},"then":{"kind":"atom","property":"P000176","value":true},"uid":"T000431","description":"By definition.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000044","value":true},{"kind":"atom","property":"P000090","value":true},{"kind":"atom","property":"P000176","value":true}]},"then":{"kind":"atom","property":"P000045","value":true},"uid":"T000432","description":"Explored in {{mathse:4812062}}; shown a few different ways assuming\n{P78}, and generalized to {P90}.","refs":[{"name":"Relationship between dispersion point and biconnected property","mathse":4812062}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000036","value":true},{"kind":"atom","property":"P000176","value":false}]},"then":{"kind":"atom","property":"P000044","value":true},"uid":"T000433","description":"{P44} holds vacuously as $X$ is {P36}\nand every pair of subsets, each with at least two points, intersects.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000111","value":true},{"kind":"atom","property":"P000140","value":true}]},"then":{"kind":"atom","property":"P000098","value":true},"uid":"T000434","description":"Shown in {{mathse:4585825}}","refs":[{"name":"Without Hausdorff, what implications can we prove about $k_\\omega$ related to other covering properties?","mathse":4585825}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000084","value":true},{"kind":"atom","property":"P000130","value":true}]},"then":{"kind":"atom","property":"P000142","value":true},"uid":"T000435","description":"Shown in {{mathse:4848174}}","refs":[{"name":"When can we upgrade $k$-space definitions under a local Hausdorff condition?","mathse":4848174}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000084","value":true},{"kind":"atom","property":"P000141","value":true}]},"then":{"kind":"atom","property":"P000142","value":true},"uid":"T000436","description":"Shown in {{mathse:4848174}}","refs":[{"name":"When can we upgrade $k$-space definitions under a local Hausdorff condition?","mathse":4848174}]},{"when":{"kind":"atom","property":"P000052","value":true},"then":{"kind":"atom","property":"P000123","value":true},"uid":"T000437","description":"Every $x\\in X$ has the singleton $\\{x\\}$ as a neighborhood homeomorphic to $\\mathbb R^0$.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000236","value":true},{"kind":"atom","property":"P000139","value":true}]},"then":{"kind":"atom","property":"P000052","value":true},"uid":"T000438","description":"A point is isolated iff it has a neighborhood homeomorphic to $\\mathbb R^0_+=\\mathbb R^0=\\{0\\}$.\nSo if this holds for one point, it holds for all the points and the space is {P52}.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000016","value":true},{"kind":"atom","property":"P000057","value":true}]},"then":{"kind":"atom","property":"P000020","value":true},"uid":"T000439","description":"Shown in {{mathse:4851117}}.\n\n----\n*Note*: This theorem is an ingredient in the proof of the stronger result {T806}.","refs":[{"name":"Do compactness and sequential compactness coincide in countable spaces?","mathse":4851117}]},{"when":{"kind":"atom","property":"P000180","value":true},"then":{"kind":"atom","property":"P000026","value":true},"uid":"T000440","description":"Immediate from the definition.","refs":[]},{"when":{"kind":"atom","property":"P000180","value":true},"then":{"kind":"atom","property":"P000081","value":true},"uid":"T000441","description":"Suppose $A\\subseteq X$ and $p\\in\\overline A$. Since $X$ is {P180}, there is a countable dense subset $D\\subseteq A$, whereby $\\overline{D}=\\overline{A}\\ni p$.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000017","value":true},{"kind":"atom","property":"P000100","value":true}]},"then":{"kind":"atom","property":"P000031","value":true},"uid":"T000442","description":"Suppose $X=\\bigcup_{n<\\omega}K_n$ with each $K_n$ compact.\nLet $\\mathscr U$ be an open cover of $X$.\nFor each $n$, $K_n\\subseteq\\bigcup\\mathscr U_n$ for some finite collection $\\mathscr U_n\\subseteq\\mathscr U$.\nFor a given $n$, the set $\\bigcup_{kx_0$, then let $U,V$ be disjoint open neighborhoods with $x_0\\in U$, $f(x_0)\\in V$, and $ux$, so $A:=\\{x\\in X\\mid f(x)>x\\}$ is open, as is $B:=\\{x\\in X\\mid f(x)\\omega_1$, $\\{\\omega_1\\}$ is a closed subset \nof $\\alpha$ that is not $G_\\delta$. To see this, if \n$\\{(\\alpha_n,\\omega_1]:n\\in\\omega\\}$ is a countable collection of open intervals\ncontaining $\\omega_1$, then $\\bigcap_{n<\\omega} (\\alpha_n,\\omega_1]\\supseteq\n(\\sup\\{\\alpha_n:n<\\omega\\},\\omega_1]\\not=\\{\\omega_1\\}$.\n\nThe fact that $\\omega_1$ ({S35}) is not {P132} follows from the Pressing Down \nLemma; see for example item C of \n.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000117","value":true},{"kind":"atom","property":"P000011","value":true}]},"then":{"kind":"atom","property":"P000132","value":true},"uid":"T000496","description":"See {{mathse:4944702}}.\n\nNote that similar ideas were used in the proof of Theorem 2.8 in {{zb:0153.52404}} ().","refs":[{"name":"In a regular space with a $\\sigma$-locally finite network, is every closed set a $G_\\delta$?","mathse":4944702},{"name":"Some generalizations of metric spaces, their metrization theorems and product spaces (A. Okuyama)","zb":"0153.52404"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000130","value":true},{"kind":"atom","property":"P000100","value":true}]},"then":{"kind":"atom","property":"P000006","value":true},"uid":"T000497","description":"If every point has a local base of compact neighborhoods and every compact set is closed, every point has a local base of closed neighborhoods. Thus the space is {P11}.\n\nThe conclusion can then be strengthened to {P6} using other theorems known to π-Base.\nSpecifically, note that {P130} and {P11} implies {P12} [(Explore)](https://topology.pi-base.org/spaces?q=Locally+Compact%2BRegular%2B%7ECompletely+regular)\nand {P100} implies {P2} [(Explore)](https://topology.pi-base.org/spaces?q=KC%2B%7Et1).","refs":[{"name":"Answer to \"Show that every locally compact Hausdorff space is regular\"","mathse":4940160}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000039","value":true},{"kind":"atom","property":"P000024","value":true}]},"then":{"kind":"atom","property":"P000016","value":true},"uid":"T000498","description":"Assuming $X$ is nonempty, let $x$ be a point of $X$. By {P24},\n$x$ has an open neighborhood $V$ with compact closure. But in a {P39}\nspace every nonempty open set is dense. So $X=\\overline V$ is compact.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000093","value":true},{"kind":"atom","property":"P000018","value":true}]},"then":{"kind":"atom","property":"P000057","value":true},"uid":"T000499","description":"Cover $X$ with countable open sets, then take a countable subcover by {P18}.\nThe union is countable and is the whole space.","refs":[]},{"when":{"kind":"atom","property":"P000191","value":true},"then":{"kind":"atom","property":"P000002","value":true},"uid":"T000500","description":"If $X$ is {P191}, every point is an intersection of open sets, which is one of the characterizations for {P2}.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000028","value":true},{"kind":"atom","property":"P000002","value":true}]},"then":{"kind":"atom","property":"P000191","value":true},"uid":"T000501","description":"Suppose $X$ is {P28} and {P2} and let $x\\in X$. Take a countable local base of open neighborhoods of $x$. Since the intersection of all neighborhoods of $x$ is the singleton $\\{x\\}$ by {P2}, the intersection of the countably many open sets in the local base is also the singleton $\\{x\\}$.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000132","value":true},{"kind":"atom","property":"P000002","value":true}]},"then":{"kind":"atom","property":"P000191","value":true},"uid":"T000502","description":"Clear from the definitions, as points are closed in a {P2} space.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000023","value":true},{"kind":"atom","property":"P000003","value":true},{"kind":"atom","property":"P000191","value":true}]},"then":{"kind":"atom","property":"P000028","value":true},"uid":"T000503","description":"See {{mathse:240480}}.","refs":[{"name":"Answer to \"$G_{\\delta}$ singletons in compact Hausdorff and first countability\"","mathse":240480}]},{"when":{"kind":"atom","property":"P000092","value":true},"then":{"kind":"atom","property":"P000098","value":true},"uid":"T000504","description":"Immediate from the definitions.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000098","value":true},{"kind":"atom","property":"P000170","value":true}]},"then":{"kind":"atom","property":"P000092","value":true},"uid":"T000505","description":"Immediate from the definitions, since {P170} means all\n{P16} subspaces are {P3}.","refs":[]},{"when":{"kind":"atom","property":"P000092","value":true},"then":{"kind":"atom","property":"P000142","value":true},"uid":"T000506","description":"Let $K_n$ witness {P92}, and let $C$ have closed intersection with every\n{P16} {P3} subspace.\nThen $C$ has closed intersection with every $K_n$, and is thus closed in the space.","refs":[]},{"when":{"kind":"atom","property":"P000092","value":true},"then":{"kind":"atom","property":"P000007","value":true},"uid":"T000507","description":"See {{mathse:4952092}} and page 113 of {{zb:0416.54027}}\n(available [here](https://topology.nipissingu.ca/tp/reprints/v02/tp02105.pdf)).","refs":[{"name":"Answer to \"Are $k_\\omega$ spaces Hausdorff?\"","mathse":4952092},{"name":"A survey of $k_\\omega$-spaces (Franklin, Smith Thomas)","zb":"0416.54027"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000011","value":true},{"kind":"atom","property":"P000018","value":true},{"kind":"atom","property":"P000051","value":true},{"kind":"atom","property":"P000191","value":true}]},"then":{"kind":"atom","property":"P000057","value":true},"uid":"T000508","description":"See {{mathse:4954574}}. Also stated (with a typo) as Corollary 2.5 in {{doi:10.1017/S1446788700021777}}.","refs":[{"name":"Answer to \"Every regular, scattered and Lindelöf space with countable pseudocharacter has countable cardinality.\"","mathse":4954574},{"name":"The Lindelöf degree of scattered spaces and their products (M. Gewand)","doi":"10.1017/S1446788700021777"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000009","value":true},{"kind":"atom","property":"P000182","value":true}]},"then":{"kind":"atom","property":"P000112","value":true},"uid":"T000509","description":"See Corollary in answer to {{mo:280359}}.\n\nTo show the Corollary from the Theorem above it, suppose $X$ is {P182}. Then $X\\times X$ is {P131} since {P182} is preserved by finite products and {T260}.","refs":[{"name":"Does second countable and functionally Hausdorff imply submetrizable?","mo":280359}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000147","value":true},{"kind":"atom","property":"P000191","value":true}]},"then":{"kind":"atom","property":"P000052","value":true},"uid":"T000510","description":"Each point is open, since it is a countable intersection of open sets in a {P147} space.","refs":[]},{"when":{"kind":"atom","property":"P000073","value":true},"then":{"kind":"atom","property":"P000192","value":true},"uid":"T000511","description":"Follows directly from the definitions.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000192","value":true},{"kind":"atom","property":"P000001","value":true}]},"then":{"kind":"atom","property":"P000073","value":true},"uid":"T000512","description":"Every nonempty irreducible closed subset of $X$ is the closure of a point, and that point is unique since in a {P1} space distinct points have distinct closures.","refs":[]},{"when":{"kind":"atom","property":"P000094","value":true},"then":{"kind":"atom","property":"P000192","value":true},"uid":"T000513","description":"See {{mathse:5017256}}.","refs":[{"name":"Answer to \"Finite $T_0$ spaces are sober\"","mathse":5017256}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000192","value":true},{"kind":"atom","property":"P000039","value":true},{"kind":"atom","property":"P000135","value":true}]},"then":{"kind":"atom","property":"P000129","value":true},"uid":"T000514","description":"If $X$ is {P192}, {P39}, and nonempty, it has a generic point $x$. \nOne of the characterizations of the {P135} property is that for each point $z\\in X$ the closure $\\operatorname{cl}\\{z\\}$ is the set of points topologically indistinguishable from $z$. So each point of $\\operatorname{cl}\\{x\\}=X$ is topologically indistinguishable from $x$, and $X$ is {P129}.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000132","value":true},{"kind":"atom","property":"P000147","value":true}]},"then":{"kind":"atom","property":"P000185","value":true},"uid":"T000515","description":"If $X$ is {P132} and {P147}, then every closed subset of $X$ is open.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000027","value":true},{"kind":"atom","property":"P000005","value":true}]},"then":{"kind":"atom","property":"P000053","value":true},"uid":"T000516","description":"This is *Urysohn's metrization theorem* (Theorem 23.1 in {{zb:1052.54001}}).","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000023","value":true},{"kind":"atom","property":"P000134","value":true},{"kind":"atom","property":"P000132","value":true}]},"then":{"kind":"atom","property":"P000028","value":true},"uid":"T000517","description":"Follows from {T503} by taking Kolmogorov quotient.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000174","value":true},{"kind":"atom","property":"P000191","value":true}]},"then":{"kind":"atom","property":"P000028","value":true},"uid":"T000518","description":"See {{mathse:4963514}}.","refs":[{"name":"Is every well-based space with countable pseudocharacter first-countable?","mathse":4963514}]},{"when":{"kind":"atom","property":"P000134","value":true},"then":{"kind":"atom","property":"P000192","value":true},"uid":"T000519","description":"The Kolmogorov quotient of a {P134} space is {P3},\nwhich in turn implies {P73} [(Explore)](https://topology.pi-base.org/spaces?q=%24T_2%24%2B%7ESober).\nThus, the unquotiented space is {P192}.","refs":[]},{"when":{"kind":"atom","property":"P000112","value":true},"then":{"kind":"atom","property":"P000106","value":true},"uid":"T000520","description":"Suppose the topology of $X$ contains a coarser {P53} topology induced by a metric $d$.\nFor each $n\\ge 1$, let $\\mathscr U_n$ be the open cover of $X$ consisting of all $d$-balls of radius $1/n$.\nThen $\\{x\\}=\\bigcap_n\\operatorname{st}(x,\\mathscr U_n)$ for all $x\\in X$.","refs":[]},{"when":{"kind":"atom","property":"P000144","value":true},"then":{"kind":"atom","property":"P000192","value":true},"uid":"T000521","description":"The Kolmogorov quotient of a {P144} space is {P82},\nwhich in turn implies {P73} [(Explore)](https://topology.pi-base.org/spaces?q=Locally+metrizable%2B%7ESober).\nThus, the unquotiented space is {P192}.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000133","value":true},{"kind":"atom","property":"P000036","value":true}]},"then":{"kind":"atom","property":"P000041","value":true},"uid":"T000522","description":"Let $X$ be a {P36} {P133}, then $X$ is Dedekind-complete and has a dense ordering (see {{mathse:4968515}}), and every order-convex subset of $X$ has the same property, so is {P36} (by Theorem 24.1 in {{zb:0951.54001}}). Thus $X$ has a basis consisting of {P36} sets.","refs":[{"name":"Cardinality of connected LOTS","mathse":4968515},{"name":"Topology (Munkres)","zb":"0951.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000133","value":true},{"kind":"atom","property":"P000037","value":true}]},"then":{"kind":"atom","property":"P000042","value":true},"uid":"T000523","description":"See discussion at {{mathse:4965472}}.","refs":[{"name":"Are path-connected LOTS also locally path-connected?","mathse":4965472}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000055","value":true},{"kind":"atom","property":"P000139","value":false},{"kind":"atom","property":"P000137","value":false}]},"then":{"kind":"atom","property":"P000058","value":false},"uid":"T000524","description":"See {{mo:22830}}, or Eric Wofsey's answer to {{mathse:2304575}}.","refs":[{"name":"Improvements of the Baire Category Theorem under (not CH)?","mo":22830},{"name":"Is the cardinality of an open set in a complete metric space $X$ with no isolated points equal to $|X|$?","mathse":2304575}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000026","value":true},{"kind":"atom","property":"P000077","value":true}]},"then":{"kind":"atom","property":"P000053","value":true},"uid":"T000525","description":"Asserted at the top of page 372 of {{doi:10.1090/S0002-9939-1987-0884482-0}}.\n\nSee also [this post on Dan Ma's Topology Blog](https://dantopology.wordpress.com/2014/06/03/sigma-products-of-separable-metric-spaces-are-monolithic/),\nnoting that \"strongly monolithic\" means that the space's\ndensity (minimum cardinality of a dense set) equals its\nweight (minimum cardinality of a basis), so {P26} implies {P27}\n(and in turn {P53} as we also have {P5}).","refs":[{"name":"A note on Gul'ko compact spaces (Gruenhage)","doi":"10.1090/S0002-9939-1987-0884482-0"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000042","value":true},{"kind":"atom","property":"P000084","value":true}]},"then":{"kind":"atom","property":"P000096","value":true},"uid":"T000526","description":"An arbitrary neighborhood $V$ of a point $x$ contains a {P3} neighborhood $W$ of $x$ by {P84}.\nBy {P42}, $W$ contains a {P37} open neighborhood $U$ of $x$.\nAnd $U\\subseteq V$ is {P95} since {T240}.","refs":[]},{"when":{"kind":"atom","property":"P000075","value":true},"then":{"kind":"atom","property":"P000130","value":true},"uid":"T000527","description":"The compact open sets form a (global) basis and therefore also can comprise local bases.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000208","value":true},{"kind":"atom","property":"P000073","value":true}]},"then":{"kind":"atom","property":"P000075","value":true},"uid":"T000528","description":"It immediately follows by the definitions that Spectral is equivalent to Sober for Noetherian spaces.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000016","value":true},{"kind":"atom","property":"P000003","value":true},{"kind":"atom","property":"P000047","value":true}]},"then":{"kind":"atom","property":"P000195","value":true},"uid":"T000529","description":"By definition.","refs":[{"name":"Stone space on Wikipedia","wikipedia":"Stone_space"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000075","value":true},{"kind":"atom","property":"P000002","value":true}]},"then":{"kind":"atom","property":"P000195","value":true},"uid":"T000530","description":"See [Lemma 5.23.8](https://stacks.math.columbia.edu/tag/0905) at the Stacks project.","refs":[{"name":"Spectral space on Wikipedia","wikipedia":"Spectral_space"}]},{"when":{"kind":"atom","property":"P000195","value":true},"then":{"kind":"atom","property":"P000075","value":true},"uid":"T000531","description":"See [Theorem 4.2](https://www.google.com/books/edition/Stone_Spaces/CiWwoLNbpykC?gbpv=1&pg=PA69) in {{mr:0861951}}.","refs":[{"name":"Stone spaces (Johnstone)","mr":861951},{"name":"Stone's representation theorem for Boolean algebras on Wikipedia","wikipedia":"Stone's_representation_theorem_for_Boolean_algebras"},{"name":"Spectral space on Wikipedia","wikipedia":"Spectral_space"}]},{"when":{"kind":"atom","property":"P000195","value":true},"then":{"kind":"atom","property":"P000048","value":true},"uid":"T000532","description":"See [Theorem 4.2](https://www.google.com/books/edition/Stone_Spaces/CiWwoLNbpykC?gbpv=1&pg=PA69) in {{mr:0861951}}.","refs":[{"name":"Stone spaces (Johnstone)","mr":861951},{"name":"Stone space on Wikipedia","wikipedia":"Stone_space"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000182","value":true},{"kind":"atom","property":"P000023","value":true},{"kind":"atom","property":"P000003","value":true}]},"then":{"kind":"atom","property":"P000027","value":true},"uid":"T000533","description":"See {{mathse:2500413}} or Theorem 3.3.5 in {{zb:0684.54001}}.","refs":[{"name":"Answer to \"Weight of locally compact Hausdorff space does not exceed its cardinality\"","mathse":2500413},{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000019","value":true},{"kind":"atom","property":"P000112","value":true}]},"then":{"kind":"atom","property":"P000053","value":true},"uid":"T000534","description":"Let $f \\colon X \\to Y$ be a continuous bijection from a {P19} space $X$ to a {P53} space $Y$.\n\nLet $A \\subseteq X$ be closed.\nSince closed subspaces and continuous images of a {P19} space are {P19}, $f(A)$ is {P19}.\nSince $Y$ is {P53}, it is {P103} [(Explore)](https://topology.pi-base.org/spaces?q=Metrizable+%2B+%7EStrongly+KC), and hence $f(A)$ is closed in $Y$.\n\nTherefore, $f$ is a closed map and $f$ is a homeomorphism, which implies $X$ itself is {P53}.","refs":[]},{"when":{"kind":"atom","property":"P000123","value":true},"then":{"kind":"atom","property":"P000122","value":true},"uid":"T000535","description":"By definition.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000122","value":true},{"kind":"atom","property":"P000036","value":true}]},"then":{"kind":"atom","property":"P000123","value":true},"uid":"T000536","description":"If a point $x$ has at the same time an open neighborhood homeomorphic to $\\mathbb R^n$ and an open neighborhood homeomorphic to $\\mathbb R^m$ with $n\\ne m$, their intersection would be homeomorphic to an open set in $\\mathbb R^n$ and to an open set in $\\mathbb R^m$. But this is not possible by invariance of domain ({{wikipedia:Invariance_of_domain}}). So the dimension $n$ at the point $x$ is completely determined by the point, and by definition of {P122} the set of points of a given dimension $n$ is open in $X$. Since $X$ is {P36}, the dimension must be the same at every point.","refs":[{"name":"Invariance of domain on Wikipedia","wikipedia":"Invariance_of_domain"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000235","value":true},{"kind":"atom","property":"P000086","value":true}]},"then":{"kind":"atom","property":"P000123","value":true},"uid":"T000537","description":"If one point $x$ has an open neighborhood $U$ homeomorphic to some $\\mathbb R^n_+$,\nsome point (maybe different from $x$) in $U$ has a neighborhood homeomorphic to $\\mathbb R^n$.\nBy homogeneity, every point has a neighborhood homeomorphic to $\\mathbb R^n$.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000173","value":true},{"kind":"atom","property":"P000167","value":true}]},"then":{"kind":"atom","property":"P000147","value":true},"uid":"T000538","description":"See {{mathse:4975331}}.","refs":[{"name":"Countable union of closed subsets of a pseudoradial and sequentially closed topological space is closed","mathse":4975331}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000154","value":true},{"kind":"atom","property":"P000049","value":true}]},"then":{"kind":"atom","property":"P000052","value":true},"uid":"T000539","description":"See {{mathse:4913523}}. The answer proves the result for {P133} but the proof generalizes easily for {P154} as remarked in the original question.","refs":[{"name":"Is there a nondiscrete linearly ordered topological space (LOTS) that is extremally disconnected?","mathse":4913523}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000003","value":true},{"kind":"atom","property":"P000131","value":true}]},"then":{"kind":"atom","property":"P000191","value":true},"uid":"T000540","description":"See {{mathse:4972410}}.","refs":[{"name":"$T_2$ + Hereditarily Lindelöf implies countable pseudocharacter","mathse":4972410}]},{"when":{"kind":"atom","property":"P000037","value":true},"then":{"kind":"atom","property":"P000189","value":true},"uid":"T000541","description":"See {{mathse:4975686}}.","refs":[{"name":"$\\sigma$-connected spaces and path-connected property","mathse":4975686}]},{"when":{"kind":"atom","property":"P000193","value":true},"then":{"kind":"atom","property":"P000013","value":true},"uid":"T000542","description":"The argument in {{zb:0712.54016}} for this result goes as follows. Suppose $E$ and $F$ are disjoint closed subsets of a shrinking space $X$. Then $\\{ X \\setminus E , X \\setminus F\\}$ is an open cover of $X$, so there exists an open cover $\\{U, V\\}$ of $X$ such that $\\overline{U} \\subseteq X \\setminus E$ and $\\overline{V} \\subseteq X \\setminus F$. Note then that $E \\subseteq X \\setminus \\overline{U}$, $F \\subseteq X \\setminus \\overline{V}$, and $\\left( X \\setminus \\overline{U} \\right) \\cap \\left( X \\setminus \\overline{V} \\right) = X \\setminus \\left( \\overline{U} \\cup \\overline{V} \\right) = \\emptyset$.\n\nSee also [Dan Ma's Topology Blog post on Spaces with shrinking properties](https://dantopology.wordpress.com/2017/01/05/spaces-with-shrinking-properties/).","refs":[{"name":"Generalized paracompactness (Y. Yasui)","zb":"0712.54016"}]},{"when":{"kind":"atom","property":"P000193","value":true},"then":{"kind":"atom","property":"P000032","value":true},"uid":"T000543","description":"This implication appears in the diagram on page 191 of {{zb:0712.54016}} and is mentioned in passing in {{zb:1059.54001}} on page 199. See also Theorem 21.3 of {{zb:1052.54001}} and Theorem 5 at [Dan Ma's Topology Blog post on Countably paracompact spaces](https://dantopology.wordpress.com/2016/12/08/countably-paracompact-spaces/).","refs":[{"name":"Generalized paracompactness (Y. Yasui)","zb":"0712.54016"},{"name":"Encyclopedia of general topology","zb":"1059.54001"},{"name":"General Topology (S. Willard)","zb":"1052.54001"}]},{"when":{"kind":"atom","property":"P000031","value":true},"then":{"kind":"atom","property":"P000194","value":true},"uid":"T000544","description":"This is evident from the definitions. The single point-finite open refinement guaranteed by metacompactness generates a sequence with the desired properties.","refs":[{"name":"On Submetacompactness (H. Junnila)","zb":"0413.54027"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000194","value":true},{"kind":"atom","property":"P000013","value":true}]},"then":{"kind":"atom","property":"P000193","value":true},"uid":"T000545","description":"See Theorem 6.2 of {{zb:0712.54016}}.","refs":[{"name":"Generalized paracompactness (Y. Yasui)","zb":"0712.54016"}]},{"when":{"kind":"atom","property":"P000196","value":true},"then":{"kind":"atom","property":"P000039","value":true},"uid":"T000546","description":"The open sets are totally ordered by set inclusion and thus no two nonempty open sets are disjoint.","refs":[]},{"when":{"kind":"atom","property":"P000196","value":true},"then":{"kind":"atom","property":"P000014","value":true},"uid":"T000547","description":"Any {P196} space is trivially {P13}\n(as there are no disjoint nonempty closed sets), so all subspaces are as well.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000039","value":true},{"kind":"atom","property":"P000014","value":true}]},"then":{"kind":"atom","property":"P000196","value":true},"uid":"T000548","description":"See condition (10) of theorem 23 at {{doi:10.5186/aasfm.1977.0321}} (reading $T_5$ as {P14}).","refs":[{"name":"On ultrapseudocompact and related spaces (T. Nieminen)","doi":"10.5186/aasfm.1977.0321"}]},{"when":{"kind":"atom","property":"P000196","value":true},"then":{"kind":"atom","property":"P000174","value":true},"uid":"T000549","description":"The topology is totally ordered by inclusion, so the set of open neighborhoods of any point is as well.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000193","value":true},{"kind":"atom","property":"P000039","value":true}]},"then":{"kind":"atom","property":"P000146","value":true},"uid":"T000550","description":"In a {P39} space $X$, the closure of every nonempty open set is $X$, so any open cover that admits a shrinking must contain $X$ (as otherwise each of its open sets could only contain the closure of $\\varnothing$).\nThus, any such open cover admits an open refinement to the partition $\\{X\\}$.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000196","value":true},{"kind":"atom","property":"P000093","value":true},{"kind":"atom","property":"P000057","value":false}]},"then":{"kind":"atom","property":"P000114","value":true},"uid":"T000551","description":"By {P93} choose a countable open neighborhood of each point. So $X$ can be covered by a family of countable sets, forming a chain under inclusion by {P196}. And the union of a chain of countable sets has cardinality at most $\\aleph_1$ (see {{mathse:342091}} for example).","refs":[{"name":"Why should the union of such a family of sets have cardinality $\\aleph_1$?","mathse":342091}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000202","value":true},{"kind":"atom","property":"P000086","value":true}]},"then":{"kind":"atom","property":"P000129","value":true},"uid":"T000552","description":"If one point in a homogeneous space has $X$ as its only neighborhood, then the same is true of every point, so the space is indiscrete.","refs":[]},{"when":{"kind":"atom","property":"P000196","value":true},"then":{"kind":"atom","property":"P000223","value":true},"uid":"T000553","description":"Since {T585}, every nonempty subset of $X$ is also {P199}.\nIn particular, every open neighborhood of every point is {P199}.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000196","value":true},{"kind":"atom","property":"P000016","value":true},{"kind":"atom","property":"P000073","value":true},{"kind":"atom","property":"P000090","value":true}]},"then":{"kind":"atom","property":"P000075","value":true},"uid":"T000554","description":"Shown in Proposition 1.6.7 of {{doi:10.1017/9781316543870}}.\n\nFurthermore, if the space is nonempty, its specialization order is order-isomorphic to a successor ordinal $\\lambda$ and the space is homeomorphic to $\\lambda$ with the left-ray topology having the collection of intervals $[0,\\alpha]$ ($\\alpha\\in\\lambda$) as a base.","refs":[{"name":"Spectral spaces (Dickmann, Schwartz, Tressl)","doi":"10.1017/9781316543870"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000039","value":true},{"kind":"atom","property":"P000040","value":true},{"kind":"atom","property":"P000176","value":true}]},"then":{"kind":"atom","property":"P000044","value":false},"uid":"T000555","description":"See {{mathse:4987074}}.","refs":[{"name":"Hyperconnected ultraconnected spaces are not biconnected","mathse":4987074}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000146","value":true},{"kind":"atom","property":"P000036","value":true},{"kind":"atom","property":"P000137","value":false}]},"then":{"kind":"atom","property":"P000202","value":true},"uid":"T000556","description":"Any clopen partition of a {P36} space $X$ must include $X$, so to admit clopen refinements every open cover must include $X$.\nThus, the union of all open sets except for $X$ cannot equal $X$ (as that would be an open cover not containing $X$), so all points outside of that union have $X$ as their only neighborhood.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000001","value":true},{"kind":"atom","property":"P000090","value":true},{"kind":"atom","property":"P000027","value":true}]},"then":{"kind":"atom","property":"P000057","value":true},"uid":"T000557","description":"In an {P90} space, the smallest basis is the set of smallest neighborhoods of points.\nWhen it is also {P1}, these are all distinct, so they are in bijection with the set of points.\nThus, such a space has a countable basis iff it is countable.","refs":[]},{"when":{"kind":"atom","property":"P000204","value":true},"then":{"kind":"atom","property":"P000175","value":true},"uid":"T000558","description":"By definition, since spaces that are not {P36} have at least two points.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000198","value":true},{"kind":"atom","property":"P000052","value":true}]},"then":{"kind":"atom","property":"P000057","value":true},"uid":"T000559","description":"Follows from the definitions.","refs":[]},{"when":{"kind":"atom","property":"P000180","value":true},"then":{"kind":"atom","property":"P000197","value":true},"uid":"T000560","description":"Follows as separable discrete spaces are countable.","refs":[]},{"when":{"kind":"atom","property":"P000197","value":true},"then":{"kind":"atom","property":"P000198","value":true},"uid":"T000561","description":"Follows from the definitions.","refs":[]},{"when":{"kind":"atom","property":"P000018","value":true},"then":{"kind":"atom","property":"P000198","value":true},"uid":"T000562","description":"Closed sets in a Lindelöf space are Lindelöf, and Lindelöf discrete spaces are countable.","refs":[]},{"when":{"kind":"atom","property":"P000197","value":true},"then":{"kind":"atom","property":"P000029","value":true},"uid":"T000563","description":"See {{mathse:4988266}}.","refs":[{"name":"Countable spread implies countable chain condition","mathse":4988266}]},{"when":{"kind":"atom","property":"P000094","value":true},"then":{"kind":"atom","property":"P000093","value":true},"uid":"T000564","description":"By definition, as finite sets are countable.","refs":[]},{"when":{"kind":"atom","property":"P000094","value":true},"then":{"kind":"atom","property":"P000090","value":true},"uid":"T000565","description":"Given a finite neighborhood of a point, only finitely many intersections with open sets are needed to arrive at a smallest neighborhood.","refs":[]},{"when":{"kind":"atom","property":"P000196","value":true},"then":{"kind":"atom","property":"P000197","value":true},"uid":"T000566","description":"The {P52} subspaces of a hereditarily connected space contain at most one point.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000196","value":true},{"kind":"atom","property":"P000094","value":true}]},"then":{"kind":"atom","property":"P000057","value":true},"uid":"T000567","description":"By {P94} choose a finite open neighborhood of each point. So $X$ can be covered by a family of finite sets, forming a chain under inclusion by {P196}. These sets must then all have distinct finite cardinalities, so the cardinality of their union can be at most $\\aleph_0$, the cardinality of the natural numbers.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000126","value":true},{"kind":"atom","property":"P000039","value":true}]},"then":{"kind":"atom","property":"P000136","value":true},"uid":"T000568","description":"Suppose by contradiction that $A$ is an infinite compact subset of $X$. Write $A=B\\cup C$ with $B,C$ infinite and disjoint.\nBecause $X$ is {P39}, it is not possible that $B$ and $C$ each contain a nonempty open subset of $X$.\nSo at least one of the two sets, say $B$, has all its subsets closed in $X$ by the {P126} property.\nTherefore $B$ has the discrete topology. But $B$ is also closed in $A$, hence compact, which is not possible.","refs":[]},{"when":{"kind":"atom","property":"P000021","value":true},"then":{"kind":"atom","property":"P000198","value":true},"uid":"T000569","description":"Immediate from the definitions.","refs":[]},{"when":{"kind":"atom","property":"P000076","value":true},"then":{"kind":"atom","property":"P000012","value":true},"uid":"T000570","description":"See Definition 1.7 of {{doi:10.1016/j.topol.2014.05.010}}: \"A topological space is proximal iff it admits a compatible uniformity (i.e., one which induces its topology) which is proximal.\"","refs":[{"name":"Proximal compact spaces are Corson compact (Clontz, Gruenhage)","doi":"10.1016/j.topol.2014.05.010"}]},{"when":{"kind":"atom","property":"P000203","value":true},"then":{"kind":"atom","property":"P000052","value":false},"uid":"T000571","description":"By definition: all points in a discrete space are isolated, but an almost discrete space has exactly one non-isolated point.","refs":[]},{"when":{"kind":"atom","property":"P000203","value":true},"then":{"kind":"atom","property":"P000126","value":true},"uid":"T000572","description":"Sets that contain the non-isolated point are closed and sets that don't are open.","refs":[]},{"when":{"kind":"atom","property":"P000203","value":true},"then":{"kind":"atom","property":"P000051","value":true},"uid":"T000573","description":"For a nonempty $Y\\subseteq X$, either $|Y|=1$ and it trivially contains an isolated point, or $Y$ must contain a point isolated in $X$.","refs":[]},{"when":{"kind":"atom","property":"P000203","value":true},"then":{"kind":"atom","property":"P000146","value":true},"uid":"T000574","description":"Any open cover must include an open neighborhood $U$ of the non-isolated point.\nThe complement of $U$ can be covered by the collection of open singletons.\nThus, any open cover admits a refinement to an open partition of the form\n$\\{U\\}\\cup\\{\\{x\\}\\mid x\\in X\\setminus U\\}$.","refs":[]},{"when":{"kind":"atom","property":"P000203","value":true},"then":{"kind":"atom","property":"P000108","value":true},"uid":"T000575","description":"Let $X$ be almost discrete. Every subspace of $X$ is either almost discrete or discrete.\nThus every subspace of $X$ is {P146}:\n- For {P203} subspaces: {T574}.\n- For {P52} subspaces, [(Explore)](https://topology.pi-base.org/spaces?q=Discrete+%2B+%7EUltraparacompact).\n\nThe result then follows, since every {P146} space is {P88}\n[(Explore)](https://topology.pi-base.org/spaces?q=Ultraparacompact+%2B+%7ECollectionwise+normal\n).","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000126","value":true},{"kind":"atom","property":"P000040","value":true},{"kind":"atom","property":"P000125","value":true}]},"then":{"kind":"atom","property":"P000203","value":true},"uid":"T000576","description":"Per Corollary 2 of {{zb:0646.54028}} ().","refs":[{"name":"Door spaces are identifiable (S. D. McCartan)","zb":"0646.54028"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000126","value":true},{"kind":"atom","property":"P000003","value":true},{"kind":"atom","property":"P000052","value":false}]},"then":{"kind":"atom","property":"P000203","value":true},"uid":"T000577","description":"Discussed after the proof of the main theorem of {{mr:923909}} ().","refs":[{"name":"Door spaces are identifiable (S. D. McCartan)","zb":"0646.54028"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000203","value":true},{"kind":"atom","property":"P000036","value":true}]},"then":{"kind":"atom","property":"P000045","value":true},"uid":"T000578","description":"Removing the non-isolated point results in a {P52} space.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000045","value":true},{"kind":"atom","property":"P000175","value":true}]},"then":{"kind":"atom","property":"P000204","value":true},"uid":"T000579","description":"$X$ is connected since it {P45}. Also, {T52} means that the dispersion point is also a cut point when $X$ satisfies {P175}.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000187","value":true},{"kind":"atom","property":"P000093","value":true}]},"then":{"kind":"atom","property":"P000028","value":true},"uid":"T000580","description":"Suppose $X$ satisfies the hypotheses.\nAny point $x\\in X$ has a countable neighborhood $V$,\nwhich is also {P187} by hereditariness.\nBy Corollary 3.4 of {{zb:0327.54019}}, $V$ is {P28}.\nThe result follows because being {P28} is a local property.","refs":[{"name":"Infinite games and generalizations of first-countable spaces (Gruenhage)","zb":"0327.54019"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000187","value":true},{"kind":"atom","property":"P000011","value":true},{"kind":"atom","property":"P000026","value":true}]},"then":{"kind":"atom","property":"P000028","value":true},"uid":"T000581","description":"Shown in Theorem 3.6 of {{doi:10.1016/0016-660X(76)90024-6}}.","refs":[{"name":"Infinite games and generalizations of first-countable spaces (Gruenhage)","doi":"10.1016/0016-660X(76)90024-6"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000187","value":true},{"kind":"atom","property":"P000003","value":true},{"kind":"atom","property":"P000197","value":true}]},"then":{"kind":"atom","property":"P000191","value":true},"uid":"T000582","description":"Let $x$ be a point of $X$. $X$ is $T_2$ and there is a winning strategy for P1 at $x$. By Theorem 3.7 of {{doi:10.1016/0016-660X(76)90024-6}}, either $x$ is a $G_{\\delta}$ set, or there exists an uncountable discrete subset $D \\subseteq X$ such that $D \\cup \\{x\\}$ is compact. As $X$ has countable spread, then $x$ is a $G_{\\delta}$ set.","refs":[{"name":"Infinite games and generalizations of first-countable spaces (Gruenhage)","doi":"10.1016/0016-660X(76)90024-6"}]},{"when":{"kind":"atom","property":"P000199","value":true},"then":{"kind":"atom","property":"P000200","value":true},"uid":"T000583","description":"It follows that $X$ is {P37} by {{mathse:715720}}. Since $X$ is contractible, every map $S^1 \\to X$ is homotopic to a constant map. This means that $X$ is simply connected.","refs":[{"name":"A contractible space is path connected.","mathse":715720},{"name":"Algebraic Topology (Hatcher)","zb":"1044.55001"}]},{"when":{"kind":"atom","property":"P000199","value":true},"then":{"kind":"atom","property":"P000137","value":false},"uid":"T000584","description":"The empty space is not homotopy equivalent to a one-point space.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000196","value":true},{"kind":"atom","property":"P000137","value":false}]},"then":{"kind":"atom","property":"P000199","value":true},"uid":"T000585","description":"See {{mathse:5000498}}.","refs":[{"name":"Are hereditarily connected spaces and ultraconnected spaces contractible?","mathse":5000498}]},{"when":{"kind":"atom","property":"P000031","value":true},"then":{"kind":"atom","property":"P000083","value":true},"uid":"T000586","description":"Evident from the definitions.","refs":[]},{"when":{"kind":"atom","property":"P000018","value":true},"then":{"kind":"atom","property":"P000083","value":true},"uid":"T000587","description":"Evident from the definitions, as a countable cover is point-countable.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000036","value":true},{"kind":"atom","property":"P000133","value":true},{"kind":"atom","property":"P000175","value":true}]},"then":{"kind":"atom","property":"P000204","value":true},"uid":"T000588","description":"$X$ is connected by assumption. And on the other hand, given three points $a < b < c$ in $X$, $X\\setminus\\{b\\} = (-\\infty, b) \\cup (b, \\infty)$ is the union of two disjoint nonempty open sets, showing that $X\\setminus\\{b\\}$ is not connected.","refs":[]},{"when":{"kind":"atom","property":"P000129","value":true},"then":{"kind":"atom","property":"P000200","value":true},"uid":"T000589","description":"Every function from some topological space to an indiscrete space $X$ is continuous.\nTherefore $X$ is {P37}, since any function $f:[0,1]\\to X$ with $f(0)=x$ and $f(1)=y$ is a continuous path from $x$ to $y$.\nAnd every map $S^1 \\to X$ has a continuous extension $D^2 \\to X$.","refs":[]},{"when":{"kind":"atom","property":"P000200","value":true},"then":{"kind":"atom","property":"P000037","value":true},"uid":"T000590","description":"By definition.","refs":[]},{"when":{"kind":"atom","property":"P000201","value":true},"then":{"kind":"atom","property":"P000199","value":true},"uid":"T000591","description":"By assumption, there exists a generic point $p \\in X$. We argue that $F : X \\times [0, 1] \\to X$, defined by $$F(x, t) = \\begin{cases} x ,& \\text{ if }t = 0\\\\ p ,& \\text{ if }t > 0, \\end{cases}$$ is continuous, hence is a homotopy between the identity map of $X$ and a constant map. Suppose $A \\subset X$ is closed. If $p \\in A$, then $A = X$ because $p$ is a generic point. So $F^{-1}(A) = X \\times [0, 1]$. Otherwise, if $p \\notin A$, then $F^{-1}(A) = A \\times \\{0\\}$. Since $F^{-1}(A)$ is closed in both cases, it follows that $F$ is continuous.\n\nThis construction appears near the end of David Gao's answer to {{mathse:4993007}}. This also appears as Lemma 2.3.2 in Peter May's [Finite spaces and larger contexts](https://math.uchicago.edu/~may/FINITE/FINITEBOOK/FINITEBOOKCollatedDraft.pdf).","refs":[{"name":"Is a path-connected space with a dispersion point necessarily contractible?","mathse":4993007}]},{"when":{"kind":"atom","property":"P000201","value":true},"then":{"kind":"atom","property":"P000026","value":true},"uid":"T000592","description":"If $p$ is a generic point, then $\\{p\\}$ is a countable dense set.","refs":[]},{"when":{"kind":"atom","property":"P000201","value":true},"then":{"kind":"atom","property":"P000039","value":true},"uid":"T000593","description":"By assumption, there exists a generic point $p \\in X$. Since every nonempty open subset contains $p$ and $\\overline{\\{p\\}} = X$, it follows that the closure of any nonempty open subset is $X$.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000192","value":true},{"kind":"atom","property":"P000039","value":true},{"kind":"atom","property":"P000137","value":false}]},"then":{"kind":"atom","property":"P000201","value":true},"uid":"T000594","description":"By definition.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000201","value":true},{"kind":"atom","property":"P000135","value":true}]},"then":{"kind":"atom","property":"P000129","value":true},"uid":"T000595","description":"Suppose $X$ has a generic point $p$. This means that $\\overline{\\{p\\}} = X$. By definition of {P135}, it follows that $\\overline{\\{p\\}}$ consists of points which are topologically indistinguishable from $p$. Therefore $X$ is indiscrete.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000201","value":true},{"kind":"atom","property":"P000086","value":true}]},"then":{"kind":"atom","property":"P000129","value":true},"uid":"T000596","description":"If $X$ is homogeneous and has a generic point, then every point is a generic point. It immediately follows that the only nonempty open set is $X$.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000039","value":true},{"kind":"atom","property":"P000139","value":true}]},"then":{"kind":"atom","property":"P000201","value":true},"uid":"T000597","description":"In a hyperconnected space, every open set is dense, so an open singleton contains a generic point.","refs":[]},{"when":{"kind":"atom","property":"P000202","value":true},"then":{"kind":"atom","property":"P000040","value":true},"uid":"T000598","description":"Every nonempty closed set must contain any point whose only neighborhood is $X$.","refs":[]},{"when":{"kind":"atom","property":"P000202","value":true},"then":{"kind":"atom","property":"P000146","value":true},"uid":"T000599","description":"If the only neighborhood of a point is $X$, any open cover must include $X$ and therefore admits a refinement to the open partition $\\{X\\}$.","refs":[]},{"when":{"kind":"atom","property":"P000202","value":true},"then":{"kind":"atom","property":"P000020","value":true},"uid":"T000600","description":"Any sequence converges to all points whose only neighborhood is $X$.","refs":[]},{"when":{"kind":"atom","property":"P000202","value":true},"then":{"kind":"atom","property":"P000199","value":true},"uid":"T000601","description":"By assumption, there exists a point $p \\in X$ whose only neighborhood is $X$. We argue that $F : X \\times [0, 1] \\to X$, defined by $$F(x, t) = \\begin{cases} x & \\text{if }t < 1\\\\ p & \\text{if }t = 1\\end{cases}$$ is continuous, hence is a homotopy between the identity map of $X$ and a constant map. Suppose $A \\subset X$ is open. If $p \\in A$, then $A = X$ and $F^{-1}(A) = X \\times [0, 1]$. Otherwise, if $p \\notin A$, then $F^{-1}(A) = A \\times [0,1)$. Since $F^{-1}(A)$ is open in both cases, it follows that $F$ is continuous.\n\nThis construction appears at the end of David Gao's answer to {{mathse:4993007}}. This also appears as Lemma 2.3.2 in Peter May's [Finite spaces and larger contexts](https://math.uchicago.edu/~may/FINITE/FINITEBOOK/FINITEBOOKCollatedDraft.pdf).","refs":[{"name":"Is a path-connected space with a dispersion point necessarily contractible?","mathse":4993007}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000037","value":true},{"kind":"atom","property":"P000045","value":true},{"kind":"atom","property":"P000201","value":false}]},"then":{"kind":"atom","property":"P000202","value":true},"uid":"T000602","description":"Shown in David Gao's answer to {{mathse:4993007}}.","refs":[{"name":"Is a path-connected space with a dispersion point necessarily contractible?","mathse":4993007}]},{"when":{"kind":"atom","property":"P000026","value":true},"then":{"kind":"atom","property":"P000209","value":true},"uid":"T000603","description":"Follows as every countable set has cardinality $\\leq\\mathfrak c$.","refs":[]},{"when":{"kind":"atom","property":"P000163","value":true},"then":{"kind":"atom","property":"P000209","value":true},"uid":"T000604","description":"Follows as every space is a dense subset of itself.","refs":[]},{"when":{"kind":"atom","property":"P000203","value":true},"then":{"kind":"atom","property":"P000073","value":true},"uid":"T000605","description":"See {{mathse:5012490}}.","refs":[{"name":"Connections between almost discrete spaces and sober/spectral spaces","mathse":5012490}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000203","value":true},{"kind":"atom","property":"P000016","value":true}]},"then":{"kind":"atom","property":"P000075","value":true},"uid":"T000606","description":"See {{mathse:5012490}}.","refs":[{"name":"Connections between almost discrete spaces and sober/spectral spaces","mathse":5012490}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000203","value":true},{"kind":"atom","property":"P000002","value":true}]},"then":{"kind":"atom","property":"P000103","value":true},"uid":"T000607","description":"Suppose $X$ satisfies the hypotheses, and let $p$ be the non-isolated point of $X$.\n\nLet $A$ be a {P19} subset of $X$.\nIf $p\\in A$, the set $A$ is closed.\nIf $p\\notin A$, the set $A$ is {P52}, hence {P78} \n[(Explore)](https://topology.pi-base.org/spaces?q=Discrete%2BCountably+compact%2B%7EFinite).\nSince $X$ is {P2}, the set $A$ is also closed in this case.","refs":[]},{"when":{"kind":"atom","property":"P000047","value":true},"then":{"kind":"atom","property":"P000073","value":true},"uid":"T000608","description":"Suppose that $C$ is a {P39} non-empty closed subset of $X$. As $X$ is {P47}, $C$ is a singleton. Therefore, $X$ is {P73}.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000201","value":true},{"kind":"atom","property":"P000001","value":true},{"kind":"atom","property":"P000090","value":true}]},"then":{"kind":"atom","property":"P000139","value":true},"uid":"T000609","description":"See {{mathse:4994238}}.","refs":[{"name":"Is the generic point of an irreducible sober topological space an isolated point?","mathse":4994238}]},{"when":{"kind":"atom","property":"P000063","value":true},"then":{"kind":"atom","property":"P000006","value":true},"uid":"T000610","description":"By the definition of {P63}.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000097","value":true},{"kind":"atom","property":"P000036","value":true},{"kind":"atom","property":"P000137","value":false}]},"then":{"kind":"atom","property":"P000199","value":true},"uid":"T000611","description":"If $X$ is a connected subset of $\\mathbb R$, it is necessarily order-convex (that is, $y\\le x\\le z$ with $y,z\\in X$ implies $x\\in X$). Given a point $a\\in X$, define the map $F:X\\times[0,1]\\to X$ by $F(x,t)=(1-t)x+ta$. This is well-defined and is a homotopy between the identity map on $X$ and a constant map.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000090","value":true},{"kind":"atom","property":"P000018","value":true}]},"then":{"kind":"atom","property":"P000152","value":true},"uid":"T000612","description":"For an {P90} space, the set of smallest neighborhoods of points is an open cover.\nBy {P18}, there is a countable subcover of these.\nThis is in turn a cover by the smallest neighborhoods of a countable set of points, so {P152} holds.","refs":[]},{"when":{"kind":"atom","property":"P000202","value":true},"then":{"kind":"atom","property":"P000152","value":true},"uid":"T000613","description":"A singleton whose only neighborhood is the entire space is a countable set as needed for {P152}.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000172","value":true},{"kind":"atom","property":"P000191","value":true}]},"then":{"kind":"atom","property":"P000080","value":true},"uid":"T000614","description":"See {{mathse:4993377}}.","refs":[{"name":"Radial space with countable pseudocharacter that isn't Fréchet Urysohn","mathse":4993377}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000110","value":true},{"kind":"atom","property":"P000001","value":true}]},"then":{"kind":"atom","property":"P000102","value":true},"uid":"T000615","description":"See {{mathse:5041858}}.","refs":[{"name":"Is every developable space semimetrizable?","mathse":5041858}]},{"when":{"kind":"atom","property":"P000102","value":true},"then":{"kind":"atom","property":"P000028","value":true},"uid":"T000616","description":"The balls $B_d(x,1/n)$ for $n=1,2,\\dots$ form a neighborhood base at the point $x$.","refs":[]},{"when":{"kind":"atom","property":"P000201","value":true},"then":{"kind":"atom","property":"P000206","value":true},"uid":"T000617","description":"Regardless of how Player 2 plays, the intersection of $U_n$ will always contain any generic point of the space.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000040","value":true},{"kind":"atom","property":"P000204","value":true}]},"then":{"kind":"atom","property":"P000107","value":true},"uid":"T000618","description":"If $p \\in X$ is a cut point, then $X \\setminus \\{p\\} = U \\cup V$, with $U, V$ disjoint, nonempty, and closed in $X \\setminus \\{p\\}$. For $X$ to be ultraconnected, $U \\cup \\{p\\}$ and $V \\cup \\{p\\}$ must be closed in $X$, so $\\{p\\}$ must also be closed in $X$.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000039","value":true},{"kind":"atom","property":"P000204","value":true}]},"then":{"kind":"atom","property":"P000139","value":true},"uid":"T000619","description":"If $p \\in X$ is a cut point, then $X \\setminus \\{p\\} = U \\cup V$, with $U, V$ disjoint, nonempty, and open in $X \\setminus \\{p\\}$. For $X$ to be hyperconnected, $U \\cup \\{p\\}$ and $V \\cup \\{p\\}$ must be open in $X$, so $\\{p\\}$ must also be open in $X$.","refs":[]},{"when":{"kind":"atom","property":"P000196","value":true},"then":{"kind":"atom","property":"P000204","value":false},"uid":"T000620","description":"Clear from the definitions, since removing a point does not disconnect the space.","refs":[]},{"when":{"kind":"atom","property":"P000107","value":true},"then":{"kind":"atom","property":"P000137","value":false},"uid":"T000621","description":"By definition.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000040","value":true},{"kind":"atom","property":"P000193","value":true},{"kind":"atom","property":"P000137","value":false}]},"then":{"kind":"atom","property":"P000202","value":true},"uid":"T000622","description":"Assume to the contrary that $X$ is not {P202}. The intersection of $\\overline{\\{x\\}}$\nfor all $x \\in X$ coincides with the intersection of all nonempty closed sets, so it is the set of all focal points of $X$,\nwhence empty by assumption. Thus, $\\{\\overline{\\{x\\}}^c: x \\in X\\}$ is an open cover of $X$. As $X$ is {P193}, there\nis an open cover $\\{V_x\\}_{x \\in X}$ of $X$ s.t. $\\overline{V_x} \\subseteq \\overline{\\{x\\}}^c$ for all $x$. Now, fix $x_0 \\in X$.\nThere exists $x \\in X$ s.t. $x_0 \\in V_x \\subseteq \\overline{V_x} \\subseteq \\overline{\\{x\\}}^c$.\nThen $\\overline{V_x}$ and $\\overline{\\{x\\}}$ are nonempty disjoint closed sets,\nwhich contradicts $X$ being {P40}.","refs":[]},{"when":{"kind":"atom","property":"P000063","value":true},"then":{"kind":"atom","property":"P000140","value":true},"uid":"T000623","description":"See Theorem 3.9.5 of {{zb:0684.54001}}.","refs":[{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"when":{"kind":"atom","property":"P000102","value":true},"then":{"kind":"atom","property":"P000104","value":true},"uid":"T000624","description":"Immediate from definitions.","refs":[]},{"when":{"kind":"atom","property":"P000104","value":true},"then":{"kind":"atom","property":"P000002","value":true},"uid":"T000625","description":"If $A = \\{a\\}$ is a singleton, then $d(x,A)=d(x,a)>0$ for any $x \\in X \\setminus A$, so $A$ is closed.","refs":[]},{"when":{"kind":"atom","property":"P000104","value":true},"then":{"kind":"atom","property":"P000228","value":true},"uid":"T000626","description":"For every $x \\in X, n \\in \\mathbb{N}$ set $V_n(x) := B_d(x, \\frac{1}{n}) = \\{y\\in X:d(x,y)<\\frac{1}{n}\\}$\nwith $d$ being the symmetric in the definition of {P104}.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000104","value":true},{"kind":"atom","property":"P000198","value":true}]},"then":{"kind":"atom","property":"P000131","value":true},"uid":"T000627","description":"See Theorem 1.2.5 in {{doi:10.2991/978-94-6239-216-8}}. Note that while the book assumes {P3}, the proof did not use that\nassumption.","refs":[{"name":"Generalized Metric Spaces and Mappings (Lin , Yun)","doi":"10.2991/978-94-6239-216-8"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000104","value":true},{"kind":"atom","property":"P000080","value":true},{"kind":"atom","property":"P000099","value":true}]},"then":{"kind":"atom","property":"P000102","value":true},"uid":"T000628","description":"See the third part of {{mathse:5016336}}.","refs":[{"name":"Answer to \"Symmetrizability and Semimetrizability of one-point compactifications\"","mathse":5016336}]},{"when":{"kind":"atom","property":"P000154","value":true},"then":{"kind":"atom","property":"P000109","value":true},"uid":"T000629","description":"See Corollary 5.6 in {{zb:0269.54009}}.\nAlso [here](http://at.yorku.ca/b/ask-a-topologist/2003/0383.htm) for the case of a {P133}.","refs":[{"name":"Monotonically normal spaces (Heath, Lutzer, Zenor)","zb":"0269.54009"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000002","value":true},{"kind":"atom","property":"P000137","value":false}]},"then":{"kind":"atom","property":"P000107","value":true},"uid":"T000630","description":"Every singleton subset of a $T_1$ space is closed.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000204","value":true},{"kind":"atom","property":"P000107","value":false}]},"then":{"kind":"atom","property":"P000139","value":true},"uid":"T000631","description":"See Theorem 3.2 of {{doi:10.1090/S0002-9939-99-04839-X}}.","refs":[{"name":"Cut-point spaces (Honari & Bahrampour)","doi":"10.1090/S0002-9939-99-04839-X"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000040","value":true},{"kind":"atom","property":"P000107","value":true}]},"then":{"kind":"atom","property":"P000202","value":true},"uid":"T000632","description":"Since $X$ is ultraconnected, a closed singleton must be a subset of every nonempty closed set. So a closed point $p \\in X$ is a focal point.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000086","value":true},{"kind":"atom","property":"P000107","value":true}]},"then":{"kind":"atom","property":"P000002","value":true},"uid":"T000633","description":"Homeomorphisms preserve closed sets.","refs":[]},{"when":{"kind":"atom","property":"P000203","value":true},"then":{"kind":"atom","property":"P000107","value":true},"uid":"T000634","description":"Every singleton subset except for one is open. The union of the open singletons is therefore the complement of a closed singleton.","refs":[]},{"when":{"kind":"atom","property":"P000205","value":true},"then":{"kind":"atom","property":"P000204","value":true},"uid":"T000635","description":"By definition.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000204","value":true},{"kind":"atom","property":"P000086","value":true}]},"then":{"kind":"atom","property":"P000205","value":true},"uid":"T000636","description":"Cut points are preserved under homeomorphism, so every point is a cut point.","refs":[]},{"when":{"kind":"atom","property":"P000205","value":true},"then":{"kind":"atom","property":"P000001","value":true},"uid":"T000637","description":"Removing one of a pair of topologically indistinguishable points can't change connectivity, so any cut point must be topologically distinguishable from all other points.","refs":[]},{"when":{"kind":"atom","property":"P000205","value":true},"then":{"kind":"atom","property":"P000016","value":false},"uid":"T000638","description":"Corollary 3.10 in {{doi:10.1090/s0002-9939-99-04839-x}}.","refs":[{"name":"Cut-Point Spaces (Honari & Bahrampour)","doi":"10.1090/s0002-9939-99-04839-x"}]},{"when":{"kind":"atom","property":"P000205","value":true},"then":{"kind":"atom","property":"P000107","value":true},"uid":"T000639","description":"By Theorem 3.7 in {{doi:10.1090/s0002-9939-99-04839-x}}, a cut point space has infinitely many closed points.","refs":[{"name":"Cut-Point Spaces (Honari & Bahrampour)","doi":"10.1090/s0002-9939-99-04839-x"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000105","value":true},{"kind":"atom","property":"P000062","value":true}]},"then":{"kind":"atom","property":"P000018","value":true},"uid":"T000640","description":"See {{mathse:5005198}}.","refs":[{"name":"Answer to \"Is every paracompact CCC space Lindelöf?\"","mathse":5005198}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000024","value":true},{"kind":"atom","property":"P000001","value":true},{"kind":"atom","property":"P000137","value":false}]},"then":{"kind":"atom","property":"P000107","value":true},"uid":"T000641","description":"Since the space $X$ is nonempty and {P24}, it has a nonempty closed {P16} subset $Y$.\nNow, as $Y$ is compact, any chain of nonempty closed subsets of $Y$ has nonempty intersection. Thus, by Zorn's lemma, there is a minimal nonempty closed subset $A$ of $Y$. Since $Y$ is closed in $X$, so is $A$, and thus $A$ is a minimal nonempty closed subset of $X$. As $X$ is {P1}, it is easy to see that $A$ must be a singleton.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000137","value":false},{"kind":"atom","property":"P000063","value":true},{"kind":"atom","property":"P000139","value":false}]},"then":{"kind":"atom","property":"P000068","value":false},"uid":"T000642","description":"See {{mathse:5004551}}.","refs":[{"name":"Answer to \"Are Čech complete spaces with no isolated points non-Rothberger?\"","mathse":5004551}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000002","value":true},{"kind":"atom","property":"P000028","value":true},{"kind":"atom","property":"P000203","value":true}]},"then":{"kind":"atom","property":"P000055","value":true},"uid":"T000643","description":"Shown in {{mathse:4987443}}.","refs":[{"name":"Answer to \"Are almost discrete metrizable spaces always completely metrizable?\"","mathse":4987443}]},{"when":{"kind":"atom","property":"P000203","value":true},"then":{"kind":"atom","property":"P000206","value":true},"uid":"T000644","description":"Shown in {{mathse:4987327}}.","refs":[{"name":"Answer to \"Are almost discrete metrizable spaces always completely metrizable?\"","mathse":4987327}]},{"when":{"kind":"atom","property":"P000063","value":true},"then":{"kind":"atom","property":"P000206","value":true},"uid":"T000645","description":"For a {P3} {P16} space, Player 2 may win by choosing $V_n$\nwith $\\overline{V_n}\\subseteq U_n$.\n\nThe result follows by Exercise 8.16 (iii) of {{zb:0819.04002}}, which says nonempty \n$G_\\delta$ subsets of {P206} spaces are {P206}.\nSee also {{wikipedia:Choquet_game}}.","refs":[{"name":"Choquet game","wikipedia":"Choquet_game"},{"name":"Classical Descriptive Set Theory (Kechris)","zb":"0819.04002"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000196","value":true},{"kind":"atom","property":"P000026","value":false}]},"then":{"kind":"atom","property":"P000206","value":true},"uid":"T000646","description":"Player 2 can win by choosing $V_n=U_n$ in each round.\nIndeed, since the countable set $\\{x_n:x<\\omega\\}$ is not dense in $X$,\nthere is a nonempty open set $W$ not containing any of the $x_n$,\nand thus not containing any of the $U_n$.\nSince $X$ is {P196}, $W$ is contained in all the $U_n$.\nIt follows that $\\bigcap_n U_n\\ne\\emptyset$.","refs":[]},{"when":{"kind":"atom","property":"P000094","value":true},"then":{"kind":"atom","property":"P000206","value":true},"uid":"T000647","description":"During round 0, Player 2 chooses any finite neighborhood $V_0$ of $x_0$ that is contained in $U_0$. Then $\\{U_n\\}_{n \\geq 1}$ is a decreasing sequence of nonempty finite sets, so it must have nonempty intersection.","refs":[]},{"when":{"kind":"atom","property":"P000207","value":true},"then":{"kind":"atom","property":"P000088","value":true},"uid":"T000648","description":"Shown in {{zb:0046.16403}}; see also {{mathse:345476}}.","refs":[{"name":"Sur une problème de M. Dieudonné (Cohen)","zb":"0046.16403"},{"name":"Answer to \"Is every $T_4$ topological space divisible?\"","mathse":345476}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000126","value":true},{"kind":"atom","property":"P000137","value":false}]},"then":{"kind":"atom","property":"P000107","value":true},"uid":"T000649","description":"If a {P126} space has no closed points, then every singleton is open, which implies that the space is {P52}.\nThis is a contradiction if the space is non-empty.","refs":[]},{"when":{"kind":"atom","property":"P000208","value":true},"then":{"kind":"atom","property":"P000016","value":true},"uid":"T000650","description":"By definition.","refs":[]},{"when":{"kind":"atom","property":"P000208","value":true},"then":{"kind":"atom","property":"P000041","value":true},"uid":"T000651","description":"See [Lemma 5.9.6](https://stacks.math.columbia.edu/tag/04MF) from the Stacks project, which makes a stronger claim.","refs":[]},{"when":{"kind":"atom","property":"P000208","value":true},"then":{"kind":"atom","property":"P000130","value":true},"uid":"T000652","description":"For each $x \\in X$, the set of open neighborhoods of $x$ is a local basis of compact neighborhoods, since every open set is compact.","refs":[]},{"when":{"kind":"atom","property":"P000030","value":true},"then":{"kind":"atom","property":"P000105","value":true},"uid":"T000653","description":"Immediate from definitions.","refs":[]},{"when":{"kind":"atom","property":"P000018","value":true},"then":{"kind":"atom","property":"P000105","value":true},"uid":"T000654","description":"Immediate from definitions.","refs":[]},{"when":{"kind":"atom","property":"P000105","value":true},"then":{"kind":"atom","property":"P000083","value":true},"uid":"T000655","description":"Immediate from definitions.","refs":[]},{"when":{"kind":"atom","property":"P000040","value":true},"then":{"kind":"atom","property":"P000207","value":true},"uid":"T000656","description":"$X^2$ is the only neighborhood of the diagonal, and $X^2\\circ X^2=X^2$.","refs":[]},{"when":{"kind":"atom","property":"P000208","value":true},"then":{"kind":"atom","property":"P000131","value":true},"uid":"T000657","description":"By definition, because a compact space is Lindelöf.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000016","value":true},{"kind":"atom","property":"P000185","value":true}]},"then":{"kind":"atom","property":"P000245","value":true},"uid":"T000658","description":"The partition generating the topology must be finite due to $X$ being {P16}, so there are only finitely many open sets.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000208","value":true},{"kind":"atom","property":"P000134","value":true}]},"then":{"kind":"atom","property":"P000185","value":true},"uid":"T000659","description":"First observe that a {P208} {P3} space is {P52}, since every subset is compact, hence closed.\n\nNow, if $X$ is {P208} and {P134}, its Kolmogorov quotient is {P208} and {P3}, hence {P52}. This is equivalent to $X$ being {P185}.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000203","value":true},{"kind":"atom","property":"P000208","value":true}]},"then":{"kind":"atom","property":"P000078","value":true},"uid":"T000660","description":"Let $p \\in X$ be the only non-isolated point. The subspace $X \\setminus \\{p\\}$ is {P52} and {P208},\nhence {P78} [(Explore)](https://topology.pi-base.org/spaces?q=Discrete+%2B+Noetherian+%2B+%7EFinite).\nTherefore $X$ is also {P78}.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000197","value":true},{"kind":"atom","property":"P000013","value":true}]},"then":{"kind":"atom","property":"P000088","value":true},"uid":"T000661","description":"Given a discrete family $(F_i)_{i \\in I}$ of nonempty closed sets in $X$,\ntake a point $x_i\\in F_i$ for each $i$.\nThe subspace $\\{x_i:i\\in I\\}$ is discrete,\nhence countable because $X$ is {P197}.\n\nSo there are only countably many $F_i$,\nand one can find a disjoint open expansion of $(F_i)_i$ by Theorem 2.1.14 of {{zb:0684.54001}}.","refs":[{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000198","value":true},{"kind":"atom","property":"P000007","value":true}]},"then":{"kind":"atom","property":"P000088","value":true},"uid":"T000662","description":"Given a discrete family $(F_i)_{i \\in I}$ of nonempty closed sets in $X$,\ntake a point $x_i\\in F_i$ for each $i$.\nEvery singleton $\\{x_i\\}$ is closed and the subspace $Y=\\{x_i:i\\in I\\}$ is discrete.\nTherefore $Y$ is closed and discrete,\nand hence countable because $X$ is {P198}.\n\nSo there are only countably many $F_i$,\nand one can find a disjoint open expansion of $(F_i)_i$ by Theorem 2.1.14 of {{zb:0684.54001}}.","refs":[{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000197","value":true},{"kind":"atom","property":"P000014","value":true}]},"then":{"kind":"atom","property":"P000108","value":true},"uid":"T000663","description":"{P197} is a hereditary property and so is {P14}. Thus, the result follows from\n{T661}.","refs":[]},{"when":{"kind":"atom","property":"P000109","value":true},"then":{"kind":"atom","property":"P000032","value":true},"uid":"T000664","description":"See Theorems 1.2 and 2.3 of Chapter 17 of {{doi:10.1016/C2009-0-12309-7}}.\n\nSee also {{mathse:5012078}} for a more detailed explanation.\n\nCompare with {T805} and {T543}, which constitute a refinement of this result.\n\n","refs":[{"name":"Handbook of Set-Theoretic Topology (Kunen & Vaughan)","doi":"10.1016/C2009-0-12309-7"},{"name":"High-level overview and detailed proof that monotonically normal spaces are countably paracompact","mathse":5012078}]},{"when":{"kind":"atom","property":"P000108","value":true},"then":{"kind":"atom","property":"P000088","value":true},"uid":"T000665","description":"By definition.","refs":[]},{"when":{"kind":"atom","property":"P000108","value":true},"then":{"kind":"atom","property":"P000014","value":true},"uid":"T000666","description":"By {T213}, every subspace of $X$ is {P13}.","refs":[]},{"when":{"kind":"atom","property":"P000109","value":true},"then":{"kind":"atom","property":"P000008","value":true},"uid":"T000667","description":"If $X$ is {P109}, it is {P13} from part (i) of Definition (1),\nand is also {P2}, hence {P7}.\nBut monotone normality is hereditary. Therefore $X$ is {P8}.","refs":[]},{"when":{"kind":"atom","property":"P000109","value":true},"then":{"kind":"atom","property":"P000108","value":true},"uid":"T000668","description":"See Theorem 3.1 in {{zb:0269.54009}}.","refs":[{"name":"Monotonically normal spaces (Heath, Lutzer, Zenor)","zb":"0269.54009"}]},{"when":{"kind":"atom","property":"P000053","value":true},"then":{"kind":"atom","property":"P000109","value":true},"uid":"T000669","description":"Let $(X,d)$ be a metric space.\nWrite $B(x,\\epsilon)$ for the open ball centered at $x$ with radius $\\epsilon$.\n\nThe space is {P2}.\n\nFor $x\\in U$ with $U$ open in $X$,\ndefine $\\mu(x,U)$ to be some open ball $B(x,\\epsilon)$, $\\epsilon>0$,\nsuch that $B(x,2\\epsilon)\\subseteq U$.\nIf $B(x,\\epsilon_1)\\cap B(y,\\epsilon_2)\\ne\\emptyset$ with $\\epsilon_2\\le\\epsilon_1$,\nnecessarily $y\\in B(x,2\\epsilon_1)$ by the triangle inequality.\nSo $\\mu(x,U)$ satisfies Definition (2) of {P109}.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000032","value":true},{"kind":"atom","property":"P000039","value":true}]},"then":{"kind":"atom","property":"P000019","value":true},"uid":"T000670","description":"Let $\\mathscr U$ be a countable open cover of $X$.\nLet $\\mathscr V$ be a locally finite open refinement of $\\mathscr U$ covering $X$.\nGiven a point $x\\in X$, there is an open neighborhood $W$ of $x$ that intersects only finitely many members of $\\mathscr V$.\nBut since $X$ is {P39}, $W$ intersects all (nonempty) members of $\\mathscr V$;\nhence $\\mathscr V$ is finite.\nThat is sufficient to conclude that $\\mathscr U$ has a finite subcover.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000203","value":true},{"kind":"atom","property":"P000010","value":false}]},"then":{"kind":"atom","property":"P000049","value":true},"uid":"T000671","description":"By {{mathse:5016854}}, an almost discrete space that is not semiregular must be\nthe disjoint union of the {S10} and a {P52} space.\n\nThe {S10|P49}, {P52} implies extremally disconnected [(Explore)](https://topology.pi-base.org/spaces?q=Discrete+%2B+%7EExtremally+disconnected),\nand the disjoint union of two extremally disconnected spaces is itself extremally disconnected.","refs":[{"name":"What is an almost discrete space that isn't semiregular like?","mathse":5016854}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000203","value":true},{"kind":"atom","property":"P000010","value":false}]},"then":{"kind":"atom","property":"P000094","value":true},"uid":"T000672","description":"By {{mathse:5016854}}, an almost discrete space that is not semiregular must be\nthe disjoint union of the {S10} and a {P52} space.\n\nThe {S10|P94}, {P52} implies locally finite [(Explore)](https://topology.pi-base.org/spaces?q=Discrete+%2B+%7ELocally+finite),\nand the disjoint union of two locally finite spaces is itself locally finite.","refs":[{"name":"What is an almost discrete space that isn't semiregular like?","mathse":5016854}]},{"when":{"kind":"atom","property":"P000085","value":true},"then":{"kind":"atom","property":"P000061","value":true},"uid":"T000673","description":"Let $U$ be a cozero set in $X$.\nIts closure $\\overline U$ is clopen, as is $X\\setminus\\overline U$.\nTherefore, the set $X\\setminus\\overline U$ is a cozero complement of $U$.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000203","value":true},{"kind":"atom","property":"P000002","value":true}]},"then":{"kind":"atom","property":"P000109","value":true},"uid":"T000674","description":"$X$ is $T_1$.\n\nFor the monotone normality operator $\\mu(x,U)$ for $x\\in U$ with $U$ open,\none can choose $\\mu(x,U)=\\{x\\}$ if $x$ is an isolated point of $X$\nand $\\mu(x,U)=U$ if $x$ is the non-isolated point,\nas suggested in the comment to {{mo:313089}}.\nIt is easy to check that this satisfies Definition 2 of {P109}.","refs":[{"name":"Answer to \"A question on monotonically normal spaces\"","mo":313089}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000196","value":true},{"kind":"atom","property":"P000016","value":true},{"kind":"atom","property":"P000073","value":true},{"kind":"atom","property":"P000058","value":true}]},"then":{"kind":"atom","property":"P000075","value":true},"uid":"T000675","description":"See part 4 of {{mathse:5007860}}.","refs":[{"name":"Answer to \"For a compact sober \"highly non-$T_1$\" space, how much \"highly connectedness\" is needed to imply it's a spectral space?\"","mathse":5007860}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000202","value":true},{"kind":"atom","property":"P000043","value":true}]},"then":{"kind":"atom","property":"P000038","value":true},"uid":"T000676","description":"$X$ has a point $p$ whose only neighborhood is $X$. As $X$ is {P43}, $p$ has a {P38}\nneighborhood. Hence, $X$ is {P38}.","refs":[]},{"when":{"kind":"atom","property":"P000094","value":true},"then":{"kind":"atom","property":"P000117","value":true},"uid":"T000677","description":"Let $\\mathcal{N} = \\{\\{x\\}:x\\in X\\}$. Then $\\mathcal{N}$ is a locally finite network for $X$.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000136","value":true},{"kind":"atom","property":"P000117","value":true}]},"then":{"kind":"atom","property":"P000118","value":true},"uid":"T000678","description":"If $\\mathcal{N}$ is a $\\sigma$-locally finite network, and if $K\\subseteq U$ where $K$ is compact and $U$ is open, then because $K$ is finite, for each $x\\in K$ take $N_x\\in\\mathcal{N}$ with $x\\in N_x\\subseteq U$, then $\\mathcal{N}^\\ast = \\{N_x : x\\in K\\}$ is a finite subfamily of $\\mathcal{N}$ such that $K\\subseteq \\bigcup \\mathcal{N}^\\ast\\subseteq U$. So $\\mathcal{N}$ is a $\\sigma$-locally finite $k$-network.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000039","value":true},{"kind":"atom","property":"P000202","value":false},{"kind":"atom","property":"P000107","value":true}]},"then":{"kind":"atom","property":"P000165","value":false},"uid":"T000679","description":"Let $p\\in X$ be a closed point and find open $U$ with $p\\in U$ and $U\\neq X$. If $X$ were {P165} then there would be open $V$ with $p\\in V\\subseteq \\overline{V}\\subseteq U$, but $\\overline{V} = X$ since $X$ is {P39}. But then $U = X$, a contradiction.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000023","value":true},{"kind":"atom","property":"P000134","value":true}]},"then":{"kind":"atom","property":"P000206","value":true},"uid":"T000680","description":"A space $X$ is {P206} if and only if the Kolmogorov quotient of $X$ is {P206}.\n\nThe Kolmogorov quotient of $X$ is {P23}, {P3} and non-empty, so strongly Choquet [(Explore)](https://topology.pi-base.org/spaces?q=weakly+locally+compact+%2B+T_2+%2B+not+empty+%2B+not+strongly+choquet).","refs":[]},{"when":{"kind":"atom","property":"P000146","value":true},"then":{"kind":"atom","property":"P000034","value":true},"uid":"T000681","description":"Let $\\mathcal V$ be an open cover of $X$. Since $X$ is {P146},\n$\\mathcal V$ has an open refinement $\\mathcal U$ that is a partition of $X$.\nAnd since the members of $\\mathcal U$ are pairwise disjoint,\n$\\mathcal U$ is a star refinement of $\\mathcal V$.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000208","value":true},{"kind":"atom","property":"P000201","value":true},{"kind":"atom","property":"P000001","value":true}]},"then":{"kind":"atom","property":"P000089","value":true},"uid":"T000682","description":"See {{mathse:5014330}}.","refs":[{"name":"Noetherian spaces with a generic point have the fixed point property","mathse":5014330}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000203","value":true},{"kind":"atom","property":"P000079","value":true}]},"then":{"kind":"atom","property":"P000080","value":true},"uid":"T000683","description":"See {{mathse:5016263}}.","refs":[{"name":"Answer to \"Symmetrizability and Semimetrizability of one-point compactifications\"","mathse":5016263}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000123","value":true},{"kind":"atom","property":"P000036","value":true},{"kind":"atom","property":"P000003","value":true}]},"then":{"kind":"atom","property":"P000086","value":true},"uid":"T000684","description":"See {{mathse:5015469}}. Note that the proof does not rely on {P27}.","refs":[{"name":"Answer to \"Are all connected manifolds homogeneous?\"","mathse":5015469}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000124","value":true},{"kind":"atom","property":"P000016","value":true},{"kind":"atom","property":"P000125","value":true}]},"then":{"kind":"atom","property":"P000199","value":false},"uid":"T000685","description":"Let $M$ be a {P16} {P199} $n$-manifold.\nSince {P199} spaces are {P36}\n[(Explore)](https://topology.pi-base.org/spaces?q=Contractible%2B%7EConnected),\n$M$ is a connected closed $n$-manifold\n(where \"closed manifold\" here means a \"compact manifold without boundary\";\nsee {{wikipedia:Closed_manifold}}).\nIt follows from the remark after Theorem 3.26 in the Poincaré duality section of {{zb:1044.55001}} that $H_n(M; \\mathbb Z_2) = \\mathbb Z_2$.\n\nBut since $M$ is {P199}, we have $H_n(M; \\mathbb Z_2) = 0$ for all $n \\geq 1$.\nThis forces $n = 0$ and therefore $M$ is {S162}.\n\nSee also {{mathse:418509}}.","refs":[{"name":"Is there a compact contractible manifold?","mathse":418509},{"name":"Algebraic Topology (Hatcher)","zb":"1044.55001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000031","value":true},{"kind":"atom","property":"P000088","value":true},{"kind":"atom","property":"P000011","value":true}]},"then":{"kind":"atom","property":"P000030","value":true},"uid":"T000686","description":"This is the Michael-Nagami theorem.\n\nAccording to Theorem 5.3.3 of {{zb:0684.54001}} (which assumes {P3} for all notions in that chapter),\nspaces that are {P31}, {P88} and {P3} are {P30}.\nBy passing to the Kolmogorov quotient, one obtains that \n{P31} {P88} {P134} spaces are {P30}.\n\nNow {P88} implies {P13}.\nAnd for {P13} spaces, the properties {P135}, {P134} and {P11} are equivalent\n[(Explore)](https://topology.pi-base.org/spaces?q=Normal%2B%24R_1%24%2B%7ERegular).\nSo one can equivalently formulate the theorem with the {P11} property.\n\n----\nOPEN QUESTION: Is the {P11} hypothesis needed?","refs":[{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000208","value":true},{"kind":"atom","property":"P000183","value":true}]},"then":{"kind":"atom","property":"P000027","value":true},"uid":"T000687","description":"If $X$ is {P208}, then each pseudobase necessarily contains every open set.\nIf, in addition, $X$ is {P183}, then there are only countably many open sets and $X$ is {P27}.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000017","value":true},{"kind":"atom","property":"P000086","value":true},{"kind":"atom","property":"P000100","value":true},{"kind":"atom","property":"P000056","value":false}]},"then":{"kind":"atom","property":"P000023","value":true},"uid":"T000688","description":"Suppose $X$ satisfies the hypotheses.\nBy {P17} and {P100}, $X$ is a countable union of compact closed sets.\nSince $X$ is not {P56}, one of the closed sets has nonempty interior.\nA point in that interior has a compact neighborhood.\nSince $X$ is {P86}, every point has a compact neighborhood.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000093","value":true},{"kind":"atom","property":"P000144","value":true},{"kind":"atom","property":"P000041","value":true}]},"then":{"kind":"atom","property":"P000185","value":true},"uid":"T000689","description":"By the known result {P57} ∧ {P121} ∧ {P36} ⇒ {P129}\n[(Explore)](https://topology.pi-base.org/spaces?q=Countable+%2B+Pseudometrizable+%2B+Connected+%2B+%7EIndiscrete),\neach subset of a space that is countable, pseudometrizable, and connected is indiscrete.\nThus, a space with such an open neighborhood of every point has a cover by\nindiscrete open sets. Distinct open and indiscrete sets are necessarily disjoint,\nhence the space has a {P185}.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000100","value":true},{"kind":"atom","property":"P000131","value":true}]},"then":{"kind":"atom","property":"P000103","value":true},"uid":"T000690","description":"Let $A \\subseteq X$ be a {P19} subset. It must be {P18} because $X$ is {P131}.\n\nSince {P18} {P19} spaces are {P16}, we know $A$ is {P16}. Thus $A$ is closed since $X$ is {P100}.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000003","value":true},{"kind":"atom","property":"P000049","value":true},{"kind":"atom","property":"P000147","value":true},{"kind":"atom","property":"P000164","value":true}]},"then":{"kind":"atom","property":"P000052","value":true},"uid":"T000691","description":"See {{mathse:5022632}}.","refs":[{"name":"Answer to \"Existence of non-discrete Tychonoff extremally disconnected P-space and measurable cardinals\"","mathse":5022632}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000036","value":true},{"kind":"atom","property":"P000145","value":true}]},"then":{"kind":"atom","property":"P000018","value":true},"uid":"T000692","description":"Follows from the fact that every star-finite open cover of a connected space is countable. See {{mo:433812}}.","refs":[{"name":"A stronger version of paracompactness","mo":433812}]},{"when":{"kind":"atom","property":"P000049","value":true},"then":{"kind":"atom","property":"P000085","value":true},"uid":"T000693","description":"Evident, as every cozero set is an open set.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000085","value":true},{"kind":"atom","property":"P000015","value":true}]},"then":{"kind":"atom","property":"P000049","value":true},"uid":"T000694","description":"Follows from the definitions since in a {P15} space open sets are cozero sets.","refs":[]},{"when":{"kind":"atom","property":"P000147","value":true},"then":{"kind":"atom","property":"P000085","value":true},"uid":"T000695","description":"Every cozero set is open and an $F_\\sigma$ set, which is closed in a {P147}.\nHence every cozero set is clopen and so is its closure.\n\nSee Figure 1 on page 346 of {{zb:1059.54001}}.","refs":[{"name":"Encyclopedia of general topology (Hart et al)","zb":"1059.54001"}]},{"when":{"kind":"atom","property":"P000060","value":true},"then":{"kind":"atom","property":"P000085","value":true},"uid":"T000696","description":"If $X$ is {P60}, the only cozero sets are $\\emptyset$ and $X$, which are clopen.","refs":[]},{"when":{"kind":"atom","property":"P000085","value":true},"then":{"kind":"atom","property":"P000217","value":true},"uid":"T000697","description":"Suppose $Z_1, Z_2$ are disjoint zero-sets. Then there exist disjoint cozero sets $U_1, U_2$ such that $Z_k\\subseteq U_k$. Then $V = \\overline{U_1}$ is a clopen set such that $Z_1\\subseteq V$ and $Z_2\\cap V = \\emptyset$. So $X$ is {P217}.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000023","value":true},{"kind":"atom","property":"P000105","value":true},{"kind":"atom","property":"P000036","value":true}]},"then":{"kind":"atom","property":"P000018","value":true},"uid":"T000698","description":"Call a space *weakly locally Lindelöf* if every point has a neighborhood that is Lindelöf.\n\nIn {{mathse:5028016}} it is shown that a {P36} weakly locally Lindelöf {P105} space is {P18}, and {P23} implies weakly locally Lindelöf.","refs":[{"name":"Answer to \"Connected, locally compact, paracompact Hausdorff space is exhaustible by compacts\"","mathse":5028016}]},{"when":{"kind":"atom","property":"P000102","value":true},"then":{"kind":"atom","property":"P000132","value":true},"uid":"T000699","description":"Let $A \\subseteq X$ be a closed subset.\n\nFor each $x \\in X$ and $\\varepsilon > 0$, the set $U(x, \\varepsilon) := \\operatorname{int} B_d(x, \\varepsilon)$ is an open neighborhood of $x$.\nDefine $G_n := \\bigcup_{x \\in A} U\\!\\left( x, \\frac 1n \\right)$. This is an open set containing $A$.\n\nIf $y \\notin A$, then there exists some integer $n\\ge 1$ such that $d(y, A) > \\frac 1n$. Consequently, $y \\notin B_d(x, \\frac 1n)$ for any $x \\in A$, which implies $y \\notin G_n$.\n\nTherefore, $A = \\bigcap_{n\\ge 1} G_n$.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000126","value":true},{"kind":"atom","property":"P000136","value":false}]},"then":{"kind":"atom","property":"P000203","value":true},"uid":"T000700","description":"See {{mathse:4995169}}.","refs":[{"name":"Does every door space with an infinite compact set have a single non-isolated point?","mathse":4995169}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000126","value":true},{"kind":"atom","property":"P000036","value":true},{"kind":"atom","property":"P000203","value":false}]},"then":{"kind":"atom","property":"P000039","value":true},"uid":"T000701","description":"According to Theorem 1 in {{zb:1400.39025}}, there are three types of connected door spaces:\n- an *excluded point topology* (like {S12}),\nwhich is almost discrete;\n- a *particular point topology* (like {S8}),\nwhich is hyperconnected;\n- a *free ultrafilter topology* (like {S145}),\nwhich is hyperconnected.\n\nSo, if a connected door space is not almost discrete, it must be hyperconnected.","refs":[{"name":"Connected door spaces and topological solutions of equations (Wu, Wang, Zhang)","zb":"1400.39025"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000043","value":true},{"kind":"atom","property":"P000052","value":false}]},"then":{"kind":"atom","property":"P000044","value":false},"uid":"T000702","description":"Suppose $X$ is {P43} and not {P52}.\nThere is at least one point with an {P38} neighborhood $V$ that is not a singleton.\nTwo distinct points in $V$ are joined by an injective path $f:[0,1]\\to X$.\nThen, $f([0,1/3])$ and $f([2/3,1])$ are disjoint connected sets, each of size at least $2$.\nThus $X$ is not {P44}.","refs":[]},{"when":{"kind":"atom","property":"P000095","value":true},"then":{"kind":"atom","property":"P000038","value":true},"uid":"T000703","description":"Evident from the definitions.","refs":[]},{"when":{"kind":"atom","property":"P000096","value":true},"then":{"kind":"atom","property":"P000043","value":true},"uid":"T000704","description":"Evident from the definitions.","refs":[]},{"when":{"kind":"atom","property":"P000095","value":true},"then":{"kind":"atom","property":"P000002","value":true},"uid":"T000705","description":"Given distinct points $a,b\\in X$, there is an arc $f:[0,1]\\to X$ joining $a$ to $b$.\nThe image $A=f([0,1])$ is homeomorphic to $[0,1]$.\nSo $a$ has an open neighborhood $V$ in $A$ not containing $b$.\nSince $V=A\\cap U$ for some open set $U$ in $X$,\n$a$ has $U$ as neighborhood in $X$ not containing $b$.","refs":[]},{"when":{"kind":"atom","property":"P000096","value":true},"then":{"kind":"atom","property":"P000002","value":true},"uid":"T000706","description":"Each point $x\\in X$ has an open neighborhood $U$ that is {P95}.\nBy {T705}, $U$ is $T_1$.\nTherefore $X$ is \"locally {P2}\", and hence {P2}\n(see {{mathse:3142995}}).","refs":[{"name":"Answer to \"Locally Euclidean space implies T1 space\"","mathse":3142995}]},{"when":{"kind":"atom","property":"P000076","value":true},"then":{"kind":"atom","property":"P000088","value":true},"uid":"T000707","description":"See Theorem 10 of {{doi:10.1016/j.topol.2014.06.014}}.","refs":[{"name":"An infinite game with topological consequences (Bell)","doi":"10.1016/j.topol.2014.06.014"}]},{"when":{"kind":"atom","property":"P000076","value":true},"then":{"kind":"atom","property":"P000032","value":true},"uid":"T000708","description":"See Corollary 4 of {{doi:10.1016/j.topol.2014.06.014}}.","refs":[{"name":"An infinite game with topological consequences (Bell)","doi":"10.1016/j.topol.2014.06.014"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000037","value":true},{"kind":"atom","property":"P000099","value":true}]},"then":{"kind":"atom","property":"P000038","value":true},"uid":"T000709","description":"See {{mathse:4862260}}.","refs":[{"name":"Answer to \"Does path-connected imply simple path-connected?\"","mathse":4862260}]},{"when":{"kind":"atom","property":"P000110","value":true},"then":{"kind":"atom","property":"P000028","value":true},"uid":"T000710","description":"Evident from the definitions.","refs":[]},{"when":{"kind":"atom","property":"P000110","value":true},"then":{"kind":"atom","property":"P000135","value":true},"uid":"T000711","description":"See Lemma in {{mathse:5041859}}.","refs":[{"name":"Answer to \"Is every developable space semimetrizable?\"","mathse":5041859}]},{"when":{"kind":"atom","property":"P000121","value":true},"then":{"kind":"atom","property":"P000110","value":true},"uid":"T000712","description":"Letting $\\mathscr U_n$ be the collection of all open balls of radius $1/n$ gives a development for $X$.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000088","value":true},{"kind":"atom","property":"P000110","value":true}]},"then":{"kind":"atom","property":"P000121","value":true},"uid":"T000713","description":"This is essentially one of Bing's metrization theorems.\n\nSuppose $X$ satisfies the hypotheses.\nSince {T711}, we can add {P135} to the hypotheses.\n\nPassing to the Kolmogorov quotient, it is equivalent to show that\n{P88} {P110} {P2} spaces are {P53}.\nThis is exactly Theorem 5.4.1 in {{zb:0684.54001}},\nwhich assumes {P2} as part of the definition of {P88}.","refs":[{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000018","value":true},{"kind":"atom","property":"P000110","value":true}]},"then":{"kind":"atom","property":"P000027","value":true},"uid":"T000714","description":"See {{mathse:148565}}.","refs":[{"name":"Lindelof developable spaces are second countable","mathse":148565}]},{"when":{"kind":"atom","property":"P000113","value":true},"then":{"kind":"atom","property":"P000110","value":true},"uid":"T000715","description":"By definition.","refs":[]},{"when":{"kind":"atom","property":"P000113","value":true},"then":{"kind":"atom","property":"P000005","value":true},"uid":"T000716","description":"By definition.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000110","value":true},{"kind":"atom","property":"P000005","value":true}]},"then":{"kind":"atom","property":"P000113","value":true},"uid":"T000717","description":"By definition.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000057","value":true},{"kind":"atom","property":"P000002","value":true}]},"then":{"kind":"atom","property":"P000106","value":true},"uid":"T000718","description":"Any set in a {P57} {P2} space is a $G_\\delta$ set,\nsince its complement is an $F_\\sigma$ set as a countable union of singletons.\n\nIf $X$ satisfies the hypotheses, $X\\times X$ is also {P57} and {P2}.\nThus the diagonal in $X\\times X$ is a $G_\\delta$ set.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000110","value":true},{"kind":"atom","property":"P000001","value":true}]},"then":{"kind":"atom","property":"P000106","value":true},"uid":"T000719","description":"Suppose the sequence of open covers $\\mathscr U_1,\\mathscr U_2,\\dots$\nis a development for $X$. For each $x\\in X$, the collection\n$\\{\\operatorname{st}(x,\\mathscr U_n):n=1,2,\\dots\\}$ is a local base at $x$.\nSince $X$ is {P1}, it is also {P2}\n[(Explore)](https://topology.pi-base.org/spaces?q=Developable%2B%24T_0%24%2B%7E%24T_1%24).\nTherefore, $\\bigcap_n\\operatorname{st}(x,\\mathscr U_n)=\\{x\\}$.","refs":[]},{"when":{"kind":"atom","property":"P000106","value":true},"then":{"kind":"atom","property":"P000191","value":true},"uid":"T000720","description":"Suppose the diagonal $\\Delta$ is a $G_\\delta$ set in $X^2$.\nIntersecting with $\\{x\\}\\times X$ shows that $\\{x\\}$ is a $G_\\delta$ set in $X$.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000133","value":true},{"kind":"atom","property":"P000106","value":true}]},"then":{"kind":"atom","property":"P000053","value":true},"uid":"T000721","description":"See Theorem 2.3 in {{zb:0555.54015}}.\n\nThe proof shows that if $X$ satisfies the hypotheses, it is {P110}.\nThe rest follows by results known to pi-base:\n{P133} spaces are {P88}\n[(Explore)](https://topology.pi-base.org/spaces?q=LOTS%2B%7ECollectionwise+normal)\nand {P1};\nhence $X$ is {P53}\n[(Explore)](https://topology.pi-base.org/spaces?q=Collectionwise+normal%2BDevelopable%2B%24T_0%24%2B%7EMetrizable).","refs":[{"name":"Generalized metric spaces (G. Gruenhage), Ch. 10 of Handbook of set-theoretic topology","zb":"0555.54015"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000133","value":true},{"kind":"atom","property":"P000125","value":true}]},"then":{"kind":"atom","property":"P000044","value":false},"uid":"T000722","description":"Suppose $X$ satisfies the hypotheses.\nIf $X$ is not {P36}, it is not {P44}.\nOtherwise, since $X$ has multiple points, it must have infinitely many points\nand one can find four points $a$ of open covers where each $\\mathscr V_n$ is a refinement of $\\mathscr U$ and, for each $x \\in X$, there is some $n$ with $\\operatorname{ord}(x,\\mathscr V_n)=1$, and hence $\\operatorname{ord}(x,\\mathscr V_n)$ is finite.","refs":[]},{"when":{"kind":"atom","property":"P000034","value":true},"then":{"kind":"atom","property":"P000115","value":true},"uid":"T000729","description":"See {{mathse:5061701}},\nnoting that {P34} spaces are precisely the {P30} {P13} spaces.","refs":[{"name":"Subparacompact spaces and paracompactness","mathse":5061701}]},{"when":{"kind":"atom","property":"P000110","value":true},"then":{"kind":"atom","property":"P000115","value":true},"uid":"T000730","description":"Let $\\mathscr U_1,\\mathscr U_2,\\dots$ be a development for $X$.\nLet $\\mathscr U$ be an open cover of $X$.\nTake $\\mathscr V_n$ to be an open cover that is a common refinement of $\\mathscr U$ and $\\mathscr U_n$.\nGiven $x\\in X$, take $U\\in\\mathscr U$ with $x\\in U$.\nBy definition of development, there is some $n$ such that $\\operatorname{st}(x,\\mathscr U_n)\\subseteq U$.\nThen, $\\operatorname{st}(x,\\mathscr V_n)\\subseteq\\operatorname{st}(x,\\mathscr U_n)\\subseteq U$.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000057","value":true},{"kind":"atom","property":"P000135","value":true}]},"then":{"kind":"atom","property":"P000115","value":true},"uid":"T000731","description":"If $X$ is {P135}, the collection $\\mathscr V=\\{\\operatorname{cl}\\{x\\}:x\\in X\\}$\nis a closed refinement of every open cover.\nIf $X$ is also countable, $\\mathscr V$ is $\\sigma$-discrete.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000090","value":true},{"kind":"atom","property":"P000036","value":true},{"kind":"atom","property":"P000115","value":true}]},"then":{"kind":"atom","property":"P000040","value":true},"uid":"T000732","description":"See {{mathse:5062920}}.","refs":[{"name":"Answer to \"Subparacompact spaces and paracompactness\"","mathse":5062920}]},{"when":{"kind":"atom","property":"P000210","value":true},"then":{"kind":"atom","property":"P000211","value":true},"uid":"T000733","description":"Follows directly from the definitions.","refs":[{"name":"Arhangel'skii sheaf amalgamations in topological groups (Tsaban & Zdomskyy)","zb":"1348.54033"}]},{"when":{"kind":"atom","property":"P000211","value":true},"then":{"kind":"atom","property":"P000212","value":true},"uid":"T000734","description":"See the second paragraph after Definition 1.1 in {{zb:1348.54033}}.","refs":[{"name":"Arhangel'skii sheaf amalgamations in topological groups (Tsaban & Zdomskyy)","zb":"1348.54033"}]},{"when":{"kind":"atom","property":"P000212","value":true},"then":{"kind":"atom","property":"P000213","value":true},"uid":"T000735","description":"Follows directly from the definitions.","refs":[{"name":"Arhangel'skii sheaf amalgamations in topological groups (Tsaban & Zdomskyy)","zb":"1348.54033"}]},{"when":{"kind":"atom","property":"P000213","value":true},"then":{"kind":"atom","property":"P000214","value":true},"uid":"T000736","description":"Follows directly from the definitions.","refs":[{"name":"Arhangel'skii sheaf amalgamations in topological groups (Tsaban & Zdomskyy)","zb":"1348.54033"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000008","value":true},{"kind":"atom","property":"P000087","value":true},{"kind":"atom","property":"P000167","value":false}]},"then":{"kind":"atom","property":"P000106","value":true},"uid":"T000737","description":"See theorem 2.3 of {{zb:1270.54037}}.","refs":[{"name":"On hereditarily normal topological groups (Buzyakova)","zb":"1270.54037"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000019","value":true},{"kind":"atom","property":"P000106","value":true}]},"then":{"kind":"atom","property":"P000016","value":true},"uid":"T000738","description":"See Theorem 2.14 in {{zb:0555.54015}}\nor .\n\n(Note: In the proof a certain uncountable closed discrete subset of $X$ is constructed.\nTo show that the set is closed one needs to use the fact that $X$ is {P2} for example,\nwhich follows from {P106}.)","refs":[{"name":"Generalized metric spaces (G. Gruenhage), Ch. 10 of Handbook of set-theoretic topology","zb":"0555.54015"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000207","value":true},{"kind":"atom","property":"P000106","value":true}]},"then":{"kind":"atom","property":"P000112","value":true},"uid":"T000739","description":"See {{mathse:5082980}}. It's a slight generalization of lemma 8.2 in {{zb:0175.19802}}.","refs":[{"name":"Strongly collectionwise normal spaces with $G_\\delta$ diagonal are submetrizable","mathse":5082980},{"name":"On stratifiable spaces (Borges)","zb":"0175.19802"}]},{"when":{"kind":"atom","property":"P000215","value":true},"then":{"kind":"atom","property":"P000162","value":true},"uid":"T000740","description":"By definition.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000191","value":true},{"kind":"atom","property":"P000197","value":true}]},"then":{"kind":"atom","property":"P000163","value":true},"uid":"T000741","description":"Theorem 4.9 of {{zb:0559.54003}}.","refs":[{"name":"Cardinal functions I (R. Hodel), Ch. 1 of Handbook of set-theoretic topology","zb":"0559.54003"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000112","value":true},{"kind":"atom","property":"P000164","value":true},{"kind":"atom","property":"P000006","value":true}]},"then":{"kind":"atom","property":"P000215","value":true},"uid":"T000742","description":"Since $X$ is {P112}, there exists a continuous bijection $f:X\\to Y$ where $Y$ is a {P53} space. Necessarily $Y$ has {P164}.\nEvery {P53} space is {P7} and {P194}\n[(Explore)](https://topology.pi-base.org/spaces?q=Metrizable%2B%7ESubmetacompact).\nSo every subspace of $Y$ satisfies the hypotheses of {T382},\nand hence is {P162}.\nBy Corollary 8.18 of {{zb:1380.46022}}\nevery subspace of $X$ is {P162}.","refs":[{"name":"Rings of Continuous Functions (Gillman & Jerison)","zb":"1380.46022"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000216","value":true},{"kind":"atom","property":"P000003","value":true},{"kind":"atom","property":"P000164","value":true}]},"then":{"kind":"atom","property":"P000215","value":true},"uid":"T000743","description":"Every subspace is {P30}, {P3} and {P164}, hence {P162}\n[(Explore)](https://topology.pi-base.org/spaces?q=Paracompact%2BT2%2BCardinality+less+than+every+measurable+cardinal%2B%7ERealcompact).","refs":[]},{"when":{"kind":"atom","property":"P000216","value":true},"then":{"kind":"atom","property":"P000030","value":true},"uid":"T000744","description":"By definition.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000030","value":true},{"kind":"atom","property":"P000015","value":true}]},"then":{"kind":"atom","property":"P000216","value":true},"uid":"T000745","description":"See {{mathse:5087267}}.","refs":[{"name":"Is a paracompact perfectly normal space hereditarily paracompact?","mathse":5087267}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000134","value":true},{"kind":"atom","property":"P000216","value":true}]},"then":{"kind":"atom","property":"P000108","value":true},"uid":"T000746","description":"Every subspace is {P134} and {P30} and so {P88}\n[(Explore)](https://topology.pi-base.org/spaces?q=R1%2BParacompact%2B%7ECollectionwise+normal).","refs":[]},{"when":{"kind":"atom","property":"P000208","value":true},"then":{"kind":"atom","property":"P000216","value":true},"uid":"T000747","description":"Every subspace is {P16} and so {P30}","refs":[]},{"when":{"kind":"atom","property":"P000028","value":true},"then":{"kind":"atom","property":"P000210","value":true},"uid":"T000748","description":"Stated on p. 94 of {{zb:0774.54019}}.\n\nLet $x\\in X$ and let $S_1, S_2, \\dots\\subseteq X$ be countably infinite sets, each converging to $x$.\nSince $X$ is {P28}, the point $x$ has a countable local base of neighborhoods\n$V_1\\supseteq V_2\\supseteq\\dots$.\nAs $S_n$ converges to $x$, the set $T_n=S_n\\cap V_n$ is cofinite in $S_n$ and also converges to $x$.\n\n*Claim:* The set $S=\\bigcup_{n=1}^\\infty T_n$ is as required in the definition of {P210}:\n- (1) $S\\cap S_n\\supseteq T_n$ is cofinite in $S_n$ for each $n$.\n- (2) To show that $S$ converges to $x$, consider a basic neighborhood $V_m$ of $x$ for some fixed $m$.\nIf $n\\ge m$, we have $T_n\\subseteq V_n\\subseteq V_m$.\nAnd for the finitely many $n 0$ so that $d(x, y) < r$ implies $d(y, A) = d(x, A) = d(x, B) = d(y, B)$, and therefore $B(x, r)\\subseteq C$.\nIt follows that $C$ is clopen, so that $A, B$ are contained in the disjoint clopen sets $f^{-1}([0, 1/2))$ and $f^{-1}((1/2, 1])$.\n\nThe result then follows since {P53} {P218} spaces are {P146}\n[(Explore)](https://topology.pi-base.org/spaces?q=Metrizable%2BUltranormal%2B%7EUltraparacompact).\n\nSee also Theorem 11 in {{doi:10.48550/arXiv.1306.6086}}.","refs":[{"name":"Ultraparacompactness and Ultranormality (J. Van Name)","doi":"10.48550/arXiv.1306.6086"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000053","value":true},{"kind":"atom","property":"P000146","value":true}]},"then":{"kind":"atom","property":"P000220","value":true},"uid":"T000772","description":"Let $d$ be a metric on $X$. Define $\\mathcal{U}_1 = \\{X\\}$ and $\\mathcal{U}_{n+1}$ to be an open partition which is a refinement of $\\{B_d(x, \\frac{1}{n})\\cap U : x\\in X, U\\in\\mathcal{U}_n\\}$. Define $\\delta(x, y) = \\inf\\{1/n : \\{x, y\\}\\subseteq U \\text{ for some }U\\in\\mathscr U_n \\}$.\n\n- (1) $\\delta(x, y) = 0$ iff $x = y$\n\n If $x = y$, then there is unique $U_n\\in \\mathcal{U}_n$ such that $x\\in U_n$, and so $\\delta(x, y) = 0$. Conversely, if $\\delta(x, y) = 0$, then there are $U_n\\in \\mathcal{U}_n$ such that $\\{x, y\\}\\subseteq U_n$ for all $n$. Then $\\text{diam } \\{x, y\\} \\leq \\text{diam } U_{n+1} \\leq 2/n$ and so $x = y$.\n\n- (2) $\\delta(x, y) = \\delta(y, x)$\n\n Clear from the definition.\n\n- (3) $\\delta(x, z)\\leq \\max(\\delta(x, y), \\delta(y, z))$\n\n If $\\delta(x, z)> \\max(\\delta(x, y), \\delta(y, z))$ then there exists $n$ such that $\\{x, y\\}\\subseteq V$ and $\\{y, z\\}\\subseteq U$ for some $V, U\\in \\mathcal{U}_n$ and $\\{x, z\\}$ is not contained in an element of $\\mathcal{U}_n$. Since $\\mathcal{U}_n$ is a partition and $y\\in U\\cap V$ it follows that $U = V$ and so $\\{x, z\\}\\subseteq U$, which is a contradiction.\n\n- (4) $\\delta$ and $d$ induce the same topology on $X$\n\n Let $x\\in X$ and $r > 0$ be given. Observe that $B_\\delta(x, \\frac{1}{n}) = U$ where $U\\in \\mathcal{U}_{n+1}$ is the unique element in the partition containing $x$. Then $B_\\delta(x, r) = B_\\delta(x, \\frac{1}{n})$ or $B_\\delta(x, r) = X$ and so $B_\\delta(x, r)$ is open, so there is $s > 0$ such that $B_d(x, s)\\subseteq B_\\delta(x, r)$. And since $\\text{diam }U \\leq \\frac{2}{n+1}$ for $U\\in\\mathcal{U}_{n+1}$, there is $n$ such that $B_\\delta(x, \\frac{1}{n})\\subseteq B_d(x, r)$. And so the metrics $d, \\delta$ are topologically equivalent.\n\nSince $\\delta$ is an ultrametric, it follows that $X$ is {P220}.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000082","value":true},{"kind":"atom","property":"P000093","value":true}]},"then":{"kind":"atom","property":"P000120","value":true},"uid":"T000773","description":"Follows from the fact that every {P57} and {P53} space is {P133} [(Explore)](https://topology.pi-base.org/spaces?q=countable%2Bmetrizable%2B%7ELOTS).","refs":[]},{"when":{"kind":"atom","property":"P000122","value":true},"then":{"kind":"atom","property":"P000235","value":true},"uid":"T000774","description":"By definition.","refs":[]},{"when":{"kind":"atom","property":"P000123","value":true},"then":{"kind":"atom","property":"P000236","value":true},"uid":"T000775","description":"By definition.","refs":[]},{"when":{"kind":"atom","property":"P000236","value":true},"then":{"kind":"atom","property":"P000235","value":true},"uid":"T000776","description":"By definition.","refs":[]},{"when":{"kind":"atom","property":"P000124","value":true},"then":{"kind":"atom","property":"P000237","value":true},"uid":"T000777","description":"By definition.","refs":[]},{"when":{"kind":"atom","property":"P000203","value":true},"then":{"kind":"atom","property":"P000216","value":true},"uid":"T000778","description":"Any subspace of a {P203} space is {P203} or {P52}. Since [almost discrete spaces are paracompact](https://topology.pi-base.org/spaces?q=almost+discrete+%2B+not+paracompact) and [discrete spaces are paracompact](https://topology.pi-base.org/spaces?q=discrete+%2B+not+paracompact), it follows that any subspace of a {P203} space is {P30}.","refs":[]},{"when":{"kind":"atom","property":"P000222","value":true},"then":{"kind":"atom","property":"P000002","value":true},"uid":"T000779","description":"By definition.","refs":[]},{"when":{"kind":"atom","property":"P000222","value":true},"then":{"kind":"atom","property":"P000086","value":true},"uid":"T000780","description":"Every bijection of $X$ to itself is a homeomorphism.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000222","value":true},{"kind":"atom","property":"P000078","value":false}]},"then":{"kind":"atom","property":"P000101","value":false},"uid":"T000781","description":"Write $X = A\\cup B$ where $A, B$ are disjoint and $|A| = |B| = |X|$.\nTake a map $f:X\\to A$ such that $f(a) = a$ for $a\\in A$ and $f\\restriction B$ is a bijection from $B$ to $A$.\nIf $F\\subseteq A$ is finite, then $|f^{-1}(F)| = 2|F|$ so that $f^{-1}(F)$ is finite. This shows that $f$ is continuous, so that $A$ is a retract of $X$. But $A$ is a proper infinite subset of $X$, and so is not closed.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000052","value":true},{"kind":"atom","property":"P000078","value":true}]},"then":{"kind":"atom","property":"P000222","value":true},"uid":"T000782","description":"By definition.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000222","value":true},{"kind":"atom","property":"P000078","value":false}]},"then":{"kind":"atom","property":"P000039","value":true},"uid":"T000783","description":"The intersection of any two non-empty open sets is a cofinite set, and so non-empty.","refs":[]},{"when":{"kind":"atom","property":"P000167","value":true},"then":{"kind":"atom","property":"P000210","value":true},"uid":"T000784","description":"Since $X$ has no injective convergent sequences, it is vacuously {P210}.","refs":[]},{"when":{"kind":"atom","property":"P000136","value":true},"then":{"kind":"atom","property":"P000210","value":true},"uid":"T000785","description":"If $A$ is an injective convergent sequence with limit $x$, then $A\\cup \\{x\\}$ is infinite and compact.\nSince $X$ has no infinite compact subset, it has no injective convergent sequence, so it is vacuously {P210}.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000216","value":true},{"kind":"atom","property":"P000039","value":true}]},"then":{"kind":"atom","property":"P000208","value":true},"uid":"T000786","description":"Every open set in $X$ is {P30} and {P39}, and so {P16}\n[(Explore)](https://topology.pi-base.org/spaces?q=Paracompact+%2B+Hyperconnected+%2B+not+Compact).","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000085","value":true},{"kind":"atom","property":"P000006","value":true}]},"then":{"kind":"atom","property":"P000167","value":true},"uid":"T000787","description":"Since $X$ is {P6}, in the proof of {{mathse:4751804}} we can take $U_n$ to be cozero sets. Then $U = \\bigcup_n U_{2n}$ and $V = \\bigcup_n U_{2n+1}$ are open disjoint cozero sets with $x\\in \\overline{U}\\cap \\overline{V}$, which contradicts that $X$ is basically disconnected.","refs":[{"name":"Answer to \"What separation is required to ensure extremally disconnected spaces are sequentially discrete?\"","mathse":4751804}]},{"when":{"kind":"atom","property":"P000110","value":true},"then":{"kind":"atom","property":"P000117","value":true},"uid":"T000788","description":"Let $(\\mathcal{U}_n)$ be a development for $X$. Since {T730}, there exists a $\\sigma$-locally finite closed refinement $\\mathcal{V}_n$ of $\\mathcal{U}_n$ for each $n$. Then $\\mathcal{N} = \\bigcup_n \\mathcal{V}_n$ is $\\sigma$-locally finite, and if $x\\in W$ with $W$ open, then there is $n$ with $\\text{St}(x, \\mathcal{U}_n)\\subseteq W$, and so $\\text{St}(x, \\mathcal{V}_n)\\subseteq \\text{St}(x, \\mathcal{U}_n)\\subseteq W$. Since $\\mathcal{V}_n$ is a cover, it follows that there is $V\\in\\mathcal{V}_n$ with $x\\in V\\subseteq W$. So $\\mathcal{N}$ is a $\\sigma$-locally finite network.","refs":[]},{"when":{"kind":"atom","property":"P000107","value":false},"then":{"kind":"atom","property":"P000021","value":true},"uid":"T000789","description":"Any closed discrete subspace of $X$ is empty, and so finite.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000093","value":true},{"kind":"atom","property":"P000002","value":true}]},"then":{"kind":"atom","property":"P000191","value":true},"uid":"T000790","description":"Let $x\\in U$ where $U$ is open and countable. Then $\\{x\\} = \\bigcap\\{U\\setminus \\{y\\} : y\\in U\\setminus\\{x\\}\\}$ so that $\\{x\\}$ is $G_\\delta$.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000019","value":true},{"kind":"atom","property":"P000005","value":true},{"kind":"atom","property":"P000191","value":true}]},"then":{"kind":"atom","property":"P000028","value":true},"uid":"T000791","description":"Since $X$ is {P5} and {P191}, there exist closed neighbourhoods $U_n$ of $x$ such that $\\{x\\} = \\bigcap_n U_n$ and $U_{n+1}\\subseteq U_n$. \nTake an open neighbourhood $U$ of $x$, then $(X\\setminus U_n)$ is a countable open cover of the countably compact space $X\\setminus U$ and so there exists $n$ such that $X\\setminus U\\subseteq X\\setminus U_n$, and so $U_n\\subseteq U$. It follows that $(U_n)$ is a countable neighbourhood basis for $x$.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000198","value":true},{"kind":"atom","property":"P000002","value":true},{"kind":"atom","property":"P000132","value":true}]},"then":{"kind":"atom","property":"P000197","value":true},"uid":"T000792","description":"Assume $X$ is not {P197} and take a discrete uncountable set $D\\subseteq X$.\nFor each $d\\in D$ take an open set $U_d$ such that $U_d\\cap D = \\{d\\}$. \nThere exist closed sets $F_n\\subseteq X$ such that $\\bigcup_n F_n = \\bigcup_{d\\in D} U_d$. Since $D$ is uncountable, there exists $n$ such that $F_n\\cap D$ is uncountable. Since $F_n\\cap D$ is uncountable and discrete, there exists a limit point $x\\in (F_n\\cap D)' \\subseteq F_n\\subseteq \\bigcup_{d\\in D} U_d$, and so there exists $d\\in D$ such that $x\\in U_d$.\nBut $U_d$ intersects $F_n\\cap D$ in only finitely many elements, which is impossible for a limit point in a {P2} space.","refs":[]},{"when":{"kind":"atom","property":"P000168","value":true},"then":{"kind":"atom","property":"P000103","value":true},"uid":"T000793","description":"Let $A$ be a {P19} subset of $X$.\nSince {P168} is a hereditary property,\n$A$ also satisfies that property.\nHence $A$ is finite since {T397}.\nTherefore $A$ is closed in $X$ (using {T221}).","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000120","value":true},{"kind":"atom","property":"P000047","value":true},{"kind":"atom","property":"P000011","value":true}]},"then":{"kind":"atom","property":"P000050","value":true},"uid":"T000794","description":"For each $x \\in X$, take an open neighborhood $U$ that is {P133}. Since $X$ is {P47}, $U$ is {P47} and thus {P50} [(Explore)](https://topology.pi-base.org/spaces?q=LOTS+%2B+Totally+disconnected+%2B+%7EZero+dimensional), hence $X$ is locally {P50}.\n\nA {P11} locally {P50} space is {P50}, see for example {{mathse:5098565}}.","refs":[{"name":"When is a locally zero-dimensional space zero-dimensional?","mathse":5098565}]},{"when":{"kind":"atom","property":"P000222","value":true},"then":{"kind":"atom","property":"P000208","value":true},"uid":"T000795","description":"The closed sets satisfy the descending chain condition, because all closed sets, except possibly $X$, are finite.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000222","value":true},{"kind":"atom","property":"P000125","value":true}]},"then":{"kind":"atom","property":"P000089","value":false},"uid":"T000796","description":"Any permutation of $X$ is a self-homeomorphism, as bijections preserve cardinality of sets and therefore openness.\nAnd if $|X|\\ge 2$, there are permutations of $X$ without fixed point.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000222","value":true},{"kind":"atom","property":"P000078","value":false}]},"then":{"kind":"atom","property":"P000142","value":false},"uid":"T000797","description":"An infinite subset $A\\subseteq X$ contains {S15} as a subspace.\nSince {S15|P3}, it follows that the compact Hausdorff subsets of $X$ are the finite subsets, which have the discrete topology.\nTherefore if $A\\subseteq X$ then $A\\cap K$ is open in $K$ for all Hausdorff compact $K$.\nTaking any $A$ which isn't open shows that $X$ is not a $k_3$-space.","refs":[]},{"when":{"kind":"atom","property":"P000222","value":true},"then":{"kind":"atom","property":"P000187","value":true},"uid":"T000798","description":"Player 1 can play the W-game at a point $x\\in X$ by choosing $U_0=X$.\nAnd at round $n$, if Player 2 chose $x_i\\in U_i$ at round $i$ for $i=0,...,n-1$,\nPlayer 1 can choose $U_n=(X\\setminus\\{x_0,\\dots,x_{n-1}\\})\\cup\\{x\\}$,\nwhich is an open neighborhood of $x$.\n\nEach element $y\\ne x$ can appear at most once in the sequence $(x_n)_n$.\nFrom this, it follows that $x_n\\to x$ and Player 1 wins.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000222","value":true},{"kind":"atom","property":"P000115","value":true}]},"then":{"kind":"atom","property":"P000057","value":true},"uid":"T000799","description":"Let $\\mathscr V$ be a locally finite collection of subsets of $X$.\nThen $\\mathscr V$ must be finite.\nIndeed, there is an open cofinite set $U$ meeting only finitely many elements of $\\mathscr V$;\nand any other element of $\\mathscr V$ is contained in the (finite) complement of $U$, and so there are only finitely many of them.\nAnd consequently, any $\\sigma$-locally finite collection of subsets of $X$ must be countable.\n\nNow let $\\mathscr U$ be a $\\sigma$-locally finite closed refinement of the open cover $\\left\\{ X \\setminus \\{p\\} \\mid p \\in X \\right\\}$.\nEvery element of $\\mathscr U$ is a closed proper subset of $X$, hence finite.\nAnd $\\mathscr U$ is countable. Therefore, $X=\\bigcup\\mathscr U$ is countable.","refs":[]},{"when":{"kind":"atom","property":"P000222","value":true},"then":{"kind":"atom","property":"P000210","value":true},"uid":"T000800","description":"Every countably infinite $A\\subseteq X$ converges to any point $x\\in X$ since every neighborhood of $x$ contains all but finitely many points of $A$. So with $S_1,S_2,\\dots\\subseteq X$, each converging to $x$ as in the definition of {P210}, the set $S := \\bigcup_{n \\in \\mathbb{N}} S_n$ will also converge to $x$ and satisfy the {P210} condition.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000027","value":true},{"kind":"atom","property":"P000057","value":true},{"kind":"atom","property":"P000135","value":true}]},"then":{"kind":"atom","property":"P000110","value":true},"uid":"T000801","description":"Let $\\mathcal B$ be a countable base for the topology of $X$.\nFor each $U\\in\\mathcal B$ and $x\\in U$, we have $\\overline{\\{x\\}}\\subseteq U$ by the {P135} property.\nSo $\\mathscr U_{U,x}:=\\{U,X\\setminus\\overline{\\{x\\}}\\}$ is an open cover of $X$\nand satisfies $\\operatorname{st}(x,\\mathscr{U})=U$.\nTherefore the $\\mathscr U_{U,x}$, of which there are countably many, form a development for $X$.","refs":[]},{"when":{"kind":"atom","property":"P000119","value":true},"then":{"kind":"atom","property":"P000195","value":true},"uid":"T000802","description":"By definition, using {T45}.","refs":[]},{"when":{"kind":"atom","property":"P000119","value":true},"then":{"kind":"atom","property":"P000049","value":true},"uid":"T000803","description":"By definition.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000023","value":true},{"kind":"atom","property":"P000011","value":true},{"kind":"atom","property":"P000197","value":true}]},"then":{"kind":"atom","property":"P000081","value":true},"uid":"T000804","description":"By taking the Kolmogorov quotient we might assume that $X$ is locally compact Hausdorff.\nThe compact case follows from the cardinal inequality $t(X)\\leq s(X)$ for compact Hausdorff $X$,\nwhere $t(X)$ is the tightness of $X$ and $s(X)$ is the spread of $X$. See theorem 7.15 of {{zb:0559.54003}}.\nThe general case follows from $t(X) \\leq t(\\alpha X) \\leq s(\\alpha X) = s(X)$ where $\\alpha X$ is the one-point compactification of $X$.","refs":[{"name":"Cardinal functions I (R. Hodel), Ch. 1 of Handbook of set-theoretic topology","zb":"0559.54003"}]},{"when":{"kind":"atom","property":"P000109","value":true},"then":{"kind":"atom","property":"P000193","value":true},"uid":"T000805","description":"See corollary 2.2 of {{zb:0769.54022}}. The proof for {P154} is corollary 2.2 in {{zb:0761.54022}}.","refs":[{"name":"Monotone normality (Balogh & Rudin)","zb":"0769.54022"},{"name":"The shrinking property and the B-property in ordered spaces (Kemoto)","zb":"0761.54022"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000016","value":true},{"kind":"atom","property":"P000131","value":true}]},"then":{"kind":"atom","property":"P000020","value":true},"uid":"T000806","description":"Established as Corollary 2.2 of Theorem 2.1 in {{zb:1126.54010}}\n(available [here](https://topology.nipissingu.ca/tp/reprints/v29/tp29201.pdf)).\nNote that the proof of Theorem 2.1 contains errors which need corrections. These are discussed in {{mo:506776}}.","refs":[{"name":"When is a compact space sequentially compact? (O. Alas, R. Wilson)","zb":"1126.54010"},{"name":"Proof of Alas and Wilson that countably compact hereditarily Lindelof spaces are sequentially compact","mo":506776}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000190","value":true},{"kind":"atom","property":"P000216","value":true}]},"then":{"kind":"atom","property":"P000057","value":true},"uid":"T000807","description":"Assume $X$ is uncountable, it then contains {S35}, but {S35|P30}.","refs":[]},{"when":{"kind":"atom","property":"P000120","value":true},"then":{"kind":"atom","property":"P000210","value":true},"uid":"T000808","description":"See {{mathse:5116648}}.","refs":[{"name":"Do the Arkhangel'skii $\\alpha_i$ properties hold for order topologies?","mathse":5116648}]},{"when":{"kind":"atom","property":"P000154","value":true},"then":{"kind":"atom","property":"P000210","value":true},"uid":"T000809","description":"By {T750} and {T808}, every {P133} is {P210}, the assertion follows as the property is hereditary.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000190","value":true},{"kind":"atom","property":"P000215","value":true}]},"then":{"kind":"atom","property":"P000057","value":true},"uid":"T000810","description":"Assume $X$ is uncountable, it then contains {S35}, but {S35|P162}.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000208","value":true},{"kind":"atom","property":"P000078","value":false}]},"then":{"kind":"atom","property":"P000044","value":false},"uid":"T000811","description":"Since $X$ is {P41} (using {T651}), every connected component of $X$ is open.\nAnd since $X$ is {P16}, it has finitely many connected components,\nwith one of them necessarily infinite.\n\nNow choose two disjoint infinite sets $X_1,X_2\\subseteq X$.\nEach $X_i$ is {P208},\nand therefore contains an infinite connected set.\nSo $X$ contains two disjoint connected sets, each of size at least $2$,\nshowing that $X$ is not {P44}.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000023","value":true},{"kind":"atom","property":"P000105","value":true},{"kind":"atom","property":"P000011","value":true}]},"then":{"kind":"atom","property":"P000145","value":true},"uid":"T000812","description":"From {{mathse:5028016}}, $X = \\bigcup_{i\\in I} X_i$ is a disjoint union of {P18} subspaces $X_i$. \nBy {T30}, each $X_i$ is {P145}, and so $X$ is {P145}.","refs":[{"name":"Answer to \"Connected, locally compact, paracompact Hausdorff space is exhaustible by compacts\"","mathse":5028016}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000167","value":true},{"kind":"atom","property":"P000051","value":true},{"kind":"atom","property":"P000170","value":true}]},"then":{"kind":"atom","property":"P000136","value":true},"uid":"T000813","description":"Let $K$ be a {P16} subset of $X$. Then\n$K$ is {P3} (since $X$ is {P170}),\nand $K$ is also {P167} and {P51}.\nTo show that $K$ is finite, without loss of generality we can assume that $X$ itself\nis {P16}, {P3}, {P167} and {P51}\nand show that $X$ is finite.\n\nIf every point of $X$ is isolated, $X$ is finite by compactness.\nOtherwise, because $X$ is {P51}, choose $x \\in X$ with Cantor–Bendixson rank $1$ (that is, $x\\in X'\\setminus X''$).\nSo $x$ is not isolated in $X$ and there is a neighbourhood $L$ of $x$\nsuch that all points in $L\\setminus\\{x\\}$ are isolated in $X$.\nSince $X$ is {P11}, we can assume $L$ is closed in $X$.\nThe neighbourhood $L$ must be infinite,\nbecause otherwise $x$ would be isolated.\n\nTake a countably infinite set $M\\subseteq L$ containing $x$.\nSince $L$ is {P203}, $M$ is closed in $L$, hence closed in $X$ and compact.\nAnd since $M$ is {P167} and {P16},\nit is not {P181}\n[(Explore)](https://topology.pi-base.org/spaces?q=167+%2B+16+%2B+181),\nwhich is a contradiction.","refs":[]},{"when":{"kind":"atom","property":"P000129","value":true},"then":{"kind":"atom","property":"P000219","value":true},"uid":"T000814","description":"Let $Y\\subset X$ with $|Y|=|X|$. Then any bijection $Y \\to X$ is a homeomorphism.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000219","value":true},{"kind":"atom","property":"P000078","value":false}]},"then":{"kind":"atom","property":"P000204","value":false},"uid":"T000815","description":"Assume $X$ has a cut point $p$. Then $|X\\setminus \\{p\\}|=|X|$, but the two spaces cannot be homeomorphic as $X$ is {P36} and $X \\setminus \\{p\\}$ is not.","refs":[]},{"when":{"kind":"atom","property":"P000222","value":true},"then":{"kind":"atom","property":"P000219","value":true},"uid":"T000816","description":"Let $Y\\subset X$ with $|Y|=|X|$. Then any bijection $Y \\to X$ is a homeomorphism.","refs":[]},{"when":{"kind":"atom","property":"P000052","value":true},"then":{"kind":"atom","property":"P000219","value":true},"uid":"T000817","description":"Let $Y\\subseteq X$ with $|Y|=|X|$. Then any bijection $Y \\to X$ is a homeomorphism.","refs":[]},{"when":{"kind":"atom","property":"P000078","value":true},"then":{"kind":"atom","property":"P000219","value":true},"uid":"T000818","description":"For a finite space $X$, the only subspace with the same cardinality is $X$ itself, which is trivially homeomorphic to $X$.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000219","value":true},{"kind":"atom","property":"P000002","value":false},{"kind":"atom","property":"P000078","value":false}]},"then":{"kind":"atom","property":"P000196","value":true},"uid":"T000819","description":"Follows from Theorem 6.1 in {{zb:1286.54032}}.","refs":[{"name":"The Toronto Problem (W. R. Brian)","zb":"1286.54032"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000011","value":true},{"kind":"atom","property":"P000028","value":true},{"kind":"atom","property":"P000118","value":true}]},"then":{"kind":"atom","property":"P000121","value":true},"uid":"T000820","description":"Follows from {T516} by taking Kolmogorov quotient.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000190","value":true},{"kind":"atom","property":"P000034","value":true}]},"then":{"kind":"atom","property":"P000017","value":true},"uid":"T000821","description":"See the second part of {{mathse:5117290}}, making use of results of {{zb:0078.14803}}.","refs":[{"name":"Answer to \"Is every orderable topology strongly collectionwise normal?\"","mathse":5117290},{"name":"Some generalizations of full normality (M. J. Mansfield)","zb":"0078.14803"}]},{"when":{"kind":"atom","property":"P000154","value":true},"then":{"kind":"atom","property":"P000207","value":true},"uid":"T000822","description":"See {{mathse:5117210}}.","refs":[{"name":"Is every orderable topology strongly collectionwise normal?","mathse":5117210},{"name":"Some generalizations of full normality (M. J. Mansfield)","zb":"0078.14803"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000245","value":true},{"kind":"atom","property":"P000001","value":true}]},"then":{"kind":"atom","property":"P000078","value":true},"uid":"T000823","description":"$\\text{Kol}(X)$ is finite, and is equal to $X$ since $X$ is {P1}. So $X$ is finite.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000226","value":true},{"kind":"atom","property":"P000002","value":true}]},"then":{"kind":"atom","property":"P000078","value":true},"uid":"T000824","description":"Since $X$ is {P226}, we can find a finite dense set $F\\subseteq X$. Since $X$ is {P2}, $F$ is closed, so $F = X$.","refs":[]},{"when":{"kind":"atom","property":"P000245","value":true},"then":{"kind":"atom","property":"P000226","value":true},"uid":"T000825","description":"Immediate from the definitions.","refs":[]},{"when":{"kind":"atom","property":"P000226","value":true},"then":{"kind":"atom","property":"P000090","value":true},"uid":"T000826","description":"For any $x \\in X$, the collection of open neighborhoods of $x$ must have a minimal element $U$,\nwhich is necessarily contained in every other neighborhood of $x$\n(otherwise, intersecting with a suitable neighborhood would yield a strictly smaller neighborhood).","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000226","value":true},{"kind":"atom","property":"P000203","value":true}]},"then":{"kind":"atom","property":"P000078","value":true},"uid":"T000827","description":"Let $p \\in X$ be the non-isolated point.\nThen $X\\setminus\\{p\\}$ is {P52} and {P226},\nhence {P78}\n[(Explore)](https://topology.pi-base.org/spaces?q=Discrete%2BArtinian%2B%7EFinite).","refs":[]},{"when":{"kind":"atom","property":"P000226","value":true},"then":{"kind":"atom","property":"P000021","value":true},"uid":"T000828","description":"Since {P226} is hereditary, every discrete subset of $X$ is finite\n[(Explore)](https://topology.pi-base.org/spaces?q=Discrete%2BArtinian%2B%7EFinite).","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000226","value":true},{"kind":"atom","property":"P000045","value":true}]},"then":{"kind":"atom","property":"P000078","value":true},"uid":"T000829","description":"Let $p\\in X$ be the dispersion point.\nThen $X\\setminus\\{p\\}$ is {P226} and {P47},\nhence {P78}\n[(Explore)](https://topology.pi-base.org/spaces?q=Totally+disconnected%2BArtinian%2B%7EFinite).","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000226","value":true},{"kind":"atom","property":"P000185","value":true}]},"then":{"kind":"atom","property":"P000016","value":true},"uid":"T000830","description":"If $X$ is a {P185}, then $X$ is {P16} iff the Kolmogorov quotient $\\text{Kol}(X)$ is finite. The assertion then follows from {T824}.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000226","value":true},{"kind":"atom","property":"P000001","value":true}]},"then":{"kind":"atom","property":"P000051","value":true},"uid":"T000831","description":"Let $A$ be a nonempty subset of $X$.\nLet $U$ be a minimal open set in $X$ with $U\\cap A\\ne\\emptyset$.\nThe set $U\\cap A$ cannot contain two distinct points;\notherwise, by the {P1} property there would be an open set $V$ in $X$ containing one point and not the other.\nThen $U\\cap V$ would be a strictly smaller open set that meets $A$, which is not possible.\nThus, the element of $U\\cap A$ is isolated in $A$.","refs":[]},{"when":{"kind":"atom","property":"P000226","value":true},"then":{"kind":"atom","property":"P000180","value":true},"uid":"T000832","description":"Immediate from the definitions.","refs":[]},{"when":{"kind":"atom","property":"P000227","value":true},"then":{"kind":"atom","property":"P000198","value":false},"uid":"T000833","description":"If $X$ has a closed discrete subset of size continuum,\nits extent satisfies $e(X)\\ge\\mathfrak c>\\aleph_0$.\n\n----\nThe converse is independent of ZFC. It is true iff CH holds.","refs":[]},{"when":{"kind":"atom","property":"P000227","value":true},"then":{"kind":"atom","property":"P000058","value":false},"uid":"T000834","description":"Follows from the definitions.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000052","value":true},{"kind":"atom","property":"P000058","value":false}]},"then":{"kind":"atom","property":"P000227","value":true},"uid":"T000835","description":"In {P52} spaces, any subset is closed and discrete.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000026","value":true},{"kind":"atom","property":"P000227","value":true}]},"then":{"kind":"atom","property":"P000013","value":false},"uid":"T000836","description":"Jones' lemma, see Corollary 2.10 in {{zb:0684.54001}}.","refs":[{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000226","value":true},{"kind":"atom","property":"P000202","value":true},{"kind":"atom","property":"P000001","value":true}]},"then":{"kind":"atom","property":"P000089","value":true},"uid":"T000837","description":"See {{mathse:5119185}}.","refs":[{"name":"Answer to \"Noetherian spaces with a generic point have the fixed point property\"","mathse":5119185}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000026","value":true},{"kind":"atom","property":"P000227","value":true}]},"then":{"kind":"atom","property":"P000032","value":false},"uid":"T000838","description":"See Exercise 5.2.C(b) in {{zb:0684.54001}}.\n\nSee also the Claim on page 359 of {{zb:1116.54006}}.","refs":[{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"},{"name":"Normality and countable paracompactness of hyperspaces of ordinals. (N. Kemoto)","zb":"1116.54006"}]},{"when":{"kind":"atom","property":"P000120","value":true},"then":{"kind":"atom","property":"P000172","value":true},"uid":"T000839","description":"Each point has a neighborhood that is {P133} and hence {P172}\n[(Explore)](https://topology.pi-base.org/spaces?q=LOTS%2B%7ERadial).","refs":[]},{"when":{"kind":"atom","property":"P000228","value":true},"then":{"kind":"atom","property":"P000079","value":true},"uid":"T000840","description":"Suppose $A\\subseteq X$ is not a closed set.\nThen $A^c=X\\setminus A$ is not open and by the {P228} property\nthere is some $x\\in A^c$ such that $A^c$ does not contain any $V_n(x)$, that is, every $V_n(x)$ meets $A$.\nChoose $x_n\\in V_n(x)\\cap A$ for each $n$.\nThe sequence $x_1,x_2,\\dots$ converges to $x$,\nsince every neighborhood of $x$ contains all $V_n(x)$ for $n$ sufficiently large.\nThus $A$ is not sequentially closed.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000219","value":true},{"kind":"atom","property":"P000227","value":true},{"kind":"atom","property":"P000065","value":true}]},"then":{"kind":"atom","property":"P000052","value":true},"uid":"T000841","description":"A closed discrete subspace of size $\\mathfrak c$ has the same cardinality as $X$ and thus by {P219} must be homeomorphic to $X$.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000235","value":true},{"kind":"atom","property":"P000036","value":true}]},"then":{"kind":"atom","property":"P000236","value":true},"uid":"T000842","description":"Suppose a point $x$ has at the same time an open neighborhood homeomorphic to an open subset of $\\mathbb R^n_+$,\nand an open neighborhood homeomorphic to an open subset of $\\mathbb R^m_+$ with $n\\ne m$.\nTheir intersection $U$ is a nonempty open set in $X$\nwhich is homeomorphic to an open set in $\\mathbb R^n_+$ and to an open set in $\\mathbb R^m_+$.\n\nEvery nonempty open set in $\\mathbb R^n_+$ contains a nonempty open set homeomorphic to an open set in $\\mathbb R^n$.\nSo there is a nonempty open set $V\\subseteq U$ homeomorphic to an open set in $\\mathbb R^n$.\nSince $V$ is also homeomorphic to an open set in $\\mathbb R^m_+$,\nthere is a nonempty open set $W\\subseteq V$ homeomorphic to an open set in $\\mathbb R^m$\n(and also to an open set in $\\mathbb R^n$).\nBut this is not possible by invariance of domain ({{wikipedia:Invariance_of_domain}}).\n\nIt follows that the dimension $n$ at the point $x$ is completely determined by the point,\nand by definition of {P235} the set of points of a given dimension $n$ is open in $X$.\nSince $X$ is {P36}, the dimension must be the same at every point.","refs":[{"name":"Invariance of domain on Wikipedia","wikipedia":"Invariance_of_domain"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000219","value":true},{"kind":"atom","property":"P000197","value":false},{"kind":"atom","property":"P000114","value":true}]},"then":{"kind":"atom","property":"P000052","value":true},"uid":"T000843","description":"An uncountable discrete subspace has the same cardinality as $X$ and thus by {P219} must be homeomorphic to $X$.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000219","value":true},{"kind":"atom","property":"P000203","value":true}]},"then":{"kind":"atom","property":"P000078","value":true},"uid":"T000844","description":"Suppose $p$ is the non-isolated point and $X$ is infinite. Then $|X\\setminus \\{p\\}|=|X|$, but $X\\setminus \\{p\\}$ is discrete while $X$ is not, contradicting {P219}.","refs":[]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000228","value":true},{"kind":"atom","property":"P000080","value":true},{"kind":"atom","property":"P000099","value":true}]},"then":{"kind":"atom","property":"P000028","value":true},"uid":"T000845","description":"See Theorem 1.10 in {{zb:0285.54022}}.\nThere, the author assumes {P3} instead of {P99} (see section 1.7), but the latter is sufficient for the proof. {P3} is only used to show that if $x_1,x_2,\\dots$ is a sequence converging to $x$, then $\\{x,x_1,x_2,\\dots\\}$ is closed in $X$. But indeed, one can use the following general fact:\n\n\"If $x_1,x_2,\\dots$ is a sequence in a {P99} space $X$ converging to a point $x\\in X$, then the set $\\{x,x_1,x_2,\\dots\\}$ is sequentially closed.\"\n\nIn combination with {T840}, the assertion follows.","refs":[{"name":"On defining a space by a weak base. (Siwiec)","zb":"0285.54022"}]},{"when":{"kind":"and","subs":[{"kind":"atom","property":"P000120","value":true},{"kind":"atom","property":"P000042","value":true},{"kind":"atom","property":"P000139","value":false}]},"then":{"kind":"atom","property":"P000241","value":true},"uid":"T000846","description":"The empty space is vacuously {P241}.\nNow assume $X\\ne\\emptyset$ and let $x\\in X$.\nThere is an open neighborhood $U$ of $x$ that is orderable ({P133}),\nand a {P37} open neighborhood $V$ of $x$ with $V\\subseteq U$.\nSince a connected set in a LOTS is order-convex,\n$V$ is also a LOTS with the order $\\le$ induced from $U$.\n\nTake a neighborhood of $x$ in $(V,\\le)$ of the form $[p,q]$ for some $p\\le x\\le q$;\nnecessarily $p \\beta$. It follows that $(x_\\lambda)_{\\lambda < \\omega_1}$ converges to $p$.\n","refs":[]},{"space":"S000017","property":"P000189","value":true,"uid":"","description":"\n\nSince closed subsets properly contained in $X$ are countable, if $X$ were not $\\sigma$-connected then it would be a countable union of countable sets, and so countable. But {S17|P65}.\n","refs":[]},{"space":"S000017","property":"P000206","value":true,"uid":"","description":"\n\nIf $(U_n)$ are non-empty open sets then $\\bigcap_n U_n$ has countable complement so is non-empty. Therefore player $2$ always wins.","refs":[]},{"space":"S000017","property":"P000219","value":true,"uid":"","description":"\n\nLet $Y\\subseteq X$ with $|Y|=|X|$. Then any bijection $f:Y\\to X$ is a homeomorphism.\n","refs":[]},{"space":"S000018","property":"P000002","value":false,"uid":"","description":"\n\nHas {S4} as subspace and {S4|P2}.\n","refs":[]},{"space":"S000018","property":"P000026","value":false,"uid":"","description":"\n\nFor any $D = \\{(x_n, a_n)\\ |\\ n \\in \\omega\\} \\subset X \\times \\{0,1\\}$, $U = \\{(y,z)\\ |\\ y \\neq x_n \\text{ for any } n\\}$ is a basic open set disjoint from $D$ and so $D$ is not dense.\n","refs":[]},{"space":"S000018","property":"P000033","value":false,"uid":"","description":"\n\nThe Kolmogorov quotient of $X$ is {S17}\nand {S17|P33}.\n","refs":[]},{"space":"S000018","property":"P000037","value":false,"uid":"","description":"\n\nThe projection of $X$ onto the first component is a continuous map onto {S17}.\nThe {P37} property is preserved by continuous images.\nBut {S17|P37}.\n","refs":[]},{"space":"S000018","property":"P000039","value":true,"uid":"","description":"\n\n$X$ is a product of two {P39} spaces,\nas {S17|P39} and {S4|P39}.\n","refs":[]},{"space":"S000018","property":"P000065","value":true,"uid":"","description":"\n\nBy construction.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000018","property":"P000071","value":false,"uid":"","description":"\n\nThe Kolmogorov quotient of $X$ is {S17}\nand {S17|P71}.\n","refs":[]},{"space":"S000018","property":"P000086","value":true,"uid":"","description":"\n\n$X$ is a product of two {P86} spaces,\nas {S17|P86} and {S4|P86}.\n","refs":[]},{"space":"S000018","property":"P000131","value":true,"uid":"","description":"\n\nThe Kolmogorov quotient of $X$ is {S17}\nand {S17|P131}.\n","refs":[]},{"space":"S000018","property":"P000135","value":true,"uid":"","description":"\n\n$X$ is {P135} as a product of two spaces with the property,\nas {S17|P135} and {S4|P135}.\n","refs":[]},{"space":"S000018","property":"P000136","value":true,"uid":"","description":"\n\n$X$ is a product of two {P136} spaces, as {S17|P136} and {S4|P136}.\n","refs":[]},{"space":"S000018","property":"P000147","value":true,"uid":"","description":"\n\nThe Kolmogorov quotient of $X$ is {S17}\nand {S17|P147}.\n","refs":[]},{"space":"S000018","property":"P000151","value":true,"uid":"","description":"\n\nThe Kolmogorov quotient of $X$ is {S17}\nand {S17|P151}.\n","refs":[]},{"space":"S000018","property":"P000165","value":false,"uid":"","description":"\n\nNote that the Kolmogorov quotient of $X$, denoted by $Y$, is {S17}\nand {S17|P165}. If $q:X\\to Y$ is the Kolmogorov quotient map and $H\\subseteq Y$ is a countable closed set and $U\\supseteq H$ an open set such that there's no open set $V$ with $H\\subseteq V\\subseteq \\text{cl}(V)\\subseteq U$, then because $q$ has finite fibers, $H_0 = q^{-1}(H)$ is a countable closed set and $U_0 = q^{-1}(U)$ an open set. If there were open $V_0\\subseteq X$ such that $H_0\\subseteq V_0\\subseteq \\text{cl}(V_0)\\subseteq U_0$, then $H\\subseteq q(V_0)\\subseteq \\text{cl}(q(V_0)) \\subseteq U$ where $V = q(V_0)$ is open, contradiction. So $X$ is not pseudonormal.\n","refs":[]},{"space":"S000018","property":"P000172","value":true,"uid":"","description":"\n\nThe Kolmogorov quotient of $X$ is {S17}\nand {S17|P172}.\n","refs":[]},{"space":"S000018","property":"P000189","value":true,"uid":"","description":"\n\nThe Kolmogorov quotient of $X$ is {S17}\nand {S17|P189}.\n","refs":[]},{"space":"S000018","property":"P000206","value":true,"uid":"","description":"\n\nThe Kolmogorov quotient of $X$ is {S17}\nand {S17|P206}.\n","refs":[]},{"space":"S000018","property":"P000229","value":true,"uid":"","description":"\n\n{S17} is the Kolmogorov quotient of this space, and {S17|P229}.\n","refs":[]},{"space":"S000019","property":"P000016","value":true,"uid":"","description":"\n\nSee item #4 for space #22 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000019","property":"P000039","value":true,"uid":"","description":"\n\nLet $U_1$ and $U_2$ be nonempty open sets in $X$. Their complements $C_1$ and $C_2$ are compact in the Euclidean topology, and so is their union $C_1\\cup C_2$, which cannot be the whole of $X$ since $\\mathbb{R}$ is not compact in the Euclidean topology. This shows that the intersection of $U_1$ and $U_2$ is not empty.\n\nSee item #3 for space #22 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000019","property":"P000043","value":true,"uid":"","description":"\n\nSee {{mathse:2272196}}.\n","refs":[{"name":"Answer to \"Is the compact complement topology locally arc connected?\"","mathse":2272196}]},{"space":"S000019","property":"P000056","value":true,"uid":"","description":"\n\nFor any $n \\in \\omega$, let $I_n$ denote the interval $(n, n+2)$. $\\overline {I_n} = [n, n+2]$ and $int (\\overline{I_n}) = \\emptyset$ since any open set in $X$ is unbounded. Clearly $X = \\bigcup_{n \\in \\omega} I_n$, so $X$ is meager.\n","refs":[]},{"space":"S000019","property":"P000068","value":false,"uid":"","description":"\n\nIt is easy to check being {P68} passes to closed subspaces. Since $[0,1]$ is a closed subspace which is\nhomeomorphic to {S158} and {S158|P68}, the result follows.\n","refs":[]},{"space":"S000019","property":"P000086","value":true,"uid":"","description":"\n\nFollows by construction (since {S25} is).\n","refs":[]},{"space":"S000019","property":"P000089","value":false,"uid":"","description":"\n\nThe map $f(x) = x+1$ is continuous and it has no fixed points.\n","refs":[]},{"space":"S000019","property":"P000095","value":true,"uid":"","description":"\n\nFor a bounded set $A\\subseteq\\mathbb R$, its subspace topology induced from $X$ coincides with the Euclidean topology induced from $\\mathbb R$.\nSo given distinct points $ax+2$.\nBut there is no Euclidean path $f:[0,1]\\to U$ from $x$ to $y$.\nSo $V$ cannot be {P95}.\n","refs":[]},{"space":"S000019","property":"P000101","value":false,"uid":"","description":"\n\nThe map $f(x) = |x|$ is a retraction of $X$ onto $A = [0, \\infty)$ which is not closed in $X$.\n","refs":[]},{"space":"S000019","property":"P000110","value":true,"uid":"","description":"\n\nLet $\\mathcal{U}_n = \\{(-\\frac{1}{2n}+x, \\frac{1}{2n}+x)\\cup (-\\infty, -n)\\cup (n, \\infty) : x\\in X\\}$, then $\\mathcal{U}_n$ is an open cover of $X$ and $\\text{St}(x, \\mathcal{U}_n) = (-\\frac{1}{n}+x, \\frac{1}{n}+x) \\cup (-\\infty, -n)\\cup (n, \\infty)$ for $|x| < n$, so that $(\\text{St}(x, \\mathcal{U}_n))_{n=1}^\\infty$ is a local base at $x\\in X$.\n","refs":[]},{"space":"S000019","property":"P000130","value":true,"uid":"","description":"\n\nThe sets $A_n = (-\\infty, -n]\\cup [x-\\frac{1}{n}, x+\\frac{1}{n}] \\cup [n, \\infty)$ form a local base of compact neighbourhoods at $x\\in X$.\n","refs":[]},{"space":"S000019","property":"P000142","value":false,"uid":"","description":"\n\nSuppose that $K\\subseteq X$ is compact Hausdorff. Then $K\\cap [a, b]$ is a compact subset of $[a, b]\\subseteq X$, where the topology on $[a, b]$ coincides with the Euclidean topology. It follows that $K$ is closed in the Euclidean topology. If $K$ were unbounded, and if $K_1, K_2\\subseteq K$ are compact sets in the Euclidean topology, then $(K\\setminus K_1)\\cap (K\\setminus K_2)\\neq\\emptyset$ so that $K$ is not Hausdorff in the subspace topology coming from $X$. It follows that compact Hausdorff subsets of $X$ are precisely the subsets compact in the Euclidean topology. The topology which those sets determine is the Euclidean topology (since {S25|P140}) and not the topology of $X$, so that $X$ is not a $k_3$-space.\n","refs":[]},{"space":"S000019","property":"P000199","value":true,"uid":"","description":"\n\nSee {{mathse:5069153}}.\n","refs":[{"name":"Answer to \"Is R with compact complement topology simply connected?\"","mathse":5069153}]},{"space":"S000019","property":"P000208","value":false,"uid":"","description":"\n\nThe closed sets different from $X$ are exactly the compact sets in {S25}. However, they don't satisfy the descending chain condition: we have $Y_1 \\supsetneq Y_2 \\supsetneq \\cdots$ where $Y_n = \\left\\{ 0, \\frac 1n, \\frac 1 {n + 1}, \\frac 1 {n + 2}, \\dots \\right\\}$.\n","refs":[]},{"space":"S000020","property":"P000016","value":true,"uid":"","description":"\n\nLet $\\mathcal{A}$ be an open cover of $X$. There must exist some set $A_p \\in \\mathcal{A}$ where $p \\in A_p$. Thus, $X \\setminus A_p$ is finite. Let $X \\setminus A_p = \\{x_1, x_2, \\dots, x_n\\}$.\n\nFor each $i$, pick $A_i \\in \\mathcal{A}$ (not necessarily distinct) with $x_i \\in A_i$. The collection $\\{A_p,A_1,\\dots,A_n\\}$ is a finite subcover.\n\nSee item #4 for space #23 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000020","property":"P000181","value":true,"uid":"","description":"\n\nBy definition.\n","refs":[]},{"space":"S000020","property":"P000190","value":true,"uid":"","description":"\n\nBy definition.\n","refs":[]},{"space":"S000020","property":"P000203","value":true,"uid":"","description":"\n\nAll points except $\\infty$ are isolated.\n","refs":[]},{"space":"S000021","property":"P000003","value":true,"uid":"","description":"\nLet $x \\ne 0 \\in H$. Then the map $f(z) = \\langle z, x \\rangle$ is\ncontinuous with respect to the weak topology and satisfies $f(0) = 0$\nand $f(x) = \\langle x,x \\rangle > 0$. So $0$ and $x$ are separated by\ndisjoint open sets. It follows, since we are in a topological vector\nspace, that any two points $x,y$ are separated by disjoint open sets.\n\nSee 28.15 of {{doi:10.1016/B978-0-12-622760-4.X5000-6}}.\n","refs":[{"name":"Handbook of Analysis and Its Foundations","doi":"10.1016/B978-0-12-622760-4.X5000-6"}]},{"space":"S000021","property":"P000012","value":true,"uid":"","description":"\nLet $x$ be a point and $E$ a closed set with respect to the weak\ntopology. Since we are in a topological vector space, we can assume\nwithout loss of generality that $x = 0$. By definition of the\ntopology, there exists a number $\\epsilon > 0$ and finitely many\nelements $x_1, \\dots, x_n \\in H$ such that the basic open set $U = \\{ y :\n|\\langle y, x_i \\rangle| < \\epsilon, i = 1, \\dots, n\\}$ is disjoint\nfrom $E$. Note $0 \\in U$. Then the function $f(y) = \\sum_{i=1}^n\n|\\langle y, x_i \\rangle|$ satisfies $f(0) = 0$ and $f \\ge \\epsilon$ on\n$E$.\n\nSee 26.30 of {{doi:10.1016/B978-0-12-622760-4.X5000-6}} for a more\ngeneral assertion that every topological vector space is completely\nregular.\n","refs":[{"name":"Handbook of Analysis and Its Foundations","doi":"10.1016/B978-0-12-622760-4.X5000-6"}]},{"space":"S000021","property":"P000023","value":false,"uid":"","description":"\nAccording to 27.17 of {{doi:10.1016/B978-0-12-622760-4.X5000-6}},\nevery locally compact Hausdorff topological vector space is finite\ndimensional. However, $H$ is infinite dimensional.\n","refs":[{"name":"Handbook of Analysis and Its Foundations","doi":"10.1016/B978-0-12-622760-4.X5000-6"}]},{"space":"S000021","property":"P000026","value":true,"uid":"","description":"\nAs noted in 28.13 of {{doi:10.1016/B978-0-12-622760-4.X5000-6}}, all open sets in the weak topology are open in the original, so any countable dense subset of $H$ with its original separable topology remains dense in the weak topology.\n","refs":[{"name":"Handbook of Analysis and Its Foundations","doi":"10.1016/B978-0-12-622760-4.X5000-6"}]},{"space":"S000021","property":"P000065","value":true,"uid":"","description":"\nIn the norm topology, $H$ is an uncountable complete separable metric\nspace. By {{doi:10.1007/978-93-86279-42-2}}, Corollary 6.5, every\nsuch space has cardinality $\\mathfrak{c}$.\n","refs":[{"name":"Classical Descriptive Set Theory","doi":"10.1007/978-1-4612-4190-4"}]},{"space":"S000021","property":"P000068","value":false,"uid":"","description":"\n\nThis space has {S25} as a closed subspace, and {S25|P68}.\n","refs":[]},{"space":"S000021","property":"P000109","value":false,"uid":"","description":"\n\nIn Theorem 6a of {{zb:0949.54034}} it is shown that a topological vector space with its weak topology is metrizable iff monotonically normal. But $X$ is not metrizable as seen in Corollary 28.18 (c) of {{zb:0943.26001}}, and so cannot be monotonically normal.\n","refs":[{"name":"Nonstratifiability of topological vector spaces","zb":"0949.54034"},{"name":"Handbook of Analysis and Its Foundations","zb":"0943.26001"}]},{"space":"S000021","property":"P000111","value":true,"uid":"","description":"\nBy the Banach-Alaoglu theorem (see Theorem 5.2.1 of\n{{doi:10.1007/978-93-86279-42-2}}), the closed unit ball (with respect\nto the inner product of $H$) is compact in the weak-* topology, which\non a Hilbert space is equivalent to the weak topology. Since $H$ is a\ntopological vector space, the same is true for every closed ball.\n\nLet $K_n$ be the closed ball centered at origin with radius $n$, then $K_n$ is compact and $H = \\bigcup_{n \\in \\omega} K_n$.\nSuppose $K$ is any (weakly) compact subset of $H$. From the fact that the continuous image of a compact set is compact, we know that $K$ is bounded in the weak topology, that is, $\\sup_{y \\in K} \\left| \\left< x, y \\right> \\right| < + \\infty$ for any $x \\in H$.\n\nTherefore, $K$ is bounded in norm by [Uniform Boundedness Principle](https://en.wikipedia.org/wiki/Uniform_boundedness_principle), thus contained in some $K_n$.\n","refs":[{"name":"Functional Analysis","doi":"10.1007/978-93-86279-42-2"}]},{"space":"S000021","property":"P000140","value":false,"uid":"","description":"\n\nA direct consequence of the assertion at the beginning of Section 5 of {{doi:10.1285/I15900932V11P273}}, that a Banach space equipped with its weak topology cannot be a $k$-space unless it is finite-dimensional.\n","refs":[{"name":"The Mackey dual of a Banach space","doi":"10.1285/I15900932V11P273"}]},{"space":"S000021","property":"P000179","value":true,"uid":"","description":"\nCorollary 7.10 of {{mr:206907}} says that dual of a separable Banach space in its weak* topology is an $\\aleph_0$-space. \nSince $\\ell^2$ is a separable Banach space with $\\ell^2$ as its dual, the weak and weak* topology on $\\ell^2$ coincide, and corollary 7.10 applies.\n","refs":[{"name":"$\\aleph_0$-spaces (E. Michael)","mr":206907}]},{"space":"S000021","property":"P000204","value":false,"uid":"","description":"\n\nIf $x\\in X$ and $x_1, x_2\\in X\\setminus\\{x\\}$, then we can find a $2$-dimensional affine subspace $V\\subseteq X$ such that $x, x_1, x_2\\in V$. Then $V\\setminus \\{x\\}$ is homeomorphic to $\\mathbb{R}^2\\setminus\\{(0, 0)\\}$ which is path-connected, so there is a path in $X\\setminus \\{x\\}$ from $x_1$ to $x_2$. It follows that $X\\setminus \\{x\\}$ is path-connected, and so connected, so that $x$ is not a cut point.\n","refs":[]},{"space":"S000021","property":"P000214","value":false,"uid":"","description":"\n\nLet $(e_k)$ be an orthonormal basis of $H$. For each $n\\ge 1$, let $S_n = \\{ne_k : k\\in \\omega\\}$.\nParseval's identity $\\sum_k \\langle e_k, x\\rangle^2 = \\|x\\|^2$ implies that $\\langle e_k, x\\rangle\\to 0$ for each $x\\in H$. It follows that each $S_n$ converges to $0$ in $X$. Any countably infinite set $S\\subseteq X$ such that $S_n\\cap S\\neq\\emptyset$ for infinitely many $n$ is necessarily norm-unbounded. Since weakly convergent sequences are weakly bounded, and therefore norm-bounded by the uniform boundedness principle, it follows that such an $S$ does not converge in $X$. Therefore $X$ is not {P214}.\n","refs":[]},{"space":"S000021","property":"P000238","value":true,"uid":"","description":"\n\nSee 28.15 of {{doi:10.1016/B978-0-12-622760-4.X5000-6}}.\n","refs":[{"name":"Handbook of Analysis and Its Foundations","doi":"10.1016/B978-0-12-622760-4.X5000-6"}]},{"space":"S000022","property":"P000002","value":true,"uid":"","description":"\n\nBy inspection, every singleton is closed.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000022","property":"P000065","value":true,"uid":"","description":"\n\nBy construction.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000022","property":"P000147","value":true,"uid":"","description":"\n\nThe point $p$ is a P-point, since its neighborhoods are the cocountable subsets of $X$ containing the point and a countable intersection of them is still such a neighborhood.\nAnd every isolated point is also a P-point.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000022","property":"P000151","value":true,"uid":"","description":"\n\nSee {{mathse:4727833}}.\n","refs":[{"name":"What kinds of selection principles hold for Fortissimo space?","mathse":4727833}]},{"space":"S000022","property":"P000172","value":true,"uid":"","description":"\n\nSee Proposition 1 in {{mathse:4833761}}, with $\\kappa=\\omega_1$.\n","refs":[{"name":"LOTS and radial properties for generalized Fort/Fortissimo spaces","mathse":4833761}]},{"space":"S000022","property":"P000203","value":true,"uid":"","description":"\n\nAll points except $p$ are isolated.\n","refs":[]},{"space":"S000023","property":"P000003","value":true,"uid":"","description":"\n\nSee item #1 for space #26 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000023","property":"P000049","value":false,"uid":"","description":"\n\nSee item #9 for space #26 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000023","property":"P000136","value":true,"uid":"","description":"\n\nAsserted in {{mathse:4566624}}.\n","refs":[{"name":"Can a hemicompact space fail to be weakly locally compact?","mathse":4566624}]},{"space":"S000023","property":"P000181","value":true,"uid":"","description":"\n\nBy definition.\n\nSee item #4 for space #26 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000023","property":"P000203","value":true,"uid":"","description":"\n\nAll points except $(0,0)$ are isolated.\n","refs":[]},{"space":"S000024","property":"P000016","value":true,"uid":"","description":"\n\nSee item #1 for space #27 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000024","property":"P000029","value":false,"uid":"","description":"\n\n$\\{\\{y\\}\\ |\\ y \\in Y\\}$ is an uncountable pairwise disjoint collection of open sets.\n","refs":[]},{"space":"S000024","property":"P000051","value":true,"uid":"","description":"\n\nSee item #6 for space #27 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000024","property":"P000061","value":false,"uid":"","description":"\n\nThe Tychonoff reflection of $X$ is {S154}.\nSince {S154|P61}, neither is $X$, by Proposition 1 in {{mathse:5108293}}.\n","refs":[{"name":"Cozero complemented property for a space and its Tychonoff reflection","mathse":5108293}]},{"space":"S000024","property":"P000065","value":true,"uid":"","description":"\n\nBy definition.\n","refs":[]},{"space":"S000024","property":"P000084","value":true,"uid":"","description":"\n\nEach point has a neighbourhood homeomorphic to {S154} and {S154|P3}.\n","refs":[]},{"space":"S000024","property":"P000101","value":false,"uid":"","description":"\n\nLet $A = \\mathbb{R}\\cup \\{\\infty_1\\}$ and $f:X\\to A$ be defined by $f(x) = x$ for $x\\in A$ and $f(\\infty_2) = \\infty_1$. Then $f$ is a continuous, and so $A$ is a retract of $X$, but $A$ is not closed in $X$.\n","refs":[]},{"space":"S000024","property":"P000115","value":false,"uid":"","description":"\n\nConsider the open cover $\\mathcal{U} = \\{X\\setminus\\{\\infty_1\\}, X\\setminus\\{\\infty_2\\}\\}$. Suppose that $\\mathcal{V}$ is a $\\sigma$-locally finite closed refinement of $\\mathcal{U}$. Since infinite closed subsets of $X$ need to contain both $\\infty_1$ and $\\infty_2$, $\\mathcal{V}$ consists of finite subsets of $X$, each of which does not contain both $\\infty_1$ and $\\infty_2$. Write $\\mathcal{V} = \\bigcup_n \\mathcal{V}_n$ where $\\mathcal{V}_n$ are locally finite. The union of those members of $\\mathcal{V}_n$ which contain $\\infty_1$ is closed, and so is finite. The same is true for the ones which contain $\\infty_2$ and the ones which contain neither. It follows that $\\bigcup\\mathcal{V}_n$ is finite, and so $X = \\bigcup_n \\bigcup\\mathcal{V}_n$ is countable, which is impossible since $X$ is uncountable.\n","refs":[]},{"space":"S000024","property":"P000120","value":false,"uid":"","description":"\n\nThe open subspace $X\\setminus \\{\\infty_1\\}$ is homeomorphic to {S154}\nand {S154|P120}.\n","refs":[]},{"space":"S000024","property":"P000130","value":true,"uid":"","description":"\n\nEach point has a neighbourhood homeomorphic to {S154} and {S154|P130}.\n","refs":[]},{"space":"S000024","property":"P000151","value":true,"uid":"","description":"\n\n$X$ is covered by two copies of {S154}\nand {S154|P151}.\n","refs":[]},{"space":"S000024","property":"P000169","value":false,"uid":"","description":"\n\nIf $\\infty_1\\in X\\setminus A$ where $A$ is finite and $\\infty_2\\in A$, then $\\text{int}(\\text{cl}(X\\setminus A)) = (X\\setminus A) \\cup \\{\\infty_2\\}$ so that $X\\setminus A$ is not a regular open set.\n","refs":[]},{"space":"S000024","property":"P000187","value":true,"uid":"","description":"\n\nEvery point has a neighbourhood homeomorphic to {S154} and {S154|P187}.\n","refs":[]},{"space":"S000024","property":"P000191","value":false,"uid":"","description":"\n\nHas {S154} as subspace and {S154|P191}.\n","refs":[]},{"space":"S000024","property":"P000210","value":true,"uid":"","description":"\n\nLet $x \\in X$.\nIf $x \\in \\{\\infty_1,\\infty_2\\}$, all neighborhoods of $x$ are cofinite, so we can follow the same argument as in {T800}.\nOtherwise $x$ is isolated, so there cannot be a countably infinite set converging to $x$. Hence, the assertion {P210} is trivially true.\n","refs":[]},{"space":"S000024","property":"P000216","value":true,"uid":"","description":"\n\nAny open subspace of this space is homeomorphic to either a subspace of {S154} or to {S24} itself.\nBut {S154|P216} and {S24|P30}, so this space is {P216}.\n","refs":[]},{"space":"S000024","property":"P000217","value":true,"uid":"","description":"\n\nIf $Z\\subseteq X$ is a zero-set, and $\\infty_i\\in Z$, then because $\\infty_1, \\infty_2$ can't be separated by open sets, we must have $\\{\\infty_1, \\infty_2\\}\\subseteq Z$.\nIt follows that if $Z_1, Z_2\\subseteq X$ are disjoint zero-sets, then $Z_i\\subseteq \\mathbb{R}$ for some $i$ and so $Z_i$ is clopen in $X$.\n","refs":[]},{"space":"S000025","property":"P000022","value":false,"uid":"","description":"\n\nThe identity map is an unbounded continuous function.\n","refs":[]},{"space":"S000025","property":"P000027","value":true,"uid":"","description":"\n\nSee item #2 for space #28 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000025","property":"P000053","value":true,"uid":"","description":"\n\nBy definition.\n","refs":[]},{"space":"S000025","property":"P000065","value":true,"uid":"","description":"\n\nBy definition.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000025","property":"P000095","value":true,"uid":"","description":"\n\nThe straight line segment between two points is an arc.\n","refs":[]},{"space":"S000025","property":"P000133","value":true,"uid":"","description":"\n\nThe standard order on $\\mathbb{R}$ induces the Euclidean topology.\n","refs":[{"name":"Topology (Munkres)","zb":"0951.54001"}]},{"space":"S000025","property":"P000155","value":true,"uid":"","description":"\n\nEvident from the definition of {P155}.\n","refs":[{"name":"Topology (Munkres)","zb":"0951.54001"}]},{"space":"S000025","property":"P000238","value":true,"uid":"","description":"\n\nThe standard addition and scalar multiplication on $\\mathbb R$ are continuous, and together satisfy the axioms of a real vector space.\n","refs":[]},{"space":"S000026","property":"P000016","value":true,"uid":"","description":"\n\nSee item #2 for space #29 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000026","property":"P000050","value":true,"uid":"","description":"\n\nFor each $\\sigma \\in 2^{<\\omega}$, let $[\\sigma] = \\{ x \\in 2^\\omega\\ |\\ x \\text{ extends } \\sigma \\text{ as a sequence} \\}$. Then $\\{ [\\sigma]\\ |\\ \\sigma \\in 2^{<\\omega}\\}$ is a clopen basis.\n","refs":[]},{"space":"S000026","property":"P000055","value":true,"uid":"","description":"\n\nSee item #2 for space #29 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000026","property":"P000065","value":true,"uid":"","description":"\n\n$|2^\\omega|=2^{\\aleph_0}=\\mathfrak c$.\n","refs":[]},{"space":"S000026","property":"P000087","value":true,"uid":"","description":"\n\nThe group with two elements together with the discrete topology is a topological group.\nThe space $X$ is {P87} as a product of such two element spaces.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000027","property":"P000023","value":false,"uid":"","description":"\n\nSee item #8 for space #30 in {{zb:0386.54001}}.\nSee also {{mathse:2934970}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Rational numbers are not locally compact","mathse":2934970}]},{"space":"S000027","property":"P000087","value":true,"uid":"","description":"\n\nThe space is a subgroup of the topological group $(\\mathbb R, +)$.\n","refs":[{"name":"Rational number on Wikipedia","wikipedia":"Rational_number"}]},{"space":"S000027","property":"P000133","value":true,"uid":"","description":"\n\nThe topology on $\\mathbb Q$ coincides with the topology induced by the usual order inherited from $\\mathbb R$.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000027","property":"P000181","value":true,"uid":"","description":"\n\nBy definition.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000028","property":"P000022","value":false,"uid":"","description":"\n\nThe inclusion map $\\mathbb{R}\\setminus\\mathbb{Q} \\hookrightarrow \\mathbb{R}$ is unbounded.\n","refs":[]},{"space":"S000028","property":"P000023","value":false,"uid":"","description":"\n\nSee item #8 for space #31 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000028","property":"P000027","value":true,"uid":"","description":"\n\nThe space is {P27} as a subspace of $\\mathbb R$ ({S25}), which is {P27}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000028","property":"P000050","value":true,"uid":"","description":"\n\nThe family $\\{ (r,s) \\cap X : r,s\\text{ rational with } r < s\\}$ is a base consisting of clopen sets.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000028","property":"P000055","value":true,"uid":"","description":"\n\nSee item #5 for space #31 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000028","property":"P000065","value":true,"uid":"","description":"\n\n$|\\mathbb{R} \\setminus \\mathbb{Q}| = |\\mathbb{R}| = \\mathfrak{c}$\n","refs":[]},{"space":"S000028","property":"P000066","value":false,"uid":"","description":"\n\nSee {{mathse:4727297}} and [this entry in Dan Ma's topology blog](https://dantopology.wordpress.com/2020/02/18/the-space-of-irrational-numbers-is-not-menger/).\n","refs":[{"name":"Answer to \"Is every Lindelöf space Menger?\"","mathse":4727297}]},{"space":"S000028","property":"P000087","value":true,"uid":"","description":"\n\nIn its Baire space $\\omega^\\omega$ incarnation, the space is a product of discrete spaces, each admitting a compatible topological group structure ({S2|P87}).\n","refs":[{"name":"Baire_space_(set_theory) on Wikipedia","wikipedia":"Baire_space_(set_theory)"}]},{"space":"S000029","property":"P000016","value":true,"uid":"","description":"\n\nCompactifications are by construction compact.\n\nSee item #1 for space #35 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000029","property":"P000041","value":false,"uid":"","description":"\n\n$X$ contains {S27} as open subset\nand {S27|P41}.\n","refs":[]},{"space":"S000029","property":"P000045","value":true,"uid":"","description":"\n\nSee item #5 for space #35 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000029","property":"P000073","value":true,"uid":"","description":"\n\nLet $A\\subseteq X$ be non-empty and irreducible. If there existed distinct $x, y\\in A\\cap \\mathbb{Q}$, then $x, y$ have disjoint neighbourhoods in $\\mathbb{Q}$ and so in $A$, and this is impossible since $A$ is irreducible. So $A$ can have at most two points so that $A$ is discrete, and because {S1|P39}, $A$ is a singleton, and so $X$ is sober. \n","refs":[]},{"space":"S000029","property":"P000080","value":true,"uid":"","description":"\n\nSee {{mathse:4709641}}.\n","refs":[{"name":"Is the one-point compactification of the rationals sequential or Frechet-Urysohn?","mathse":4709641}]},{"space":"S000029","property":"P000084","value":false,"uid":"","description":"\n\nSuppose the point $\\infty$ has a Hausdorff open neighborhood $U$ and let $x\\in U\\cap\\mathbb Q$. The points $\\infty$ and $x$ have disjoint open neighborhoods in $U$, which are also open in $X$. Say $\\infty\\in X\\setminus K$ for some compact $K\\subseteq\\mathbb Q$ and $x\\in V$ with $V\\cap(X\\setminus K)=\\emptyset$. So $x\\in V\\subseteq K\\subseteq\\mathbb Q$, showing that $K$ is a compact neighborhood of $x$. That is impossible since $\\mathbb Q$ is nowhere locally compact.\n","refs":[]},{"space":"S000029","property":"P000089","value":true,"uid":"","description":"\n\nShown in {{mathse:5070817}}.\n","refs":[{"name":"Does the one-point compactification of $\\mathbb{Q}$ have the fixed point property?","mathse":5070817}]},{"space":"S000029","property":"P000100","value":true,"uid":"","description":"\n\nSee Example 1, p. 43 in {{doi:10.2307/2312998}}.\n\nAlso {{mathse:2524171}}.\n","refs":[{"name":"When are Compact and Closed Equivalent? (N. Levine)","doi":"10.2307/2312998"},{"name":"Are compact subsets closed in the one-point compactification of $\\mathbb Q$?","mathse":2524171}]},{"space":"S000029","property":"P000138","value":false,"uid":"","description":"\n\nIf $f:\\mathbb{Q}\\to\\mathbb{Q}$ is any homeomorphism of $\\mathbb{Q}$, then the map $\\tilde{f}:X\\to X$ defined by $\\tilde{f}\\restriction_\\mathbb{Q} = f$ and $\\tilde{f}(\\infty) = \\infty$ is a homeomorphism of $X$. By writing $\\mathbb{Q} = \\bigcup_n Q_n$ where $Q_n$ are open, pairwise disjoint and homeomorphic to $\\mathbb{Q}$, for example $Q_0 = (-\\infty, \\sqrt{2})\\cap \\mathbb{Q}$ and $Q_n = (\\sqrt{2}n, \\sqrt{2}(n+1))\\cap\\mathbb{Q}$, and $h_n:\\mathbb{Q}\\to Q_n$ are some fixed homeomorphisms, we see that any bijection $g:\\mathbb{N}\\to\\mathbb{N}$ induces a homeomorphism $\\hat{g}:\\mathbb{Q}\\to\\mathbb{Q}$ by the formula $\\hat{g}(x) = h_{g(n)}(h_n^{-1}(x))$ for $x\\in Q_n$. Since there are continuum many such bijections (see {{mo:29475}}), there are continuum many homeomorphisms of $\\mathbb{Q}$, and so of $X$.\n","refs":[{"name":"Easy proof of the uncountability of bijections on natural numbers","mo":29475}]},{"space":"S000029","property":"P000169","value":false,"uid":"","description":"\n\nEvery neighborhood of $\\infty$ is dense, and therefore\nevery regular open neighborhood of $\\infty$ is the entire\nspace.\n","refs":[]},{"space":"S000029","property":"P000181","value":true,"uid":"","description":"\n\nBy definition.\n","refs":[]},{"space":"S000029","property":"P000183","value":true,"uid":"","description":"\n\nSince {S27|P183} we can find a countable pseudobase $\\mathcal{N}$ for $\\mathbb{Q}$. Let $\\mathcal{N}^\\ast = \\mathcal{N}\\cup \\{X\\setminus N: N\\in\\mathcal{N}\\}$ and note that $\\mathcal{N}^\\ast$ is countable. Suppose $K\\subseteq U$ where $K$ is compact and $U\\subseteq X$ is open. If $\\infty\\notin K$ there exists some $N\\in\\mathcal{N}$ such that $K\\subseteq N\\subseteq U\\cap \\mathbb{Q}\\subseteq U$ since $\\mathcal{N}$ is a pseudobase for $\\mathbb{Q}$. If $\\infty\\in K$ then there is some $N\\in\\mathcal{N}$ with $X\\setminus U\\subseteq N\\subseteq X\\setminus K$ since $X\\setminus U$ is compact and $K$ is closed because {S29|P100}, and so $K\\subseteq X\\setminus N\\subseteq U$. So $\\mathcal{N}^\\ast$ is a $k$-network for $X$.\n","refs":[]},{"space":"S000029","property":"P000214","value":false,"uid":"","description":"\n\nSee {{mathse:5111069}}.\n","refs":[{"name":"Which of the Arkhangel'skii $\\alpha_i$ properties hold for the one-point compactification of the rationals?","mathse":5111069}]},{"space":"S000029","property":"P000216","value":false,"uid":"","description":"\n\nAny neighbourhood of $\\infty$ is dense in $X$ (see Lemma 2 in {{mathse:5070821}}).\nLet $U$ be an open neighbourhood of $\\infty$ and assume $U$ is {P30}.\nAny open cover $\\mathscr U$ of $U$ has a locally finite open refinement $\\mathscr V$.\nLet $V\\in\\mathscr V$ contain $\\infty$.\nSince $V$ is dense in $X$, the cover $\\mathscr V$ is necessarily finite,\nand therefore $\\mathscr U$ has a finite subcover. That is, $U$ is compact.\nNow, {S29|P100}, which implies $U$ is closed in $X$.\nSince $U$ is dense in $X$, necessarily $U=X$.\nSo any other $U$, of the form $X\\setminus K$ for some nonempty compact set $K\\subseteq\\mathbb Q$, is not {P30}.\n","refs":[{"name":"Answer to \"Does the one-point compactification of $\\mathbb{Q}$ have the fixed point property?\"","mathse":5070821}]},{"space":"S000029","property":"P000243","value":true,"uid":"","description":"\n\n{S27} is an open dense subspace of $X$\nand {S27|P243}.\n","refs":[]},{"space":"S000030","property":"P000026","value":true,"uid":"","description":"\n\nThe countable product of {P26} spaces is {P26}, and\n{S25|P26}.\n\nOr, see item #2 for space #36/#37 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000030","property":"P000055","value":true,"uid":"","description":"\n\n$X$ is a countable product of copies of {S25}\nand {S25|P55}.\n","refs":[]},{"space":"S000030","property":"P000066","value":false,"uid":"","description":"\n\n{S28} is homeomorphic to $\\mathbb Z^\\omega$, which can be viewed as a closed subspace of {S30}.\nAnd {P66} is preserved by closed subspaces.\nBut {S28|P66}.\n","refs":[]},{"space":"S000030","property":"P000184","value":false,"uid":"","description":"\n\nEvery Euclidean space $\\mathbb R^n$ embeds into $X$.\nTherefore, as explained in {{mathse:5009117}}, $X$ cannot itself be embedded into a fixed $\\mathbb R^m$.\n","refs":[{"name":"The Hilbert cube cannot be embedded in some Euclidean space $\\mathbb R^n$","mathse":5009117}]},{"space":"S000030","property":"P000204","value":false,"uid":"","description":"\n\nIf $x\\in X$ and $x_1, x_2\\in X\\setminus\\{x\\}$, then we can find a $2$-dimensional affine subspace $V\\subseteq X$ such that $x, x_1, x_2\\in V$. Then $V\\setminus \\{x\\}$ is homeomorphic to $\\mathbb{R}^2\\setminus\\{(0, 0)\\}$ which is path-connected, so there is a path in $X\\setminus \\{x\\}$ from $x_1$ to $x_2$. It follows that $X\\setminus \\{x\\}$ is path-connected, and so connected, so that $x$ is not a cut point.\n","refs":[]},{"space":"S000030","property":"P000238","value":true,"uid":"","description":"\n\n$\\mathbb{R}^\\omega$ is a product of topological vector spaces, namely $\\mathbb{R}$. And {S25|P238}.\n","refs":[]},{"space":"S000031","property":"P000016","value":true,"uid":"","description":"\n\n$X$ is compact as a product of compact spaces.\n","refs":[{"name":"Weak Hausdorff space not KC","mathse":632320}]},{"space":"S000031","property":"P000041","value":false,"uid":"","description":"\n\nBy Problem 27F.2 of {{zb:1052.54001}}, the product of two nonempty spaces is {P41} iff each factor is. Since {S29} is not {P41}, neither is $X$.\n","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"space":"S000031","property":"P000044","value":false,"uid":"","description":"\n\nSince {S29|P36}, the sets $\\mathbb{Q}^*\\times \\{x\\}$ are connected and disjoint. So $X$ is not biconnected.\n","refs":[]},{"space":"S000031","property":"P000057","value":true,"uid":"","description":"\n\nBy definition.\n","refs":[{"name":"Weak Hausdorff space not KC","mathse":632320}]},{"space":"S000031","property":"P000073","value":true,"uid":"","description":"\n\nIf $A\\subseteq X$ is irreducible and non-empty, then $A\\subseteq \\pi_1(A)\\times \\pi_2(A)$ where $\\pi_k$ are projections onto the $k$th coordinate, so $\\pi_k(A)$ are non-empty and irreducible in {S29}. Since {S29|P73}, $\\pi_k(A)$ are singletons, and so $A$ is a singleton. \n","refs":[]},{"space":"S000031","property":"P000084","value":false,"uid":"","description":"\n\n$X$ has {S29} as a subspace and {S29|P84}.\n","refs":[]},{"space":"S000031","property":"P000086","value":false,"uid":"","description":"\n\nEvery neighbourhood of $(\\infty, \\infty)$ is dense in $X$ while there are neighbourhoods of $x\\in \\mathbb{Q}\\times \\mathbb{Q}$ which are not dense, so $X$ cannot be homogeneous. \n","refs":[]},{"space":"S000031","property":"P000089","value":true,"uid":"","description":"\n\nShown in {{mathse:5070817}}.\n","refs":[{"name":"Does the one-point compactification of $\\mathbb{Q}$ have the fixed point property?","mathse":5070817}]},{"space":"S000031","property":"P000101","value":false,"uid":"","description":"\n\nThe diagonal of any non-{P3} space fails to be closed, but\nis always a retract of the square under the map $(x,y)\\mapsto(x,x)$.\n\nSee {{mathse:4762250}}.\n","refs":[{"name":"Are retracts always closed in a compact weak Hausdorff space?","mathse":4762250}]},{"space":"S000031","property":"P000130","value":false,"uid":"","description":"\n\nBy Theorem 18.6 of {{zb:1052.54001}}, the product of two nonempty spaces is {P130} iff each factor is. Since {S29} is not {P130}, neither is $X$.\n","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"space":"S000031","property":"P000138","value":false,"uid":"","description":"\n\nFor any continuous $f:\\mathbb{Q}^*\\to \\mathbb{Q}^*$, $f\\times f: X \\to X$ is also continuous.\nAs {S29|P138}, this gives us uncountably many continuous self-maps of $X$. \n","refs":[]},{"space":"S000031","property":"P000143","value":true,"uid":"","description":"\n\nThe space {S29} has the {P143} property. And the property is preserved by products, as shown in {{mathse:632320}}.\n","refs":[{"name":"Weak Hausdorff space not KC","mathse":632320}]},{"space":"S000031","property":"P000169","value":false,"uid":"","description":"\n\nEvery neighborhood of $(\\infty, \\infty)$ is dense (consequence of Lemma 2 in {{mathse:5070821}}).\nTherefore every regular open neighborhood of $(\\infty, \\infty)$ is the entire space.\n","refs":[{"name":"Answer to \"Does the one-point compactification of $\\mathbb Q$ have the fixed point property?\"","mathse":5070821}]},{"space":"S000031","property":"P000183","value":true,"uid":"","description":"\n\n$X$ is the square of {S29} and {S29|P183}.\n","refs":[]},{"space":"S000031","property":"P000204","value":false,"uid":"","description":"\n\nSince {S29|P36}, from {{mathse:214317}} it follows that if you remove a point from $X$ then it remains connected.\n","refs":[{"name":"$(X\\times Y)\\setminus (A\\times B)$ is connected for $A,B$ proper subsets of connected $X,Y$","mathse":214317}]},{"space":"S000031","property":"P000214","value":false,"uid":"","description":"\n\n$X$ has {S29} as a subspace and {S29|P214}.\n","refs":[]},{"space":"S000031","property":"P000216","value":false,"uid":"","description":"\n\n$X$ has {S29} as a subspace and {S29|P216}.\n","refs":[]},{"space":"S000031","property":"P000243","value":true,"uid":"","description":"\n\n{S29|P243},\nand $X$ is a product of two copies of that space.\n","refs":[]},{"space":"S000032","property":"P000016","value":true,"uid":"","description":"\n\nThe Hilbert Cube is compact by Tychonoff Product Theorem\n\nSee item #4 for space #38 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000032","property":"P000026","value":true,"uid":"","description":"\n\nThe set $D = \\{(q_0,\\ldots,q_n,0,0,\\ldots,0,\\ldots)\\mid n \\in \\mathbb{N}, \\forall_{i \\le n} q_i \\in \\mathbb{Q} \\}$ is countable (as a union of countable sets that are equinumerous to $\\mathbb{Q}^n$ for all finite $n$...) and dense, as basic product open sets only depend on finitely many coordinates.\n\nSee item #3 for space #38 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000032","property":"P000038","value":true,"uid":"","description":"\n\nSee item #5 for space #38 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000032","property":"P000043","value":true,"uid":"","description":"\n\nSee item #5 for space #38 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000032","property":"P000053","value":true,"uid":"","description":"\n\n$X$ is a countable power of {S158} and {S158|P53}.\n","refs":[]},{"space":"S000032","property":"P000086","value":true,"uid":"","description":"\n\nSee {{mathse:2194}}.\n","refs":[{"name":"Why is the Hilbert Cube homogeneous?","mathse":2194}]},{"space":"S000032","property":"P000089","value":true,"uid":"","description":"\n\nSince we can embed $[0,1]^\\omega$ as a compact subspace of the topological vector space $\\mathbb{R}^\\omega$ ({S30}), the property follows from the Schauder fixed-point theorem (see {{wikipedia:Schauder fixed-point theorem}}).\n","refs":[{"name":"Homogeneous topological space with the fixed-point property","mathse":884045},{"name":"Schauder fixed-point theorem on Wikipedia","wikipedia":"Schauder_fixed-point_theorem"}]},{"space":"S000032","property":"P000120","value":false,"uid":"","description":"\n\nIt contains a non-open subspace homeomorphic to {S170},\nwhich is a contradiction with Lemma 3.5 in {{zb:1458.54023}}.\n","refs":[{"name":"Locally ordered topological spaces (P. Pikul)","zb":"1458.54023"}]},{"space":"S000032","property":"P000184","value":false,"uid":"","description":"\n\nEvery Euclidean space $\\mathbb R^n$, which is homeomorphic to $(0,1)^n$, embeds into $X$.\nTherefore, as explained in {{mathse:5009117}}, $X$ cannot itself be embedded into a fixed $\\mathbb R^m$.\n","refs":[{"name":"The Hilbert cube cannot be embedded in some Euclidean space $\\mathbb R^n$","mathse":5009117}]},{"space":"S000032","property":"P000199","value":true,"uid":"","description":"\n\nSame as {S103|P199}.\n","refs":[]},{"space":"S000033","property":"P000016","value":false,"uid":"","description":"\n\nSee item #7 for space #40 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000033","property":"P000057","value":true,"uid":"","description":"\n\nSee item #5 for space #40 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000033","property":"P000190","value":true,"uid":"","description":"\n\nBy definition.\n","refs":[]},{"space":"S000033","property":"P000203","value":true,"uid":"","description":"\n\nAll points except $\\omega\\in X$ are isolated.\n","refs":[]},{"space":"S000034","property":"P000016","value":true,"uid":"","description":"\n\nSee item #6 for space #41 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000034","property":"P000126","value":false,"uid":"","description":"\n\nThe set $A=\\{\\alpha\\in X:\\omega\\le\\alpha<\\omega+\\omega\\}$ is not open\n(because it does not contain a neighborhood of $\\omega$)\nand not closed (because it does not contain its accumulation point $\\omega+\\omega$).\n","refs":[]},{"space":"S000034","property":"P000181","value":true,"uid":"","description":"\n\nBy construction.\n","refs":[]},{"space":"S000034","property":"P000190","value":true,"uid":"","description":"\n\nBy definition.\n","refs":[]},{"space":"S000035","property":"P000019","value":true,"uid":"","description":"\n\nSee item #8 for space #42 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000035","property":"P000062","value":false,"uid":"","description":"\n\nConsider the open cover $\\mathcal{O}=\\{[0,\\alpha)\\mid \\alpha \\in X\\}$.\nEvery countable subset of $\\omega_1$ has an upper bound in $\\omega_1$.\nSo if $\\mathcal U$ is a countable subcollection of $\\mathcal O$, there is some $\\alpha\\in\\omega_1$ such that\n$\\bigcup\\mathcal U\\subseteq[0,\\alpha]$, which is not dense in $X$.\n","refs":[]},{"space":"S000035","property":"P000076","value":true,"uid":"","description":"\n\n$X$ is a closed subspace of {S149} and {S149|P76}.\n","refs":[]},{"space":"S000035","property":"P000082","value":true,"uid":"","description":"\n\nFor $\\alpha \\in X$ the open neighborhood $[0,\\alpha + 1)\\subseteq X$ is {P57} and {P190} and thus {P53} [(Explore)](https://topology.pi-base.org/spaces?q=Countable+%2B+Ordinal+space+%2B+%7EMetrizable).\n","refs":[]},{"space":"S000035","property":"P000093","value":true,"uid":"","description":"\n\nEvery $\\alpha\\in\\Omega$ has $[0,\\alpha]$ as a countable neighborhood.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000035","property":"P000114","value":true,"uid":"","description":"\n\nBy construction.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000035","property":"P000186","value":true,"uid":"","description":"\n\n$X$ is a subspace of {S149} and {S149|P186}.\n","refs":[]},{"space":"S000035","property":"P000190","value":true,"uid":"","description":"\n\nBy definition.\n","refs":[]},{"space":"S000036","property":"P000016","value":true,"uid":"","description":"\n\n$[0,\\omega_1]$ is compact\n\nSee item #8 for space #43 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000036","property":"P000103","value":false,"uid":"","description":"\n\nThe subspace $[0,\\omega_1)$ is countably compact ({S35|P19})\nbut not closed in $[0,\\omega_1]$.\n\nSee example 2.1 in {{zb:1224.54011}}.\n","refs":[{"name":"On minimal strongly KC-spaces (Sun, Xu, Li)","zb":"1224.54011"}]},{"space":"S000036","property":"P000114","value":true,"uid":"","description":"\n\nBy construction.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000036","property":"P000151","value":true,"uid":"","description":"\n\nIn the first round, after Player I picked some open cover, Player II can then pick a cocountable neighborhood $U$ of the point $\\omega_1 \\in X$. Choose an enumeration $X\\setminus U = \\{x_1,x_2,\\dots\\}$. Player II then has a winning strategy by picking a neighborhood of $x_n$ in the $n+1$-th step of the game.\n","refs":[]},{"space":"S000036","property":"P000190","value":true,"uid":"","description":"\n\nBy definition.\n","refs":[]},{"space":"S000037","property":"P000020","value":true,"uid":"","description":"\n\nIf a sequence of elements of $X$ contains the same value infinitely many times, it has a convergent subsequence. Otherwise, the sequence will have a subsequence with values all in $[0,\\omega_1)$. That subsequence will in turn have a convergent subsequence since {S35} is {P20}.\n","refs":[{"name":"Relationship between weak Hausdorff and US properties","mathse":4267169}]},{"space":"S000037","property":"P000029","value":false,"uid":"","description":"\n\nThere are uncountably many isolated points, namely all the countable successor ordinals.\n","refs":[{"name":"Relationship between weak Hausdorff and US properties","mathse":4267169}]},{"space":"S000037","property":"P000051","value":true,"uid":"","description":"\n\nThe reasoning is similar to showing {S36} is {P51}. Let $A$ be a nonempty subset of $X$. If $A$ contains an element different from $\\omega_1$ and $\\omega'_1$, the minimum element of $A$ is isolated in $A$. Otherwise, $A$ is a subspace of $\\{\\omega_1,\\omega'_1\\}$, which has the discrete topology, and every element of $A$ is isolated in $A$.\n","refs":[{"name":"Relationship between weak Hausdorff and US properties","mathse":4267169}]},{"space":"S000037","property":"P000061","value":true,"uid":"","description":"\n\nThe Tychonoff reflection of $X$ is {S36},\nwith the reflection map $\\varphi:X\\to [0,\\omega_1]$\nsending $\\omega'_1$ to $\\omega_1$ and every other element to itself.\n\nSince {S36|P61}\nand the reflection map $\\varphi$ is open,\nthe result follows from Proposition 2 in {{mathse:5108293}}.\n","refs":[{"name":"Cozero complemented property for a space and its Tychonoff reflection","mathse":5108293}]},{"space":"S000037","property":"P000085","value":false,"uid":"","description":"\n\n$X$ contains $\\omega+1$ ({S20}) as clopen subspace and {S20|P85}.\n","refs":[]},{"space":"S000037","property":"P000099","value":true,"uid":"","description":"\n\nSee Example 1 in {{mathse:4268177}}.\n","refs":[{"name":"Answer to \"Relationship between weak Hausdorff and US properties\"","mathse":4268177}]},{"space":"S000037","property":"P000101","value":false,"uid":"","description":"\n\nThe subspace $A=[0,\\omega_1]$ is not closed since $\\{\\omega'_1\\}$ is not open, but $A$ is a retract by virtue of the continuous function $f$ defined by $f(a)=a$ for all $a \\in A$ and $f(\\omega'_1) = \\omega_1$.\n","refs":[]},{"space":"S000037","property":"P000114","value":true,"uid":"","description":"\n\nBy construction.\n","refs":[{"name":"Relationship between weak Hausdorff and US properties","mathse":4267169}]},{"space":"S000037","property":"P000115","value":false,"uid":"","description":"\n\nAssume by contradiction that $X$ is {P115}.\nLet $\\mathcal{U} = \\{\\omega_1\\cup \\{\\omega_1\\}, \\omega_1\\cup \\{\\omega_1'\\}\\}$ and take a sequence $\\mathcal{V}_n$ of open covers of $X$, each refining $\\mathcal{U}$, such that for each $x\\in X$ there exists $n$ with $\\text{ord}(x, \\mathcal{V}_n) = 1$.\nFor each $n$, $\\mathcal V_n$ contains a neighborhood of $\\omega_1$ and a neighborhood of $\\omega'_1$, necessarily distinct; therefore, for some $\\alpha_n<\\omega_1$ the interval $(\\alpha_n,\\omega_1)$ is contained in two different members of $\\mathcal V_n$.\nLet $x = \\sup_n \\alpha_n + 1 < \\omega_1$.\nThen $\\text{ord}(x, \\mathcal{V}_n)\\geq 2$ for all $n$, contradiction.\n","refs":[]},{"space":"S000037","property":"P000120","value":true,"uid":"","description":"\n\nBy definition the sets $X\\setminus\\{\\omega_1\\}$ and $X\\setminus\\{\\omega'_1\\}$ are open and homeomorphic to $\\omega_1+1$.\n","refs":[]},{"space":"S000037","property":"P000130","value":true,"uid":"","description":"\n\n$X$ is covered by the open sets $[0,\\omega_1]$ and $[0,\\omega'_1]$, each homeomorphic to {S36} and hence {P130}.\n","refs":[{"name":"Relationship between weak Hausdorff and US properties","mathse":4267169}]},{"space":"S000037","property":"P000151","value":true,"uid":"","description":"\n\n$X$ is union of two subspaces homeomorphic to {S36} and {S36|P151}.\n","refs":[]},{"space":"S000037","property":"P000169","value":false,"uid":"","description":"\n\nIf $\\omega_1\\in U$, where $U$ is regular open, then $(\\alpha, \\omega_1]\\subseteq U$ for some $\\alpha < \\omega_1$, implying $\\operatorname{int}(\\operatorname{cl}((\\alpha, \\omega_1])) = (\\alpha, \\omega_1]\\cup \\{\\omega_1'\\}\\subseteq U$. So there is no regular open neighbourhood of $\\omega_1$ missing $\\omega_1'$.\n","refs":[]},{"space":"S000037","property":"P000174","value":true,"uid":"","description":"\n\nEvery point has a neighborhood homeomorphic to {S36} and {S36|P174}.\nTherefore the space is {P174} since that is a local property.\n","refs":[]},{"space":"S000037","property":"P000216","value":false,"uid":"","description":"\n\n$X$ contains {S36} as a subspace and {S36|P216}.\n","refs":[]},{"space":"S000037","property":"P000217","value":true,"uid":"","description":"\n\nThe Tychonoff reflection of $X$ is {S36},\nwith the reflection map $\\varphi:X\\to [0,\\omega_1]$\nsending $\\omega'_1$ to $\\omega_1$ and every other element to itself.\n\nSince {S36|P217},\nthe result follows from point 1 of Proposition in {{mathse:5108583}}.\n","refs":[{"name":"Answer to \"Cozero complemented property for a space and its Tychonoff reflection\"","mathse":5108583}]},{"space":"S000038","property":"P000019","value":true,"uid":"","description":"\n\nEvery infinite countable set $S\\subseteq X$ is bounded in $X$,\nand thus is contained in some interval $[p,q]$ with $px$.\nThe interval $[p,q)$ is a neighborhood of $x$ that is homeomorphic to {S210}.\n","refs":[]},{"space":"S000039","property":"P000016","value":true,"uid":"","description":"\n\nSee item #5 for space #46 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000039","property":"P000036","value":true,"uid":"","description":"\n\nSee item #6 for space #46 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000039","property":"P000037","value":false,"uid":"","description":"\n\nSee item #6 for space #46 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000039","property":"P000046","value":false,"uid":"","description":"\n\n$X$ contains {S38} as a subspace and {S38|P46}.\n","refs":[]},{"space":"S000039","property":"P000061","value":true,"uid":"","description":"\n\nSee the \"Related spaces\" section at the end of {{mathse:5106636}}.\n","refs":[{"name":"Answer to \"Is the long ray cozero complemented?\"","mathse":5106636}]},{"space":"S000039","property":"P000065","value":true,"uid":"","description":"\n\nBy construction, as $\\aleph_1 \\cdot \\mathfrak c = \\mathfrak c$.\n","refs":[]},{"space":"S000039","property":"P000103","value":false,"uid":"","description":"\n\n$X$ contains {S38} as a subspace and {S38|P19} but not closed in $X$.\n","refs":[]},{"space":"S000039","property":"P000133","value":true,"uid":"","description":"\n\nBy definition.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000039","property":"P000174","value":true,"uid":"","description":"\n\n$X$ is well-based at every point of its open subset {S38}, since {S38|P174}. And $X$ is well-based at $\\langle\\omega_1,0\\rangle$, as it is the maximum point of its order (see {{mathse:4886508}}).\n","refs":[{"name":"Answer to \"Are linearly ordered topological spaces well-based?\"","mathse":4886508}]},{"space":"S000039","property":"P000215","value":false,"uid":"","description":"\n\n$X$ contains {S38} as a subspace and {S38|P162}.\n","refs":[]},{"space":"S000040","property":"P000009","value":true,"uid":"","description":"\n\nAny totally-ordered set is clearly this, so we consider when $p$ is one of the points in question. That is, let $q$ be a point on the long line and $p$ the special point added. Let $r$ be the least ordinal greater than $q$. Define the function $f:X\\to [0,1]$ as follows for $t\\le q$, define $f(q)=0$. Now note that $[q, r]$ is homeomorphic with $[0,1]$, so for $q \\le t \\le r$, let $f(t)$ be such a homeomorphism with $f(q)=0$ and $f(r)=1$. Then define $f(t)=1$ for $t > r$ or $t=p$.\n","refs":[]},{"space":"S000040","property":"P000020","value":true,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #47 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000040","property":"P000029","value":false,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #47 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000040","property":"P000037","value":false,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #47 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000040","property":"P000041","value":false,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #47 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000040","property":"P000046","value":false,"uid":"","description":"\n\nGiven any point $x$ in the long line, the interval from $x$ to the least ordinal greater than $x$ is homeomorphic to $[0,1]$, hence is path connected.\n","refs":[]},{"space":"S000040","property":"P000056","value":false,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #47 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000040","property":"P000062","value":true,"uid":"","description":"\n\nProof from {{mathse:4874239}}:\ntake an open cover $\\mathcal U$, and choose $\\alpha<\\omega_1$ where $U_\\alpha\\subseteq V\\in \\mathcal U$. \nWe note that $\\alpha\\times [0,1)\\cup\\{\\langle \\alpha,0\\rangle\\}\\cong[0,1]$ is compact, so choose \n$\\mathcal F\\subseteq\\mathcal U$ finite covering it. Then $\\mathcal F\\cup\\{V\\}$ covers \n$X\\setminus(\\omega_1 \\times \\{0\\})$, which is dense.\n","refs":[{"name":"Is there a first countable, $T_1$, weakly Lindelof, sequentially compact space which is not also compact?","mathse":4874239}]},{"space":"S000040","property":"P000065","value":true,"uid":"","description":"\n\nBy construction, as $\\aleph_1 \\cdot \\mathfrak c = \\mathfrak c$.\n","refs":[]},{"space":"S000040","property":"P000081","value":false,"uid":"","description":"\n\nThe point $\\infty\\in cl(Y)$, but $\\infty\\not\\in cl(C)$ for any countable\n$C\\subseteq Y$.\n","refs":[]},{"space":"S000040","property":"P000117","value":false,"uid":"","description":"\n\n$X$ contains {S153} as a subspace and {S153|P117}.\n","refs":[]},{"space":"S000040","property":"P000120","value":true,"uid":"","description":"\n\nIt suffices to observe that the neighbourhood $U_1$ of $\\infty$ is {P133}\nwith lexicographic order on $U_1\\cap Y$ and $\\infty$ as the maximal element.\n","refs":[]},{"space":"S000040","property":"P000204","value":true,"uid":"","description":"\n\n$X$ is connected. See item #2 for space #47 in {{zb:0386.54001}}.\n\nAnd, $X \\setminus \\{\\langle 1,0 \\rangle\\}$ is disconnected.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000040","property":"P000205","value":false,"uid":"","description":"\n\n$\\langle 0,0 \\rangle$ is not a cut point.\n","refs":[]},{"space":"S000041","property":"P000016","value":true,"uid":"","description":"\n\nSee item #2 for space #48 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000041","property":"P000028","value":true,"uid":"","description":"\n\nFix a point $\\left\\in [0,1]\\times [0,1]$. Consider all open intervals $(\\left,\\left)$ containing $\\left$ with all of $a,b,c,d$ rational, or $a=p$ or $c=p$. This is a countable basis at $\\left$.\n\nSee item #4 for space #48 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000041","property":"P000036","value":true,"uid":"","description":"\n\nSee item #5 for space #48 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000041","property":"P000037","value":false,"uid":"","description":"\n\nThe path components of $X$ are the sets of the form $\\{p\\}\\times [0,1]$ for $p\\in [0,1]$. This is because given any two points $\\left$ and $\\left$ with $p < p'$, there exists an uncountable collection of disjoint open sets in the interval from $\\left$ to $\\left$.\n\nSee item #5 for space #48 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000041","property":"P000046","value":false,"uid":"","description":"\n\nFix any $p\\in [0,1]$. Then the map $f:[0,1]\\to X$ given by $f(t)=\\left$ is a non-constant path in the lexicographic square.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000041","property":"P000061","value":false,"uid":"","description":"\n\nThe set $U = \\bigl( \\left[ 0, 1 \\right] \\cap \\mathbb Q \\bigr) \\times (0, 1)$ is an $F_\\sigma$ open set in $X$, therefore a cozero set, since {S41|P13} (see Corollary 1.5.13 in {{zb:0684.54001}}).\n\nSuppose $V$ is a cozero set disjoint from $U$ such that $U \\cup V$ is dense.\nLet $A := \\operatorname{int}(X \\setminus U) = \\bigl( \\left[ 0, 1 \\right] \\setminus \\mathbb Q \\bigr) \\times (0, 1)$.\nThen we have $V \\subseteq A \\subseteq \\operatorname{cl} V$.\n\nLet $p \\colon X \\to [0, 1]$ be the projection mapping to the first coordinate. Then $p(V) = \\left[ 0, 1 \\right] \\setminus \\mathbb Q$.\n\nWrite $V = \\bigcup_{n \\in \\omega} F_n$ where $F_n$ is closed in $X$, then $p(V) = \\bigcup_{n \\in \\omega} p(F_n)$. So there is at least one $F_n$ such that $p(F_n)$ is infinite; thus $p(F_n)$ contains an accumulation point $r \\in [0, 1]$.\nThen either $\\left< r, 0 \\right> \\in F_n$ or $\\left< r, 1 \\right> \\in F_n$, contradicting $F_n \\subseteq V \\subseteq A$.\n\nTherefore, $X$ is not {P61}.\n","refs":[{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"space":"S000041","property":"P000076","value":false,"uid":"","description":"\n\n$X$ contains the closed subspace\n$\\bigl( (0, 1] \\times \\{0\\} \\bigr) \\cup \\bigl( [0, 1) \\times \\{1\\} \\bigr)$,\nwhich is homeomorphic to {S93}.\n\n{S93|P76}, so neither is $X$\nsince the {P76} property is hereditary with respect to closed sets.\n","refs":[]},{"space":"S000041","property":"P000133","value":true,"uid":"","description":"\n\nBy definition.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000042","property":"P000001","value":true,"uid":"","description":"\n\nGiven distinct $x,y \\in \\mathbb{R}$, either $x < y$ (whence $B_x$ is an open neighborhood of $y$ not containing $x$), or $y < x$ (whence $B_y$ is an open neighborhood of $x$ not containing $y$).\n\nSee item #4 for space #50 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000042","property":"P000027","value":true,"uid":"","description":"\n\nSee item #6 for space #50 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000042","property":"P000038","value":true,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #50 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000042","property":"P000043","value":true,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #50 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000042","property":"P000056","value":true,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #50 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000042","property":"P000065","value":true,"uid":"","description":"\n\nBy definition.\n","refs":[]},{"space":"S000042","property":"P000086","value":true,"uid":"","description":"\n\nFor any $a,b\\in\\mathbb R$, $x\\mapsto x-a+b$ is a self-homeomorphism taking $a$ to $b$.\n","refs":[]},{"space":"S000042","property":"P000089","value":false,"uid":"","description":"\n\nFor any non-zero $a$, $x\\mapsto x+a$ is a self-homeomorphism with no fixed points.\n","refs":[]},{"space":"S000042","property":"P000130","value":true,"uid":"","description":"\n\nEach of the rays $R_a = [a, \\to)$ is compact, since every open set containing $a$ must cover $R_a$. Each point $x$ then has a local base $\\{R_a \\mid a < x\\}$ of compact neighborhoods.\n","refs":[]},{"space":"S000042","property":"P000152","value":true,"uid":"","description":"\n\nSee {{mathse:3200061}}: contain $-n$ from the $n$th cover.\n","refs":[{"name":"Answer to Rothberger game and Meager set.","mathse":3200061}]},{"space":"S000042","property":"P000196","value":true,"uid":"","description":"\n\nThe open sets form a chain under inclusion.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000043","property":"P000026","value":true,"uid":"","description":"\n\nThe set $\\mathbb Q$ is dense in $X$.\n\nSee item #4 for space #51 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000043","property":"P000050","value":true,"uid":"","description":"\n\nEvery basic open set $[a,b)$ is both open and closed.\n\nSee item #7 for space #51 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000043","property":"P000063","value":false,"uid":"","description":"\n\n{S43} admits a compactification homeomorphic to {S93}\nand is not a $G_\\delta$ subset therein.\nSee {{mathse:5011073}}.\n","refs":[{"name":"Answer to \"Is every Baire space Cech-complete?\"","mathse":5011073}]},{"space":"S000043","property":"P000065","value":true,"uid":"","description":"\n\nBy definition.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000043","property":"P000066","value":false,"uid":"","description":"\n\nSee Example 3 in .\n\nAlso shown in Lemma 13 of {{doi:10.1285/i15900932v22n2p167}}.\n","refs":[{"name":"Combinatorics of open covers (IX): Basic properties (Babinkostova & Scheepers)","doi":"10.1285/i15900932v22n2p167"}]},{"space":"S000043","property":"P000076","value":false,"uid":"","description":"\n\nThe square of {S43} is {S76}, and {S76|P76}.\nSo {S43} is not {P76}.\n\nSee Example 3 of {{doi:10.1016/j.topol.2014.06.014}}.\n","refs":[{"name":"An infinite game with topological consequences (Bell)","doi":"10.1016/j.topol.2014.06.014"}]},{"space":"S000043","property":"P000086","value":true,"uid":"","description":"\n\nEvery translation $x\\mapsto x+a$ is a homeomorphism of $X$ onto itself.\n","refs":[]},{"space":"S000043","property":"P000102","value":false,"uid":"","description":"\n\nAssume $d$ is a semimetric on $X$. For $x\\in X$ pick $r(x)>0$\nsuch that $B_d(x,r(x))\\subseteq [x,+\\infty)$. For $n\\in \\N$ define\n$A_n=\\{x\\in\\R: r(x) > 1/n \\}$. Then $A_m$ is uncountable for some $m\\in \\N$.\nObserve that for distinct $x,y\\in A_m$ either\n$y\\notin [x,+\\infty)$ or $x\\notin [y,+\\infty)$,\nimplying $d(x,y)=d(y,x)>1/m$.\n\nSince semimetric balls form local bases at every point of $X$, their interiors are not empty,\nhence for every $a\\in A_m$ there exists an interval from the canonical base $[a,a+\\varepsilon_a)\\subset B_d(a,1/m)$ not containing other points of $A_m$. These intervals form an uncountable disjoint family, contradicting the fact that\n{S43|P29}.\n","refs":[]},{"space":"S000043","property":"P000149","value":false,"uid":"","description":"\n\nNot every power of the Sorgenfrey line is Lindelöf, since {S76|P18}.\n","refs":[]},{"space":"S000043","property":"P000154","value":true,"uid":"","description":"\n\nBy construction.\n","refs":[]},{"space":"S000043","property":"P000166","value":true,"uid":"","description":"\n\nThe topology contains the topology for {S25}.\n","refs":[]},{"space":"S000043","property":"P000182","value":false,"uid":"","description":"\n\nSuppose $\\mathscr N$ is a network for $X$. For every $x\\in X$ there is some $U_x\\in\\mathscr N$ such that $x\\in U_x\\subseteq[x,x+1)$. As $x$ is the smallest element of $U_x$, all the $U_x$ are distinct and $\\mathscr N$ is uncountable.\n","refs":[]},{"space":"S000043","property":"P000206","value":true,"uid":"","description":"\n\nPlayer 2 has the following winning strategy:\n\nFor every $n<\\omega$ it is enough to pick $V_n:=[x_n,y_n)$, where $[x_n,y_n]\\subset U_n$, $y_n>x_n$.\n\nThen $\\bigcap_{n=0}^\\infty U_n \\supset \\bigcap_{n=0}^\\infty [x_n,y_n]$, which is nonempty\nas a nested intersection of nonempty compact sets in the Euclidean topology.\n","refs":[]},{"space":"S000044","property":"P000027","value":true,"uid":"","description":"\n\nThe topology on $X$ is itself countable.\n\nSee item #5 for space #52 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000044","property":"P000033","value":false,"uid":"","description":"\n\nThe cover $\\{U_n\\}$ has no finite refinement and $\\frac{1}{4}$ is in every set in the cover.\n\nSee item #8 for space #52 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000044","property":"P000038","value":true,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #52 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000044","property":"P000043","value":true,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #52 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000044","property":"P000065","value":true,"uid":"","description":"\n\nBy definition.\n","refs":[]},{"space":"S000044","property":"P000139","value":false,"uid":"","description":"\n\nEach nonempty open set in $X$ is an open interval of $\\mathbb{R}$ and thus has infinitely many elements.\n","refs":[]},{"space":"S000044","property":"P000196","value":true,"uid":"","description":"\n\nThe open sets form a chain under inclusion.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000044","property":"P000226","value":true,"uid":"","description":"\n\n$X$ is a product of {P226} spaces, since {S199|P226} and {S194|P226}.\n","refs":[]},{"space":"S000045","property":"P000013","value":false,"uid":"","description":"\n\n$\\{-1\\}$ and $\\{1\\}$ are disjoint closed sets, but any neighborhood of either must contain 0.\n\nSee item #1 for space #53 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000045","property":"P000016","value":true,"uid":"","description":"\n\nGiven any open cover of $X$, any two sets containing -1 and 1 must cover $X$.\n\n\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000045","property":"P000027","value":true,"uid":"","description":"\n\nThe sets $[-1,b),(a,b),(a,1]$ for $a<0\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000045","property":"P000038","value":true,"uid":"","description":"\n\nGiven distinct points $x,y \\in X$ there is an injective path $[0,1] \\rightarrow [-1,1]$ from $x$ to $y$ in the usual Euclidean topology.\nSince the topology on $X$ is coarser, this map is also an injective path in $X$.\n\n\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000045","property":"P000043","value":true,"uid":"","description":"\n\nEvery open subset of $X$ is {P38}\nby the same argument as in {S45|P38}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000045","property":"P000073","value":true,"uid":"","description":"\n\nSee the first part of {{mathse:5007860}}.\n","refs":[{"name":"Answer to \"For a compact sober \"highly non-$T_1$\" space, how much \"highly connectedness\" is needed to imply it's a spectral space?\"","mathse":5007860}]},{"space":"S000045","property":"P000075","value":false,"uid":"","description":"\n\nSee the first part of {{mathse:5007860}}.\n","refs":[{"name":"Answer to \"For a compact sober \"highly non-$T_1$\" space, how much \"highly connectedness\" is needed to imply it's a spectral space?\"","mathse":5007860}]},{"space":"S000045","property":"P000089","value":true,"uid":"","description":"\n\nLet $f:X\\to X$ be a continuous function.\nSince $x\\mapsto -x$ is a homeomorphism, without loss of generality we can assume $f(0)\\geq 0$.\nIf $f(0)=0$ we are done. Otherwise consider $c=f(0)>0$.\nThen $0\\notin f^{-1}([-1,c))$, hence the open set $f^{-1}([-1,c))$ is empty and consequently $f(X)\\subset [0,1]$.\nThe restriction $f|_{[0,1]}$ is then a continuous self map of $[0,1]$ with the subspace topology,\nwhich is homeomorphic to {S159}.\nThen use {S159|P89}.\n","refs":[]},{"space":"S000045","property":"P000130","value":true,"uid":"","description":"\n\nFor $t\\in[0,1]$ consider its neighbourhood $U_\\varepsilon=[-\\varepsilon,t+\\varepsilon]\\cap X$ for $\\varepsilon>0$.\n$U_\\varepsilon$ is covered by union of arbitrary neighbourhoods of its least and greatest element, hence\nit is compact (every open cover has an at most two-element subcover). Sets $U_\\varepsilon$ ($\\varepsilon>0$) form a neighbourhood basis for $t$.\n\nSince $t\\mapsto -t$ is a homeomorphism, every point has a local base of compact sets.\n","refs":[]},{"space":"S000045","property":"P000139","value":false,"uid":"","description":"\n\nIt is evident from the definition that no open set is a singleton.\n","refs":[]},{"space":"S000045","property":"P000152","value":true,"uid":"","description":"\n\nThe space satisfies the *topologically countable* condition equivalent to {P152}.\nIt suffices to take the set $\\{x_n:n<\\omega\\}=\\{-1,1\\}$.\nEvery neighbourhood of $-1$ contains $A_-:=[-1,0]$, every neighbourhood of $1$ contains $A_+:=[0,1]$ and $X=A_+\\cup A_-$.\n","refs":[]},{"space":"S000045","property":"P000201","value":true,"uid":"","description":"\n\nEvery nonempty open set contains $0$.\n\n\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000045","property":"P000216","value":false,"uid":"","description":"\n\nThe open cover $\\{(-t,t): t\\in(0,1) \\}$ of the subspace $(-1,1)$ has no finite open refinement and since $0$ is a generic point, any infinite refinement cannot be locally finite at $0$.\n","refs":[]},{"space":"S000046","property":"P000027","value":true,"uid":"","description":"\n\nThe topology on $X$ has a countable subbase, hence a countable base.\n\nSee item #5 for space #54 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000046","property":"P000031","value":true,"uid":"","description":"\n\nThe open cover $\\{S_n\\}$ is a refinement of every open cover of $X$.\nAnd it is point-finite as every $x > 1$ is contained in only one $S_n$ and every $x < 1$ is contained in finitely many $S_n$.\n\nSee item #7 for space #54 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000046","property":"P000032","value":false,"uid":"","description":"\n\nThe open cover $\\{S_n\\}$ has no proper refinement but $S_1 = (0,1) \\cup (1,2)$ intersects every set in the cover, so $\\{S_n\\}$ is not locally finite.\n\nSee item #6 for space #54 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000046","property":"P000038","value":true,"uid":"","description":"\n\n{S46|P199}, hence is {P37}\n[(Explore)](https://topology.pi-base.org/spaces?q=Contractible%2B%7EPath+connected).\n\nNow, using the fact that every point has $\\mathfrak c$ many points topologically indistinguishable from it,\none can construct an injective path between any two distinct points of $X$, as shown in {{mathse:5006475}}.\n","refs":[{"name":"Situation where a non-$T_0$ path-connected space is injectively path-connected","mathse":5006475}]},{"space":"S000046","property":"P000039","value":true,"uid":"","description":"\n\nSee item #2 for space #54 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000046","property":"P000043","value":true,"uid":"","description":"\n\n{S46|P90}, hence is {P42}\n[(Explore)](https://topology.pi-base.org/spaces?q=Alexandrov%2B%7ELocally+path+connected).\n\nNow given any open path connected set $V\\subseteq X$ (from a base of such sets),\n$V$ contains all the points of $X$ topologically indistinguishable from any of its points.\nUsing the fact that every point has $\\mathfrak c$ many points topologically indistinguishable from it,\none can construct an injective path in $V$ between any two distinct points of $V$, as shown in {{mathse:5006475}}.\n","refs":[{"name":"Situation where a non-$T_0$ path-connected space is injectively path-connected","mathse":5006475}]},{"space":"S000046","property":"P000056","value":true,"uid":"","description":"\n\nEach interval $U_n=(0,1/n)$ for $n\\ge 2$ is open in $X$ and dense since $X$ is {P39}.\nSo its complement $U_n^c=X\\setminus (0,1/n)$ is nowhere dense.\nSince the intersection of all the $U_n$ is empty,\n$X$ is the countable union of the nowhere dense sets $U_n^c$.\n","refs":[]},{"space":"S000046","property":"P000065","value":true,"uid":"","description":"\n\nClearly $|X| = \\mathfrak{c}$.\n","refs":[]},{"space":"S000046","property":"P000086","value":false,"uid":"","description":"\n\nLet $[x]$ be the set of points topologically indistinguishable from $x$. Then $[1/2] = [1/2, 1) \\cup (1, 2)$ is closed, while $[1/3] = [1/3, 1/2)$ is not closed.\n","refs":[]},{"space":"S000046","property":"P000090","value":true,"uid":"","description":"\n\nEach point $x\\in X$ has a smallest neighborhood, namely, one of the intervals $(0,1/n)$ for some $n\\ge 2$ or one of the sets $S_n$ for some $n\\ge 1$.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000046","property":"P000165","value":true,"uid":"","description":"\n\nFor each point $x\\in X$ there are uncountably many points topologically indistinguishable from it. So every nonempty closed set is uncountable and the space is vacuously {P165}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000046","property":"P000199","value":true,"uid":"","description":"\n\nSee {{mathse:5003024}}.\n","refs":[{"name":"Is the interlocking interval topology contractible?","mathse":5003024}]},{"space":"S000047","property":"P000001","value":true,"uid":"","description":"\n\nThe {S10|P1}, and the topological sum of a collection of $T_0$ spaces is $T_0$.\n\nSee item #2 for space #55 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000047","property":"P000002","value":false,"uid":"","description":"\n\nThe {S10|P2} and thus neither is $X$.\n\nSee item #2 for space #55 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000047","property":"P000014","value":true,"uid":"","description":"\n\nThe {S10|P14}, and the topological sum of a collection of completely normal spaces is completely normal.\n\nSee item #2 for space #55 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000047","property":"P000021","value":false,"uid":"","description":"\n\nThe subset $\\{2n-1:n=1,2,\\dots\\}$ is infinite and without limit point, as it is closed and discrete.\n","refs":[]},{"space":"S000047","property":"P000024","value":true,"uid":"","description":"\n\nFor each point, the summand $\\{2n-1,2n\\}$ that it belongs to is a closed and compact neighborhood.\n","refs":[]},{"space":"S000047","property":"P000049","value":true,"uid":"","description":"\n{S10|P49}, and a topological sum of extremally disconnected spaces is extremally disconnected.\n","refs":[]},{"space":"S000047","property":"P000094","value":true,"uid":"","description":"\nThe {S10|P94}, and the topological sum of a collection of locally finite spaces is locally finite.\n","refs":[]},{"space":"S000047","property":"P000126","value":false,"uid":"","description":"\n\n$\\{2, 3\\}$ is neither open nor closed.\n","refs":[]},{"space":"S000047","property":"P000146","value":true,"uid":"","description":"\n\nThe partition of $X$ by the open sets $\\{2n-1,2n\\}$ is a refinement of any open cover.\n\nSee item #3 for space #55 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000047","property":"P000181","value":true,"uid":"","description":"\n\nBy construction.\n","refs":[]},{"space":"S000047","property":"P000216","value":true,"uid":"","description":"\n\n{P216} is preserved by disjoint unions and {S10|P216}.\n","refs":[]},{"space":"S000048","property":"P000001","value":true,"uid":"","description":"\n\nBy inspection.\n","refs":[]},{"space":"S000048","property":"P000013","value":false,"uid":"","description":"\n\n$\\{0\\}$ and $\\{1\\}$ are disjoint closed sets but there are no nonempty disjoint open sets in $X$.\n","refs":[]},{"space":"S000048","property":"P000027","value":true,"uid":"","description":"\n\nBy construction, it is easy to see that there are only countably many open sets.\n","refs":[]},{"space":"S000048","property":"P000042","value":true,"uid":"","description":"\n\nEvery nonempty open subset of $X$ has a generic point, namely $p$. And {P201} implies\n{P37} [(Explore)](https://topology.pi-base.org/spaces?q=Has+a+generic+point+%2B+not+Path+connected).\n","refs":[]},{"space":"S000048","property":"P000044","value":false,"uid":"","description":"\n\n$\\{0,p\\}$ and $\\omega\\setminus\\{0\\}$ form a partition of $X$ into two connected subspaces, each with at least two points.\n","refs":[]},{"space":"S000048","property":"P000138","value":false,"uid":"","description":"\n\nAny $f: X \\to X$ s.t. $f(p) = p$ and restricts to a permutation of $\\omega$ is easily seen to be continuous. Since there are continuum many permutations of $\\omega$, $X$ has at least continuum many self-maps.\n","refs":[]},{"space":"S000048","property":"P000139","value":false,"uid":"","description":"\n\n{S15|P139}, and $p$ is assumed to be non-isolated in $X$.\n","refs":[]},{"space":"S000048","property":"P000181","value":true,"uid":"","description":"\n\nBy definition.\n","refs":[]},{"space":"S000048","property":"P000192","value":true,"uid":"","description":"\n\nA nonempty closed subset $A \\subset X$, by definition, is either the entirety of $X$, which has $p$ as a generic point; or is a\nnonempty finite subset of $\\omega$. In the latter case, $A$ has discrete topology, so it is {P39} iff it is a\nsingleton, which, of course, has a generic point as well.\n","refs":[]},{"space":"S000048","property":"P000201","value":true,"uid":"","description":"\n\nBy definition, because $p$ is contained in every nonempty open set.\n","refs":[]},{"space":"S000048","property":"P000208","value":true,"uid":"","description":"\n\nThe closed sets satisfy the descending chain condition, because all closed sets are finite except $X$.\n","refs":[]},{"space":"S000049","property":"P000001","value":true,"uid":"","description":"\n\nFor any two points $x1$ be an integer and let $\\varphi:X\\to X$ denote multiplication by $n$. Then $\\varphi$ has no fixed points. And $\\varphi$ is continuous since if $U_m$ is a basic open set then $\\varphi^{-1}(U_m)$ is open again. Indeed, $\\varphi^{-1}(U_m) = U_{m/n}$ if $n \\mid m$ and it is empty otherwise.\n","refs":[]},{"space":"S000049","property":"P000094","value":true,"uid":"","description":"\nEach point $n\\in X$ has a finite neighborhood $U_n$.\n","refs":[]},{"space":"S000049","property":"P000181","value":true,"uid":"","description":"\n\nBy definition.\n\nSee item #6 for space #57 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000049","property":"P000199","value":true,"uid":"","description":"\n\nSee {{mathse:4999893}}.\n","refs":[{"name":"Is the divisor topology from Steen-Seebach contractible?","mathse":4999893}]},{"space":"S000049","property":"P000226","value":false,"uid":"","description":"\n\nNote that for $x \\in X$, $\\overline{\\{x\\}}=\\{kx\\mid k \\in \\mathbb{N}_{>0}\\}$. Therefore $X$ cannot have a finite dense set, as $x_1\\cdots x_n + 1 \\notin \\overline{\\{x_1,\\dots, x_n\\}}$.\n","refs":[]},{"space":"S000050","property":"P000010","value":true,"uid":"","description":"\n\nIt is easy to check any regular open $U \\subsetneq \\mathbb{Q}$ is still regular open in $X$. And, of course, $X$ is regular open in itself. The result now easily follows from {S27|P10} and\n{S27|P202}.\n","refs":[]},{"space":"S000050","property":"P000014","value":true,"uid":"","description":"\n\nNote that as {S50|P202}, it is\n{P13} [(Explore)](https://topology.pi-base.org/spaces?q=Has+a+point+with+a+unique+neighborhood+%2B+not+Normal).\nNow, any open subset of $X$ is either $X$ itself, which is normal; or an open subset of $\\mathbb{Q}$, which is normal\nsince {S27|P14}.\n","refs":[]},{"space":"S000050","property":"P000027","value":true,"uid":"","description":"\n\n{S27|P27} and any open basis of {S27} together with $X$ forms an open basis of $X$.\n","refs":[]},{"space":"S000050","property":"P000041","value":false,"uid":"","description":"\n\nBeing {P41} passes to open subspaces. {S27} is an open subspace of $X$ and\n{S27|P41}.\n","refs":[]},{"space":"S000050","property":"P000045","value":true,"uid":"","description":"\n\nSince {S50|P202}, it is\n{P36} [(Explore)](https://topology.pi-base.org/spaces?q=Has+a+point+with+a+unique+neighborhood+%2B+not+Connected).\nAnd, $X\\setminus\\{\\infty\\}$ is {S27}. {S27|P47}.\n","refs":[]},{"space":"S000050","property":"P000056","value":true,"uid":"","description":"\n\nIt is easy to see that $\\{q,\\infty\\}$ is nowhere dense for all $q \\in \\mathbb{Q}$, and $X$ is the countable union of all such sets.\n","refs":[]},{"space":"S000050","property":"P000089","value":true,"uid":"","description":"\n\nThe Alexandrov coreflection of $X$ is {S12}. And {S12|P89}.\nSee below for a more elementary proof.\n\nLet $f : X \\to X$ be continuous, and assume for contradiction that $f$ has no fixed point. Then $f(\\infty) \\ne \\infty$, so $f(\\infty) \\in \\mathbb{Q}$, and $f(f(\\infty)) \\ne f(\\infty)$, so $f(\\infty) \\in \\mathbb{Q} \\setminus \\{f(f(\\infty))\\}$. Since the preimage of an open set by a continuous function is open, $$f^{-1}(\\mathbb{Q} \\setminus \\{f(f(\\infty))\\}) = \\{ x \\mid f(x) \\in \\mathbb{Q} \\setminus \\{f(f(\\infty))\\} \\}$$ is an open set, containing $\\infty$ but not containing $f(\\infty)$, contradicting the fact that the only open set containing $\\infty$ is $X$.\n","refs":[]},{"space":"S000050","property":"P000130","value":false,"uid":"","description":"\n\nBeing {P130} passes to open subspaces. {S27} is an open subspace of $X$ and\n{S27|P130}.\n","refs":[]},{"space":"S000050","property":"P000138","value":false,"uid":"","description":"\n\nSee {{mathse:5098358}}.\n","refs":[{"name":"Do the rational numbers $\\mathbb{Q}$ extended by a focal point have countably many continuous self-maps?","mathse":5098358}]},{"space":"S000050","property":"P000181","value":true,"uid":"","description":"\n\nBy construction.\n","refs":[]},{"space":"S000050","property":"P000192","value":true,"uid":"","description":"\n\nAny nonempty closed subset $A$ of $X$ has to contain $\\infty$. If $A = \\{\\infty\\}$, then $A$ clearly has a generic point. Otherwise,\nassume $\\{\\infty\\} \\subsetneq A$ and $A$ is {P39}. We see that $A$ cannot contain more than one point of\n$\\mathbb{Q}$, as {S27|P3} and $\\mathbb{Q}$ is open in $X$. Thus, $A = \\{q,\\infty\\}$ for some\n$q \\in \\mathbb{Q}$, and $q$ is a generic point of $A$.\n","refs":[]},{"space":"S000050","property":"P000202","value":true,"uid":"","description":"\n\nBy definition, the only neighborhood of $\\infty$ is $X$.\n","refs":[]},{"space":"S000050","property":"P000216","value":true,"uid":"","description":"\n\nLet $A \\subseteq X$ with its subspace topology.\nIf $\\infty \\in A$, then any open cover of $A$ contains $A$ itself, as $X$ is the only neighborhood of $\\infty$ in $X$. In particular $A$ is compact and therefore paracompact.\nIf $\\infty \\notin A$, we can view $A$ as a subspace of {S27}. So as {S27|P216}, $A$ is paracompact.\n","refs":[]},{"space":"S000051","property":"P000010","value":true,"uid":"","description":"\n\nConsidering the basis from the description, one may find the following:\n$$\\operatorname{int}\\operatorname{cl}\\{2i+1\\}=\\operatorname{int}\\{2i,2i+1,2i+2\\}=\\{2i+1\\}$$\n$$\\operatorname{int}\\operatorname{cl}\\{2i-1,2i,2i+1\\}=\\operatorname{int}\\{2i-2,2i-1,2i,2i+1,2i+2\\}=\\{2i-1,2i,2i+1\\}$$\n","refs":[]},{"space":"S000051","property":"P000021","value":false,"uid":"","description":"\n\nThe set $\\{2i\\mid i\\in\\mathbb Z\\}$ is closed, discrete, and infinite.\n","refs":[]},{"space":"S000051","property":"P000024","value":true,"uid":"","description":"\n\nConsidering the basis from the description, the closure of $\\{2i+1\\}$ is $\\{2i,2i+1,2i+2\\}$,\nthe closure of $\\{2i-1,2i,2i+1\\}$ is $\\{2i-2,2i-1,2i,2i+1,2i+2\\}$,\nand these sets are finite and therefore compact.\n","refs":[]},{"space":"S000051","property":"P000089","value":false,"uid":"","description":"\n\n$x\\mapsto x+2$ is a continuous map with no fixed points.\n","refs":[]},{"space":"S000051","property":"P000094","value":true,"uid":"","description":"\n\nBy construction.\n","refs":[]},{"space":"S000051","property":"P000145","value":true,"uid":"","description":"\n\nAs the {S51|P90}, the set of smallest neighborhoods (the basis given in the description)\nis a refinement of any open cover, and this collection is star-finite.\n","refs":[]},{"space":"S000051","property":"P000181","value":true,"uid":"","description":"\n\nBy construction.\n","refs":[]},{"space":"S000051","property":"P000199","value":true,"uid":"","description":"\n\nLet $I = [0, 1]$.\nWe view {S51} $K$ as the quotient space of {S25} with the quotient map $p \\colon \\mathbb R \\to K$ defined by $p(i) = 2 i$ and $p(x) = 2 i + 1$ if $i < x < i + 1$, for all $i \\in \\mathbb Z$.\n\nDefine a continuous map $G \\colon \\mathbb R \\times I \\to \\mathbb R$ with \\[ G(x, t) = \\operatorname{sgn} x \\cdot \\min\\left\\{ \\left|x\\right|, \\frac t {1 - t} \\right\\}, \\]\nthen $p \\circ G$ is a continuous map from $\\mathbb R \\times I \\to K$.\n\nMoreover, we have \\[ p(x) = p(y) \\implies p \\bigl( G(x, t) \\bigr) = p \\bigl( G(y, t) \\bigr), \\quad \\forall x, y \\in \\mathbb R, \\forall t \\in I, \\]\nthis induces a map $F \\colon K \\times I \\to K$ such that $F(p(x), t) = p \\bigl( G(x, t) \\bigr)$.\n\nBy Whitehead's theorem (Theorem 3.3.17 in {{zb:0684.54001}}), $p \\times \\mathrm{id}_I \\colon \\mathbb R \\times I \\to K \\times I$ is also a quotient map, and $F \\circ (p \\times \\mathrm{id}_I) = p \\circ G$ is continuous, so $F$ is continuous.\n\nActually $F$ gives a homotopy from $K$ to $\\{0\\}$ since $F(x, 1) = x$ and $F(x, 0) = 0$.\nTherefore, $K$ is {P199}.\n","refs":[{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"space":"S000051","property":"P000205","value":true,"uid":"","description":"\n\nBy inspection, the space is connected, and after removing a point $p$, the sets\n$\\{x\\mid xp\\}$ are open sets partitioning $\\mathbb Z\\setminus\\{p\\}$.\n","refs":[{"name":"Cut-point spaces (Honari & Bahrampour)","doi":"10.1090/s0002-9939-99-04839-x"}]},{"space":"S000052","property":"P000003","value":true,"uid":"","description":"\n\nSee item #4 for space #60 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000052","property":"P000004","value":false,"uid":"","description":"\n\nSee item #5 for space #60 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000052","property":"P000010","value":true,"uid":"","description":"\n\nSee item #12 for space #60 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000052","property":"P000027","value":true,"uid":"","description":"\n\nThis is trivial, as the base used for defining the topology, $\\{U_a(b) : a,b \\in X, gcd(a,b)=1\\}$, is clearly countable, being dependent on two parameters $a,b$ from the (countable) set $X$.\n\nSee item #6 for space #60 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000052","property":"P000041","value":false,"uid":"","description":"\n\nSee item #8 for space #60 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000052","property":"P000044","value":false,"uid":"","description":"\n\nSee {{mathse:5110702}}.\n","refs":[{"name":"The Golomb space and the Kirch space are not biconnected","mathse":5110702}]},{"space":"S000052","property":"P000060","value":true,"uid":"","description":"\n\nSee item #7 for space #60 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000052","property":"P000082","value":false,"uid":"","description":"\n\nBy Proposition 3.2 in {{zb:1474.54066}}, for all coprime $a,b \\in X$, the subspace $U'_a(b)\\subseteq X$ is not regular and thus not metrizable. \n","refs":[{"name":"On continuous self-maps and homeomorphisms of the Golomb space (T. Banakh et al.)","zb":"1474.54066"}]},{"space":"S000052","property":"P000086","value":false,"uid":"","description":"\n\nSee Lemma 5.4 in {{zb:1474.54066}}.\n","refs":[{"name":"On continuous self-maps and homeomorphisms of the Golomb space (T. Banakh et al.)","zb":"1474.54066"}]},{"space":"S000052","property":"P000089","value":false,"uid":"","description":"\n\nThe map $x \\mapsto 2x$ has no fixed points and is continuous;\nsee Theorem 4.4 in {{zb:1474.54066}}\n","refs":[{"name":"On continuous self-maps and homeomorphisms of the Golomb space (T. Banakh et al.)","zb":"1474.54066"}]},{"space":"S000052","property":"P000138","value":false,"uid":"","description":"\n\nSee Theorem 4.9 in {{zb:1474.54066}}.\n","refs":[{"name":"On continuous self-maps and homeomorphisms of the Golomb space (T. Banakh et al.)","zb":"1474.54066"}]},{"space":"S000052","property":"P000181","value":true,"uid":"","description":"\n\nBy definition.\n","refs":[]},{"space":"S000052","property":"P000204","value":false,"uid":"","description":"\n\nThe closures of any two base elements $U_a(b)$ and $U_c(d)$ intersect in infinitely many points.\n\nHowever if there were a cut point $p \\in X$, there would have to be base elements $U_{a'}(b'), U_{c'}(d')$, such that $\\overline{U_{a'}(b')} \\cap \\overline{U_{c'}(d')} \\subseteq \\{p\\}$.\n","refs":[]},{"space":"S000053","property":"P000003","value":true,"uid":"","description":"\n\nSee item #4 for space #61 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000053","property":"P000004","value":false,"uid":"","description":"\n\nSee item #5 for space #61 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000053","property":"P000010","value":false,"uid":"","description":"\n\nSee Corollary 4.4 of {{zb:1348.11010}}.\n","refs":[{"name":"Regular open arithmetic progressions in connected topological spaces on the set of positive integers (Szczuka, Paulina)","zb":"1348.11010"}]},{"space":"S000053","property":"P000027","value":true,"uid":"","description":"\n\nBy definition the space has a countable sub-basis. By taking finite intersections of sub-basis-elements one obtains a countable basis.\n\nSee item #6 for space #61 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000053","property":"P000041","value":true,"uid":"","description":"\n\nSee item #9 for space #61 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000053","property":"P000044","value":false,"uid":"","description":"\n\nSee {{mathse:5110702}}.\n","refs":[{"name":"The Golomb space and the Kirch space are not biconnected","mathse":5110702}]},{"space":"S000053","property":"P000060","value":true,"uid":"","description":"\n\nSee item #7 for space #61 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000053","property":"P000086","value":false,"uid":"","description":"\n\nSee {{zb:1479.54043}}.\n","refs":[{"name":"The Kirch space is topologically rigid","zb":"1479.54043"}]},{"space":"S000053","property":"P000089","value":false,"uid":"","description":"\n\nThe map $x \\mapsto 2x$ has no fixed points and is continuous.\n","refs":[]},{"space":"S000053","property":"P000138","value":false,"uid":"","description":"\n\nProposition 4.3 in {{zb:1474.54066}} that each progressive function is continuous still holds for {S53}, and the argument in Theorem 4.9 implies that $C(X)$ is of cardinality continuum.\n","refs":[{"name":"On continuous self-maps and homeomorphisms of the Golomb space (T. Banakh et al.)","zb":"1474.54066"}]},{"space":"S000053","property":"P000181","value":true,"uid":"","description":"\n\nBy definition.\n","refs":[]},{"space":"S000053","property":"P000204","value":false,"uid":"","description":"\n\nSimilar to the proof that {S52|P204}.\n","refs":[]},{"space":"S000054","property":"P000002","value":false,"uid":"","description":"\n\nHas {S4} as a subspace and {S4|P2}.\n","refs":[]},{"space":"S000054","property":"P000011","value":true,"uid":"","description":"\n\n\nSee item #2 for space #62 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000054","property":"P000022","value":false,"uid":"","description":"\n\nSee item #3 for space #62 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000054","property":"P000024","value":true,"uid":"","description":"\n\nFor any $(a,b) \\in X$, $(a-1,a+1) \\times \\{0,1\\}$ is a neighborhood of $(a,b)$ with compact closure.\n","refs":[]},{"space":"S000054","property":"P000027","value":true,"uid":"","description":"\n\nLet $\\{U_n\\}$ be a countable base for $\\mathbb{R}$. Then $\\{U_n \\times \\{0,1\\}\\}$ is a countable base for $X$.\n","refs":[]},{"space":"S000054","property":"P000038","value":true,"uid":"","description":"\n\nSee item #4 for space #62 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000054","property":"P000043","value":true,"uid":"","description":"\n\nConsider two distinct points $(a,i),(b,j)\\in U\\times\\{0,1\\} \\subset X$ with $U$ some open interval in $\\mathbb R$.\n\nIf $a\\ne b$, let $f':[0,1]\\to U$ be an injective path from $a$ to $b$ and let $g:[0,1] \\rightarrow \\{0,1\\}$ be any function with $g(0)=i$ and $g(1)=j$. Then $f=f'\\times g$ is an injective path in $X$ from $(a,i)$ to $(b,j)$.\n\nIf $a=b$, one can construct an injective path in $U\\times\\{0,1\\}$ from $(a,0)$ to $(a,1)$\nby concatenating two paths built as in the previous paragraph,\none from $(a,0)$ to a point $(c,0)$ with $c\\in U$, $c\\ne a$,\nfollowed by a path from $(c,0)$ to $(a,1)$, with the two paths intersecting only in $(c,0)$.\n","refs":[]},{"space":"S000054","property":"P000065","value":true,"uid":"","description":"\n\nBy construction, as $2 \\cdot \\mathfrak c = \\mathfrak c$.\n","refs":[]},{"space":"S000054","property":"P000068","value":false,"uid":"","description":"\n\nIt is easy to check being {P68} passes to the Kolmogorov quotient. Since the Kolmogorov quotient of the space is\n{S25} and {S25|P68}, the result follows.\n","refs":[]},{"space":"S000054","property":"P000087","value":true,"uid":"","description":"\n\nThe space is the product of two spaces satisfying the property: {S25|P87}, and {S4|P87}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000054","property":"P000199","value":true,"uid":"","description":"\n\nThis follows because {S25|P199} and {S4|P199}, and a product of contractible spaces is contractible. See {{mathse:4463999}}.\n","refs":[{"name":"The cartesian product of contractible spaces is contractible","mathse":4463999}]},{"space":"S000055","property":"P000010","value":false,"uid":"","description":"\n\nSee item #4 for space #63 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000055","property":"P000018","value":true,"uid":"","description":"\n\n$X$ is Lindelöf: let $\\left\\{U_i, i \\in I\\right\\}$ be a basic open cover of $X$. There are two cases:\n\n1: One of the $U_i$ has countable complement, then we only need countably many extra $U_j$ to cover all of $X$, and so we find a countable subcover, or\n\n2: All $U_i$ are standard Euclidean open sets. But then we can use that $\\mathbb{R}$ in the usual topology is Lindelöf (as it's second countable e.g.) to see that we again have a countable subcover.\n\nSo the countable complement extension topology is Lindelöf.\n\nSee item #7 for space #63 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000055","property":"P000028","value":false,"uid":"","description":"\n\nSee item #8 for space #63 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000055","property":"P000036","value":true,"uid":"","description":"\n\nSee item #9 for space #63 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000055","property":"P000065","value":true,"uid":"","description":"\n\nBy definition.\n","refs":[]},{"space":"S000055","property":"P000068","value":false,"uid":"","description":"\n\nIt is easy to check being {P68} passes to coarser topologies. Since the topology on the space is finer than\n{S25} and {S25|P68}, the result follows.\n","refs":[]},{"space":"S000055","property":"P000166","value":true,"uid":"","description":"\n\nSee item #1 for space #63 in {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000055","property":"P000168","value":true,"uid":"","description":"\nBy construction, as {S17|P168}.\n","refs":[]},{"space":"S000056","property":"P000010","value":false,"uid":"","description":"\n\nNo neighbourhood of $0$ contained in the open set $U_0=\\mathbb R\\setminus K$ has a closure contained in $U_0$.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000056","property":"P000017","value":true,"uid":"","description":"\n\nSee item #3 for space #64 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000056","property":"P000022","value":false,"uid":"","description":"\n\nThe topology is finer than the Euclidean, hence the identity map $\\mathrm{id}:X\\to\\mathbb R$ is continuous and unbounded.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000056","property":"P000037","value":false,"uid":"","description":"\n\nSee {{mathse:2960540}}.\n","refs":[{"name":"Please clarify proof that $\\mathbb{R}_K$ is not path connected.","mathse":2960540}]},{"space":"S000056","property":"P000041","value":false,"uid":"","description":"\n\nSee {{mathse:1903761}}.\n","refs":[{"name":"Some properties about $K$-topology","mathse":1903761}]},{"space":"S000056","property":"P000046","value":false,"uid":"","description":"\n\nThe subspace $(-\\infty,0)$ is homeomorphic to {S25}\nand {S25|P46}.\n","refs":[]},{"space":"S000056","property":"P000061","value":true,"uid":"","description":"\n\nIn order to find the Tychonoff reflection of $X$, we first determine the completely regular reflection of $X$, with topology generated by the cozero sets of $X$. \n\nNote that each cozero set in a topological space is a union of regular open sets. In the current space $X$, \na regular open set containing a point $x\\in X$ also contains\na Euclidean neighborhood of $x$.\nFor the non-obvious case of $x=0$, it is because\n$\\operatorname{int}_X\\text{cl}_X((-\\varepsilon,\\varepsilon)\\setminus K) =\n\\operatorname{int}_X[-\\varepsilon,\\varepsilon] = (-\\varepsilon,\\varepsilon)$.\nSo the topology generated by the cozero sets of $X$ is contained in the Euclidean topology of $\\mathbb R$.\nOn the other hand, {S25|P12}.\nSo the completely regular reflection of $X$ is {S25}.\nAnd since $\\mathbb R$ is even Tychonoff, it is the Tychonoff reflection of $X$.\n\nThe reflection map $\\varphi$ is the identity function on $\\mathbb R$ as a set;\nit is open at each point $x$ of the dense subset $X\\setminus\\{0\\}\\subseteq X$ (i.e., sends neighborhoods of $x$ to neighborhoods of $f(x)$). Since {S25|P61},\nthe desired result then follows from the Remark after Proposition 2 in {{mathse:5108293}}.\n","refs":[{"name":"Cozero complemented property for a space and its Tychonoff reflection","mathse":5108293}]},{"space":"S000056","property":"P000068","value":false,"uid":"","description":"\n\nIt is easy to check being {P68} passes to coarser topologies. Since the topology on the space is finer than\n{S25} and {S25|P68}, the result follows.\n","refs":[]},{"space":"S000056","property":"P000089","value":false,"uid":"","description":"\n\nThe function $f:X\\to X$ given by $f(x)=x-1$ for $x\\leq 0$ and $f(x)=-1$ for $x\\geq 0$ is continuous and has no fixed point.\n","refs":[]},{"space":"S000056","property":"P000110","value":true,"uid":"","description":"\n\nFor every point $x\\in X\\setminus\\{0\\}$ fix a local base consisting of sets $U^n_x:= B_e(x,2^{-n})\\setminus\\{0\\}$ for $n\\in\\mathbb N$, where $B_e$ stands for an Euclidean ball. For $0$ take neighbourhoods $U^n_0:=B_e(0,2^{-n})\\setminus K$.\nDefine the open covers $\\mathscr V_n:=\\{U_x^n:x\\in X\\}$ ($n\\in\\mathbb N$).\nFor $x\\neq 0$ we have $U_x^n\\subset \\mathrm{st}(x,\\mathscr V_n)\\subset B_e(x,2^{1-n})$ and $\\mathrm{st}(0,\\mathscr V_n)=U^n_0$. Therefore the open covers form a development for $X$.\n","refs":[]},{"space":"S000056","property":"P000120","value":true,"uid":"","description":"\n\nThe open sets $X\\setminus\\{0\\}$ and $X\\setminus K$ have their topologies induced by the standard order from $\\mathbb R$.\n","refs":[]},{"space":"S000056","property":"P000166","value":true,"uid":"","description":"\n\nSee item #1 for space #64 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000056","property":"P000189","value":true,"uid":"","description":"\n\n$X$ is {P36} by {{mathse:3331141}}.\n\nThen see {{mathse:4975867}}.\n","refs":[{"name":"Proving that $\\mathbb R_K$ is connected","mathse":3331141},{"name":"Answer to \"$\\sigma$-connected spaces and path-connected property\"","mathse":4975867}]},{"space":"S000056","property":"P000205","value":true,"uid":"","description":"\n\n{S56|P36}.\nThe subsets $(-\\infty,x)$ and $(x,+\\infty)$ are open for any $x\\in X$.\n","refs":[]},{"space":"S000056","property":"P000206","value":true,"uid":"","description":"\n\nNote that $X\\setminus\\{0\\}$ has the Euclidean topology.\n\nIf player 1 picks $x_n\\neq 0$, player 2 wins the game played on the subspace $(0,+\\infty)$ or $(-\\infty,0)$,\nhomeomorphic to {S25} ({S25|P206}).\nIf $x_n\\equiv 0$, then player 1 loses as well.\n","refs":[]},{"space":"S000057","property":"P000024","value":true,"uid":"","description":"\n\nSee item #4 for space #65 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000057","property":"P000026","value":true,"uid":"","description":"\n\nSee item #3 for space #65 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000057","property":"P000051","value":true,"uid":"","description":"\n\nSee item #2 for space #65 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000057","property":"P000065","value":true,"uid":"","description":"\n\nThe space's point-set is $\\mathbb R$.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000057","property":"P000085","value":false,"uid":"","description":"\n\nPick $x\\in\\mathbb R\\setminus\\mathbb Q$ and an enumeration of\nrationals $\\mathbb Q=\\{q_n:n<\\omega\\}$ such that $q_{2n}=x_n$ is the particular sequence converging to $x$.\n\nDefine the function $f:X\\to \\mathbb R$ as\n$$f(t)=\\begin{cases}\n0&\\text{for } t\\in X\\setminus \\{q_{4n}:n<\\omega\\}\\\\\n\\frac1n &\\text{for } t=q_{4n}\n\\end{cases},\\quad t\\in X.$$\nIt is continuous at every isolated point. For any $x\\in X\\setminus \\mathbb Q$ and $n<\\omega$ some neighbourhood of $x$ is contained in $\\{x\\}\\cup\\{q_k:k>4n\\}\\subset f^{-1}((-\\frac1n,\\frac1n))$, hence $f$ is also continuous at $x$.\n\nThe closure $A=\\text{cl}(f^{-1}((0,+\\infty)))$ contains $x=\\lim_{n\\to\\infty} q_{4n}$ but does not\ncontain the isolated points $\\{q_{4n+2}:n<\\omega\\}$.\nHence $x$ does not belong to the interior of $A$ and $A$ is not open.\n","refs":[]},{"space":"S000057","property":"P000093","value":true,"uid":"","description":"\nBy construction.\n","refs":[]},{"space":"S000057","property":"P000110","value":true,"uid":"","description":"\n\nFix an enumeration of rationals $\\mathbb Q=\\{q_n:n<\\omega\\}$ and denote $\\mathbb Q_n=\\{q_k: k\\leq n\\}$.\n\nA development for $X$ can be formed out of the open covers\n$\\mathscr V_n=\\{ U_n(x)\\setminus \\mathbb Q_n : x\\in\\mathbb R\\setminus\\mathbb Q\\}\\cup\\{\\{q_k\\}:k<\\omega \\}$.\n","refs":[]},{"space":"S000057","property":"P000117","value":true,"uid":"","description":"\n\nNote that the family of all singletons is $\\sigma$-locally finite and is obviously a network. Indeed, $\\{\\{x\\}: x\\in\\mathbb R\\setminus \\mathbb Q\\}$ and $\\{ \\{q\\} \\}$ for $q\\in\\mathbb Q$ are locally finite families.\n","refs":[]},{"space":"S000057","property":"P000120","value":true,"uid":"","description":"\n\nThe set of rationals is open and discrete, hence {P133}.\nThe local basis elements $U_n(x)$ of irrationals are homeomorphic to {S20}, and {S20|P133}.\n\nIt is worth noting that every neighbourhood from the canonical basis is also orderable by the order induced from $\\mathbb R$.\n","refs":[]},{"space":"S000057","property":"P000166","value":true,"uid":"","description":"\n\nSee item #1 for space #65 in {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000057","property":"P000227","value":true,"uid":"","description":"\n\n$X \\setminus \\mathbb Q$ is a closed and discrete subset of cardinality continuum.\n","refs":[]},{"space":"S000058","property":"P000010","value":false,"uid":"","description":"\n\nThe closure of any neighbourhood of a rational number contains an entire interval. Therefore the interior of the closure of such a neighbourhood cannot be countable (contained in $\\mathbb Q$).\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000058","property":"P000022","value":false,"uid":"","description":"\n\nSince the topology is finer than {S25},\nthe identity map to {S25} is continuous, and unbounded.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000058","property":"P000027","value":true,"uid":"","description":"\n\nSee item #9 for space #66 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000058","property":"P000041","value":false,"uid":"","description":"\n\nSee item #8 for space #66 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000058","property":"P000044","value":false,"uid":"","description":"\n\nSets $(-\\infty,0)$ and $(0,+\\infty)$ are disjoint and {P36} as homeomorphic to $X$.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000058","property":"P000046","value":true,"uid":"","description":"\n\nSee item #8 for space #66 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000058","property":"P000056","value":true,"uid":"","description":"\n\nNote that the set $\\mathbb R\\setminus\\mathbb Q$ is closed and has empty interior in $X$. Together with singletons of rational numbers it forms a countable partition of $X$ into nowhere dense sets.\n","refs":[]},{"space":"S000058","property":"P000066","value":false,"uid":"","description":"\n\n{P66} is hereditary on closed subsets.\n$X$ contains {S28} as a closed subspace and {S28|P66}.\n","refs":[]},{"space":"S000058","property":"P000086","value":false,"uid":"","description":"\n\nEvery rational number has a countable neighbourhood and neighbourhoods of irrational numbers are uncountable.\n","refs":[]},{"space":"S000058","property":"P000089","value":false,"uid":"","description":"\n\nThe map $x\\mapsto x+1$ is a self-homeomorphism without a fixed point.\n","refs":[]},{"space":"S000058","property":"P000110","value":true,"uid":"","description":"\n\nSee {{mathse:5045688}}.\n","refs":[{"name":"Indiscrete rational extension of $\\mathbb R$ is developable","mathse":5045688}]},{"space":"S000058","property":"P000144","value":false,"uid":"","description":"\n\nEvery neighbourhood of an irrational number contains an interval.\nAn open interval in $X$ is homeomorphic to $X$ and hence not {P121}\n(e.g. {S58|P10}).\n","refs":[]},{"space":"S000058","property":"P000166","value":true,"uid":"","description":"\n\nThe topology contains the topology for {S25}.\n","refs":[]},{"space":"S000058","property":"P000189","value":false,"uid":"","description":"\n\nThe family $\\{\\mathbb R\\setminus\\mathbb Q\\}\\cup\\{\\{q\\}:q\\in\\mathbb Q\\}$\nis a partition of $X$ into countably many mutually disjoint closed subsets.\n","refs":[]},{"space":"S000058","property":"P000205","value":true,"uid":"","description":"\n\nTo show $X$ is {P36}, suppose by contradiction that $X=A\\cup B$ is a partition of $X$ into open subsets, with $a0$, implying $A=\\overline{A}\\supset [a,x+\\varepsilon]$\nand contradicting the definition of $x$.\n\nAnd every $x\\in X$ is a cut point, since the two open sets $(-\\infty,x)$ and $(x,+\\infty)$ partition $X\\setminus \\{x\\}$.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000059","property":"P000010","value":false,"uid":"","description":"\n\nTake an open set $U\\subset \\mathbb R\\setminus\\mathbb Q$.\nThere exists an open interval $V$ such that $V\\setminus\\mathbb Q\\subset U$.\nThen $\\mathrm{int}(\\overline{U})\\supset\\mathrm{int}(\\overline{V\\setminus\\mathbb Q})=V$,\nbut $V\\not\\subset U$.\n","refs":[]},{"space":"S000059","property":"P000017","value":false,"uid":"","description":"\n\nCompact sets are nowhere dense in $\\mathbb R$, which isn't meager.\n\nSee item #10 for space #67 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000059","property":"P000022","value":false,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #67 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000059","property":"P000027","value":true,"uid":"","description":"\n\nOpen intervals with rational endpoints, with or without the rationals removed, form a countable base.\n\nSee item #9 for space #67 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000059","property":"P000031","value":true,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #67 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000059","property":"P000036","value":true,"uid":"","description":"\n\nSee item #8 for space #67 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000059","property":"P000041","value":false,"uid":"","description":"\n\nSee item #8 for space #67 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000059","property":"P000044","value":false,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #67 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000059","property":"P000046","value":true,"uid":"","description":"\n\nSee item #8 for space #67 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000059","property":"P000068","value":false,"uid":"","description":"\n\nIt is easy to check being {P68} passes to coarser topologies. Since the topology on the space is finer than\n{S25} and {S25|P68}, the result follows.\n","refs":[]},{"space":"S000059","property":"P000071","value":true,"uid":"","description":"\n\nSee Example 5.8 of {{doi:10338.dmlcz/146790}}.","refs":[{"name":"Applications of limited information strategies in Menger's game","doi":"10338.dmlcz/146790"}]},{"space":"S000059","property":"P000082","value":false,"uid":"","description":"\n\nSuppose an open interval $U$ is a neighbourhood of $x\\in\\mathbb Q$ with $U$ metrizable by a metric $d$.\nFor $y\\in U\\setminus \\mathbb Q$ there exists $r>0$\nsuch that $B(y,r)\\subset U\\setminus\\mathbb Q$,\nwhere $B(y,r)$ is the ball with respect to $d$.\nSince $B(y,r/2)$ is a neighbourhood of $y$, it contains $V\\setminus\\mathbb Q$ for some open interval $V\\subset U$.\nTherefore $\\overline{V\\setminus\\mathbb Q}=V\\subset\\overline{B(y,r/2)}\\subset B(y,r)$,\nwhich is impossible because $B(y,r)$ does not contain rational numbers.\n","refs":[]},{"space":"S000059","property":"P000166","value":true,"uid":"","description":"\n\nThe topology contains the topology for {S25}.\n","refs":[]},{"space":"S000060","property":"P000010","value":false,"uid":"","description":"\n\nFor any nonempty set $U\\subset \\mathbb Q$ open in $X$ there exist numbers $a0$).\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000060","property":"P000110","value":true,"uid":"","description":"\n\nFix an enumeration $\\{q_n:n<\\omega\\}=\\mathbb Q$.\nFor $n<\\omega$ and $x\\in\\mathbb R$ define the open set\n$U_n(x):=\\{x\\}\\cup\\{q_k: k>n,\\ |q_k-x|<1/n\\}$.\nThe family of open covers $\\mathscr U_n:=\\{U_n(x): x\\in X\\}$, for all $n<\\omega$,\nis then a development for $X$.\nIndeed: for $x\\notin\\mathbb Q$, $\\operatorname{st}(x,\\mathscr U_n)=U_n(x)$.\nAnd for $x=q_l\\in\\mathbb Q$ and $n>l$, $\\operatorname{st}(x,\\mathscr U_n)=U_n(x)$.\n\n","refs":[]},{"space":"S000060","property":"P000118","value":true,"uid":"","description":"\n\nLet $\\mathscr U$ be a countable base for the Euclidean topology on $\\mathbb Q$.\nThen the family $\\mathcal N=\\{\\{x\\}:x\\in\\mathbb R\\setminus\\mathbb Q\\}\n\\cup\\bigcup_{U\\in\\mathscr U}\\{U\\}$ is $\\sigma$-locally finite.\n\nSuppose $K\\subset X$ is compact and $U\\supseteq K$ is open.\nFor each $x\\in K$ there exists $V_x\\in\\mathscr U$\nsuch that $\\{x\\}\\cup V_x$ is an open subset of $U$.\nBy compactness, $K$ is covered by finitely many $\\{x\\}\\cup V_x$.\nSo $K$ can be covered by finitely many elements of $\\mathscr U$ and finitely many singletons in $\\mathcal N$, all contained in $U$.","refs":[]},{"space":"S000060","property":"P000166","value":true,"uid":"","description":"\n\nSee item #1 for space #68 in {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000060","property":"P000189","value":false,"uid":"","description":"\n\nThe space can be partitioned into a countable family of nonempty closed sets, namely the set $\\mathbb R\\setminus \\mathbb Q$ and singletons of rational numbers.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000060","property":"P000205","value":true,"uid":"","description":"\n\n$X$ is {P36}: see item #4 for space #68 in {{zb:0386.54001}}.\n\nFor every $x\\in\\mathbb R$ the subsets $(-\\infty,x)$ and $(x,+\\infty)$ are open hence $x$ is a cut point.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000060","property":"P000227","value":true,"uid":"","description":"\n\nThe subspace $\\mathbb R\\setminus\\mathbb Q$ is closed discrete with cardinality $\\mathfrak c$.\n","refs":[]},{"space":"S000061","property":"P000010","value":false,"uid":"","description":"\n\nFor any nonempty set $U\\subset \\mathbb R\\setminus\\mathbb Q$ open in $X$ there exist numbers $a n \\}$.\nEach $F_n$ is closed in $X$ and the $F_n$ are nested with empty intersection.\nMoreover, $F_n$ is dense in $\\mathbb R$ with respect to the usual Euclidean topology.\n\nNote that the subspace topology on $D=\\mathbb R\\setminus\\mathbb Q$ coincides with the Euclidean topology induced from $\\mathbb R$.\nSo the subspace $D$ is exactly {S28}.\n\nNow, suppose $U_n$ is an open set containing $F_n$.\nThe intersection $U_n \\cap D$ is open in $D$ and it is easy to check it is dense in $D$.\n\n\nSince {S28|P64}, $\\bigcap_{n \\in \\omega} (U_n \\cap D)\\ne\\varnothing$,\nand hence $\\bigcap_{n \\in \\omega}U_n\\ne\\varnothing$.\nThis proves that the space is not {P33}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000061","property":"P000044","value":false,"uid":"","description":"\n\nSets $(-\\infty,0)$ and $(0,+\\infty)$ are disjoint and {P36} as homeomorphic to $X$.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000061","property":"P000046","value":true,"uid":"","description":"\n\nSee item #8 for space #69 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000061","property":"P000068","value":false,"uid":"","description":"\n\nIt is easy to check being {P68} passes to coarser topologies. Since the topology on the space is finer than\n{S25} and {S25|P68}, the result follows.\n","refs":[]},{"space":"S000061","property":"P000071","value":true,"uid":"","description":"\n\nFirst we show that $I_n:=[n,n+1]\\setminus\\mathbb Q$ is relatively compact in $X$.\nFix an open cover $\\mathscr U$ of $X$. For each $x\\in X$ there exists some $U_x\\in \\mathscr U$ and $r_x>0$ such that $(x-r_x,x+r_x)\\setminus \\mathbb Q \\subset U_x$. Then $\\{ (x-r_x,x+r_x): x\\in X\\}$ is an open cover of {S25};\ntherefore there is a finite set $F\\subset\\mathbb R$ such that\n$[n,n+1]\\subseteq \\bigcup_{x\\in F} (x-r_x,x+r_x)$ and therefore\n$I_n \\subseteq \\bigcup_{x\\in F}U_x$.\n\nA countable cover of $X$ by relatively compact subsets can be obtained\nas $\\{ I_n: n\\in\\mathbb Z\\}\\cup\\{\\{q\\}:q\\in\\mathbb Q\\}$.\n","refs":[]},{"space":"S000061","property":"P000086","value":true,"uid":"","description":"\n\nSee {{mathse:5015387}}.\n","refs":[{"name":"Answer to \"Pointed rational extension of reals is a homogeneous space\"","mathse":5015387}]},{"space":"S000061","property":"P000089","value":false,"uid":"","description":"\n\nThe homeomorphism $x\\mapsto x+1$ has no fixed point.\n","refs":[]},{"space":"S000061","property":"P000120","value":true,"uid":"","description":"\n\nThe topology on every open set of the form $\\{x\\}\\cup D$ coincides with the order topology induced by the standard order.\n","refs":[]},{"space":"S000061","property":"P000166","value":true,"uid":"","description":"\n\nThe topology is finer than {S25}.\nSee item #1 for space #69 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000061","property":"P000205","value":true,"uid":"","description":"\n\nThe space is {P36} by item #8 for space #69 in {{zb:0386.54001}}.\n\nOn the other hand, for any $x\\in X$ the sets $(-\\infty,x)$ and $(x,+\\infty)$ are open, hence $X\\setminus\\{x\\}$ is not {P36}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000061","property":"P000206","value":true,"uid":"","description":"\n\nWe use the fact that the topology induced on $\\mathbb R\\setminus\\mathbb Q\\subseteq X$ coincides with the standard one,\ninduced by the Euclidean metric.\n\nPlayer 2 in their first move picks $V_0:= U_0\\cap((\\mathbb R\\setminus\\mathbb Q) \\cup\\{x_0\\})$. If\n$x_n= x_0$ for each $n\\in\\omega$, the game is won by Player 2. Otherwise, at some point $V_n\\subseteq \\mathbb R\\setminus\\mathbb Q$\nand the assertion follows, since {S28|P206}. \n","refs":[]},{"space":"S000062","property":"P000003","value":true,"uid":"","description":"\n\nThe topology is finer than {S25}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000062","property":"P000027","value":true,"uid":"","description":"\n\nThe family $\\{[p,q]:p,q\\in\\mathbb Q,\\ p\\leq q\\}$ is a countable basis for the topology on $X$.\n\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000062","property":"P000050","value":true,"uid":"","description":"\n\nThe family $\\{[p,q]:p,q\\in\\mathbb Q,\\ p\\leq q\\}$ is a base for the topology consisting of clopen sets.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000062","property":"P000065","value":true,"uid":"","description":"\n\nThe space's point-set is $\\mathbb R$.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000062","property":"P000066","value":false,"uid":"","description":"\n\nThe set $\\mathbb R\\setminus\\mathbb Q$ is a closed subspace of $X$\nbut {S28|P66}.\n","refs":[]},{"space":"S000062","property":"P000139","value":true,"uid":"","description":"\n\nBy definition, each point of $D$ is open.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000062","property":"P000206","value":true,"uid":"","description":"\n\nWe describe the winning strategy for player 2. It relies on Cantor's intersection theorem and is based on the one suitable for {S25}.\n\nIf $x_n\\in\\mathbb Q$ for some $n\\in\\omega$, then the move $V_n=\\{x_n\\}$ of player 2 essentially finishes the game.\n\nFor $x_n\\notin\\mathbb Q$ we know that there exists a closed interval $I_n\\subset U_n$, with $x_n$ in the (Euclidean) interior and length not exceeding $\\frac 1n$. Player 2 can pick $V_n=\\mathrm{int}(I_n)$. Then $\\bigcap_{n=1}^\\infty U_n=\n\\bigcap_{n=1}^\\infty \\overline{V_n}$ and the latter is the intersection of descending\nclosed subsets of {S25}\nwith their diameters decreasing to $0$, which in a complete metric space is nonempty. \n","refs":[]},{"space":"S000063","property":"P000051","value":false,"uid":"","description":"\n\nThe subspace $\\mathbb Q\\subset X$ has no isolated point.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000063","property":"P000061","value":false,"uid":"","description":"\n\nThe subset $A=\\mathbb Q+\\sqrt{2}$ is open and an $F_\\sigma$ set in $X$, therefore a cozero set, since {S63|P13} (see Corollary 1.5.13 in {{zb:0684.54001}}).\n\nSuppose $B$ is a cozero set disjoint from $A$ such that $A\\cup B$ is dense. Since $\\mathbb Q\\subset \\overline{A}$, we have $B\\subseteq\\mathbb R\\setminus\\mathbb Q$.\nBy the density of $A\\cup B$ we know that $A\\cup B=\\mathbb R\\setminus \\mathbb Q$ and consequently, the irrationals are\n$F_\\sigma$. Since neighbourhoods of rational numbers always contain Euclidean neighbourhoods, it would imply $\\mathbb Q$ is a $G_\\delta$ subset of {S25},\ncontradicting the Baire category theorem.\n","refs":[{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"space":"S000063","property":"P000063","value":false,"uid":"","description":"\n\nThe space contains {S27} as a closed subspace.\nAnd the {P63} property is preserved by closed subspaces.\nBut {S27|P63}.\n","refs":[]},{"space":"S000063","property":"P000065","value":true,"uid":"","description":"\n\nThe space's point-set is $\\mathbb R$.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000063","property":"P000076","value":false,"uid":"","description":"\n\nThe product of {S63} and {S28} is {S77}, and {S77|P76}.\nBut {S28|P76}, so {S63} is not {P76}.\n","refs":[]},{"space":"S000063","property":"P000093","value":false,"uid":"","description":"\n\nNo neighbourhood of a rational number is countable.\n","refs":[]},{"space":"S000063","property":"P000139","value":true,"uid":"","description":"\n\nBy definition, each point of $D$ is open.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000063","property":"P000146","value":true,"uid":"","description":"\n\nLet $\\mathcal U$ be any open cover of $X$.\nFix an enumeration $\\{q_n:n<\\omega\\}$ of rational numbers.\nWe proceed by induction on $n<\\omega$. There are two possibilities:\n- if $q_n\\in \\bigcup_{k0,\\ x^2+y^2<4^{-n}\\}$\nand $U_n(0^*):=\\{0^*\\}\\cup\\{(x,y)\\in\\mathbb R^2: y<0,\\ x^2+y^2<4^{-n}\\}$.\nThe families $\\mathscr V_n:=\\{ U_n(x): x\\in X\\}$ can be readily verified to form a development.\n","refs":[]},{"space":"S000066","property":"P000204","value":false,"uid":"","description":"\n\nThe space remains {P37} after the removal of any finite\nnumber of points, as shown in {{mathse:5012402}}.\n","refs":[{"name":"Answer to \"Does the double origin plane have a cut point?\"","mathse":5012402}]},{"space":"S000066","property":"P000206","value":true,"uid":"","description":"\n\nThe space contains precisely two points with neighbourhoods not homeomorphic to {S176}.\n\nCase 1. If player 1 picks $x_n\\in\\mathbb R^2\\setminus\\{0\\}$, then\nplayer 2 wins the game, since {S176|P206}.\n\nCase 2. If $x_n$ is one of the zero points, player 2 can\nchoose $V_n$ to exclude the other zero. Then either\n$(x_m)_{m\\geq n}$ is constant (and player 2 wins) or at some point we have Case 1.\n ","refs":[]},{"space":"S000067","property":"P000003","value":true,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #75 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000067","property":"P000004","value":false,"uid":"","description":"\n\nSee item #2 for space #75 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000067","property":"P000010","value":false,"uid":"","description":"\n\nSee item #3 for space #75 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000067","property":"P000036","value":true,"uid":"","description":"\n\nSee item #4 for space #75 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000067","property":"P000044","value":false,"uid":"","description":"\n\nPick an irrational number $\\phi$ and consider subspaces $X_1=\\{(x,y)\\in X: x<\\phi\\}$ and $X_2=\\{(x,y)\\in X: x>\\phi\\}$.\nThey are both connected, since the argument from item #4 (cf. item #2) for space #75 in {{zb:0386.54001}} remains valid.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000067","property":"P000057","value":true,"uid":"","description":"\n\nBy definition.\n\nSee item #4 for space #75 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000067","property":"P000082","value":true,"uid":"","description":"\n\nThe open subset $\\mathbb Q{\\times}\\{0\\}$ is homeomorphic to a metrizable space $\\mathbb Q$.\n\nConsider $(x,y)\\in X$ with $y>0$. Observe that for $0<\\epsilon0$,\nwhich is homeomorphic to {S76} and {S76|P13},\nhence is not metrizable.\n","refs":[]},{"space":"S000076","property":"P000086","value":true,"uid":"","description":"\n\nEvery translation $(x, y) \\mapsto (x, y) + (a, b)$ is a homeomorphism of $X$ onto itself.\n","refs":[]},{"space":"S000076","property":"P000093","value":false,"uid":"","description":"\n\n$X$ contains {S43} as a subspace and {S43|P93}.\n","refs":[]},{"space":"S000076","property":"P000102","value":false,"uid":"","description":"\n\n$X$ contains {S43} as a subspace and {S43|P102}.\n","refs":[]},{"space":"S000076","property":"P000132","value":true,"uid":"","description":"\n\nShown in {{mr:287515}}.\nSee also .\n","refs":[{"name":"A property of the Sorgenfrey line","mr":287515}]},{"space":"S000076","property":"P000165","value":false,"uid":"","description":"\n\nWitnessed by the standard proof that the space is not {P13}:\nThe countable closed set $\\{(q,-q):q\\in\\mathbb Q\\}$ and closed set\n$\\{(p,-p):p\\in\\mathbb R\\setminus\\mathbb Q\\}$ cannot be separated by\nopen sets.\n\nSee item #2 for space #84 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000076","property":"P000166","value":true,"uid":"","description":"\n\nThe topology contains the topology for {S176}.\n","refs":[]},{"space":"S000076","property":"P000177","value":false,"uid":"","description":"\n\n$X$ contains {S43} as a subspace and {S43|P177}.\n","refs":[]},{"space":"S000076","property":"P000227","value":true,"uid":"","description":"\n\nThe closed set $\\{(x,-x):x\\in\\mathbb R\\}$ is a discrete subspace with cardinality $\\mathfrak c$.\n","refs":[]},{"space":"S000077","property":"P000013","value":false,"uid":"","description":"\n\nSee item #1 for space #85 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000077","property":"P000028","value":true,"uid":"","description":"\n\n{S63|P28} and {S28|P28}, hence so is their product.\n","refs":[]},{"space":"S000077","property":"P000031","value":true,"uid":"","description":"\n\nSee item #2 for space #85 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000077","property":"P000050","value":true,"uid":"","description":"\n\n{S63|P50} and {S28|P50}, hence so is their product.\n","refs":[]},{"space":"S000077","property":"P000062","value":false,"uid":"","description":"\n\nIf $X$ is {P62}, then so is each of its factors.\nHowever, {S63|P62}.\n","refs":[]},{"space":"S000077","property":"P000063","value":false,"uid":"","description":"\n\n$X$ contains {S63} as a closed subspace and {S63|P63}.\n","refs":[]},{"space":"S000077","property":"P000093","value":false,"uid":"","description":"\n\n$X$ contains {S63} as a subspace and {S63|P93}.\n","refs":[]},{"space":"S000077","property":"P000132","value":false,"uid":"","description":"\n\n$X$ contains {S63} as a subspace and {S63|P132}.\n","refs":[]},{"space":"S000077","property":"P000139","value":false,"uid":"","description":"\n\nIf $X$ {P139}, then so does each of its factors.\nHowever, {S28|P139}.\n","refs":[]},{"space":"S000077","property":"P000144","value":false,"uid":"","description":"\n\n$X$ contains {S63} as a subspace and {S63|P144}.\n","refs":[]},{"space":"S000077","property":"P000166","value":true,"uid":"","description":"\n\nThe Euclidean topology of $\\mathbb R \\times (\\mathbb R \\setminus \\mathbb Q)$ is a coarser topology which is {P26} and {P53}.\n","refs":[]},{"space":"S000077","property":"P000206","value":true,"uid":"","description":"\n\n{S63|P206} and {S28|P206}, hence so is their product.\n","refs":[]},{"space":"S000077","property":"P000227","value":true,"uid":"","description":"\n\n$X$ contains {S63} as a closed subspace, and {S63|P227}.\n","refs":[]},{"space":"S000078","property":"P000014","value":false,"uid":"","description":"\n\nThe {S79} is a subspace which is not {P13}.\n\nSee item #2 for space #86 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000078","property":"P000016","value":true,"uid":"","description":"\n\nSee item #1 for space #86 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000078","property":"P000029","value":false,"uid":"","description":"\n\nThere are uncountably many isolated points.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000078","property":"P000049","value":false,"uid":"","description":"\n\nSee item #5 for space #86 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000078","property":"P000050","value":true,"uid":"","description":"\n\nSee item #5 for space #86 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000078","property":"P000051","value":true,"uid":"","description":"\n\nSee item #5 for space #86 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000078","property":"P000081","value":false,"uid":"","description":"\n\n{S36} is homeomorphic to a subspace of $X$\nand {S36|P81}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000078","property":"P000103","value":false,"uid":"","description":"\n\n$X$ contains a copy of {S36} as a subspace and {S36|P103}.\n","refs":[]},{"space":"S000078","property":"P000114","value":true,"uid":"","description":"\n\nBy construction.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000078","property":"P000172","value":false,"uid":"","description":"\n\nSee {{mathse:4789562}}.\n","refs":[{"name":"Radial and pseudoradial properties for Tychonoff plank and variants","mathse":4789562}]},{"space":"S000078","property":"P000173","value":true,"uid":"","description":"\n\nSee {{mathse:4789562}}.\n","refs":[{"name":"Radial and pseudoradial properties for Tychonoff plank and variants","mathse":4789562}]},{"space":"S000078","property":"P000210","value":true,"uid":"","description":"\n\n{S36|P210} and {S20|P210},\nhence so is their product.\n","refs":[]},{"space":"S000078","property":"P000215","value":false,"uid":"","description":"\n\n$X$ contains a copy of {S35} as a subspace and {S35|P215}.\n","refs":[]},{"space":"S000079","property":"P000006","value":true,"uid":"","description":"\n\n$X$ is a subspace of {S78}\nand {S78|P6}.\n","refs":[]},{"space":"S000079","property":"P000022","value":true,"uid":"","description":"\n\nSee item #4 for space #87 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000079","property":"P000029","value":false,"uid":"","description":"\n\nThere are uncountably many isolated points, e.g. $\\langle\\alpha+1,0\\rangle$ for $\\alpha<\\omega_1$.\n","refs":[]},{"space":"S000079","property":"P000032","value":false,"uid":"","description":"\n\nSee {{mathse:4819068}}.\n","refs":[{"name":"Paracompactness properties of the deleted Tychonoff plank","mathse":4819068}]},{"space":"S000079","property":"P000033","value":true,"uid":"","description":"\n\nSee {{mathse:4819068}}.\n","refs":[{"name":"Paracompactness properties of the deleted Tychonoff plank","mathse":4819068}]},{"space":"S000079","property":"P000051","value":true,"uid":"","description":"\n\nSee item #5 for space #87 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000079","property":"P000061","value":true,"uid":"","description":"\n\nSuppose $f:X\\to[0,1]$ is continuous and $A=f^{-1}((0,1])$.\n\nFor each $n<\\omega$ there exists $\\alpha_n<\\omega_1$\nsuch that $f|_{[\\alpha_n,\\omega_1]}$ is constant (the intersection of\npreimages of countably many neighbourhoods of $f(\\langle\\omega_1,n\\rangle)$ has nonempty interior).\nTaking $\\alpha_\\omega=\\sup_{n<\\omega}\\alpha_n$ we know that\non the clopen set $X_1:=(\\alpha_\\omega,\\omega_1]{\\times}[0,\\omega]\\setminus\\{\\langle\\omega_1,\\omega\\rangle\\}$ the function $f$ is constant\non each row. Therefore a cozero set therein equals $(\\alpha_\\omega,\\omega_1]{\\times}A_0\\cap X$ for some cozero subset $A_0$ of\n{S20}. It then admits\nan appropriate cozero complement in $X_1$\nfor {S20|P61}\n\nOn the other hand, the clopen subspace $X_0:=X\\setminus X_1$ is countable\nand {P12} (since {S79|P12}),\nand therefore {P61}\n[(Explore)](https://topology.pi-base.org/spaces?q=countable%2BCompletely+regular%2B%7ECozero+complemented).\n\nSince $A\\cap X_0$ admits a cozero complement in $X_0$ and\n$A\\cap X_1$ admits a cozero complement in $X_1$, $A$ has a cozero complement in $X$.\n","refs":[]},{"space":"S000079","property":"P000062","value":true,"uid":"","description":"\n\nThe space contains a dense {P17} subspace, namely the countable union of the horizontal slices $(\\omega_1+1)\\times\\{n\\}$, each one of which is compact.\n\nGiven an open cover of such a space, each of the compact sets is covered by a finite subcollection of the cover. The union of all these subcollections is a countable subcollection whose union is dense in the space.\n","refs":[]},{"space":"S000079","property":"P000083","value":false,"uid":"","description":"\n\nThe closed subspace $[0,\\omega_1)\\times\\{\\omega\\}$ is homeomorphic\nto {S35} and {S35|P83}.\n","refs":[]},{"space":"S000079","property":"P000103","value":false,"uid":"","description":"\n\nThe subset $[0,\\omega_1)\\times\\{0\\}$ is not closed but is homeomorphic to\n{S35} and {S35|P19}.\n","refs":[]},{"space":"S000079","property":"P000114","value":true,"uid":"","description":"\n\nBy construction.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000079","property":"P000120","value":true,"uid":"","description":"\n\n$X$ can be covered by the open and orderable subsets of the form\n- $[0,\\omega_1]{\\times}\\{n\\}$ for $n<\\omega$, \n- $[0,\\alpha]{\\times}[0,\\omega]=[0,\\alpha+1){\\times}[0,\\omega]$\nfor $\\alpha<\\omega_1$.\n\nThe first ones are homeomorphic to {S36},\nand the latter are compact Hausdorff and countable,\ntherefore {P133} [(Explore)](https://topology.pi-base.org/spaces?q=countable%2BT_2%2Bcompact%2B%7ELOTS).\n","refs":[]},{"space":"S000079","property":"P000130","value":true,"uid":"","description":"\n\nThe space is {P130} as an open subspace of {S78}, which is {P130}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000079","property":"P000165","value":false,"uid":"","description":"\n\nConsider the countable closed subset $A=\\{\\omega_1\\}\\times[0,\\omega)$ and its neighbourhood $U=[0,\\omega_1]\\times[0,\\omega)$.\nEvery neighbourhood $V$ of $A$ contains a set of the form\n$V_\\lambda=\\bigcup_{n<\\omega} (\\lambda_n,\\omega_1]\\times\\{n\\}$\nfor some choice of $\\lambda_n<\\omega_1$.\nThen, for $\\alpha=\\sup\\{\\lambda_n:n<\\omega\\}<\\omega_1$\nwe have $\\{\\alpha\\}\\times[0,\\omega)\\subset V_\\lambda$\nand $\\langle\\alpha,\\omega\\rangle\\in \\overline V\\setminus U$.\n","refs":[]},{"space":"S000079","property":"P000174","value":true,"uid":"","description":"\n\nSee {{mathse:4789562}}.\n","refs":[{"name":"Radial and pseudoradial properties for Tychonoff plank and variants","mathse":4789562}]},{"space":"S000079","property":"P000194","value":false,"uid":"","description":"\n\nThe closed subspace $[0,\\omega_1)\\times\\{\\omega\\}$ is homeomorphic\nto {S35} and {S35|P194}.\n","refs":[]},{"space":"S000079","property":"P000198","value":true,"uid":"","description":"\n\n$X$ is covered by the countably many subspaces\n$[0,\\omega_1]\\times\\{n\\}$ for $n<\\omega$ and $[0,\\omega_1)\\times\\{\\omega\\}$,\nwith each of them {P198}\nsince {S36|P198}\nand {S35|P198}.\n","refs":[]},{"space":"S000079","property":"P000217","value":true,"uid":"","description":"\n\nThe Stone-Čech compactification of $X$ is homeomorphic to {S78}\n(see Proposition 4.4. in {{zb:0292.54001}}) and {S78|P50}.\n","refs":[{"name":"The Stone-Čech Compactification (R.C. Walker)","zb":"0292.54001"}]},{"space":"S000080","property":"P000004","value":true,"uid":"","description":"\n\nSee discussion at {{mathse:1715435}}, using the fact that no two points of $S$ have the same $y$-coordinate.\n","refs":[{"name":"Is Arens Square a Urysohn space?","mathse":1715435},{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000080","property":"P000009","value":false,"uid":"","description":"\n\nSee item #3 for space #80 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000080","property":"P000010","value":true,"uid":"","description":"\n\nSee item #4 for space #80 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000080","property":"P000011","value":false,"uid":"","description":"\n\nSee item #2 for space #80 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000080","property":"P000022","value":false,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #80 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000080","property":"P000028","value":true,"uid":"","description":"\n\nBy construction.\n","refs":[{"name":"Answer to \"Is Arens Square a Urysohn space?\"","mathse":1716095}]},{"space":"S000080","property":"P000047","value":true,"uid":"","description":"\n\nSee item #6 for space #80 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000080","property":"P000057","value":true,"uid":"","description":"\n\nBy construction.\n","refs":[{"name":"Answer to \"Is Arens Square a Urysohn space?\"","mathse":1716095}]},{"space":"S000080","property":"P000139","value":false,"uid":"","description":"\n\nBy construction.\n","refs":[{"name":"Answer to \"Is Arens Square a Urysohn space?\"","mathse":1716095}]},{"space":"S000081","property":"P000006","value":false,"uid":"","description":"\n\nFor all $\\alpha , n$ there is no open neighborhood $V$ of $( \\omega_1 , 0 )$ such that $\\overline{V} \\subseteq U(\\alpha,n)$.\n","refs":[]},{"space":"S000081","property":"P000009","value":true,"uid":"","description":"\n\nSee item #1 for space #88 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000081","property":"P000010","value":true,"uid":"","description":"\n\nSee item #2 for space #88 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000081","property":"P000011","value":false,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #88 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000081","property":"P000018","value":false,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #88 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000081","property":"P000019","value":false,"uid":"","description":"\n\nSee item #3 for space #88 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000081","property":"P000028","value":false,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #88 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000081","property":"P000029","value":false,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #88 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000081","property":"P000031","value":false,"uid":"","description":"\n\nSee item #3 for space #88 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000081","property":"P000036","value":false,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #88 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000081","property":"P000041","value":false,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #88 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000081","property":"P000046","value":false,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #88 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000081","property":"P000056","value":false,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #88 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000081","property":"P000065","value":true,"uid":"","description":"\n\nThe space's cardinality is $\\aleph_1\\cdot\\mathfrak c=\\mathfrak c$.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000081","property":"P000198","value":true,"uid":"","description":"\n\n$X$ is the union of finitely many subspaces, each with {P198}:\n* $[0, \\omega_1] \\times [-1, 0)$, because it is {P18} as the product of a {P16} space and a {P18} space (see for example {{mathse:3652735}}), and {T562}.\n* $[0, \\omega_1] \\times (0, 1]$ for the same reason.\n* $[0, \\omega_1) \\times \\{0\\}$, because it is homeomorphic to {S35} and {S35|P198}.\n* $\\{\\left< \\omega_1, 0 \\right>\\}$, because {S162|P198}.\n","refs":[{"name":"If $X$ is Lindelof and $Y$ is compact, then $X \\times Y$ is Lindelof","mathse":3652735}]},{"space":"S000081","property":"P000229","value":true,"uid":"","description":"\n\nThe subspace topology on $[0, \\omega_1) \\times [-1, 1] \\subseteq X$ is the standard product topology. The subspace topology on $\\{\\omega_1\\} \\times [-1, 1] \\subseteq X$ is a (topological) disjoint union of $\\{\\omega_1\\} \\times [-1, 0)$ and $\\{\\omega_1\\} \\times [0, 1]$ with their standard topologies. $X$ has a finer topology than $[0, \\omega_1] \\times [-1, 1]$ with the product topology.\n\nIt follows that the path components of $X$ are $\\{\\omega_1\\} \\times [-1, 0)$, $\\{\\omega_1\\} \\times [0, 1]$, and $\\{\\alpha\\} \\times [-1, 1]$ for each $\\alpha < \\omega_1$, all with their standard topologies. Thus each path component is homeomorphic to {S210} or {S158}.\n\nThe result follows since {S210|P229} and {S158|P229}.\n","refs":[]},{"space":"S000082","property":"P000001","value":true,"uid":"","description":"\n\n$X$ is the direct sum of copies of {S42},\nand {S42|P1}.\n","refs":[]},{"space":"S000082","property":"P000022","value":false,"uid":"","description":"\n\nA function that sends each copy of {S42} to an integer within $\\mathbb R$ is continuous, \nand with infinitely-many copies this can be unbounded.\n","refs":[{"name":"Example: limit point compact + ¬countably compact+ ¬Lindelöf","mathse":3155216}]},{"space":"S000082","property":"P000032","value":false,"uid":"","description":"\n\n$X$ contains a closed copy of {S42}, but {S42|P32}.\n","refs":[]},{"space":"S000082","property":"P000036","value":false,"uid":"","description":"\n\nEach copy of {S42} is clopen by construction.\n","refs":[]},{"space":"S000082","property":"P000043","value":true,"uid":"","description":"\n\n{P43} is preserved by disjoint unions and {S42|P43}.\n","refs":[]},{"space":"S000082","property":"P000049","value":true,"uid":"","description":"\n\n{P49} is preserved by disjoint unions and {S42|P49}.\n","refs":[]},{"space":"S000082","property":"P000054","value":true,"uid":"","description":"\n\n{P54} is preserved by disjoint unions and {S42|P54}.\n","refs":[]},{"space":"S000082","property":"P000056","value":true,"uid":"","description":"\n\n{P56} is preserved by disjoint unions and {S42|P56}.\n","refs":[]},{"space":"S000082","property":"P000062","value":false,"uid":"","description":"\n\nFor each countable subcollection of the open cover using\neach of the uncountable copies of {S42}, its union is \nclosed and fails to contain uncountably-many copies\nof {S42}.\n","refs":[{"name":"Example: limit point compact + ¬countably compact+ ¬Lindelöf","mathse":3155216}]},{"space":"S000082","property":"P000065","value":true,"uid":"","description":"\n\nBy construction, as $\\mathfrak c \\cdot \\mathfrak c = \\mathfrak c$.\n","refs":[]},{"space":"S000082","property":"P000073","value":false,"uid":"","description":"\n\n$X$ contains a closed copy of {S42}, but {S42|P73}.\n","refs":[]},{"space":"S000082","property":"P000086","value":true,"uid":"","description":"\n\nEach component {S42} is {P86} by {S42|P86}.\n","refs":[]},{"space":"S000082","property":"P000090","value":false,"uid":"","description":"\n\nThe space contains a copy of {S42}, and {S42|P90}.\n","refs":[]},{"space":"S000082","property":"P000093","value":false,"uid":"","description":"\n\n$X$ contains a copy of {S42}, but {S42|P93}.\n","refs":[]},{"space":"S000082","property":"P000105","value":true,"uid":"","description":"\n\n{P105} is preserved by disjoint unions and {S42|P105}.\n","refs":[]},{"space":"S000082","property":"P000108","value":true,"uid":"","description":"\n\n{P108} is preserved by disjoint unions and {S42|P108}.\n","refs":[]},{"space":"S000082","property":"P000130","value":true,"uid":"","description":"\n\n{P130} is preserved by disjoint unions and {S42|P130}.\n","refs":[]},{"space":"S000082","property":"P000207","value":true,"uid":"","description":"\n\n{P207} is preserved by disjoint unions and {S42|P207}.\n","refs":[]},{"space":"S000082","property":"P000229","value":true,"uid":"","description":"\n\n{S42|P37} and so each path component of $X$ is a copy of {S42}. The assertion follows since {S42|P229}.\n","refs":[]},{"space":"S000083","property":"P000022","value":false,"uid":"","description":"\n\nThere are real-valued continuous functions on $X$ that are unbounded. For example $f$ defined by $f(x)=x$ for $x\\ne 0$ and $f(0_1)=f(0_2)=0$.\n","refs":[{"name":"Non-Hausdorff manifold on Wikipedia","wikipedia":"Non-Hausdorff_manifold"}]},{"space":"S000083","property":"P000024","value":true,"uid":"","description":"\n\nEach of the sets $A_n=([-n,n]\\setminus\\{0\\})\\cup\\{0_1,0_2\\}$ is compact and closed in $X$, and each point is in the interior of some $A_n$.\n","refs":[{"name":"Non-Hausdorff manifold on Wikipedia","wikipedia":"Non-Hausdorff_manifold"}]},{"space":"S000083","property":"P000038","value":false,"uid":"","description":"\n\nThere is no injective path joining one origin to the other.\nOne can prove this by the same arguments used to show that\n{S65|P38}.\nSee for example {{mathse:1853792}}.\n","refs":[{"name":"Proof about a topological space being arc connected","mathse":1853792}]},{"space":"S000083","property":"P000061","value":true,"uid":"","description":"\n\nThe Tychonoff reflection of $X$ is {S25},\nwith the reflection map $\\varphi:X\\to\\mathbb R$ sending the two \"origins\" to $0$\nand every other element to itself.\n\nSince {S25|P61}\nand the reflection map $\\varphi$ is open,\nthe result follows from Proposition 2 in {{mathse:5108293}}.\n","refs":[{"name":"Cozero complemented property for a space and its Tychonoff reflection","mathse":5108293}]},{"space":"S000083","property":"P000068","value":false,"uid":"","description":"\n\nIt is easy to check being {P68} passes to closed subspaces. Since $[1,2]$ is a closed subspace which is\nhomeomorphic to {S158} and {S158|P68}, the result follows.\n","refs":[]},{"space":"S000083","property":"P000089","value":false,"uid":"","description":"\n\nFor any non-zero $a$, $(x,i)\\mapsto (x+a,i)$ is a self-homeomorphism of $\\mathbb R\\times\\{1,2\\}$ with no fixed points, which remains a continuous function with no fixed points after selecting a representative element for each equivalence class.\n","refs":[]},{"space":"S000083","property":"P000101","value":false,"uid":"","description":"\n\nThe map that sends $0_2$ to $0_1$ and leaves every other point fixed is a retraction onto a non-closed subspace.\n","refs":[{"name":"Non-Hausdorff manifold on Wikipedia","wikipedia":"Non-Hausdorff_manifold"}]},{"space":"S000083","property":"P000110","value":true,"uid":"","description":"\n\n$X$ is homeomorphic to a subspace of {S85}\nand {S85|P110}.\n","refs":[]},{"space":"S000083","property":"P000145","value":true,"uid":"","description":"\n\nSuppose $\\mathcal U$ is an open cover of $X$. The subspace $Y=(\\mathbb R\\setminus\\{0\\})\\cup\\{0_1\\}$ is open in $X$ and homeomorphic to $\\mathbb R$. Intersecting every element of $\\mathcal U$ with $Y$ gives an open cover of $Y$, and, since {S25|P145}, we can choose a star finite open refinement $\\mathcal V$ of that cover of $Y$. \n\nTake some $U_1\\in\\mathcal V$ containing $0_1$ and some $U_2\\in\\mathcal U$ containing $0_2$.\nThe set $U:=(U_1\\cap U_2)\\cup\\{0_2\\}\\subseteq U_2$ is open in $X$.\nWe have $U\\cap Y\\subseteq U_1$, hence $U$ meets finitely many elements of $\\mathcal V$,\nand $\\mathcal V':=\\mathcal V\\cup\\{U\\}$ is a star-finite open refinement of $\\mathcal U$.\n","refs":[]},{"space":"S000083","property":"P000155","value":true,"uid":"","description":"\n\nEach point is contained in an open set homeomorphic to $\\mathbb R$, namely $X\\setminus\\{0_1\\}$ or $X\\setminus\\{0_2\\}$.\n\nSee Exercise 5 of section 36 in {{zb:0951.54001}}.\n","refs":[{"name":"Topology (Munkres)","zb":"0951.54001"}]},{"space":"S000083","property":"P000169","value":false,"uid":"","description":"\n\nEvery regular open neighborhood of $0_1$ contains $0_2$, since $0_2$ is in the interior of the closure of every neighborhood of $0_1$.\n","refs":[{"name":"Non-Hausdorff manifold on Wikipedia","wikipedia":"Non-Hausdorff_manifold"}]},{"space":"S000083","property":"P000200","value":false,"uid":"","description":"\n\n$\\pi_1(X) \\cong \\mathbb{Z}$. See {{mo:25472}}.\n","refs":[{"name":"Fundamental group of the line with the double origin.","mo":25472}]},{"space":"S000083","property":"P000204","value":true,"uid":"","description":"\n\n$X$ is {P36}.\nEvery point other than one of the origins is a cut point.\n","refs":[]},{"space":"S000083","property":"P000205","value":false,"uid":"","description":"\n\nNeither of the origins is a cut point.\n","refs":[]},{"space":"S000083","property":"P000216","value":true,"uid":"","description":"\n\n$X$ embeds as a subspace of {S209} and {S209|P216}.\n","refs":[]},{"space":"S000084","property":"P000022","value":false,"uid":"","description":"\n\nThere are real-valued continuous functions on $X$ that are unbounded. For example $f$ defined by $f(x)=x$ for $x\\ne 0$ and $f(0_\\alpha)=0$ for every origin $0_\\alpha$.\n","refs":[{"name":"Non-Hausdorff manifold on Wikipedia","wikipedia":"Non-Hausdorff_manifold"}]},{"space":"S000084","property":"P000024","value":false,"uid":"","description":"\n\nThe closure of any neighborhood $U$ of any of the origins contains all the other origins. That closure $\\overline U$ cannot be compact, as the set of all origins is infinite, closed and discrete. So the origin points don't have closed compact neighborhoods.\n","refs":[{"name":"Is the line with multiple origins locally compact?","mathse":2404301}]},{"space":"S000084","property":"P000027","value":true,"uid":"","description":"\n\nA countable base for the topology is given by a countable base for the open subspace $\\mathbb R\\setminus\\{0\\}$ together with the countably many intervals $(-1/n,0)\\cup\\{0_\\alpha\\}\\cup(0,1/n)$ around each of the countably many origins $0_\\alpha$.\n","refs":[{"name":"Non-Hausdorff manifold on Wikipedia","wikipedia":"Non-Hausdorff_manifold"}]},{"space":"S000084","property":"P000032","value":false,"uid":"","description":"\n\nSee {{mathse:4756135}}.\n","refs":[{"name":"Paracompactness properties of the line with multiple origins","mathse":4756135}]},{"space":"S000084","property":"P000038","value":false,"uid":"","description":"\n\nThere is no injective path joining one origin to another.\nOne can prove this by the same arguments used to show that\n{S65|P38}.\nSee for example {{mathse:1853792}}.\n","refs":[{"name":"Proof about a topological space being arc connected","mathse":1853792}]},{"space":"S000084","property":"P000061","value":true,"uid":"","description":"\n\nThe Tychonoff reflection of $X$ is {S25},\nwith the reflection map $\\varphi:X\\to\\mathbb R$ sending the \"origins\" to $0$\nand every other element to itself.\n\nSince {S25|P61}\nand the reflection map $\\varphi$ is open,\nthe result follows from Proposition 2 in {{mathse:5108293}}.\n","refs":[{"name":"Cozero complemented property for a space and its Tychonoff reflection","mathse":5108293}]},{"space":"S000084","property":"P000068","value":false,"uid":"","description":"\n\nIt is easy to check being {P68} passes to closed subspaces. Since $[1,2]$ is a closed subspace which is\nhomeomorphic to {S158} and {S158|P68}, the result follows.\n","refs":[]},{"space":"S000084","property":"P000089","value":false,"uid":"","description":"\n\nFor any non-zero $a$, $(x,i)\\mapsto (x+a,i)$ is a self-homeomorphism of $\\mathbb R\\times S$ with no fixed points, which remains a continuous function with no fixed points after selecting a representative element for each equivalence class.\n","refs":[]},{"space":"S000084","property":"P000101","value":false,"uid":"","description":"\n\nThe map that sends all the origins to one specific origin $0_\\alpha$ and leaves every other point fixed is a retraction onto a non-closed subspace.\n","refs":[{"name":"Non-Hausdorff manifold on Wikipedia","wikipedia":"Non-Hausdorff_manifold"}]},{"space":"S000084","property":"P000110","value":true,"uid":"","description":"\n\n$X$ is homeomorphic to a subspace of {S85}\nand {S85|P110}.\n","refs":[]},{"space":"S000084","property":"P000155","value":true,"uid":"","description":"\n\nAny point is contained in some open subspace homeomorphic to {S25}.\n","refs":[{"name":"Non-Hausdorff manifold on Wikipedia","wikipedia":"Non-Hausdorff_manifold"}]},{"space":"S000084","property":"P000169","value":false,"uid":"","description":"\n\nThe interior of the closure of any neighborhood of one of the origins $0_\\alpha$ contains all the other origins. So there is no regular open neighborhood of $0_\\alpha$ avoiding the other origins.\n","refs":[{"name":"Non-Hausdorff manifold on Wikipedia","wikipedia":"Non-Hausdorff_manifold"}]},{"space":"S000084","property":"P000200","value":false,"uid":"","description":"\n\nChoose two of the origins, $0_1$ and $0_2$. The map sending every origin except for $0_1$ to $0_2$, and fixing all other points, is a retraction onto {S83}. Since {S83|P200}, it follows by Proposition 1.17 of {{zb:1044.55001}} that $X$ is not simply connected.\n","refs":[{"name":"Algebraic Topology (Hatcher)","zb":"1044.55001"}]},{"space":"S000084","property":"P000204","value":true,"uid":"","description":"\n\n$X$ is {P36}.\nEvery point other than one of the origins is a cut point.\n","refs":[]},{"space":"S000084","property":"P000205","value":false,"uid":"","description":"\n\nNone of the origins is a cut point.\n","refs":[]},{"space":"S000085","property":"P000022","value":false,"uid":"","description":"\n\nThere are real-valued continuous functions on $X$ that are unbounded. For example $f$ defined by $f(x)=x$ for $x\\ne 0$ and $f(0_\\alpha)=0$ for every origin $0_\\alpha$.\n","refs":[{"name":"Non-Hausdorff manifold on Wikipedia","wikipedia":"Non-Hausdorff_manifold"}]},{"space":"S000085","property":"P000024","value":false,"uid":"","description":"\n\nThe closure of any neighborhood $U$ of any of the origins contains all the other origins. That closure $\\overline U$ cannot be compact, as the set of all origins is infinite, closed and discrete. So the origin points don't have closed compact neighborhoods.\n","refs":[{"name":"Is the line with multiple origins locally compact?","mathse":2404301}]},{"space":"S000085","property":"P000026","value":true,"uid":"","description":"\n\nThe set $\\mathbb Q\\setminus\\{0\\}$ is dense in $X$.\n","refs":[{"name":"Non-Hausdorff manifold on Wikipedia","wikipedia":"Non-Hausdorff_manifold"}]},{"space":"S000085","property":"P000032","value":false,"uid":"","description":"\n\nSee {{mathse:4756135}}.\n","refs":[{"name":"Paracompactness properties of the line with multiple origins","mathse":4756135}]},{"space":"S000085","property":"P000038","value":false,"uid":"","description":"\n\nThere is no injective path joining one origin to another.\nOne can prove this by the same arguments used to show that\n{S65|P38}.\nSee for example {{mathse:1853792}}.\n","refs":[{"name":"Proof about a topological space being arc connected","mathse":1853792}]},{"space":"S000085","property":"P000059","value":true,"uid":"","description":"\n\nBy construction.\n","refs":[]},{"space":"S000085","property":"P000061","value":true,"uid":"","description":"\n\nThe Tychonoff reflection of $X$ is {S25},\nwith the reflection map $\\varphi:X\\to\\mathbb R$ sending the \"origins\" to $0$\nand every other element to itself.\n\nSince {S25|P61}\nand the reflection map $\\varphi$ is open,\nthe result follows from Proposition 2 in {{mathse:5108293}}.\n","refs":[{"name":"Cozero complemented property for a space and its Tychonoff reflection","mathse":5108293}]},{"space":"S000085","property":"P000089","value":false,"uid":"","description":"\n\nFor any non-zero $a$, $(x,i)\\mapsto (x+a,i)$ is a self-homeomorphism of $\\mathbb R\\times S$ with no fixed points, which remains a continuous function with no fixed points after selecting a representative element for each equivalence class.\n","refs":[]},{"space":"S000085","property":"P000101","value":false,"uid":"","description":"\n\nThe map that sends all the origins to one specific origin $0_\\alpha$ and leaves every other point fixed is a retraction onto a non-closed subspace.\n","refs":[{"name":"Non-Hausdorff manifold on Wikipedia","wikipedia":"Non-Hausdorff_manifold"}]},{"space":"S000085","property":"P000110","value":true,"uid":"","description":"\n\nFor each integer $n\\ge 1$, let $\\mathscr U_n$ be the open cover of $X$ consisting of all open intervals $(a,b)\\subseteq\\mathbb R$ of length $1/n$ and not containing $0$, together with all the intervals\n$((-1/n,1/n)\\setminus\\{0\\})\\cup\\{0_\\alpha\\}$ for $\\alpha\\in S$.\nThese open covers form a development for $X$.\n","refs":[]},{"space":"S000085","property":"P000118","value":true,"uid":"","description":"\n\nAs $\\mathbb R\\setminus\\{0\\}$ is second countable, take a countable base $\\mathcal A$ of open sets for $\\mathbb R\\setminus\\{0\\}$.\nLet $\\mathcal B$ be the collection of deleted open neighborhoods of $0$ of the form\n$B_n=(-1/n,1/n)\\setminus\\{0\\}$ for $n=1,2,\\dots$.\nAnd let $\\mathcal C=\\{\\{0_\\alpha\\}:\\alpha\\in S\\}$, which is a locally finite family of sets.\nSince $\\mathcal A$ and $\\mathcal B$ are countable families of sets, they are $\\sigma$-locally finite.\nSo the union $\\mathcal N=\\mathcal A\\cup\\mathcal B\\cup\\mathcal C$ is $\\sigma$-locally finite.\n\nClaim: $\\mathcal N$ is a $k$-network.\nIndeed, consider a compact set $K\\subseteq U$ with $U$ open in $X$.\nThe set of origin points in $K$ is closed in $X$ and discrete, hence compact and discrete;\nso the index set $T=\\{\\alpha\\in S:0_\\alpha\\in K\\}$ is finite.\n\nIf $T\\ne\\emptyset$, $U$ is a neighborhood of some $0_\\alpha$ with $\\alpha\\in T$.\nSo $U$ contains some set $B_n\\in\\mathcal B$.\nThe compact set $K'=K\\setminus(B_{n+1}\\cup\\{0_\\alpha:\\alpha\\in T\\})\\subseteq\\mathbb R\\setminus\\{0\\}$\nis covered by finitely many elements of $\\mathcal A$, each contained in $U$.\nTogether with $B_n\\in \\mathcal B$ and the singletons $\\{0_\\alpha\\}\\in\\mathcal C$ for $\\alpha\\in T$,\nthis provides a finite $\\mathcal N'\\subseteq\\mathcal N$ with $K\\subseteq\\bigcup\\mathcal N'\\subseteq U$.\n\nIf $T=\\emptyset$, $K$ can be covered by finitely many elements of $\\mathcal A$, each contained in $U$.\n","refs":[]},{"space":"S000085","property":"P000155","value":true,"uid":"","description":"\n\nAny point is contained in some open subspace homeomorphic to {S25}.\n","refs":[{"name":"Non-Hausdorff manifold on Wikipedia","wikipedia":"Non-Hausdorff_manifold"}]},{"space":"S000085","property":"P000163","value":false,"uid":"","description":"\n\nBy construction, $|X| = \\mathfrak{c}+2^\\mathfrak{c} = 2^\\mathfrak{c}$.\n","refs":[{"name":"Non-Hausdorff manifold on Wikipedia","wikipedia":"Non-Hausdorff_manifold"}]},{"space":"S000085","property":"P000169","value":false,"uid":"","description":"\n\nThe interior of the closure of any neighborhood of one of the origins $0_\\alpha$ contains all the other origins. So there is no regular open neighborhood of $0_\\alpha$ avoiding the other origins.\n","refs":[{"name":"Non-Hausdorff manifold on Wikipedia","wikipedia":"Non-Hausdorff_manifold"}]},{"space":"S000085","property":"P000200","value":false,"uid":"","description":"\n\nSimilar to {S84|P200}.\n","refs":[]},{"space":"S000085","property":"P000204","value":true,"uid":"","description":"\n\n$X$ is {P36}.\nEvery point other than one of the origins is a cut point.\n","refs":[]},{"space":"S000085","property":"P000205","value":false,"uid":"","description":"\n\nNone of the origins is a cut point.\n","refs":[]},{"space":"S000085","property":"P000227","value":true,"uid":"","description":"\n\nThe set of origins is a closed and discrete subspace of cardinality $2^\\mathfrak c$.\n","refs":[]},{"space":"S000086","property":"P000022","value":false,"uid":"","description":"\n\nThe map $f:X\\to\\mathbb R$ defined by $f(r,0)=f(r,1)=r$ is continuous and unbounded.\n","refs":[{"name":"Manifolds: Hausdorffness versus homogeneity (Baillif & Gabard)","doi":"10.1090/S0002-9939-07-09100-9"}]},{"space":"S000086","property":"P000024","value":false,"uid":"","description":"\n\nNo point has a closed compact neighborhood. Indeed, let $U$ be a neighborhood of some point. The set $U$ contains a non-trivial interval $I=[a,b]\\times\\{0\\}\\subseteq A$. The closure $\\overline I=[a,b]\\times\\{0,1\\}$ is not compact because it contains $[a,b]\\times\\{1\\}$, which is infinite, discrete and closed in $X$. So the closure of $U$ is not compact.\n","refs":[{"name":"Manifolds: Hausdorffness versus homogeneity (Baillif & Gabard)","doi":"10.1090/S0002-9939-07-09100-9"}]},{"space":"S000086","property":"P000026","value":true,"uid":"","description":"\n\nThe set $\\mathbb Q\\times\\{0\\}$ is dense in $X$.\n","refs":[{"name":"Manifolds: Hausdorffness versus homogeneity (Baillif & Gabard)","doi":"10.1090/S0002-9939-07-09100-9"}]},{"space":"S000086","property":"P000033","value":false,"uid":"","description":"\n\nSee {{mathse:4771463}}.\n","refs":[{"name":"Paracompactness of the Everywhere doubled line","mathse":4771463}]},{"space":"S000086","property":"P000037","value":true,"uid":"","description":"\n\nAny two points with at least one of them in $A$ are contained in a subspace of $X$\nhomeomorphic to {S83},\nand {S83|P37}.\n\nSo, given two points $x$ and $y$, choose a point $z\\in A$ with different first coordinate than $x$ and $y$. Then take a path from $x$ to $z$, followed by a path from $z$ to $y$.\n","refs":[{"name":"Manifolds: Hausdorffness versus homogeneity (Baillif & Gabard)","doi":"10.1090/S0002-9939-07-09100-9"}]},{"space":"S000086","property":"P000038","value":false,"uid":"","description":"\n\nThere is no injective path joining a point $(r,1)\\in B$ to the corresponding point $(r,0)\\in A$.\nOne can prove this by the same arguments used to show that\n{S65|P38}.\nSee for example {{mathse:1853792}}.\n","refs":[{"name":"Proof about a topological space being arc connected","mathse":1853792}]},{"space":"S000086","property":"P000061","value":true,"uid":"","description":"\n\nThe Tychonoff reflection of $X$ is {S25},\nwith the reflection map $\\varphi:X\\to\\mathbb R$ sending $(x,i)$ to $x$.\n\nSince {S25|P61}\nand the reflection map $\\varphi$ is open,\nthe result follows from Proposition 2 in {{mathse:5108293}}.\n","refs":[{"name":"Cozero complemented property for a space and its Tychonoff reflection","mathse":5108293}]},{"space":"S000086","property":"P000065","value":true,"uid":"","description":"\n\nBy construction.\n","refs":[{"name":"Manifolds: Hausdorffness versus homogeneity (Baillif & Gabard)","doi":"10.1090/S0002-9939-07-09100-9"}]},{"space":"S000086","property":"P000086","value":true,"uid":"","description":"\n\nSee Proposition 3.1 in {{doi:10.1090/S0002-9939-07-09100-9}},\nor {{mathse:4772218}}.\n","refs":[{"name":"Manifolds: Hausdorffness versus homogeneity (Baillif & Gabard)","doi":"10.1090/S0002-9939-07-09100-9"},{"name":"Answer to Paracompactness of the Everywhere doubled line","mathse":4772218}]},{"space":"S000086","property":"P000089","value":false,"uid":"","description":"\n\nFor any non-zero $a$, $\\langle x,i\\rangle\\mapsto \\langle x+a,i\\rangle$ is a self-homeomorphism with no fixed points.\n","refs":[]},{"space":"S000086","property":"P000101","value":false,"uid":"","description":"\n\nThe map that sends points $x=(r,1)\\in B$ to $(r,0)\\in A$ and leaves points of $A$ fixed is a retraction of $X$ onto $A$, but $A$ is not closed in $X$.\n","refs":[{"name":"Manifolds: Hausdorffness versus homogeneity (Baillif & Gabard)","doi":"10.1090/S0002-9939-07-09100-9"}]},{"space":"S000086","property":"P000104","value":false,"uid":"","description":"\n\nIf $d:X^2\\to[0,+\\infty)$ was a symmetric generating the topology, then for every $x\\in\\mathbb R$ there would exist $r_x>0$ such that $B_d((x,0),r_x)\\subseteq \\mathbb R\\times\\{0\\}$.\nLet $A_n:=\\{x\\in\\mathbb R: r_x>1/n\\}$.\nSince $\\mathbb R = \\bigcup_{n=1}^\\infty A_n$ is uncountable, so is $A_m$ for some $m\\geq 1$.\n\nAn uncountable subset of $\\mathbb R$ has an accumulation point (cf. {{mathse:310113}}), hence there exists a sequence of distinct values $(a_n)_{n=1}^\\infty\\subset A_m$ such that $a_n\\to a\\in \\mathbb R$.\nLet $S:=\\overline{\\{(a_n,0):n\\in\\mathbb N\\}}=\\{(a_n,0):n\\in\\mathbb N\\}\\cup\\{(a,0),(a,1)\\}$ and $S_0:=S\\setminus\\{(a,1)\\}$.\nSince $S$ is closed, every point $u\\in X\\setminus S$ satisfies $d(u,S_0)\\ge d(u,S)>0$.\nBy definition of $A_m$, we also have $d((a_n,0),(a,1))\\geq 1/m$, hence $d((a,1),S_0)>0$.\nSince every point not in $S_0$ has a positive distance to $S_0$, $S_0$ is closed, which is a contradiction.\n","refs":[{"name":"Accumulation points of uncountable sets","mathse":310113}]},{"space":"S000086","property":"P000106","value":true,"uid":"","description":"\n\nWe can construct a $G_\\delta$-diagonal sequence $\\mathscr U_1,\\mathscr U_2, \\dots$ by letting\n$\\mathscr U_n=\\{B(\\langle x,1\\rangle,1/n):x\\in\\mathbb R\\}$, where\n\n$\\quad B(\\langle x,1\\rangle,\\frac1n)=\\{\\langle x,1\\rangle\\}\\cup\\big(((x-\\frac1n,x)\\cup(x,x+\\frac1n))\\times\\{0\\}\\big)$\n\nis a basic open neighborhood around the point\n$\\langle x,1\\rangle$ in the top line as described in the definition of the space.\n\nEach $\\mathscr U_n$ is an open cover of $X$.\nOne easily checks that for a point $\\langle x,1\\rangle$ in the top line,\n$\\operatorname{st}(\\langle x,1\\rangle,\\mathscr U_n)=B(\\langle x,1\\rangle,\\frac1n)$;\nand for a point $\\langle x,0\\rangle$ in the bottom line,\n$\\operatorname{st}(\\langle x,0\\rangle,\\mathscr U_n)\\subseteq((x-\\frac2n,x+\\frac2n)\\times\\{0,1\\})\n\\setminus\\{\\langle x,1\\rangle\\}$.\nFrom there it follows that \n$\\bigcap_n\\operatorname{st}(\\langle x,i\\rangle,\\mathscr U_n)=\\{\\langle x,i\\rangle\\}$\nfor all $\\langle x,i\\rangle\\in X$.\n","refs":[]},{"space":"S000086","property":"P000117","value":true,"uid":"","description":"\n\nLet $\\mathcal B$ be a countable base of the Euclidean topology on $\\mathbb R$.\nAs a $\\sigma$-locally finite network $\\mathcal N$ take the union of the countably many families\n$\\{B{\\times}\\{0\\} \\}$ for $B\\in \\mathcal B$ and\n$\\{\\{(x,1)\\}: x\\in \\mathbb R\\}$. Each of these families is locally finite,\nas every point of $X$ admits a neighbourhood intersecting at most one set from each of those families.\n\n$\\mathcal N$ is a refinement of a base of the topology on $X$,\nhence it is a network.\n","refs":[]},{"space":"S000086","property":"P000118","value":false,"uid":"","description":"\n\nSee {{mathse:5034748}}.\n","refs":[{"name":"Does the Everywhere doubled line have a $\\sigma$-locally finite $k$-network?","mathse":5034748}]},{"space":"S000086","property":"P000155","value":true,"uid":"","description":"\n\nAny point is contained in some open subspace homeomorphic to {S25}.\n","refs":[{"name":"Manifolds: Hausdorffness versus homogeneity (Baillif & Gabard)","doi":"10.1090/S0002-9939-07-09100-9"}]},{"space":"S000086","property":"P000169","value":false,"uid":"","description":"\n\nThe interior of the closure of any neighborhood of a point $x$ contains the other point with the same first coordinate. So there is no regular open neighborhood of a point that avoids this other point.\n","refs":[{"name":"Manifolds: Hausdorffness versus homogeneity (Baillif & Gabard)","doi":"10.1090/S0002-9939-07-09100-9"}]},{"space":"S000086","property":"P000200","value":false,"uid":"","description":"\n\nThe map $(x,i)\\mapsto(x,0)$ for $x\\in\\mathbb R\\setminus\\{0\\}$, $i\\in\\{0,1\\}$ and fixing the two \"zero points\" is a retraction onto {S83}. Since {S83|P200}, it follows by Proposition 1.17 of {{zb:1044.55001}} that $X$ is not simply connected.\n","refs":[{"name":"Algebraic Topology (Hatcher)","zb":"1044.55001"}]},{"space":"S000086","property":"P000205","value":false,"uid":"","description":"\n\nThe subset $\\mathbb R\\times\\{0\\}$ is dense and homeomorphic to {S25}, hence $X\\setminus\\{(0,1)\\}$ is connected.\nTherefore $(0,1)$ is not a cut point.\n","refs":[]},{"space":"S000086","property":"P000227","value":true,"uid":"","description":"\n\nThe subspace $\\mathbb{R}{\\times}\\{1\\}$ is closed and discrete with cardinality $\\mathfrak c$.\n","refs":[]},{"space":"S000087","property":"P000022","value":false,"uid":"","description":"\n\nConsider $f:X\\to\\mathbb R$ given by\n\n- $f(n,m) = n$ for $n\\in [0,\\omega)$, $m\\in [0,\\omega]$;\n- $f(\\alpha,m)=0$ for $\\alpha\\in[\\omega,\\omega_1]$, $m \\in [0,\\omega]$.\n\nIt can be readily verified that $f$ is continuous but not bounded.\n","refs":[]},{"space":"S000087","property":"P000031","value":true,"uid":"","description":"\n\nSee item #3 for space #89 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000087","property":"P000032","value":false,"uid":"","description":"\n\nSee item #4 for space #89 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000087","property":"P000049","value":false,"uid":"","description":"\n\nClosure of the open set $\\{ (0,2n) : n\\in\\mathbb N\\}$ contains one additional point and is not open.\n","refs":[]},{"space":"S000087","property":"P000050","value":true,"uid":"","description":"\n\n{P50} is a hereditary property. The space $X$ is a subspace of {S191} and {S191|P50}.\n","refs":[]},{"space":"S000087","property":"P000051","value":true,"uid":"","description":"\n\n{P51} is a hereditary property. The space $X$ is a subspace of {S191} and {S191|P51}.\n","refs":[]},{"space":"S000087","property":"P000062","value":true,"uid":"","description":"\n\nGiven an open cover $\\mathscr U$ of $X$, choose a countable subcollection $\\mathscr A$ covering the countable set $\\{\\omega_1\\}\\times[0,\\omega)$.\nThe union $\\bigcup\\mathscr A$ contains all but countably many columns of $[0,\\omega_1]\\times[0,\\omega)$.\nTherefore the closure of $\\bigcup\\mathscr A$ has countable complement $B$ in $X$.\nOne can then extend $\\mathscr A$ to a countable subcollection of $\\mathscr U$ that covers $B$.\nThe union of the extended subcollection is dense in $X$.\n","refs":[]},{"space":"S000087","property":"P000103","value":true,"uid":"","description":"\n\nAssume $A$ is a {P19} subset of $X$. It cannot contain an infinite discrete subset that is closed in $A$.\nFrom that, one can deduce the following facts.\nIf a column of $A$ (set of the form $A\\cap (\\{\\alpha\\}{\\times}[0,\\omega])$ for some $\\alpha<\\omega_1$) is infinite,\nit necessarily contains the limit point $(\\alpha,\\omega)$ of that column.\nAlso, there have to be at most finitely many non-empty such columns (picking a single point from each column leads to a closed and discrete subset).\nAnd finally, $A$ has a finite intersection with $\\{\\omega_1\\}{\\times}[0,\\omega)$.\nIt follows that $A$ is closed as a finite union of closed sets.\n","refs":[]},{"space":"S000087","property":"P000114","value":true,"uid":"","description":"\n\nBy construction.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000087","property":"P000120","value":true,"uid":"","description":"\n\nSee Example 5.12 in {{zb:1458.54023}}.\n","refs":[{"name":"Locally ordered topological spaces (P. Pikul)","zb":"1458.54023"}]},{"space":"S000087","property":"P000136","value":false,"uid":"","description":"\n\nThe subset $\\{0\\}\\times[0,\\omega]$ is compact and infinite.\n","refs":[]},{"space":"S000087","property":"P000140","value":false,"uid":"","description":"\n\nThe closed subspace $[0,\\omega_1]\\times\\{0\\}$ is homeomorphic to {S155}\nand {S155|P140}.\n","refs":[]},{"space":"S000087","property":"P000174","value":true,"uid":"","description":"\n\nSee {{mathse:4821179}}.\n","refs":[{"name":"Radial and pseudoradial properties for Dieudonné plank and variant","mathse":4821179}]},{"space":"S000087","property":"P000198","value":false,"uid":"","description":"\n\nThe set $[0,\\omega_1)\\times\\{\\omega\\}$ is uncountable, closed and discrete.\n","refs":[]},{"space":"S000088","property":"P000005","value":true,"uid":"","description":"\n\nSee item #2 for space #90 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000088","property":"P000009","value":false,"uid":"","description":"\n\nSee item #3 for space #90 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000088","property":"P000022","value":true,"uid":"","description":"\n\nNote that {S79|P22}.\nTherefore given a continuous function\n$f:X\\to\\mathbb R$, for each $n\\in\\mathbb Z$ one can define\n$c_n=\\sup\\{|f(x)|: x\\in q(Y\\times\\{n\\})\\}$, where $q:Z\\to X$ is the quotient map.\nBut since $\\lim_{n\\to\\pm\\infty} c_n= |f(a^\\pm)|$ we obtain the global bound for $f$.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000088","property":"P000029","value":false,"uid":"","description":"\n\nThere exist uncountably many isolated points.\nCompare with {S79|P29}.\n","refs":[]},{"space":"S000088","property":"P000031","value":false,"uid":"","description":"\n\nThe space contains as a closed subspace a copy of\n{S35} and {S35|P31}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000088","property":"P000032","value":false,"uid":"","description":"\n\nThe space contains a closed subspace homeomorphic to {S79}\nand {S79|P32}.\n\nErroneously asserted as true in the General Reference Chart for space #90 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000088","property":"P000033","value":true,"uid":"","description":"\n\nConsider open subspaces $A_n=q([0,\\omega_1)\\times[0,\\omega]\\times\\{2n,2n+1\\})$\nand $B_n=q([0,\\omega_1]\\times[0,\\omega)\\times\\{2n,2n-1\\})$ for $n\\in\\mathbb Z$.\nEach of them is {P33}\n(for $A_n$: use the fact that the product of a {P19} space with a {P16} space is {P19};\nfor $B_n$: because it is Hausdorff and {P17}\n[(Explore)](https://topology.pi-base.org/spaces?q=%24%5Csigma%24-compact%2Bt2%2B%7Emetacompact))\nand each point belongs to at most two such subspaces.\nTherefore given a countable open cover $\\mathscr U$ of $X$, for each $n\\in\\mathbb Z$\nwe can find point-finite open refinements $\\mathscr V^A_n$ and $\\mathscr V^B_n$\nof the covers $\\{U\\cap A_n:U\\in\\mathscr U\\}$ and $\\{U\\cap B_n:U\\in\\mathscr U\\}$ respectively. Picking two more open sets covering the limit points, $a^\\pm\\in U^\\pm\\in \\mathscr U$, we obtain a point-finite refinement\n$\\bigcup_{n\\in\\mathbb Z}\\mathscr V^A_n\\cup \\bigcup_{n\\in\\mathbb Z}\\mathscr V^B_n\\cup\\{U^+,U^-\\}$ of $\\mathscr U$.\n","refs":[]},{"space":"S000088","property":"P000051","value":true,"uid":"","description":"\n\nSee item #7 for space #90 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000088","property":"P000062","value":true,"uid":"","description":"\n\n$X$ has a {P17} dense subspace\n$\\bigcup_{n\\in\\mathbb Z}q( [0,\\omega_1]\\times\\omega\\times\\{n\\})$,\nwhere $q:Z\\to X$ is the quotient map.\nGiven an open cover of $X$, each of the compact sets is covered by a finite subcollection of the cover.\nThe union of all these subcollections is countable and has a dense union in $X$.\n","refs":[]},{"space":"S000088","property":"P000085","value":false,"uid":"","description":"\n\nThe clopen subspace $q(\\{0\\}\\times[0,\\omega]\\times\\{0,1\\})$ is homeomorphic\nto {S20}\nand {S20|P85}.\n","refs":[]},{"space":"S000088","property":"P000103","value":false,"uid":"","description":"\n\nThe subspace $q([0,\\omega_1)\\times\\{0\\}\\times\\{0\\})$ is not closed but is homeomorphic\nto {S35} and {S35|P19}.\n","refs":[]},{"space":"S000088","property":"P000114","value":true,"uid":"","description":"\n\nBy construction.\n","refs":[]},{"space":"S000088","property":"P000115","value":false,"uid":"","description":"\n\n$X$ has a closed subspace homeomorphic to {S35}\nand {S35|P115}.\n","refs":[]},{"space":"S000088","property":"P000165","value":false,"uid":"","description":"\n\nThe space contains a closed subspace homeomorphic to {S79}\nand {S79|P165}.\n","refs":[]},{"space":"S000088","property":"P000174","value":true,"uid":"","description":"\n\nClearly the points $a^\\pm$ admit totally ordered local bases.\nA point of the form $q((\\omega_1,k,2n))=q((\\omega_1,k,2n-1))$ has a well-ordered base of neighbourhoods\n$q((\\lambda,\\omega_1]\\times\\{k\\}\\times\\{2n-1,2n\\})$ for $\\lambda<\\omega_1$.\nAny other point belongs to a first countable open subspace\n$q([0,\\alpha]\\times[0,\\omega]\\times\\{2n,2n+1\\})\\cong [0,\\alpha]\\times[0,\\omega]$ for some $\\alpha<\\omega_1$, $n\\in\\mathbb Z$, and we apply {T102}.\n","refs":[]},{"space":"S000088","property":"P000198","value":true,"uid":"","description":"\n\n$X$ is the union of two singletons and\ncountably many spaces homeomorphic to {S79}\nand {S79|P198}.\n","refs":[]},{"space":"S000089","property":"P000005","value":true,"uid":"","description":"\n\nSee item #6 for space #91 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000089","property":"P000006","value":false,"uid":"","description":"\n\nSee item #6 for space #91 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000089","property":"P000009","value":true,"uid":"","description":"\n\nSee item #6 for space #91 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000089","property":"P000021","value":false,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #91 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000089","property":"P000028","value":false,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #91 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000089","property":"P000029","value":false,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #91 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000089","property":"P000031","value":false,"uid":"","description":"\n\nThe {P31} property is hereditary with respect to closed sets. And the space contains as a closed subspace a copy of {S35}, which is not {P31}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000089","property":"P000032","value":false,"uid":"","description":"\n\nThe {P32} property is hereditary with respect to closed sets. And the space contains as a closed subspace a copy of {S79}, which is not {P32}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000089","property":"P000048","value":true,"uid":"","description":"\n\nSee item #7 for space #91 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000089","property":"P000050","value":false,"uid":"","description":"\n\nSee item #7 for space #91 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000089","property":"P000051","value":true,"uid":"","description":"\n\nSee item #7 for space #91 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000089","property":"P000103","value":false,"uid":"","description":"\n\n$X$ contains a copy of {S79} as a subspace and {S79|P103}.\n","refs":[]},{"space":"S000089","property":"P000114","value":true,"uid":"","description":"\n\nBy construction.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000089","property":"P000174","value":true,"uid":"","description":"\n\n$X$ is a subspace of {S88} and {S88|P174}.\n","refs":[]},{"space":"S000090","property":"P000005","value":true,"uid":"","description":"\n\nSee item #3 for space #92 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000090","property":"P000013","value":false,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #92 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000090","property":"P000016","value":false,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #92 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000090","property":"P000040","value":false,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #92 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000090","property":"P000060","value":true,"uid":"","description":"\n\nIf $x,y \\in X$ and $\\lambda = \\Gamma(x,y)$ then $\\psi(a^+_\\lambda)=x$ and $\\psi(a^-_\\lambda)=y$. But $\\hat{f}$ is continuous on $A$ and hence on $A_\\lambda$ so $\\hat{f}(a^+_\\lambda) = \\hat{f}(a^-_\\lambda)$, whence $f(x) = f(\\psi(a^+_\\lambda)) = \\hat{f}(a^+_\\lambda) = \\hat{f}(a^-_\\lambda) = f(\\psi(a^-_\\lambda)) = f(y)$.\n\nSee item #5 for space #92 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000090","property":"P000114","value":true,"uid":"","description":"\n\nBy construction.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000091","property":"P000022","value":false,"uid":"","description":"\n\nDefine $f:X\\to \\mathbb R$ as follows:\n$$ f((x,t))=\\begin{cases}\nn & \\text{ if } x=\\frac1n \\geq t, (n\\in\\mathbb N)\\\\ \n0&\\text{ otherwise }\n\\end{cases},\\quad (x,t)\\in X.$$\nNote that in a fixed row or column there are only finitely many points\nwhere the value of $f$ is different than at the limit point, hence the function is continuous.\nIt is unbounded, since $f((\\frac1n,0))=n$.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000091","property":"P000023","value":true,"uid":"","description":"\n\nFor each point either the row is clopen and homeomorphic to {S154}\nor the column is clopen and homeomorphic to {S20}.\n{S154|P16} and {S20|P16}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000091","property":"P000031","value":true,"uid":"","description":"\n\nAny open cover $\\mathscr U$ has an open refinement $\\mathscr V$ consisting of all the singletons of isolated points\nand exactly one base neighbourhood for each of the points $(0,t)$ and $(x,0)$.\nThe cover $\\mathscr V$ is point-finite.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000091","property":"P000032","value":false,"uid":"","description":"\n\nConsider the countable open cover $\\mathscr U:=\\{ L_i: i\\geq 1\\}\\cup \\{X\\setminus (\\{0\\}{\\times}(0,1])\\}$.\nLet $\\mathscr V$ be an open refinement of $\\mathscr U$. For each $i\\geq 1$ there exists\n$V_i\\in \\mathscr V$ such that $V_i$ is cofinite in $L_i$.\nThe intersection $A=(0,1]\\cap\\bigcap_{i\\geq 1}\\pi_1(V_i)$ is non-empty (with $\\pi_1$ the projection onto the first coordinate).\nFor $x\\in A$, every neighbourhood of $(x,0)$ intersects infinitely many $V_i$'s,\nhence $\\mathscr V$ is not locally finite.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000091","property":"P000050","value":true,"uid":"","description":"\n\nSee item #1 for space #93 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000091","property":"P000051","value":true,"uid":"","description":"\n\nEvery basic open set contains at most one point non-isolated in $X$.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000091","property":"P000061","value":false,"uid":"","description":"\n\nThe space contains a clopen copy of {S154}\nand {S154|P61}.\n","refs":[]},{"space":"S000091","property":"P000062","value":true,"uid":"","description":"\n\nEach $L_i$ for $i\\geq 1$ is compact ({S154|P16}),\nhence their union, i.e. $X\\setminus L_0$, admits a countable subcover (see {T122}), and $\\overline{X\\setminus L_0}=X$.\n","refs":[]},{"space":"S000091","property":"P000065","value":true,"uid":"","description":"\n\nThe space's cardinality is bounded between $|(0,1)|=\\mathfrak c$ and $|\\mathbb R^2|=\\mathfrak c$.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000091","property":"P000120","value":false,"uid":"","description":"\n\nThe space contains an open subspace homeomorphic\nto {S154}\nand {S154|P120}.\n","refs":[]},{"space":"S000091","property":"P000162","value":false,"uid":"","description":"\n\nSee {{mathse:4732529}}.","refs":[{"name":"Thomas plank is not realcompact","mathse":4732529}]},{"space":"S000091","property":"P000165","value":false,"uid":"","description":"\n\nThe closed sets $L_0$ and $\\{(0,\\frac1n):n=1,2,\\ldots\\}$ do not admit disjoint neighbourhoods and the latter is countable.\nSee item #2 for space #93 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000091","property":"P000174","value":false,"uid":"","description":"\n\nThe space contains a copy of {S154}\nand {S154|P174}.\n","refs":[]},{"space":"S000091","property":"P000186","value":true,"uid":"","description":"\n\n$X$ is a subspace of the product of {S20}\nand {S154}.\n{S20|P186}\nand {S154|P186}.\n","refs":[]},{"space":"S000091","property":"P000210","value":true,"uid":"","description":"\n\n{S154|P210} and\n{S20|P210},\nhence so is their product, which contains $X$ as a homeomorphic subspace.\n","refs":[]},{"space":"S000091","property":"P000227","value":true,"uid":"","description":"\n\nThe subspace $L_0$ is closed and discrete with cardinality $\\mathfrak c$.\n","refs":[]},{"space":"S000092","property":"P000003","value":true,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #94 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000092","property":"P000009","value":false,"uid":"","description":"\n\nSee item #4 for space #94 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000092","property":"P000011","value":true,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #94 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000092","property":"P000065","value":true,"uid":"","description":"\n\nBy construction, as {S91|P65} and this space is formed from a countable number of copies of it.\n","refs":[]},{"space":"S000093","property":"P000016","value":true,"uid":"","description":"\n\nThe lexicographic order on $Y=[0,1]\\times\\{0,1\\}$ is complete since both orders on $[0,1]$ and $\\{0,1\\}$ are; hence $Y$ is compact.\nAnd $X$ is compact as a closed subset of $Y$.\n\nSee item #4 for space #95 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000093","property":"P000026","value":true,"uid":"","description":"\n\nSee item #5 for space #95 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000093","property":"P000050","value":true,"uid":"","description":"\n\nSee item #6 for space #95 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000093","property":"P000065","value":true,"uid":"","description":"\n\nBy construction.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000093","property":"P000086","value":true,"uid":"","description":"\n\nThe map $F:X\\to X$, $\\langle x,t\\rangle\\mapsto\\langle 1-x,1-t\\rangle$ is a homeomorphism interchanging the subsets $B=[0,1)\\times\\{1\\}$ and $A=(0,1]\\times\\{0\\}$.\n\nFor $c\\in(0,1)$ define the map $S_c:X\\to X$ by the formula\n$$S_c(\\langle x,t\\rangle)=\\begin{cases} \\langle x-c,t\\rangle & \\text{for } x>c\n\\text{ or }\\langle x,t\\rangle=\\langle c,1\\rangle\\\\\n\\langle x+1-c,t\\rangle & \\text{for } x 0$ such that $\\{ (x,1) : 0 \\leq x < b \\} \\subseteq U$. As $( \\frac{b}{2} , 0 ) \\in V$ there is an $a < \\frac{b}{2}$ such that $\\{ (x,0) : a < x \\leq \\frac{b}{2} \\} \\cup \\{ (x,1) : a < x < \\frac{b}{2} \\} \\subseteq V$. But then $(x,1) \\in U \\cap V$ for all $a < x < \\frac{b}{2}$.\n\nSee item #2 for space #96 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000094","property":"P000017","value":false,"uid":"","description":"\n\nSee item #5 for space #96 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000094","property":"P000018","value":true,"uid":"","description":"\n\nSee item #5 for space #96 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000094","property":"P000026","value":true,"uid":"","description":"\n\nSee item #5 for space #96 in {{zb:0386.54001}}.\n$\\{ (x,1) : x \\in \\mathbb{Q}, 0 \\leq x < 1 \\}$ is a countable dense subset.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000094","property":"P000027","value":false,"uid":"","description":"\n\nSee item #5 for space #96 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000094","property":"P000028","value":true,"uid":"","description":"\n\nSee item #5 for space #96 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000094","property":"P000031","value":true,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #96 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000094","property":"P000048","value":true,"uid":"","description":"\n\nSee item #3 for space #96 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000094","property":"P000049","value":false,"uid":"","description":"\n\nSee item #8 for space #96 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000094","property":"P000051","value":false,"uid":"","description":"\n\nSee item #7 for space #96 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000094","property":"P000065","value":true,"uid":"","description":"\n\nThe space is the union of two subintervals of $\\mathbb R$.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000094","property":"P000068","value":false,"uid":"","description":"\n\nIt is easy to check being {P68} passes to closed subspaces and coarser topologies. Note that the subset\n$\\{(x,0): \\frac{1}{2}\\leq x \\leq 1\\}$ is closed, and the topology on it is homeomorphic to a topology finer than\n{S158}. As {S158|P68}, the result follows.\n","refs":[]},{"space":"S000094","property":"P000120","value":false,"uid":"","description":"\n\nThe open subspace $B\\setminus\\{(0,1)\\}$ is homeomorphic to {S43}\nand {S43|P120}.","refs":[]},{"space":"S000095","property":"P000003","value":true,"uid":"","description":"\n\nWe will prove that the Concentric Circles Topology is $T_2$ by cases. There are four cases.\nCase one both a and b are on $C_1$ thus if we find the radial distance from a to b on $C_1$ and then take half the distance and call that distance $\\epsilon$ the create $V_{\\epsilon}(a)$ and $V_{\\epsilon}(b)$ these two sets are disjoint and open on $C_1$ when also projected onto $C_2$.\nCase two a is on $C_1$ and b is the radial projection of a onto $C_2$. find any open set on $C_1 $ where a is the midpoint and the set containing b on $C_2$ these two sets are disjoint and open.\nCase three a is on $C_1$ and b is not the radial midpoint of a onto $C_2$. first we find the pre-image j of b in $C_1$ find the radial distance between a and j then multiply that by two and call that $\\epsilon$ the take the epsilon neighborhood around j and the set containing b on $C_2$ these are two disjoint open sets containing a and b respectively.\nCase four both a and b are on $C_2$ this is trivial and just take the sets only containing a and b respectively.\nTherefore the concentric circles Topology is $T_2$.\n","refs":[]},{"space":"S000095","property":"P000014","value":true,"uid":"","description":"\n\nGiven Separated sets $A$ and $B$, now see that $O_A = A \\cap C_1$ and $O_B = B \\cap C_1$ are subsets of $\\mathbb{R}^2$ and since $O_A$ and $O_B$ are separated thus we can find disjoint neighborhoods in $\\mathbb{R}^2$ $U_A$ and $U_B$ in $C_1$ respectively. Now given any point a in $O_A$, a has an open set $U_a$ which is disjoint from B and can be chosen such that $U_a \\cap C_1 \\subset U_A$ and similarly for any point b in $B \\cap C_1$ we can find a disjoint open set $U_b$ from $A$ and $U_b \\cap C_1 \\subset U_B$. Now since $A \\cap C_2$ and $B \\cap C_2$ are open we can get open sets $(A \\cap C_2) \\cup (\\bigcup\\limits_{a \\in A} U_a)$ and $(B \\cap C_2) \\cup (\\bigcup\\limits_{b \\in B} U_b)$ which are disjoint and contain $A$ and $B$ respectively.\n\nSee item #2 for space #97 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000095","property":"P000015","value":false,"uid":"","description":"\n\n\nAsserted in the General Reference Chart for space #97 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000095","property":"P000016","value":true,"uid":"","description":"\n\nTo prove the concentric circles is compact it is sufficient to prove that $C_1$ is compact. We know that $C_1 \\subset \\mathbb{R}^2$ thus by the Heine-Borel Theorem of Higher Order Reals we must only show that $C_1$ is closed and bounded in $\\mathbb{R}^2$. To show $C_1$ is bounded we can see that the distance from $(0,0)$ in $\\mathbb{R}^2$ is equal to $1$ thus $C_1$ is bounded by $1$. To show that $C_1$ is closed it is enough to show that the complement of $C_1$ is open thus we will construct the complement of $C_1$ since $C_1 = \\left\\{x \\in \\mathbb{R}^2 : |x| = 1 \\right\\}$ thus\n\n$C_1 {}^c = \\left\\{x \\in \\mathbb{R}^2 : |x| < 1 \\right\\} \\cup \\left\\{x \\in \\mathbb{R}^2 : |x| > 1 \\right\\}$\n\nwhich are open sets in $\\mathbb{R}^2$ therefore $C_1$ is compact. Thus the Concentric Circles Topology is compact.\n\nSee item #3 for space #97 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000095","property":"P000020","value":true,"uid":"","description":"\n\nSee item #4 for space #97 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000095","property":"P000026","value":false,"uid":"","description":"\n\nSee item #6 for space #97 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000095","property":"P000027","value":false,"uid":"","description":"\n\nAs each $x \\in C_2$ is isolated, any base for $X$ must contain all singletons $\\{ x \\}$ for $x \\in C_2$. As $C_2$ is uncountable, $X$ has no countable base.\n\nSee item #5 for space #97 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000095","property":"P000028","value":true,"uid":"","description":"\n\nSee item #5 for space #97 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000095","property":"P000029","value":false,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #97 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000095","property":"P000036","value":false,"uid":"","description":"\n\nSince concentric circles are Hausdorff, singletons are closed. Any singleton in $C_2$ is also open. Therefore the space is disconnected.\n","refs":[]},{"space":"S000095","property":"P000041","value":false,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #97 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000095","property":"P000046","value":false,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #97 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000095","property":"P000065","value":true,"uid":"","description":"\n\nAs each circle has cardinality $\\mathfrak c$, the union of the two also has cardinality $\\mathfrak c$.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000095","property":"P000068","value":false,"uid":"","description":"\n\nIt is easy to check being {P68} passes to closed subspaces. Since $C_1$ is a closed subspace which is\nhomeomorphic to {S170} and {S170|P68}, the result follows.\n","refs":[]},{"space":"S000095","property":"P000120","value":false,"uid":"","description":"\n\nBy definition, the subspace $C_1$ is not open but homeomorphic to {S170},\nwhich is a contradiction with Lemma 3.5 in {{zb:1458.54023}}.\n","refs":[{"name":"Locally ordered topological spaces (P. Pikul)","zb":"1458.54023"}]},{"space":"S000095","property":"P000139","value":true,"uid":"","description":"\n\nEvery point of $C_2$ is an isolated point.\n","refs":[]},{"space":"S000095","property":"P000229","value":true,"uid":"","description":"\n\nThe path components of $X$ are $C_1$, which is homeomorphic to {S170}, and the singletons $\\{x\\}$ for $x\\in C_2$.\nThe result follows since {S170|P229} and {S162|P229}.\n","refs":[]},{"space":"S000096","property":"P000002","value":true,"uid":"","description":"\n\nSee item #2 for space #98 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000096","property":"P000023","value":false,"uid":"","description":"\n\nSee item #6 for space #98 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000096","property":"P000028","value":false,"uid":"","description":"\n\nSuppose $U_n$ is a countable family of local neighbourhoods for $1$. For every $n$ pick some number $a_n \\in U_n$ with $a_n > 10^n$. Then define $A = \\{a_n: n \\in \\omega\\}$, which has density $0$, so $\\{1\\} \\cup \\mathbb{N}\\setminus A$ is a neighbourhood of $1$ that contains no $U_n$.\n\nSee item #5 for space #98 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000096","property":"P000049","value":false,"uid":"","description":"\n\nLet $A$ be the even numbers $>1$, $B$ the odd numbers $>1$. Both are open and $1$ is in the closure of both (every set of density $1$ intersects loads of even and odd numbers).\n\nSee item #9 for space #98 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000096","property":"P000057","value":true,"uid":"","description":"\n\nThis is true by definition (defined on the integers).\n\nSee item #4 for space #98 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000096","property":"P000136","value":true,"uid":"","description":"\n\nAsserted in {{mathse:4566624}}.\n","refs":[{"name":"Can a hemicompact space fail to be weakly locally compact?","mathse":4566624}]},{"space":"S000096","property":"P000203","value":true,"uid":"","description":"\n\nAll points but one are isolated by definition.\n","refs":[]},{"space":"S000097","property":"P000010","value":true,"uid":"","description":"\n\nThe given basis for the space consists of regular open sets:\nthe closure of a neighborhood of $x$ includes $y$, but not in its\ninterior; likewise the closure of a neighborhood of $y$ includes $x$,\nbut not in its interior; and every point in $\\omega^2$ is clopen.\n","refs":[]},{"space":"S000097","property":"P000016","value":true,"uid":"","description":"\n\nSee item #2 for space #99 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000097","property":"P000051","value":true,"uid":"","description":"\n\nSee item #6 for space #99 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000097","property":"P000057","value":true,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #99 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000097","property":"P000084","value":true,"uid":"","description":"\n\nThe sets $\\{x\\}\\cup\\omega^2$ and $\\{y\\}\\cup\\omega^2$ are\n{P3} and open.\n","refs":[]},{"space":"S000097","property":"P000100","value":true,"uid":"","description":"\n\nSee item #3 for space #99 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000097","property":"P000187","value":false,"uid":"","description":"\n\nNote that the {P187} property is hereditary.\nSo the result follows as the subspace $\\omega^2\\cup\\{x\\}$ is homeomorphic to\n{S131}\nand {S131|P187}.\n\nSee {{mathse:4973385}}.\n","refs":[{"name":"Answer to \"Are W-spaces with countable pseudocharacter first countable?\"","mathse":4973385}]},{"space":"S000097","property":"P000214","value":false,"uid":"","description":"\n\n$X$ contains {S131} as a subspace and {S131|P214}.\n","refs":[]},{"space":"S000097","property":"P000243","value":true,"uid":"","description":"\n\n{S131} is an open dense subspace of $X$\nand {S131|P243}.\n","refs":[]},{"space":"S000098","property":"P000003","value":true,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #100 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000098","property":"P000004","value":false,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #100 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000098","property":"P000010","value":true,"uid":"","description":"\n\nSee item #2 for space #100 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000098","property":"P000022","value":true,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #100 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000098","property":"P000027","value":true,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #100 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000098","property":"P000041","value":false,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #100 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000098","property":"P000056","value":false,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #100 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000098","property":"P000057","value":true,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #100 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000099","property":"P000003","value":true,"uid":"","description":"\n\nSee item #1 for space #101 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000099","property":"P000014","value":false,"uid":"","description":"\n\nSee item #3 for space #101 in {{zb:0386.54001}}.\n(Note that the first edition from 1970 erroneously marked this property as true.)\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000099","property":"P000016","value":true,"uid":"","description":"\n\nSee item #2 for space #101 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000099","property":"P000020","value":true,"uid":"","description":"\n\nSee item #4 for space #101 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000099","property":"P000028","value":false,"uid":"","description":"\n\nWe will prove that for each $x \\in [0,1]$ there doesn't exist a countable base at point $(x,x)$. Let's assert opposite, that is, that there exists $x \\in [0,1]$ such that there exists a countable base $\\mathcal{B}=\\{B_{n}:n \\in \\mathbb{N}\\}$ at point $(x,x)$. For each $n \\in \\mathbb{N}$ let $A_{n}= \\{x_{0},\\dots ,x_k\\}$ where $B_n=\\{(t,y) \\in X: |y-x|<\\epsilon , t \\notin \\{x_0,\\dots,x_k\\}\\}$. Let $A=\\bigcup\\limits_{n \\in \\mathbb{N}}A_n$. Then $cardA\\leq\\aleph_0$.\nSuppose $z \\in [0,1]$ is an arbitrary element. Then $U=([0,1]\\setminus z)\\times[0,1]$ is an open set which contains $(x,x)$. So there exists $B \\in \\mathcal{B}$ such that $ (x,x) \\in B \\subseteq U$. That implies $z\\notin B$ so it must be $z \\in A_n\\subseteq A$. We conclude $[0,1]\\subseteq A$, but that is a contradiction with $card[0,1]=\\mathfrak{c}>cardA=\\aleph_0$.\n\nSee item #4 for space #101 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000099","property":"P000029","value":false,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #101 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000099","property":"P000038","value":true,"uid":"","description":"\n\nSee item #5 for space #101 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000099","property":"P000041","value":false,"uid":"","description":"\n\nSee item #5 for space #101 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000099","property":"P000065","value":true,"uid":"","description":"\n\nBy construction, as $| [0,1] \\times [0,1] | = \\mathfrak{c} \\cdot \\mathfrak{c} = \\mathfrak{c}$.\n","refs":[]},{"space":"S000099","property":"P000086","value":false,"uid":"","description":"\n\nPoints on the diagonal don't have a countable local base, but points off the diagonal do. (See {S99|P28}.)\n","refs":[]},{"space":"S000100","property":"P000001","value":true,"uid":"","description":"\n\nSee {{mathse:5001004}}.\n","refs":[{"name":"Answer to \"Are hereditarily connected spaces and ultraconnected spaces contractible?\"","mathse":5001004}]},{"space":"S000100","property":"P000027","value":true,"uid":"","description":"\n\nSee {{mathse:5001004}}.\n","refs":[{"name":"Answer to \"Are hereditarily connected spaces and ultraconnected spaces contractible?\"","mathse":5001004}]},{"space":"S000100","property":"P000039","value":true,"uid":"","description":"\n\nSee {{mathse:5001004}}.\n","refs":[{"name":"Answer to \"Are hereditarily connected spaces and ultraconnected spaces contractible?\"","mathse":5001004}]},{"space":"S000100","property":"P000040","value":true,"uid":"","description":"\n\nSee {{mathse:5001004}}.\n","refs":[{"name":"Answer to \"Are hereditarily connected spaces and ultraconnected spaces contractible?\"","mathse":5001004}]},{"space":"S000100","property":"P000056","value":true,"uid":"","description":"\n\nSee {{mathse:5001004}}.\n","refs":[{"name":"Answer to \"Are hereditarily connected spaces and ultraconnected spaces contractible?\"","mathse":5001004}]},{"space":"S000100","property":"P000086","value":false,"uid":"","description":"\n\nThe order $\\le$ on elements of $X$ is determined by the topology, as it coincides with the specialization order:\n$x\\le y$ iff $x\\subseteq\\overline{\\{y\\}}$.\n\nThe element $x_1\\in X$ with length $l(x_1)=-1$ and range $\\{0\\}$ has a single immediate successor for that order, namely, the element with length $0$ and range $\\{0\\}$.\nBut for example the element $x_2\\in X$ with length $-2$ and range $\\{0\\}$ has two immediate successors, extending $x_2$ to length $-1$ with two possible values.\n","refs":[{"name":"Answer to \"Are hereditarily connected spaces and ultraconnected spaces contractible?\"","mathse":5001004}]},{"space":"S000100","property":"P000090","value":false,"uid":"","description":"\n\nSee {{mathse:5001004}}.\n","refs":[{"name":"Answer to \"Are hereditarily connected spaces and ultraconnected spaces contractible?\"","mathse":5001004}]},{"space":"S000100","property":"P000130","value":true,"uid":"","description":"\n\nSee {{mathse:5001004}}.\n","refs":[{"name":"Answer to \"Are hereditarily connected spaces and ultraconnected spaces contractible?\"","mathse":5001004}]},{"space":"S000100","property":"P000181","value":true,"uid":"","description":"\n\nBy construction.\n","refs":[]},{"space":"S000100","property":"P000199","value":false,"uid":"","description":"\n\nSee {{mathse:5001004}}.\n","refs":[{"name":"Answer to \"Are hereditarily connected spaces and ultraconnected spaces contractible?\"","mathse":5001004}]},{"space":"S000100","property":"P000200","value":true,"uid":"","description":"\n\nSee {{mathse:5001004}}.\n","refs":[{"name":"Answer to \"Are hereditarily connected spaces and ultraconnected spaces contractible?\"","mathse":5001004}]},{"space":"S000101","property":"P000006","value":true,"uid":"","description":"\n\n{S2|P6}, hence so is every power of it.\n","refs":[]},{"space":"S000101","property":"P000022","value":false,"uid":"","description":"\n\nFollows as {S2|P22};\nsee {{mathse:4991560}}.\n","refs":[{"name":"Answer to \"Does every product of copies of $\\mathbb Z$ fail to be pseudocompact?\"","mathse":4991560}]},{"space":"S000101","property":"P000023","value":false,"uid":"","description":"\n\nLet $x$ be any point and let $U$ be a neighborhood of $x$. Let $V\\subset U$ be a basic open set. That is, $V$ is a product of subsets of $\\mathbb{Z}^+$ which is all of $\\mathbb{Z}^+$ in all but finitely many coordinates. Let $\\alpha$ be one of the coordinates where $V$ is all of $\\mathbb{Z}^+$ in that coordinate. Then define for each $n\\in \\mathbb{Z}^+$ $A_n$ to be the set of all $u\\in U$ such that $u_\\alpha=n$. Then we see that the collection $\\{A_n\\}$ is a covering of $U$ which has no finite subcovering, hence $U$ is not compact.\n","refs":[]},{"space":"S000101","property":"P000026","value":true,"uid":"","description":"\n\nSee item #3 for space #103\nin {{doi:10.1007\\/978-1-4612-6290-9_6}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000101","property":"P000044","value":false,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #103\nin {{doi:10.1007\\/978-1-4612-6290-9_6}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000101","property":"P000050","value":true,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #103\nin {{doi:10.1007\\/978-1-4612-6290-9_6}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000101","property":"P000051","value":false,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #103\nin {{doi:10.1007\\/978-1-4612-6290-9_6}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000101","property":"P000087","value":true,"uid":"","description":"\n\nThe space is a product of spaces satisfying the property: {S2|P87}.\n","refs":[]},{"space":"S000101","property":"P000140","value":false,"uid":"","description":"\n\nSee {{mathse:4644785}}.\n","refs":[{"name":"The product of uncountably many copies of the real line is not a k-space","mathse":4644785}]},{"space":"S000101","property":"P000163","value":false,"uid":"","description":"\n\nBy construction, as $\\aleph_0^{\\mathfrak c} = 2^{\\mathfrak c} > \\mathfrak c$.\n","refs":[]},{"space":"S000101","property":"P000191","value":false,"uid":"","description":"\n\nPer definition of product topology, it is easy to see that any nonempty $G_\\delta$ set must contain a subset of the form $Y \\times \\omega^{\\mathfrak{c} \\setminus I}$ where $I \\subset \\mathfrak{c}$ is countable and $Y \\subset \\omega^I$ is nonempty. In particular, no $G_\\delta$ set is a singleton.\n","refs":[]},{"space":"S000101","property":"P000214","value":false,"uid":"","description":"\nSee {{mathse:5106079}}.\n","refs":[{"name":"Do the Arkhangel'skii $\\alpha_i$ properties hold for uncountable products?","mathse":5106079}]},{"space":"S000101","property":"P000227","value":true,"uid":"","description":"\n\nSee Exercise 3.1.H(a) in {{zb:0684.54001}}.\n\nSee also .\n","refs":[{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"space":"S000102","property":"P000023","value":false,"uid":"","description":"\n\nAny neighbourhood of a point in $X$ contains a closed copy of {S000003},\nand {S3|P16}.\n\nSee item #4 for space #104 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000102","property":"P000029","value":false,"uid":"","description":"\n\nThe family $\\{\\{y\\}\\times \\mathbb{R}^\\omega : y\\in \\mathbb{R} \\}$ is a partition of $X$ into continuum many pairwise disjoint non-empty open sets.\n","refs":[]},{"space":"S000102","property":"P000055","value":true,"uid":"","description":"\n\n$X$ is a countable product of copies of {S3}\nand {S3|P55}.\n","refs":[]},{"space":"S000102","property":"P000087","value":true,"uid":"","description":"\n\n$X$ is a product of copies of {S3}\nand {S3|P87}.\n","refs":[]},{"space":"S000102","property":"P000093","value":false,"uid":"","description":"\n\nA basic open set in the product topology fixes finitely many coordinates\nand imposes no restriction on the other coordinates. So it is uncountable.\n","refs":[]},{"space":"S000102","property":"P000166","value":true,"uid":"","description":"\n\n{S30} has a coarser topology than $X$.\n{S30|P26}\nand {S30|P53}.\n","refs":[]},{"space":"S000102","property":"P000218","value":true,"uid":"","description":"\n\nSee Example 7.3.14 in {{zb:0684.54001}}.\n","refs":[{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"space":"S000102","property":"P000227","value":true,"uid":"","description":"\n\nThe subspace $\\mathbb R \\times \\{0\\}^\\omega$ is closed discrete with cardinality $\\mathfrak c$.\n","refs":[]},{"space":"S000103","property":"P000003","value":true,"uid":"","description":"\n\nA product of Hausdorff spaces is Hausdorff and {S158|P3}.\n","refs":[]},{"space":"S000103","property":"P000014","value":false,"uid":"","description":"\n\nBy Theorem 2.3.23. in {{zb:0684.54001}}, the space $X$ contains a homeomorphic copy of any {P6} space admitting\na base of cardinality at most continuum. In particular a copy\nof {S57} and {S57|P13}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"space":"S000103","property":"P000016","value":true,"uid":"","description":"\n\n$I^I$ is a product of compact spaces and thus is compact.\n\nSee item #1 for space #105 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000103","property":"P000020","value":false,"uid":"","description":"\n\nSee item #5 for space #105 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000103","property":"P000026","value":true,"uid":"","description":"\n\nContinuum-sized product of separable spaces is separable.\nSee Theorem 16.4 c) in {{zb:0205.26601}}.\n","refs":[{"name":"General Topology (Willard, 1970)","zb":"0205.26601"}]},{"space":"S000103","property":"P000042","value":true,"uid":"","description":"\n\nEvery point has a neighbourhood basis consisting of sets\nhomeomorphic to $I^I$ and {S103|P37}.\n","refs":[]},{"space":"S000103","property":"P000081","value":false,"uid":"","description":"\n\nBy Theorem 2.3.23. in {{zb:0684.54001}}, the space $X$ contains a homeomorphic copy of any {P6} space admitting\na base of cardinality at most continuum. In particular a copy\nof {S36} and {S36|P81}.\n","refs":[{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"space":"S000103","property":"P000086","value":true,"uid":"","description":"\n\nSince $|I|=|\\omega\\times I|$, the space is homeomorphic to the product\nof continuum many copies of {S32}.\nThe assertion follows from the fact that {S32|P86}.\n","refs":[]},{"space":"S000103","property":"P000089","value":true,"uid":"","description":"\n\n$X$ is a compact convex subset of a locally convex topological vector space $\\mathbb R^I$. Therefore Schauder-Tychonoff fixed point theorem (Theorem 5.28 in {{zb:0867.46001}}) applies.\n","refs":[{"name":"Functional analysis (W. Rudin, 1991)","zb":"0867.46001"}]},{"space":"S000103","property":"P000103","value":false,"uid":"","description":"\n\nBy Theorem 2.3.23. in {{zb:0684.54001}}, the space $X$ contains a homeomorphic copy of any {P6} space admitting\na base of cardinality at most continuum. In particular a copy\nof {S36} and {S36|P103}.\n","refs":[{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"space":"S000103","property":"P000163","value":false,"uid":"","description":"\n\nThe space is of cardinality $|I^I|=2^{\\mathfrak c}$.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000103","property":"P000172","value":false,"uid":"","description":"\n\n\nConsider the set of characteristic functions of all finite subsets of $I$,\ni.e. $A=\\{f\\in X: f(I)\\subseteq\\{0,1\\},\\ |f^{-1}(1)|<\\aleph_0\\}$.\nThe constant function $\\chi_I\\equiv 1$ belongs to the closure of $A$, since every base neighbourhood of $\\chi_I$\ncontains an element of $A$.\n\nSuppose $(f_\\alpha)_{\\alpha<\\lambda}\\subset A$ is a transfinite sequence converging to $\\chi_I$.\nFor each $x\\in I$ there exists $\\alpha_x<\\lambda$ such that\n$f_\\alpha(x)>\\frac12$ for all $\\alpha\\in(\\alpha_x,\\lambda)$.\nSince $I$ is uncountable, there exists $x\\in I$ such that $L_x=\\{y\\in I: \\alpha_y\\leq \\alpha_x\\}$ is infinite.\nThen for $\\alpha>\\alpha_x$ and each of the infinitely many $y\\in L_x$ we have $f_\\alpha(y)>\\frac12$, while $f_\\alpha\\in A$ has finite support, a contradiction.\n","refs":[]},{"space":"S000103","property":"P000197","value":false,"uid":"","description":"\n\nThe set of all characteristic functions of the singletons,\ni.e. $\\{f\\in X: f(I)=\\{0,1\\},\\ |f^{-1}(1)|=1\\}$,\nis discrete and uncountable.\n","refs":[]},{"space":"S000103","property":"P000199","value":true,"uid":"","description":"\n\nThis follows because {S158|P199} and a product of contractible spaces is contractible. See {{mathse:4463999}}.\n","refs":[{"name":"The cartesian product of contractible spaces is contractible","mathse":4463999}]},{"space":"S000103","property":"P000214","value":false,"uid":"","description":"\nSee {{mathse:5106079}}.\n","refs":[{"name":"Do the Arkhangel'skii $\\alpha_i$ properties hold for uncountable products?","mathse":5106079}]},{"space":"S000103","property":"P000215","value":false,"uid":"","description":"\n\nBy Theorem 2.3.23. in {{zb:0684.54001}}, the space $X$ contains a homeomorphic copy of any {P6} space admitting\na base of cardinality at most continuum. In particular a copy\nof {S35} and {S35|P162}.\n","refs":[{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"space":"S000104","property":"P000006","value":true,"uid":"","description":"\n\nThis holds because both factor spaces are also \$T_{3\\frac{1}{2}}\$.\n\nSee item #1 for space #106 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000104","property":"P000013","value":false,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #106 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000104","property":"P000016","value":false,"uid":"","description":"\n\nThe collection of open sets $[0,b)\\times I^I$ is an open covering which has no finite (in fact, no countable) subcovering.\n\nSee item #2 for space #106 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000104","property":"P000019","value":true,"uid":"","description":"\n\n$[ 0 , \\Omega ) \\times I^I$ is the product of the compact space $I^I$ and the countably compact space $[ 0 , \\Omega )$, and is therefore countably compact.\n\nSee item #2 for space #106 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000104","property":"P000020","value":false,"uid":"","description":"\n\n{S103} is homeomorphic to the closed subspace $\\{0\\} \\times I ^ I \\subseteq X$, and {S103|P20}.\n","refs":[]},{"space":"S000104","property":"P000023","value":true,"uid":"","description":"\n\nLet $(a,b)$ be a point in the space. That is, $a$ is a countable ordinal and $b\\in I^I$. Then the set $[0,a+1]\\times I^I$ is a compact neighborhood of $(a,b)$. It is compact because it is the product of compact sets. It is a neighborhood because $[0,a+1)\\times I^I$ is an open set containing $(a,b)$.\n\nSee item #3 for space #106 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000104","property":"P000029","value":false,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #106 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000104","property":"P000036","value":false,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #106 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000104","property":"P000041","value":false,"uid":"","description":"\n\nThe point $\\omega_0\\times 0$ has no connected neighborhood. Indeed, let $U$ be an open neighborhood of $\\omega_0\\times 0$. Then $U$ contains a set of the form $(a,b)\\times V$, with $a < \\omega_0 < b$. Let $A=([0,a+2)\\times I^I)\\cap U$ and $B=((a+1,\\Omega)\\times I^I)\\cap U$. Each is open since it is the intersection of two open sets. They are clearly disjoint, and $U=A\\cup B$, so they form a separation of $U$, hence $U$ is not connected. \n\nIt is noted that this works replacing $\\omega_0$ with any limit ordinal and relies only on the fact that $[0,\\Omega)$ is not locally connected.\n","refs":[]},{"space":"S000104","property":"P000046","value":false,"uid":"","description":"\n\n{S103} is homeomorphic to the subspace $\\{0\\} \\times I ^ I \\subseteq X$, and {S103|P46}.\n","refs":[]},{"space":"S000104","property":"P000056","value":false,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #106 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000104","property":"P000059","value":true,"uid":"","description":"\n\n$| [ 0, \\Omega ) \\times I^I | = | [0,\\Omega ) | \\cdot | I^I | = \\aleph_1 \\cdot |I|^{|I|} = \\aleph_1 \\cdot ( 2^{\\aleph_0} )^{2^{\\aleph_0}} = \\aleph_1 \\cdot 2^{\\aleph_0 \\cdot 2^{\\aleph_0}} = \\aleph_1 \\cdot 2^{2^{\\aleph_0}} = 2^{2^{\\aleph_0}} = 2^\\mathfrak{c}$.\n","refs":[]},{"space":"S000104","property":"P000081","value":false,"uid":"","description":"\n\n{S103} is homeomorphic to the subspace $\\{0\\} \\times I ^ I \\subseteq X$, and {S103|P81}.\n","refs":[]},{"space":"S000104","property":"P000103","value":false,"uid":"","description":"\n\n$X$ contains {S103} as a subspace and {S103|P103}.\n","refs":[]},{"space":"S000104","property":"P000163","value":false,"uid":"","description":"\n\n$| [ 0, \\Omega ) \\times I^I | = | [0,\\Omega ) | \\cdot | I^I | = \\aleph_1 \\cdot |I|^{|I|} = \\aleph_1 \\cdot ( 2^{\\aleph_0} )^{2^{\\aleph_0}} = \\aleph_1 \\cdot 2^{\\aleph_0 \\cdot 2^{\\aleph_0}} = \\aleph_1 \\cdot 2^{2^{\\aleph_0}} = 2^{2^{\\aleph_0}} = 2^\\mathfrak{c} > \\mathfrak{c}$.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000104","property":"P000172","value":false,"uid":"","description":"\n\n$X$ contains {S103} as a subspace and {S103|P172}.\n","refs":[]},{"space":"S000104","property":"P000209","value":true,"uid":"","description":"\n\n{S35|P209} and {S103|P209}, hence so is their product.\n","refs":[]},{"space":"S000104","property":"P000214","value":false,"uid":"","description":"\nSee {{mathse:5106079}}.\n","refs":[{"name":"Do the Arkhangel'skii $\\alpha_i$ properties hold for uncountable products?","mathse":5106079}]},{"space":"S000105","property":"P000003","value":true,"uid":"","description":"\n\nFollows as {S105} is a subspace of {S103}, which is {P3}.\n","refs":[]},{"space":"S000105","property":"P000008","value":false,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #107 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000105","property":"P000016","value":true,"uid":"","description":"\n\nLet $d$ be the sup-norm metric on $I^I$, let $f\\in I^I\\setminus X$, and find $x,y\\in I$ such that $x\\geq y$ but $f(x)< f(y)$. Then, for every $g\\in X$, $d(f,g) \\geq \\frac{1}{2}(f(y)-f(x))$ and so $f$ admits an open neighbourhood in $I^I\\setminus X$. It follows that $X$ is a closed subset of a compact space and therefore compact itself.\n\nSee item #1 for space #107 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000105","property":"P000026","value":true,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #107 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000105","property":"P000028","value":true,"uid":"","description":"\n\nSee item #2 for space #107 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000105","property":"P000058","value":false,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #107 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000105","property":"P000199","value":true,"uid":"","description":"\n\nLet $\\iota:X\\to X$ be the identity and $z:X\\to X$ be the constant map to the zero function $0\\in X$. Then $H(f, t) := (1-t)f$ defines a continuous function $H : X \\times [0, 1] \\to X$ satisfying $H(f,0)=\\iota(f)$ and $H(f,1)=z(f)$, showing the identity $\\iota$ and constant map $z$ are [homotopic](https://en.wikipedia.org/wiki/Homotopy#Formal_definition).\n","refs":[]},{"space":"S000107","property":"P000008","value":false,"uid":"","description":"\n\nSee {{doi:10.4064/fm-88-2-127-132}} and {{mathse:3039316}}.\n","refs":[{"name":"van Douwen, Eric; The box product of countably many metrizable spaces need not be normal","doi":"10.4064/fm-88-2-127-132"},{"name":"Is Rω endowed with the box topology completely normal (or hereditarily normal)?","mathse":3039316}]},{"space":"S000107","property":"P000022","value":false,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #109 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000107","property":"P000023","value":false,"uid":"","description":"\n\nSee item #5 for space #109 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000107","property":"P000029","value":false,"uid":"","description":"\n\nSee item #7 for space #109 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000107","property":"P000036","value":false,"uid":"","description":"\n\nSee item #3 for space #109 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000107","property":"P000046","value":false,"uid":"","description":"\n\nContains copy of {S25} and {S25|P37}.\n","refs":[]},{"space":"S000107","property":"P000065","value":true,"uid":"","description":"\n\nBy construction, as $\\mathfrak{c}^{\\aleph_0} = \\mathfrak{c}$.\n","refs":[]},{"space":"S000107","property":"P000081","value":false,"uid":"","description":"\n\nSee {{mathse:5132326}}.\n","refs":[{"name":"Answer to \"Box topology on $\\mathbb{R}^\\omega$ is not Frechet-Urysohn\"","mathse":5132326}]},{"space":"S000107","property":"P000087","value":true,"uid":"","description":"\n\nSee {{mathse:2322269}}. A product of topological groups when given the box topology is still a topological group, i.e., multiplication and inverse are continuous (because they are so componentwise). And the space {S25} is {P87}.\n","refs":[{"name":"Topological group with box topology","mathse":2322269}]},{"space":"S000107","property":"P000103","value":true,"uid":"","description":"\n\nSee {{mathse:5132370}}.\n","refs":[{"name":"Is the countable box product of real lines $\\mathbb{R}^\\omega$ Strongly KC?","mathse":5132370}]},{"space":"S000107","property":"P000120","value":false,"uid":"","description":"\n\nIt contains {S170} as a non-open subspace,\nwhich is a contradiction with Lemma 3.5 in {{zb:1458.54023}}.\n","refs":[{"name":"Locally ordered topological spaces (P. Pikul)","zb":"1458.54023"}]},{"space":"S000107","property":"P000140","value":false,"uid":"","description":"\n\nSee {{mo:510109}}.\n","refs":[{"name":"Answer to \"Is the countable box product of real lines a Čech-complete space?\"","mo":510109}]},{"space":"S000107","property":"P000162","value":true,"uid":"","description":"\n\nThe identity function $\\text{Id}:\\mathbb{R}^\\omega\\to \\mathbb{R}^\\omega$, $\\text{Id}(x) = x$ from {S107} to $\\mathbb{R}^\\omega$ with product topology is a continuous bijection. Since every subspace of $\\mathbb{R}^\\omega$ with product topology is realcompact ($\\mathbb{R}^\\omega$ is {P5} and {P131}, and see {T384}), {S107} is realcompact from corollary 8.18 in {{zb:1380.46022}}.\n\n","refs":[{"name":"Rings of Continuous Functions (Gillman and Jerison)","zb":"1380.46022"}]},{"space":"S000107","property":"P000166","value":true,"uid":"","description":"\n\n{S30} has a coarser topology than $X$. The assertion then follows since {S30|P26} and {S30|P53}.\n","refs":[]},{"space":"S000107","property":"P000187","value":false,"uid":"","description":"\n\nNote that the {P187} property is hereditary.\n\nLet $f_{m,n}:\\omega\\to\\mathbb R$ be defined by\n$f_{m,n}(m)=\\frac{1}{n+1}$ and $f_{m,n}(k)=0$ otherwise.\n\nThe result follows as the subspace\n$\\{\\vec 0\\}\\cup\\{f_{m,n}:m,n<\\omega\\}$ is homeomorphic to\n{S131}\nand {S131|P187}.\n\nSee {{mathse:4973385}}.\n","refs":[{"name":"Answer to \"Are W-spaces with countable pseudocharacter first countable?\"","mathse":4973385}]},{"space":"S000107","property":"P000191","value":true,"uid":"","description":"\n\nEach $f:\\omega\\to\\mathbb R$ is the countable intersection of\n$\\bigcap_{m<\\omega}\\prod_{n<\\omega}(f(n)-\\frac{1}{m+1},f(n)+\\frac{1}{m+1})$.\n\nSee {{mathse:1644041}}.\n","refs":[{"name":"Answer to \"Familiar spaces in which every one point set is Gδ but space is not first countable\"","mathse":1644041}]},{"space":"S000107","property":"P000206","value":true,"uid":"","description":"\n\nBox product of {S25}, and {S25|P206}.\n","refs":[]},{"space":"S000107","property":"P000214","value":false,"uid":"","description":"\n\n$X$ contains {S131} as a subspace (see {S107|P187}) and {S131|P214}.\n","refs":[]},{"space":"S000107","property":"P000227","value":true,"uid":"","description":"\n\n$\\left\\{ 0, 1 \\right\\}^\\omega$ is a closed and discrete subset of cardinality $\\mathfrak c$.\n","refs":[]},{"space":"S000107","property":"P000234","value":false,"uid":"","description":"\n\nBy {{mathse:5012784}} the connected component of an arbitrary point $x\\in X$ is $A = \\{y : y_n = x_n\\text{ for all but finitely many }n\\}$. Since $\\text{int}(A) = \\emptyset$, it follows that $A$ is not open.\n","refs":[{"name":"Answer to \"Is $\\ell^\\infty$ with box topology connected?\"","mathse":5012784}]},{"space":"S000108","property":"P000008","value":false,"uid":"","description":"\n\nsee Steen, Seebach \"Counterexamples in Topology\" (1978), General Reference Chart, p. 200\n\nAsserted in the General Reference Chart for space #111 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000108","property":"P000016","value":true,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #111 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000108","property":"P000020","value":false,"uid":"","description":"\n\nThe sequence $x_n = n$ has no convergent subsequence.\n","refs":[]},{"space":"S000108","property":"P000026","value":true,"uid":"","description":"\n\n$\\mathbb{N}$ (or, more precisely, the family of principal ultrafilters on $\\mathbb{N}$) is a countable dense subset.\n\nGiven any nonempty $U \\subset \\mathbb{N}$, for each $n \\in U$ we have that $n$ (or, more precisely, the principal ultrafilter determined by $n$) belongs to $\\overline{U}$.\n\nSee item #8 for space #111 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000108","property":"P000028","value":false,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #111 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000108","property":"P000049","value":true,"uid":"","description":"\n\nSince {S2|P49}. See proposition 2 of {{mathse:5025114}}.\n","refs":[{"name":"Answer to \"Which spaces admit only extremally disconnected compactifications?\"","mathse":5025114}]},{"space":"S000108","property":"P000050","value":true,"uid":"","description":"\n\nSee item #6 for space #111 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000108","property":"P000051","value":false,"uid":"","description":"\n\nSee item #7 for space #111 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000108","property":"P000059","value":true,"uid":"","description":"\n\n$X$ has cardinality $2^{\\mathfrak c}$.\n\nSee item #3 for space #111 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000108","property":"P000081","value":false,"uid":"","description":"\n\nIt has {S109} as a subspace and {S109|P81}. \n","refs":[]},{"space":"S000108","property":"P000139","value":true,"uid":"","description":"\n\nEvery point of $\\omega$ is isolated in $\\beta\\omega$.\n\nSee item #1 for space #111 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000108","property":"P000163","value":false,"uid":"","description":"\n\n$X$ has cardinality $2^{\\mathfrak c}$.\n\nSee item #3 for space #111 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000108","property":"P000167","value":true,"uid":"","description":"\n\nSee {{mo:138161}}.\n","refs":[{"name":"Non-trivial convergent sequence in Stone-Čech compactification of N","mo":138161}]},{"space":"S000108","property":"P000215","value":false,"uid":"","description":"\n\n$X$ contains {S109} as a subspace and {S109|P162}.\n","refs":[]},{"space":"S000108","property":"P000243","value":true,"uid":"","description":"\n\nThe set $\\{\\{n\\}: n < \\omega\\}$ is a countable $\\pi$-base for $\\beta \\omega$.\n","refs":[]},{"space":"S000109","property":"P000006","value":true,"uid":"","description":"\n\n$X$ is a subspace of {S108} and {S108|P6}.\n","refs":[]},{"space":"S000109","property":"P000016","value":false,"uid":"","description":"\n\nBy definition $X$ is a proper subset of $\\beta\\omega$, and it is dense because it contains $\\omega$. But if it were compact, then necessarily $X = \\beta\\omega$.\n","refs":[]},{"space":"S000109","property":"P000021","value":true,"uid":"","description":"\n\nLet $S\\subseteq X$ be countably infinite. Then $S = S_\\xi$ for some $\\xi < 2^\\mathfrak{c}$ and so $x_\\xi \\in \\overline{S}\\setminus S$. So $X$ does not contain an infinite closed discrete subspace.\n","refs":[]},{"space":"S000109","property":"P000026","value":true,"uid":"","description":"\n\n$\\omega$ is a countable dense subset of $X$.\n","refs":[]},{"space":"S000109","property":"P000049","value":true,"uid":"","description":"\n\n{S109} is dense in {S108} and {S108|P49}.\n","refs":[]},{"space":"S000109","property":"P000081","value":false,"uid":"","description":"\n\n$X$ has uncountable tightness, see {{mo:505981}}.\n","refs":[{"name":"Tightness of Novak space","mo":505981}]},{"space":"S000109","property":"P000139","value":true,"uid":"","description":"\n\nEvery point of $\\omega$ is isolated in $X$.\n","refs":[]},{"space":"S000109","property":"P000197","value":false,"uid":"","description":"\n\nLet $\\mathcal{A}$ be an almost disjoint family of size $\\mathfrak{c}$ on $\\mathbb{N}$ (see {{mathse:162387}} and {{wikipedia:Almost_disjoint_sets}}). Every element of $\\mathcal{A}$ is a countably infinite set, and so there exists $A\\subseteq \\lambda$ with $|A| = \\mathfrak{c}$ such that $\\mathcal{A} = \\{S_\\xi : \\xi \\in A\\}$. Let $Y = \\{x_\\xi : \\xi \\in A\\}$. If $\\xi\\in A$ then $\\overline{S_\\xi}\\subseteq X$ is open and $\\overline{S_\\xi}\\cap Y = \\{x_\\xi\\}$ since $(\\overline{S_\\xi}\\setminus S_\\xi)\\cap(\\overline{S_\\eta}\\setminus S_\\eta) = \\emptyset$ for distinct $\\xi, \\eta\\in A$\n(here note that necessarily $\\overline{S_\\xi}\\setminus S_\\xi\\subseteq \\beta\\omega\\setminus \\omega$ for $\\xi\\in A$; compare with Example 3.6.18 in {{zb:0684.54001}}).\nSo $Y\\subseteq X$ is a discrete set of size continuum.\n","refs":[{"name":"Can a countable set contain uncountably many infinite subsets such that the intersection of any two of the subsets is finite?","mathse":162387},{"name":"Almost disjoint sets on Wikipedia","wikipedia":"Almost_disjoint_sets"},{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"space":"S000110","property":"P000003","value":true,"uid":"","description":"\n\nSee item #1 for space #113 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000110","property":"P000011","value":false,"uid":"","description":"\n\nSee item #3 for space #113 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000110","property":"P000022","value":true,"uid":"","description":"\n\nFor any continuous $f: X \\to \\mathbb{R}$, the restriction $f\\restriction_{\\omega\\cup \\{F\\}}$ is continuous for any $F\\in M$. Since the topologies on $\\omega\\cup \\{F\\}$ as a subspace of $X$ and of $\\beta \\omega$ coincide, and $\\omega$ is dense in $\\beta\\omega$, it follows from theorem I.8.5.1 of {{doi:10.1007/978-3-642-61701-0}} that $f$ is continuous as a function $f:\\beta\\omega\\to \\mathbb{R}$. Since {S108|P22}, $f$ is bounded.\n","refs":[{"name":"General Topology. Chapters 1-4 (Bourbaki)","doi":"10.1007/978-3-642-61701-0"}]},{"space":"S000110","property":"P000026","value":true,"uid":"","description":"\n\nSee item #5 for space #113 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000110","property":"P000032","value":false,"uid":"","description":"\n\nSee {{mathse:5016607}}.\n","refs":[{"name":"Is the strong ultrafilter topology countably paracompact?","mathse":5016607}]},{"space":"S000110","property":"P000049","value":true,"uid":"","description":"\n\nSee item #2 for space #113 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000110","property":"P000051","value":true,"uid":"","description":"\n\nSee item #4 for space #113 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000110","property":"P000093","value":true,"uid":"","description":"\n\nEvery $n \\in \\omega$ has $\\{n\\}$ as a countable neighborhood and every $F \\in M$ has $\\omega \\cup \\{F\\}$ as a countable neighborhood.\n","refs":[]},{"space":"S000110","property":"P000103","value":true,"uid":"","description":"\n\nBy essentially the same proof as in {S110|P136}, any {P19}\nsubset $K$ of $X$ is finite. As {S110|P2}, $K$ is closed.\n","refs":[]},{"space":"S000110","property":"P000117","value":true,"uid":"","description":"\n\nFor any $n \\in \\omega$, $\\mathcal{N}_n = \\{\\{n\\}\\} \\cup \\{\\{F\\}: F \\in M\\}$ is a locally finite family. Indeed, for any $m \\in \\omega$, $\\{m\\}$ is an open neighborhood that only intersects at most one element of $\\mathcal{N}_n$. And, for any $G \\in M$, $\\omega \\cup \\{G\\}$ is an open neighborhood that only intersects two elements of $\\mathcal{N}_n$. Thus, $\\mathcal{N} = \\bigcup_{n \\in \\omega} \\mathcal{N}_n = \\{\\{x\\}: x \\in X\\}$ is a $\\sigma$-locally finite family and clearly a network.\n","refs":[]},{"space":"S000110","property":"P000132","value":true,"uid":"","description":"\n\nBy construction, every subset of $M$ is closed in $X$. And so is each of the countably many singletons $\\{n\\}$ for\n$n \\in \\omega$ since {S110|P2}. Thus, every subset of $X$ is $F_\\sigma$, and so every\nsubset of $X$ is $G_\\delta$ as well.\n","refs":[]},{"space":"S000110","property":"P000136","value":true,"uid":"","description":"\n\nLet $K \\subset X$ be compact. If $K$ contains infinitely many elements of $M$, then $\\{\\omega \\cup \\{F\\}: F \\in M \\cap K\\}$ is an open cover with no finite subcover. Thus, $K$ contains finitely many elements of $M$, which we can list as $F_1, \\cdots, F_n$.\n\nNow, suppose $K$ contains infinitely many elements of $\\omega$. We inductively choose $A_i \\in F_i$ for $1 \\leq i \\leq n$ as follows: We let $B_1=\\omega\\cap K$ and shall choose our inductive assumption as $B_i = (\\omega \\cap K) \\setminus (A_1 \\cup \\cdots \\cup A_{i-1})$ being infinite. Write $B_i = C_i \\cup D_i$ for two infinite, disjoint $C_i$ and $D_i$. If $C_i \\in F_i$, let $A_i = C_i$. Otherwise, let $A_i = \\omega \\setminus C_i$, which necessarily belongs to $F_i$. We note that, either way, $B_{i+1} = B_i \\setminus A_i$ is infinite.\n\nThus, $B_{n+1} = (\\omega \\cap K) \\setminus (A_1 \\cup \\cdots \\cup A_n)$ is infinite and $\\{A_i \\cup \\{F_i\\}\\}_{i=1}^n \\cup \\{\\{x\\}: x \\in B_{n+1}\\}$ is an open cover of $K$ with no finite subcover, also a contradiction. Hence, $K$ can contain only finitely many elements of $M$ and only finitely many elements of $\\omega$, so $K$ is finite.\n","refs":[]},{"space":"S000110","property":"P000163","value":false,"uid":"","description":"\n\n{S110} is equinumerous with {S108}, which is of size $2^\\mathfrak{c} > \\mathfrak{c}$.\n","refs":[]},{"space":"S000110","property":"P000194","value":true,"uid":"","description":"\n\nLet $\\mathscr{U}$ be an open cover of $X$. For each $F \\in M$, there exists $A_F \\in F$ s.t. $A_F \\cup \\{F\\}$ is contained in some element of $\\mathscr{U}$. Thus, for each $k \\in \\omega$, we may consider the refinement $\\mathscr{V}_k = \\{\\{n\\}: n \\in \\omega\\} \\cup \\{(A_F \\setminus \\{0, \\cdots, k\\}) \\cup \\{F\\}: F \\in M\\}$. For any $F \\in M$ and $k \\in \\omega$, $F$ is contained in exactly one element of $\\mathscr{V}_k$, namely $(A_F \\setminus \\{0, \\cdots, k\\}) \\cup \\{F\\}$. For any $n \\in \\omega$, $n$ is contained in exactly one element of $\\mathscr{V}_n$, namely $\\{n\\}$. Hence, $\\mathscr{V}_k$ form a $\\theta$-sequence of open refinements of $\\mathscr{U}$.\n","refs":[]},{"space":"S000110","property":"P000227","value":true,"uid":"","description":"\n\nBy construction, $M$ is a closed discrete subspace of $X$. There are $2^\\mathfrak{c}$ free ultrafilters on $\\omega$, i.e., $|M| = 2^\\mathfrak{c}$.\n","refs":[]},{"space":"S000111","property":"P000003","value":true,"uid":"","description":"\n\nLet $x,y \\in X$ be distinct.\n\n* If $x,y \\in \\omega$, then $\\{ x \\}$ and $\\{ y \\}$ are disjoint open neighborhoods of $x,y$, respectively.\n* If $x = F, y \\in \\omega$, then $\\omega \\setminus \\{ y \\}$ is an open neighborhood of $x$, and $\\{ y \\}$ is an open neighborhood of $y$, and these are disjoint.\n\nSee item #1 for space #114 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000111","property":"P000022","value":false,"uid":"","description":"\n\nSince $F$ is a free ultrafilter, there is some infinite $A \\in F$ with infinite complement in $\\omega$. Let $f: X \\rightarrow \\mathbb{R}$ by $f(x)=0$ if $x \\in A$ and $f(x)=x$ otherwise. Then $f$ is continuous and unbounded.\n","refs":[]},{"space":"S000111","property":"P000049","value":true,"uid":"","description":"\n\nSee item #2 for space #114 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000111","property":"P000050","value":true,"uid":"","description":"\n\nSee item #2 for space #114 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000111","property":"P000057","value":true,"uid":"","description":"\n\nBy construction.\n\nAsserted in the General Reference Chart for space #114 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000111","property":"P000136","value":true,"uid":"","description":"\n\nSee for example {{mathse:4681271}}.\n","refs":[{"name":"Answer to \"Is the Single ultrafilter topology compact?\"","mathse":4681271}]},{"space":"S000111","property":"P000203","value":true,"uid":"","description":"\n\nAll points except $F\\in X$ are isolated.\n","refs":[]},{"space":"S000111","property":"P000243","value":true,"uid":"","description":"\n\n$X$ is a dense subspace of {S108} and {S108|P243}.\n","refs":[]},{"space":"S000112","property":"P000022","value":false,"uid":"","description":"\n\nLet $\\pi: \\mathbb{R}^2 \\rightarrow \\mathbb{R}$ be projection onto the second coordinate. $\\pi|_X$ is clearly continuous and unbounded.\n","refs":[]},{"space":"S000112","property":"P000023","value":true,"uid":"","description":"\n\n$X$ is a closed subset of $\\mathbb R^2$, thus {P23}.\n\n**Note:** This trait is erroneously asserted to be false in the General Reference Chart for space #115 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000112","property":"P000036","value":false,"uid":"","description":"\n\nThe rectangle $R_1$ is clopen in $X$.\n","refs":[]},{"space":"S000112","property":"P000041","value":false,"uid":"","description":"\n\nLet $x\\in L_1$ and $U$ a neighborhood of $x$. Then we see that there is an index $N$ such that $U\\cap R_n\\ne \\emptyset$ for all $n\\ge N$. Then $A=U\\cap R_N$ and $B=U\\setminus R_N$ form a separation of $U$.\n\nSee item #1 for space #115 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000112","property":"P000046","value":false,"uid":"","description":"\n\nThe subspace $L_1$ is {P37}.\n","refs":[]},{"space":"S000112","property":"P000086","value":false,"uid":"","description":"\n\n$X$ is {P155} (thus {P41}) at some points in $X$ (e.g., points in $R_1$), while {S112|P41}.\n","refs":[]},{"space":"S000112","property":"P000097","value":false,"uid":"","description":"\n\nThe space contains a subspace homeomorphic to {S170}, and {S170|P97}.\n","refs":[]},{"space":"S000112","property":"P000120","value":false,"uid":"","description":"\n\nLet $x \\in L_1$ and suppose $U$ is an open neighborhood of $x$ that is {P133}.\nTake $V$ some open vertical segment around $x$ within $U \\cap L_1$.\n$V$ is connected, so $V$ is necessarily order-convex in $(U, \\le)$. Since $x$ is a cut point of $V$, \nthere must be points of $V$ on both sides of $x$, i.e., some $a, b \\in V$ with $a < x < b$ in $(U, \\le)$.\nThe interval $(a, b)$ is an open neighborhood of $x$ in $U$ and hence in $X$ and is contained in $V$, hence disjoint from $X \\setminus L_1$.\nBut that is impossible since $x$ is in the closure of $X \\setminus L_1$.\n","refs":[]},{"space":"S000112","property":"P000139","value":false,"uid":"","description":"\n\nEach point is contained in either a straight line or the boundary of a rectangle. Neither has isolated points.\n","refs":[]},{"space":"S000112","property":"P000184","value":true,"uid":"","description":"\n\nConstructed as a subspace of {S176}.\n","refs":[]},{"space":"S000113","property":"P000017","value":true,"uid":"","description":"\n\n$Y=X\\setminus\\{(0,0)\\}$ is homeomorphic to $[0,1)$ and\n{S210|P17}.\nFurthermore, $\\{(0,0)\\}$ is compact.\n","refs":[]},{"space":"S000113","property":"P000023","value":false,"uid":"","description":"\n\nSee item #1 for space #116 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000113","property":"P000037","value":false,"uid":"","description":"\n\nSee item #4 for space #116 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000113","property":"P000041","value":false,"uid":"","description":"\n\nSee item #4 for space #116 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000113","property":"P000046","value":false,"uid":"","description":"\n\nThe subspace $Y=\\{(x, \\sin\\frac{1}{x}):x \\in (0,1]\\}$ is homeomorphic\nto $[0,1)$ and {S210|P46}.\n","refs":[]},{"space":"S000113","property":"P000089","value":true,"uid":"","description":"\n\nSee {{mathse:5051764}}.\n\nNote that the proof uses the fact that $X$ is {P36}.\nFor this, see for example {S113|P204}.\n","refs":[{"name":"Does the topologist's sine curve have the fixed point property?","mathse":5051764}]},{"space":"S000113","property":"P000116","value":true,"uid":"","description":"\n\nBy Theorem 3.11 of {{doi:10.1007/978-1-4612-4190-4}}, and because {S114|P116}, it suffices\nto show {S113} is $G_\\delta$ in {S114}.\nThe complement of the space in {S114} is $\\{(0,y):y\\in[-1,0)\\cup(0,1]\\}$, which is $F_\\sigma$ as,\n\n$$\\{(0,y):y\\in[-1,0)\\cup(0,1]\\} = \\bigcup_{n=1}^\\infty (\\{0\\} \\times ([-1,-1/n] \\cup [1/n,1]))$$\n","refs":[{"name":"Classical Descriptive Set Theory (Kechris)","doi":"10.1007/978-1-4612-4190-4"}]},{"space":"S000113","property":"P000184","value":true,"uid":"","description":"\n\nConstructed as a subspace of {S176}.\n","refs":[]},{"space":"S000113","property":"P000189","value":true,"uid":"","description":"\n\n$X$ is {P189} because the dense subspace\n$Y=\\{(x, \\sin\\frac{1}{x}):x \\in (0,1]\\}$ is homeomorphic\nto $[0,1)$ and {S210|P189}.\n","refs":[]},{"space":"S000113","property":"P000204","value":true,"uid":"","description":"\n\n$X$ is {P36} by item #3 for space #116 in {{zb:0386.54001}}.\n\nAny point $(x_0,y_0)\\in X$ with $0x_0\\}$.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000113","property":"P000205","value":false,"uid":"","description":"\n\n$X\\setminus\\{(0,0)\\}$ is connected, so $(0,0)$ is not a cut point.\n","refs":[]},{"space":"S000113","property":"P000229","value":true,"uid":"","description":"\n\nThe path components of $X$ are $\\{\\langle x, \\sin \\frac{1}{x}\\rangle : x \\in (0, 1]\\}$, which is homeomorphic to {S210}, and $\\{\\langle 0, 0 \\rangle\\}$.\n\nThe result follows since {S210|P229} and {S162|P229}.\n","refs":[]},{"space":"S000114","property":"P000016","value":true,"uid":"","description":"\n\nSee item #3 for space #117 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000114","property":"P000037","value":false,"uid":"","description":"\n\nSee item #4 for space #117 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000114","property":"P000041","value":false,"uid":"","description":"\n\nSee item #4 for space #117 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000114","property":"P000046","value":false,"uid":"","description":"\n\nThe subspace $Y=\\{(x, \\sin\\frac{1}{x}):x \\in (0,1]\\}$ is homeomorphic\nto $[0,1)$ and {S210|P46}.\n","refs":[]},{"space":"S000114","property":"P000089","value":true,"uid":"","description":"\n\nSee {{mathse:5051764}}.\n","refs":[{"name":"Does the topologist's sine curve have the fixed point property?","mathse":5051764}]},{"space":"S000114","property":"P000184","value":true,"uid":"","description":"\n\nConstructed as a subspace of {S176}.\n","refs":[]},{"space":"S000114","property":"P000204","value":true,"uid":"","description":"\n\n$X$ is {P36} by item #3 for space #117 in {{zb:0386.54001}}.\n\nAny point $(x_0,y_0)\\in X$ with $0x_0\\}$.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000114","property":"P000229","value":true,"uid":"","description":"\n\nThe path components of $X$ are $\\{\\langle x, \\sin \\frac{1}{x}\\rangle : x \\in (0, 1]\\}$, which is homeomorphic to {S210}, and $\\{0\\} \\times [-1, 1]$.\n\nThe result follows since {S210|P229} and {S158|P229}.\n","refs":[]},{"space":"S000115","property":"P000016","value":true,"uid":"","description":"\n\n$X$ is a closed and bounded subset of the plane.\n\nAsserted in the General Reference Chart for space #118 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000115","property":"P000038","value":true,"uid":"","description":"\n\nSee item #6 for space #118 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000115","property":"P000041","value":false,"uid":"","description":"\n\nSee item #6 for space #118 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000115","property":"P000089","value":false,"uid":"","description":"\n\nDefine $f : X \\to X$ as\n$$f((x, y)) =\\begin{cases}\n(\\frac{2}{\\pi},\\, 1) \n & \\text{if } x \\leq \\frac{2}{5\\pi}, \\\\\n(-x + \\frac{12}{5\\pi},\\, \\sin(1 / (-x + \\frac{12}{5\\pi})))\n & \\text{if } \\frac{2}{5\\pi} \\leq x \\leq \\frac{2}{\\pi},\\ y = 1, \\\\\n(-x + \\frac{12}{5\\pi},\\, 1)\n & \\text{if } \\frac{2}{5\\pi} \\leq x \\leq \\frac{2}{\\pi},\\ y = \\sin(\\frac{1}{x}),\\\\\n(\\frac{2}{5\\pi},1) & \\text{if } \\frac{2}{\\pi} \\leq x. \\\\\n\\end{cases}$$\nThen $f$ has no fixed point and is continuous from gluing lemma.\n","refs":[]},{"space":"S000115","property":"P000184","value":true,"uid":"","description":"\n\nConstructed as a subspace of {S176}.\n","refs":[]},{"space":"S000115","property":"P000200","value":false,"uid":"","description":"\n\nLet $p,q\\in X$ be two consecutive local maxima on the graph of $f(x)=\\sin(1/x)$.\nThe loop $L\\subseteq X$ consisting of the line segment between $p$ and $q$ and the portion of the graph of $f$ below it is homeomorphic to a circle $S^1$.\nHence $\\pi_1(L)$ is non-trivial, since {S170|P200}.\n\nOn the other hand, $L$ is a retract of $X$ (by a map sending points to left of $p$ to $p$ and points to the right of $q$ to $q$).\nBy Proposition 1.17 of {{zb:1044.55001}}, $\\pi_1(L)$ injects into $\\pi_1(X)$, which is therefore non-trivial.\n","refs":[{"name":"Algebraic Topology (Hatcher)","zb":"1044.55001"}]},{"space":"S000115","property":"P000204","value":true,"uid":"","description":"\n\n$X$ is {P36} by item #6 for space #118 in {{zb:0386.54001}}.\n\nIt has a cut point $p := (\\frac{2}{\\pi}, 1)$ as $(X \\cap (-\\infty,\\frac{2}{\\pi})\\times\\mathbb{R}) \\cup (X \\cap (\\frac{2}{\\pi},\\infty)\\times\\mathbb{R}) = X \\setminus \\{p\\}$.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000116","property":"P000017","value":true,"uid":"","description":"\n\n$L_n \\subseteq X$ is compact for all $n \\in \\omega$, $(\\frac{1}{2},1] \\times \\{0\\} \\subseteq X$ is $\\sigma$-compact and $X = \\bigcup_{n \\in \\omega} L_n \\cup \\big((\\frac{1}{2},1] \\times \\{0\\}\\big)$.\n","refs":[]},{"space":"S000116","property":"P000023","value":false,"uid":"","description":"\n\n$(0,0)\\in X$ has no compact neighborhood.\n\nErroneously asserted as true in the General Reference Chart\nfor space #119 in {{zb:0386.54001}}.\n","refs":[]},{"space":"S000116","property":"P000037","value":false,"uid":"","description":"\n\nSee item #3 for space #119 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000116","property":"P000041","value":false,"uid":"","description":"\n\nSee item #2 for space #119 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000116","property":"P000046","value":false,"uid":"","description":"\n\nThe space admits some non-constant paths, along each of the segments $L_n$ for example.\n","refs":[]},{"space":"S000116","property":"P000055","value":true,"uid":"","description":"\n\nViewing $X$ as a subspace of {S176}, we have \n\n$\\quad \\quad X=\\overline{X} \\cap \\bigcap_{n \\geq 2}\\mathbb{R}^2\\setminus \\left(\\left[\\frac{1}{n}, \\frac{1}{2}\\right]\\times\\{0\\}\\right).$\n\nTherefore $X$ is a $G_\\delta$ subset of {S176} and the assertion follows since {S176|P55}.\n","refs":[]},{"space":"S000116","property":"P000089","value":false,"uid":"","description":"\n\nDefine $f:X \\to X$ as\n\n$\\quad p \\mapsto (1/2 + \\operatorname{dist}(p, v) / 3, 0)$\n\nwhere $v = \\left(\\frac{1}{2},0\\right)$.\nThis is well defined, continuous and has no fixed point.\n","refs":[]},{"space":"S000116","property":"P000184","value":true,"uid":"","description":"\n\nConstructed as a subspace of {S176}.\n","refs":[]},{"space":"S000116","property":"P000189","value":true,"uid":"","description":"\n\nThe dense subset $\\bigcup_{n\\in\\omega}L_n$ is {P37}\nand {T541}. Then $X$ is $\\sigma$-connected as the property extends onto the closure of a set (see Fact 3 in {{mathse:4975867}}).\n","refs":[{"name":"Answer to \"$\\sigma$-connected spaces and path-connected property\"","mathse":4975867}]},{"space":"S000116","property":"P000204","value":true,"uid":"","description":"\n\nBy item #1 for space #119 in {{zb:0386.54001}} $X$ is connected and $X \\setminus \\{(\\frac{1}{2}, \\frac{1}{2})\\}$ is disconnected.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000116","property":"P000205","value":false,"uid":"","description":"\n\n$X\\setminus \\{(1,1)\\}$ is connected by the same argument showing that $X$ is connected, as in item #1 for space #119 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000116","property":"P000229","value":true,"uid":"","description":"\n\nLet $L_n \\subset \\mathbb{R}^2$ be the closed line segment from $\\langle 0, 0\\rangle$ to $\\langle 1, \\frac{1}{n + 1} \\rangle$. The path components of $X$ are $\\bigcup_{n \\in \\omega} L_n$ and $\\big((\\frac{1}{2},1] \\times \\{0\\}\\big)$ with their topologies as subsets of $\\mathbb{R}^2$. \n\n$\\bigcup_{n \\in \\omega} L_n$ is star-convex and therefore {P199} (see {{wikipedia:Star_domain}}); $\\big((\\frac{1}{2},1] \\times \\{0\\}\\big)$ is homeomorphic to {S210} and {S210|P199}.\nHence both path components of $X$ are {P229}\n[(Explore)](https://topology.pi-base.org/spaces?q=Contractible%2B%7ESemilocally+simply+connected).\n","refs":[{"name":"Star domain on Wikipedia","wikipedia":"Star_domain"}]},{"space":"S000117","property":"P000016","value":true,"uid":"","description":"\n\n$X$ is a closed and bounded subset of the plane.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000117","property":"P000041","value":false,"uid":"","description":"\n\nSee item #2 for space #120 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000117","property":"P000089","value":true,"uid":"","description":"\n\nFor any continuous map $f:X \\to X$,\nthere is some $n \\in \\omega\\cup\\{\\infty\\}$ such that $f((0,0)) \\in L_n$.\nBy looking locally at path components and by intermediate value theorem,\none can see that $f$ has a fixed point on $L_n$.\n","refs":[]},{"space":"S000117","property":"P000184","value":true,"uid":"","description":"\n\nConstructed as a subspace of {S176}.\n","refs":[]},{"space":"S000117","property":"P000199","value":true,"uid":"","description":"\n\nThis follows because $X$ is a star-shaped set in $\\mathbb{R}^2$. See {{wikipedia:Star_domain}}. Explicitly, the map $F : \\mathbb{R}^2 \\times [0, 1] \\to \\mathbb{R}^2$, $F(x, t) = (1 - t)x$ restricts to a homotopy from the identity map of $X$ to a constant map defined on $X$.\n","refs":[{"name":"Star domain on Wikipedia","wikipedia":"Star_domain"}]},{"space":"S000117","property":"P000204","value":true,"uid":"","description":"\n\nBy item #3 for space #120 in {{zb:0386.54001}}, $X$ is (path-)connected and $X \\setminus \\{(\\frac{1}{2}, \\frac{1}{2})\\}$ is disconnected.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000118","property":"P000001","value":true,"uid":"","description":"\n\nSee item #1 for space #121 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000118","property":"P000002","value":false,"uid":"","description":"\n\nSee item #1 for space #121 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000118","property":"P000014","value":true,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #121 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000118","property":"P000041","value":false,"uid":"","description":"\n\nSee item #2 for space #121 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000118","property":"P000044","value":false,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #121 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000118","property":"P000056","value":true,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #121 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000118","property":"P000181","value":true,"uid":"","description":"\n\nBy definition.\n","refs":[]},{"space":"S000118","property":"P000202","value":true,"uid":"","description":"\n\nBy definition, the only neighborhood of the origin is $X$ itself.\n","refs":[]},{"space":"S000118","property":"P000204","value":true,"uid":"","description":"\n\nThe origin is a cut point since the complement of the origin can be written as a disjoint union of nonempty open sets, e.g., $([1, \\infty) \\cap \\omega) \\times (\\{0\\} \\cup \\{\\frac{1}{n}\\}_{n=2}^\\infty)$ and $([1, \\infty) \\cap \\omega) \\times \\{0\\}$; and moreover, {S118|P36}.\n","refs":[]},{"space":"S000119","property":"P000017","value":true,"uid":"","description":"\n\nFor each $n=1,2,\\dots$, the union of the line segment joining the points $(0,1)$ and $(n, \\frac{1}{n+1})$ with the half-line $y = \\frac{1}{n+1}$, $x \\leq n$ is homeomorphic to {S210} and therefore {P17}, as well as the line $y=0$, which is homeomorphic to {S25}.\n\nHence, $X$ is a countable union of {P17} spaces and therefore {P17} itself.\n","refs":[]},{"space":"S000119","property":"P000023","value":false,"uid":"","description":"\n\n$(0,1) \\in X$ has no compact neighborhood.\n\nErroneously asserted as true in the General Reference Chart and in\nitem #2 for space #122 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000119","property":"P000037","value":false,"uid":"","description":"\n\nAssume $\\gamma:[0,1]\\to X$ is a path joining\n$(0,0)$ with $(0,1)$. Denote by $\\pi_1$ and $\\pi_2$ projections onto the first and second coordinate, respectively.\nThere exists a sequence $(t_n)_{n=2}^\\infty\\subset(0,1]$ such that $t_n\\to 0$ and\n$\\pi_2(\\gamma(t_n))=\\frac{n+2}{(n+1)^2}$. It can be readily verified that then $\\pi_1(\\gamma(t_n))\\geq n\\frac{n^2+n+1}{(n+1)^2}$ proving that $\\gamma(t_n)$ does not converge to $(0,0)=\\gamma(0)$.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000119","property":"P000041","value":false,"uid":"","description":"\n\nSee item #3 for space #122 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000119","property":"P000046","value":false,"uid":"","description":"\n\nBy construction, the space admits nontrivial paths.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000119","property":"P000055","value":true,"uid":"","description":"\n\nViewing $X$ as a subspace of $Y=$ {S176},\nwe have $X=\\overline{X}\\setminus((0,\\infty)\\times\\{1\\})$,\nwith $\\overline X$ a $G_\\delta$ set in $Y$\nsince {S176|P132},\nand $(0,\\infty)\\times\\{1\\}$ a $\\sigma$-compact, hence $F_\\sigma$ set in $Y$.\nTherefore $X$ is a $G_\\delta$ set in $Y$ and the assertion follows\nsince {S176|P55}.\n","refs":[]},{"space":"S000119","property":"P000089","value":false,"uid":"","description":"\n\nAn example of a continuous self-map with no fixed point is\n$X\\ni (x,y)\\mapsto (x+1,0)\\in X$.\n","refs":[]},{"space":"S000119","property":"P000184","value":true,"uid":"","description":"\n\nConstructed as a subspace of {S176}.\n","refs":[]},{"space":"S000119","property":"P000189","value":true,"uid":"","description":"\n\nThe dense subset $\\{(x,y)\\in X: y>0\\}$ is {P37}\nand {T541}. Then $X$ is $\\sigma$-connected as the property extends onto the closure of a set (see Fact 3 in {{mathse:4975867}}).\n","refs":[{"name":"Answer to \"$\\sigma$-connected spaces and path-connected property\"","mathse":4975867}]},{"space":"S000119","property":"P000204","value":true,"uid":"","description":"\n\n$X$ is {P36} according to\nitem #1 for space #122 in\n{{zb:0386.54001}}.\n\n\nThe set $(-\\infty,1)\\times\\{\\frac{1}{2}\\}$ is a nontrivial clopen set in\n$X\\setminus\\{(1,\\frac{1}{2})\\}$, hence $(1,\\frac{1}{2})$ is a cut point.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000119","property":"P000205","value":false,"uid":"","description":"\n\nThe set $\\{(x,y)\\in X: y>0\\}$ is dense in $X$\nand connected, hence $(x,0)$ is not a cut point for $x\\in\\mathbb R$.\n","refs":[]},{"space":"S000120","property":"P000017","value":true,"uid":"","description":"\n\nEach of the sets $D_n=A_n\\cup B_n\\cup C_n$ is compact.\nSee item #1 for space #123 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000120","property":"P000023","value":false,"uid":"","description":"\n\nLet $x$ be an endpoint of one of the sets $B_n$.\nFor an arbitrary neighborhood $U$ of $x$, there is a large enough integer $m$ and a sequence of points $x_k\\in A_k\\cap U$ for $k\\ge m$ such that $(x_k)_{k\\ge m}$ converges in $\\mathbb R^3$ to a point on the $y$-axis that does not belong to $X$.\nThen that sequence has no accumulation point in $X$, proving that $U$ is not compact.\n\n\n**Note:** This trait is erroneously asserted to be true in the General Reference Chart for space #123 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000120","property":"P000041","value":false,"uid":"","description":"\n\nSuppose $x\\in X$ belongs to one of the $B_n$ and $V$ is a bounded neighborhood of $x$.\nFor $k$ large enough, the intersection $A_k\\cap V$ is nonempty clopen in $V$, which shows that $V$ is not connected.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000120","property":"P000046","value":false,"uid":"","description":"\n\nFor example, the set $A_1\\subset X$ is path connected with more than one point.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000120","property":"P000089","value":false,"uid":"","description":"\n\nA continuous map $f:X\\to X$ without a fixed point can be defined by setting\n$f:D_n\\ni (x,y,z)\\mapsto (\\frac{n}{n+1}x,y+2,\\frac{n^2}{(n+1)^2}z)\\in\nD_{n+1}$,\nwhere $n\\geq 1$ and $D_n=A_n\\cup B_n\\cup C_n$.\n\nTo show the continuity of $f$, consider a convergent sequence $p_n\\to p$ in $X$.\nIf $p\\neq(0,y,0)$ for $y\\in \\mathbb R$, then $p_n$ is eventually contained in $D_n$ for $D_n\\ni p$, and $f$ is continuous when restricted to $D_n$.\nIf $p=(0,y,0)$, then the first and the third coordinates of $p_n$ tend to zero and the same is true for $f(p_n)$. Convergence in the second coordinate is evident.\n","refs":[]},{"space":"S000120","property":"P000184","value":true,"uid":"","description":"\n\nConstructed as a subspace of $\\mathbb R^3$.\n","refs":[]},{"space":"S000120","property":"P000189","value":false,"uid":"","description":"\n\nThe sets $D_n=A_n\\cup B_n\\cup C_n$ for $n\\ge 1$ partition the space into countably many nonempty closed sets.\n","refs":[]},{"space":"S000120","property":"P000204","value":true,"uid":"","description":"\n\nThe space is {P36} according to\nitem #2 for space #123 in {{zb:0386.54001}}.\n\nThe set $\\{(1,y,0)\\in\\mathbb R^3:0\\leq y< 1\\}$ is clopen in $X\\setminus\\{(1,1,0)\\}$, proving that $(1,1,0)$ is a cut point.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000120","property":"P000205","value":false,"uid":"","description":"\n\nIt can be readily verified that $(1,0,0)$ is not a cut point.\n","refs":[]},{"space":"S000120","property":"P000206","value":true,"uid":"","description":"\n\nObserve that every point of $X$ not of the form $(0,n+\\frac{1}{2},0)$ (endpoint of some segment $B_k$)\nadmits a neighbourhood in $X$ that is closed in $\\mathbb R^3$ (hence {P55}).\nIts interior in $X$ is then {P55} as well.\n\nTherefore Player 1 cannot pick $x_n$ of this form,\nbecause the next move of Player 2 would reduce the game\nto the case of a completely metrizable space $V_n$, which is {P206}\n[(Explore)](https://topology.pi-base.org/spaces?q=Completely+metrizable%2B%7EEmpty%2B%7EStrongly+Choquet).\n(Let us call this the \"C-scenario\".)\n\nOn the other hand, Player 2 can always make sure that precisely one $B_k$-endpoint lies in\n$V_0$, therefore forcing Player 1 to either lose by picking a constant sequence\n$\\{x_n\\}_{n\\geq 1}$ or by triggering the C-scenario.\n","refs":[]},{"space":"S000121","property":"P000036","value":true,"uid":"","description":"\n\nSee item #3 for space #124 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000121","property":"P000046","value":true,"uid":"","description":"\n\nThe space does not contain any closed connected subset of $\\mathbb R^2$ with at least two points.\n\nSee item #1 for space #124 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000121","property":"P000065","value":true,"uid":"","description":"\n\nBy construction: this space contains a distinct point chosen from $C_\\alpha$ for each $\\alpha<\\mathfrak c$.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000121","property":"P000184","value":true,"uid":"","description":"\n\nConstructed as a subspace of {S176}.\n","refs":[]},{"space":"S000122","property":"P000003","value":true,"uid":"","description":"\n\nSee item #1 for space #125 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000122","property":"P000004","value":false,"uid":"","description":"\n\nIf $U_1, U_2$ are non-empty open sets, then $\\overline{U_1}\\cap\\overline{U_2}\\neq\\emptyset$, see item #3 for space #125 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000122","property":"P000010","value":false,"uid":"","description":"\n\nSee {{mathse:4895698}}.\n","refs":[{"name":"What separation axioms are held by Gustin's sequence space?","mathse":4895698}]},{"space":"S000122","property":"P000028","value":true,"uid":"","description":"\n\nEach point has a base of neighborhoods indexed by $i\\in\\mathbb Z$.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000122","property":"P000036","value":true,"uid":"","description":"\n\nSee item #3 for space #125 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000122","property":"P000057","value":true,"uid":"","description":"\n\nBy definition.\n","refs":[]},{"space":"S000122","property":"P000082","value":true,"uid":"","description":"\n\nSee {{mathse:5029225}}.\n","refs":[{"name":"Answer to \"Is Gustin's sequence space locally connected?\"","mathse":5029225}]},{"space":"S000122","property":"P000204","value":false,"uid":"","description":"\n\nBy the last remark in {{mathse:5009961}}, if $F\\subseteq X$ is a finite set then $X\\setminus F$ is connected. In particular if $x\\in X$ then $X\\setminus \\{x\\}$ is connected.\n","refs":[{"name":"Answer to \"What separation axioms are held by Gustin's sequence space?\"","mathse":5009961}]},{"space":"S000123","property":"P000004","value":true,"uid":"","description":"\n\nSee item #5 for space #126 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000123","property":"P000010","value":false,"uid":"","description":"\n\nIf $V_\\epsilon(r,2)$ is a neighborhood of the point $r,2$, $\\overline {V_\\epsilon (r,2)}$ contains points of the form $(s,1)$ which are interior points of $\\overline {V_\\epsilon (r,2)}$. Thus $\\operatorname{int}(\\overline {V_\\epsilon (r,2)}) \\neq V_\\epsilon (r,2)$ and $X$ is not semiregular.\n\nSee item #7 for space #126 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000123","property":"P000027","value":true,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #126 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000123","property":"P000041","value":false,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #126 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000123","property":"P000045","value":true,"uid":"","description":"\n\n$X$ is {P36} by item #2 for space #126 in {{zb:0386.54001}}.\nAnd the point $\\omega$ is a dispersion point of $X$, since $X\\setminus\\{\\omega\\}$ is {S124} and {S124|P47}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000123","property":"P000057","value":true,"uid":"","description":"\n\nSee item #4 for space #126 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000124","property":"P000005","value":false,"uid":"","description":"\n\nSee item #5 for space #127 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000124","property":"P000027","value":true,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #127 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000124","property":"P000048","value":true,"uid":"","description":"\n\nSee item #3 for space #127 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000124","property":"P000049","value":false,"uid":"","description":"\n\nSee item #6 for space #127 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000124","property":"P000051","value":false,"uid":"","description":"\n\nSee item #6 for space #127 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000124","property":"P000057","value":true,"uid":"","description":"\n\nBy definition.\n","refs":[]},{"space":"S000124","property":"P000139","value":false,"uid":"","description":"\n\nSee item #6 for space #127 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000125","property":"P000022","value":false,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #128 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000125","property":"P000023","value":false,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #128 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000125","property":"P000041","value":false,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #128 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000125","property":"P000045","value":true,"uid":"","description":"\n\n$X$ is {P36} by item #1 for space #128 in {{zb:0386.54001}}.\nAnd the point $p$ is a dispersion point of $X$ by item #2,\nshowing that {S126|P47}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000125","property":"P000065","value":true,"uid":"","description":"\n\nThe space contains a copy of the irrationals and is contained within $\\mathbb R^2$.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000125","property":"P000184","value":true,"uid":"","description":"\n\nConstructed as a subspace of {S176}.\n","refs":[]},{"space":"S000126","property":"P000016","value":false,"uid":"","description":"\n\n$X$ is a subspace of $\\mathbb{R}^2$ that is not closed in $\\mathbb{R}^2$.\n","refs":[]},{"space":"S000126","property":"P000047","value":true,"uid":"","description":"\n\nLet $Y$ be any subset of Cantor's Teepee containing more than one point. Our goal is to produce two sets $A$ and $B$ such that\n\n* $A$ and $B$ are separated open sets in the subspace topology induced by $Y$, and\n* $Y = A \\cup B$.\n\nSince Cantor's Teepee is itself a subspace of the usual topology on $\\mathbb{R}^2$, we may revise these conditions to read\n\n* $A$ and $B$ are separated open sets in the usual topology on $\\mathbb{R}$, and\n* $Y \\subseteq A \\cup B$.\n\nNow, suppose $Y$ contains points $p \\in X_a$ and $q \\in X_c$ with $a \\neq c$ (that is, $Y$ witnesses more than one \"strand\" of the teepee). Choose any real number $b$ such that $a < b < c$ and $b$ lies in a deleted interval of the Cantor set. To obtain the separated open sets $A$ and $B$, take any open set containing $Y$ (or even the entire teepee) and \"split\" it along the line connecting $(b,0)$ and $(1/2,1/2)$ (see figure).\n\n![enter image description here](http://i.stack.imgur.com/S2yxj.png)\n\nSuppose instead that all points of $Y$ belong to a single strand. The $y$-coordinates of points belonging to this strand are either all rational or all irrational. In either case, we can easily separate the strand by passing our open sets through a point $r$ with $y$-coordinate of the opposite type (see figure).\n\n![enter image description here](http://i.stack.imgur.com/qF3Lx.png)\n\nTherefore every subset of Cantor's teepee with more than one point is disconnected, which is to say Cantor's teepee is totally disconnected.\n\nSee item #2 for space #129 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000126","property":"P000048","value":false,"uid":"","description":"\n\nSee item #3 for space #129 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000126","property":"P000065","value":true,"uid":"","description":"\n\nBy construction.\n","refs":[]},{"space":"S000126","property":"P000139","value":false,"uid":"","description":"\n\nEach point lies in a copy of $\\mathbb Q$ or $\\mathbb R\\setminus \\mathbb Q$, therefore no point is isolated.\n\nSee item #5 for space #129 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000126","property":"P000184","value":true,"uid":"","description":"\n\nConstructed as a subspace of {S176}.\n","refs":[]},{"space":"S000127","property":"P000016","value":true,"uid":"","description":"\n\nConsider the construction of a pseudo-arc as the intersection of (closed) chains in $\\mathbb{R}^2$. Each chain is closed and bounded in $\\mathbb{R}^2$ and is thus compact; then so is their intersection.\n\nSee item #1 for space #130 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000127","property":"P000044","value":false,"uid":"","description":"\n\nBy Theorem 11.15 and Theorem 11.17 of {{doi:10.1201/9781315274089}}, pseudo-arc, as an indecomposable {P188}, can be partitioned into uncountably many composants.\n\nFor each composant, it follows from 5.20(a) that it is {P36} and dense, thus contains at least two points.\n","refs":[{"name":"Continuum Theory: An introduction (Nadler)","doi":"10.1201/9781315274089"}]},{"space":"S000127","property":"P000046","value":true,"uid":"","description":"\n\nLet $\\alpha$ be a continuous function from $[0,1]$ to $X$. If $\\alpha(0) \\ne \\alpha(1)$ then since $X$ is Hausdorff we may assume without loss of generality that $\\alpha$ is injective. But then $\\alpha([0,1]) = \\alpha([0, 1/2]) \\cup \\alpha([1/2, 1])$, contradicting the fact that $X$ is hereditarily indecomposable. Thus $\\alpha$ must be constant.\n","refs":[]},{"space":"S000127","property":"P000086","value":true,"uid":"","description":"\n\nAsserted in first paragraph in {{wikipedia:Pseudo-arc}}. See also Theorem 13 of {{zb:0035.39103}}.\n","refs":[{"name":"Pseudo-arc on Wikipedia","wikipedia":"Pseudo-arc"},{"name":"A homogeneous indecomposable plane continuum (R. H. Bing)","zb":"0035.39103"}]},{"space":"S000127","property":"P000089","value":true,"uid":"","description":"\n\nSee the Corollary of {{zb:0054.07003}}.\n","refs":[{"name":"A fixed point theorem for pseudo-arcs and certain other metric continua (Hamilton, O. H.)","zb":"0054.07003"},{"name":"When does topological homogeneity imply algebraic homogeneity? Pseudo-arc and Hilbert cube","mo":163083}]},{"space":"S000127","property":"P000184","value":true,"uid":"","description":"\n\nConstructed as a subspace of {S176}.\n","refs":[]},{"space":"S000128","property":"P000029","value":true,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #131 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000128","property":"P000044","value":true,"uid":"","description":"\n\nSee item #4 for space #131 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000128","property":"P000045","value":false,"uid":"","description":"\n\nSee item #4 for space #131 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000128","property":"P000129","value":false,"uid":"","description":"\n\nThe space is non-trivial by definition.\n","refs":[]},{"space":"S000128","property":"P000184","value":true,"uid":"","description":"\n\nConstructed as a subspace of {S176}.\n","refs":[]},{"space":"S000129","property":"P000042","value":true,"uid":"","description":"\n\nThe balls with respect to the metric provided in item #3 for space #132 in {{zb:0386.54001}} are {P37}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000129","property":"P000055","value":true,"uid":"","description":"\n\nDefine $h:X\\to[0,+\\infty)$ as $h(x)=\\frac{1}{\\|x\\|}-1$ and $r:X\\to \\mathbb S^1$ as $r(x):=\\frac{1}{\\|x\\|}x$. These are clearly continuous maps on $X$ (the topology is finer than Euclidean). Then consider a metric $d$ given by:\n- $d(p,q)=|h(p)-h(q)|$, if $r(p)=r(q)$;\n- $d(p,q)=h(p)+h(q)+\\|r(p)-r(q)\\|$, otherwise.\n\nSymmetry and reflexivity of $d$ is immediate. Triangle inequality requires consideration of few cases but is elementary as well.\n\nBy continuity of $r$ and $h$ we obtain that convergence in the topology on $X$ implies convergence with respect to $d$.\n\nAssume $d(x_n,x)\\to 0$ for some $x\\in X$ and $(x_n)\\subset X$. If $h(x)>0$, then clearly $r(x_n)=r(x)$ for sufficiently large $n$ and $\\|x_n\\|\\to \\|x\\|$,\nimplying convergence $x_n \\to x$ in $X$.\nFor $h(x)=0$, we have $h(x_n)\\to 0$ and $r(x_n)\\to r(x)=x$ in the Euclidean topology. But since $x$ inherits Euclidean Neighbourhoods we obtain convergence $x_n\\to x$ in this case as well.\nHence $d$ is indeed a metric providing the topology on $X$.\n\nTo prove completeness of $d$, observe that\na Cauchy sequence $(x_n)\\subset X$ with respect to $d$ either satisfies $h(x_n)\\to 0$ or is eventually contained in one of the\nopen radii. Since any radius (i.e. $(0,1]\\cdot p$ with $p\\in \\mathbb S^1$) is isometric with $[0,+\\infty)$, a Cauchy sequence contained therein has a limit.\nNow focus on the case $h(x_n)\\to 0$. Clearly,\nalso $(r(x_n))$ has to be a Cauchy sequence.\nSince Euclidean metric on $\\mathbb S^1$ is complete, there exists $z:=\\lim_{n\\to \\infty} r(x_n)$. Since also $h(x_n)\\to 0$, we obtain\n$d(x_n,z)\\to 0$, which ends the proof.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000129","property":"P000086","value":false,"uid":"","description":"\n\n{S129|P23})\nbut the point $(0,1/2)$ has a neighbourhood which is a (compact) segment.\n","refs":[]},{"space":"S000129","property":"P000089","value":false,"uid":"","description":"\n\nThe map $(x,y)\\mapsto (-x,-y)$ is a homeomorphism and has no fixed point.\n","refs":[]},{"space":"S000129","property":"P000166","value":true,"uid":"","description":"\n\nThe topology on $X$ is clearly finer than the Euclidean topology. The set $X$ with\nthe Euclidean topology is separable.\n","refs":[]},{"space":"S000129","property":"P000200","value":false,"uid":"","description":"\n\nThe map $X\\ni p\\mapsto \\frac{1}{\\|p\\|}p \\in \\mathbb S^1$ is clearly a retraction of $X$ onto the unit circle.\nHence the fundamental group of $X$ contains a subgroup isomorphic to $\\pi_1(\\mathbb S^1)\\simeq\\mathbb Z$ and therefore is nontrivial.\n","refs":[]},{"space":"S000129","property":"P000205","value":true,"uid":"","description":"\n\nThe space is {P36}; see item #2 for space #132 in {{zb:0386.54001}}.\n\nAnd for every $p\\in X$ the open segment (radius) $\\{\\lambda p: 0< \\lambda < 1 \\}$ is both a closed and open subset of $X\\setminus\\{p\\}$. Hence $p$ is a cut point.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000129","property":"P000227","value":true,"uid":"","description":"\n\n$A=\\{x\\in X: \\|x\\|=1/2\\}$ is a closed discrete subset of $X$ of cardinality $\\mathfrak c$.\n","refs":[]},{"space":"S000130","property":"P000009","value":true,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #133 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000130","property":"P000010","value":false,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #133 in\n{{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000130","property":"P000021","value":false,"uid":"","description":"\n\nThe set $C$ is an infinite set without limit points.\n","refs":[]},{"space":"S000130","property":"P000036","value":true,"uid":"","description":"\n\nSee item #3 for space #133 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000130","property":"P000065","value":true,"uid":"","description":"\n\nBy definition.\n","refs":[]},{"space":"S000130","property":"P000068","value":false,"uid":"","description":"\n\nIt is easy to check being {P68} passes to coarser topologies. Since the topology on the space is finer than\n{S25} and {S25|P68}, the result follows.\n","refs":[]},{"space":"S000131","property":"P000050","value":true,"uid":"","description":"\n\nEvery singleton $\\{x\\}$ with $x\\ne\\infty$ is clopen. And so is every neighborhood of $\\infty$.\n","refs":[{"name":"Answer to \"Can a Fréchet-Urysohn hemicompact Hausdorff space fail to be locally compact?\"","mathse":4844426}]},{"space":"S000131","property":"P000063","value":false,"uid":"","description":"\n\nLet $\\mathcal{U}_1, \\mathcal{U}_2, \\dots$ be an arbitrary sequence of open covers of $S_\\omega$.\n\nFor $n\\in\\omega$ fix an open set $U_n\\in \\mathcal U_n$ containing $\\infty$, and $f_n \\colon \\omega \\to \\omega$ satisfying $\\left\\{ (i, j) \\in \\omega^2 \\middle| j \\geq f_n(i) \\right\\} \\subset U_n$.\n\nWithout loss of generality, suppose $f_n(i) > f_{n-1}(i)$.\n\nDefine $F_n = \\left\\{ (i, j) \\in \\omega^2 \\middle| f_n(i) \\leq j \\leq f_i(i) \\right\\}$, then $F_n$ is a closed set satisfying $F_n \\subseteq U_n$ and $F_n \\subseteq F_{n-1}$, and $F_n \\neq \\varnothing$ since $\\bigl( n, f_n(n) \\bigr) \\in F_n$.\n\nNow let $(i, j)$ be an arbitrary point in $\\omega^2$. We know that $(i, j) \\notin F_{i+1}$ because $f_{i+1}(i) > f_i(i)$.\nHence, $\\bigcap_{n \\in \\omega} F_n = \\varnothing$, we conclude that $S_\\omega$ is not {P63}.\n","refs":[]},{"space":"S000131","property":"P000080","value":true,"uid":"","description":"\n\nIf $x\\in\\overline A\\setminus A$ for some $A\\subseteq X$, necessarily $x=\\infty$ and some column of $\\omega\\times\\omega$ contains infinitely many points of $A$. So some sequence in $A$ converges to $x$.\n","refs":[{"name":"Answer to \"Can a Fréchet-Urysohn hemicompact Hausdorff space fail to be locally compact?\"","mathse":4844426}]},{"space":"S000131","property":"P000111","value":true,"uid":"","description":"\n\nThe sets $K_n=([0,n]\\times\\omega)\\cup\\{\\infty\\}$ are compact and every compact set in $X$ is contained in one of the $K_n$.\n\nSee {{mathse:4844426}}.\n","refs":[{"name":"Answer to \"Can a Fréchet-Urysohn hemicompact Hausdorff space fail to be locally compact?\"","mathse":4844426}]},{"space":"S000131","property":"P000183","value":true,"uid":"","description":"\n\nEach spine $C_m=(\\{m\\}\\times\\omega)\\cup\\{\\infty\\}$ is a closed subspace of $X$ homeomorphic to a convergent sequence ({S20});\nand {S20|P183}.\nAnd every compact subset of $X$ is contained in the union of a finite number of these spines.\n\nTherefore, taking a countable $k$-network from each of the (countably many) spines and forming their union gives a countable $k$-network for $X.$\n","refs":[{"name":"Answer to \"Can a Fréchet-Urysohn hemicompact Hausdorff space fail to be locally compact?\"","mathse":4844426}]},{"space":"S000131","property":"P000203","value":true,"uid":"","description":"\n\nAll points except $\\infty$ are isolated.\n","refs":[]},{"space":"S000131","property":"P000214","value":false,"uid":"","description":"\n\nFor $n \\in \\omega$, let $S_n:=\\{n\\}\\times \\omega$. Then $S_n$ converges to $\\infty$.\nBut every $S \\subseteq \\bigcup_{n \\in \\omega}S_n$ with $|S \\cap S_n|\\geq 1$ for infinitely many $n$ has points on infinitely many columns on $\\omega \\times \\omega$ and thus cannot converge to $\\infty$.\n","refs":[]},{"space":"S000132","property":"P000048","value":true,"uid":"","description":"\n\nSee item #6 for space #136 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000132","property":"P000056","value":true,"uid":"","description":"\n\nSee Section 3 of {{doi:10.2307/2309171}}.\n","refs":[{"name":"A Topology for Sequences of Integers II (R. L. Duncan)","doi":"10.2307/2309171"}]},{"space":"S000132","property":"P000065","value":true,"uid":"","description":"\n\nIt is easily seen that for every real number $\\alpha\\in[0,1]$ there is an element $x\\in X$ with asymptotic density $\\delta(x)=\\alpha$. So $|X|\\geq \\mathfrak c$.\nAnd on the other hand, $|X|\\leq\\aleph_0^{\\aleph_0}=2^{\\aleph_0}=\\mathfrak c$.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000132","property":"P000066","value":false,"uid":"","description":"\n\nSimilar to the proof of {S28|P66}, it is easily seen that for each finite sequence $t$, $\\left[ t \\right] := \\left\\{ x \\in X \\mid x \\text{ extends } t \\right\\}$ is open in $X$.\n\nNow define the open cover $\\mathcal U_n = \\left\\{ \\left[ t \\right] \\mid t \\in \\omega^n \\right\\}$.\nGiven any finite subcollections $\\mathcal F_n \\subseteq \\mathcal U_n$, we can choose $x_n$ such that $\\left[ \\left< x_1, \\dots, x_n \\right> \\right] \\notin \\mathcal F_n$ and $x_n > 2 x_{n - 1}$.\n\nThen $x \\in X$ since $\\delta(x) = 0$ and $x \\notin \\bigcup_{n < \\omega} \\mathcal F_n$.\n","refs":[]},{"space":"S000132","property":"P000184","value":true,"uid":"","description":"\n\nIt is asserted in Section 4 of {{doi:10.2307/2309919}} that $S$ is homeomorphic to a certain subset of the plane, namely the graph of a function $\\varphi(x)$.\n","refs":[{"name":"A Topology for Sequences of Integers II (R. L. Duncan)","doi":"10.2307/2309171"}]},{"space":"S000133","property":"P000023","value":false,"uid":"","description":"\n\nThe particular point does not have a compact neighborhood. Indeed, let $U$ be a neighborhood of $p$. Then there is some $V_n\\subseteq U$. The collection $\\mathcal{A} = \\{\\{x\\}: x\\in U\\setminus V_{n+1}\\}\\cup \\{V_{n+1}\\}$ is an open covering of $U$ with infinitely many elements and no proper subcover.\n\nSee item #3 for space #139 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000133","property":"P000050","value":true,"uid":"","description":"\n\nSee item #4 for space #139 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000133","property":"P000051","value":true,"uid":"","description":"\n\nEvery point which is not the particular point is isolated.\n\nSee item #4 for space #139 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000133","property":"P000055","value":true,"uid":"","description":"\n\nNote that the space is {P53} by\nconsidering the post office metric on $\\mathbb R$.\n\nSo suppose that $\\langle x_n \\rangle$ is a Cauchy sequence that \ndoes not converge to $0$. Then there is $\\varepsilon>0$ and a \nsubsequence $x_{n_i}$ such that \n$d(x_{n_i},0)=\\|x_{n_i}\\|\\ge \\varepsilon$.\nLet $N$ be an index such that $d(x_n,x_m)<\\varepsilon$ for all \n$n,m\\ge N$. Let $I$ be a natural number with $n_I\\ge N$ and let \n$n\\ge N$ be arbitrary. We have \n$\\|x_n\\|+\\|x_{n_I}\\|\\ge d(x_{n_I},0)\\ge \\varepsilon$, and \ntherefore we must conclude that \n$d(x_n,x_{n_I})\\not=\\|x_n\\|+\\|x_{n_I}\\|$ and thus $x_{n}=x_{n_I}$.\nHence $\\langle x_n \\rangle$ is constant for $n\\ge N$ and therefore\nconvergent.\n","refs":[]},{"space":"S000133","property":"P000065","value":true,"uid":"","description":"\n\nBy construction.\n","refs":[]},{"space":"S000133","property":"P000093","value":false,"uid":"","description":"\n\nEvery neighborhood of the particular point\ncontains some uncountable $V_n$.\n","refs":[{"name":"Answer to \"Must scattered spaces with points $G_\\delta$ be locally countable?\"","mathse":4945909}]},{"space":"S000133","property":"P000166","value":true,"uid":"","description":"\n\nTake the incarnation of $X$ as $\\mathbb R$ with the post office metric.\nIts topology contains the topology for {S25}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000133","property":"P000203","value":true,"uid":"","description":"\n\nAll points except $p$ are isolated.\n","refs":[]},{"space":"S000133","property":"P000227","value":true,"uid":"","description":"\n\nThe subspace $V_0 \\setminus V_1$ is closed discrete with cardinality $\\mathfrak c$.\n","refs":[]},{"space":"S000134","property":"P000042","value":true,"uid":"","description":"\n\nEvery open half line emanating from the origin is\nisometric to {S25}\nand {S25|P42}.\n\nIt remains to show that the origin admits a local base of path connected neighbourhoods.\n\nA ball centered at zero $B_r(\\vec 0, \\varepsilon)$ coincides with the Euclidean ball. Every\npoint $\\vec x\\in B_r(\\vec 0, \\varepsilon)$ can be joined with the origin\nby a path $t\\mapsto (1-t)\\vec x$.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000134","property":"P000055","value":true,"uid":"","description":"\n\nThe metric $r$ is complete.\nEvery line passing through the origin is isometric to {S25},\nhence a Cauchy sequence eventually contained in such a line has a limit by completeness\nof the Euclidean metric.\n\nAssume $(x_n)$ is a Cauchy sequence not contained in a line, i.e. for\nevery $n$ there exists $n'>n$ such that $x_n$ and $x_{n'}$ are not collinear with $\\vec 0$.\nThen $r(x_n,x_{n'}) = d_e(x_n,\\vec 0)+d_e(x_{n'},\\vec 0)\\geq r(x_n,\\vec 0)$. The assumption $r(x_n,x_m)\\to 0$ (for $n,m\\to \\infty$) implies $r(x_n,\\vec 0)\\to 0$, hence $x_n\\to\\vec 0$.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000134","property":"P000065","value":true,"uid":"","description":"\n\nBy definition.\n","refs":[]},{"space":"S000134","property":"P000086","value":false,"uid":"","description":"\n\nRemoving the origin partitions the space into continuum many connected components (the open rays emanating from the origin), whereas removing any point $p$ other than the origin partitions the space into two connected components (the open ray emanating from $p$\n away from the origin, and the rest of the space).\n","refs":[]},{"space":"S000134","property":"P000089","value":false,"uid":"","description":"\n\nAn example of a continuous map $f:X\\to X$ with no fixed point can be defined as follows\n\n$f( (x,y))= \\begin{cases} (x+1,0) & \\mathrm{if}\\ x>0\\land y=0,\\\\\n(1,0) & \\mathrm{otherwise.}\\end{cases}$\n","refs":[]},{"space":"S000134","property":"P000166","value":true,"uid":"","description":"\nThe topology is finer than that of the {S176}.\n","refs":[]},{"space":"S000134","property":"P000199","value":true,"uid":"","description":"\n\nThe map $X\\times[0,1]\\ni (p,t)\\mapsto tp\\in X$ is a homotopy between a constant map and the identity map.\n","refs":[]},{"space":"S000134","property":"P000205","value":true,"uid":"","description":"\n\nThe space is {P36}.\n\nIn the radial metric every open half-ray starting at the origin is open, hence their disjoint union, $\\mathbb R^2\\setminus\\{(0,0)\\}$, is not {P36}.\n\nFor any $p\\in \\mathbb R^2\\setminus\\{(0,0)\\}$ the set\n$R_p=\\{\\lambda p: \\lambda >1\\}$ is open\nand $R_p\\cup\\{p\\}=\\{\\lambda p: \\lambda \\geq 1\\}$ is closed. Therefore $\\mathbb R^2\\setminus\\{p\\}$ is not {P36} in the radial metric topology.\n","refs":[]},{"space":"S000134","property":"P000227","value":true,"uid":"","description":"\n\nEvery circle $C_a=\\{(x,y)\\mid x^2+y^2=a^2\\}$ is closed, as the space is finer than the {S176}, and discrete, as balls $B_\\epsilon(x)$ contain at most one point of $C_a$ whenever $\\epsilon\\le a$.\n","refs":[]},{"space":"S000135","property":"P000063","value":false,"uid":"","description":"\n\n{S135} contains {S131} as a closed subspace, and {S131|P63}.\n","refs":[]},{"space":"S000135","property":"P000080","value":true,"uid":"","description":"\n\nTake $A\\subseteq X$ and $p\\in \\overline A\\setminus A$. If $p\\neq(0,0)$ then it admits a countable base of neighbourhoods and picking elements of $A$ from each of those neighbourhoods we obtain the required sequence.\nFor $p=(0,0)$, there has to exist at least one ray\ncontaining elements of $A$ arbitrarily close to $p$. Otherwise a neighbourhood of the origin disjoint from $A$ could be found.\nHence there exists a sequence in $A$ converging to $p$.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000135","property":"P000086","value":false,"uid":"","description":"\n\n{S135|P23}. But every point\nexcept the origin has a locally compact neighbourhood (homeomorphic to {S25}).\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000135","property":"P000089","value":false,"uid":"","description":"\n\nConsider the map $f:X\\to X$ given by\n- $f((x,0))= (x+1,0)$ if $x\\geq 0$\n- $f((x,y))= (1,0)$ if $y\\neq 0$ or $x\\leq 0$.\n\n$f$ is continuous but has no fixed point.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000135","property":"P000117","value":true,"uid":"","description":"\n\nSee {{mathse:5136509}}.\n","refs":[{"name":"Is $\\mathbb R^2$ with the radial interval topology an $\\aleph$-space or a $\\sigma$-space?","mathse":5136509}]},{"space":"S000135","property":"P000118","value":false,"uid":"","description":"\n\nSee {{mathse:5136509}}.\n","refs":[{"name":"Is $\\mathbb R^2$ with the radial interval topology an $\\aleph$-space or a $\\sigma$-space?","mathse":5136509}]},{"space":"S000135","property":"P000166","value":true,"uid":"","description":"\n\nThe Euclidean topology on the plane is coarser than the topology on $X$.\n","refs":[]},{"space":"S000135","property":"P000187","value":false,"uid":"","description":"\n\nP2 has a winning strategy at $(0,0)$. It is enough to pick\neach of the points $p_n$ on a different line passing through the origin. Then there exists a neighbourhood of $(0,0)$ not containing any of those points.\n","refs":[]},{"space":"S000135","property":"P000199","value":true,"uid":"","description":"\n\nThe map $X\\times[0,1] \\ni (\\vec x,t)\\mapsto (1-t)\\vec x\\in X$ is a homotopy between the identity and a constant map.\n","refs":[]},{"space":"S000135","property":"P000205","value":true,"uid":"","description":"\n\n$X$ is {P36}, for example because {S135|P199}.\n\nFor every $p\\in X\\setminus\\{(0,0)\\}$ the proper subset\n$\\{\\lambda p: \\lambda >1\\}$ is clopen in $X\\setminus\\{p\\}$.\nIn the subspace $X\\setminus\\{(0,0)\\}$ every half-line\nstarting at the origin is clopen.\nHence for every $p\\in X$ the space $X\\setminus\\{p\\}$ is not connected.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000135","property":"P000206","value":true,"uid":"","description":"\n\nObserve that there are two cases: either the origin belongs to $U_n$ for every $n< \\omega$ and the game is lost for the first player or for some $n$, the origin is not in $U_n$. Then the second player can pick $V_n$ contained in a line passing through the origin.\nAfter that, the game is played on the real line and\n{S25|P206}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000135","property":"P000214","value":false,"uid":"","description":"\n\n$X$ contains {S131} as a subspace and {S131|P214}.\n","refs":[]},{"space":"S000135","property":"P000227","value":true,"uid":"","description":"\n\nThis topology is finer than {S134},\n(cf. item #1 for space #141 in {{zb:0386.54001}})\nand {S134|P227}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000135","property":"P000240","value":true,"uid":"","description":"\n\nIt is a $1$-dimensional {P240} with $X_0 = \\left\\{ x \\in X \\mid \\left\\|x\\right\\|_2 \\in \\mathbb N \\right\\} \\subset X_1 = X$,\nwhere the sets $1$-cells are segments connecting $(n \\cos \\theta, n \\sin \\theta)$ and $((n + 1) \\cos \\theta, (n + 1) \\sin \\theta)$ for each $n \\in \\mathbb N$ and $0 \\leq \\theta < 2 \\pi$.\n","refs":[]},{"space":"S000136","property":"P000008","value":true,"uid":"","description":"\n\nSee exercise 5.5.3 a) in {{zb:0684.54001}}. See also .\n","refs":[{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"space":"S000136","property":"P000015","value":false,"uid":"","description":"\n\nAsserted in Example 12 of the Metrization Theory chapter in {{zb:0386.54001}}.\n\nSee also .\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000136","property":"P000019","value":false,"uid":"","description":"\n\nClosed subspaces of {P19} spaces are {P19}, but the closed subspace {S137} of this space is not {P19}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000136","property":"P000031","value":false,"uid":"","description":"\n\nSee item #5 for space #142 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000136","property":"P000032","value":true,"uid":"","description":"\n\nBy example 7 of {{doi:10.1016/s0924-6509(08)x7010-2}} {S136} is shrinking (e.g. corollary 6 in ), hence countably paracompact. Also see for another proof that {S136} is countably paracompact.\n","refs":[{"name":"Topics in General Topology","doi":"10.1016/s0924-6509(08)x7010-2"}]},{"space":"S000136","property":"P000059","value":false,"uid":"","description":"\n\nBy definition, the space has cardinality $2^{2^\\mathfrak{c}}$.\n","refs":[]},{"space":"S000136","property":"P000088","value":false,"uid":"","description":"\n\nSee Example G on p. 184 in {{doi:10.4153/CJM-1951-022-3}}, also mentioned\nin Example 12 of the Metrization Theory chapter in {{zb:0386.54001}}.\n\nSee also .\n\n","refs":[{"name":"Bing, R.H., Metrization of Topological Spaces","doi":"10.4153/CJM-1951-022-3"},{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000136","property":"P000139","value":true,"uid":"","description":"\n\nEvery point in $X\\setminus M$ is isolated by definition.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000136","property":"P000147","value":false,"uid":"","description":"\n\n$X$ contains {S137} as a subspace and {S137|P147}.\n","refs":[]},{"space":"S000136","property":"P000164","value":true,"uid":"","description":"\n\n$|X| = 2^{2^\\mathfrak{c}}$ is smaller than every measurable cardinal.","refs":[]},{"space":"S000136","property":"P000191","value":false,"uid":"","description":"\n\nAny $G_\\delta$-set $A$ with $x_r\\in A$ contains a set of the form $\\{x\\in X : x(\\lambda) = x_r(\\lambda)\\text{ for all } \\lambda\\in \\Lambda\\}$ where $\\Lambda\\subseteq 2^\\mathbb{R}$ is countable. If $\\lambda_0\\in 2^{\\mathbb{R}}\\setminus \\Lambda$ is arbitrary, let $x\\in X$ be any point such that $x(\\lambda) = x_r(\\lambda)$ for all $\\lambda\\in \\Lambda$ and $x(\\lambda_0)\\neq x_r(\\lambda_0)$, then $x\\in A\\setminus \\{x_r\\}$. It follows $\\{x_r\\}$ is not a $G_\\delta$-set for $r\\in\\mathbb{R}$.\n","refs":[]},{"space":"S000136","property":"P000227","value":true,"uid":"","description":"\n\n$X$ contains {S137} as a closed subspace, and {S137|P227}.\n\nExplicitly, $M$ is a closed and discrete subset of cardinality $\\mathfrak c$.\n\nSee item #1 for space #142 in {{zb:0386.54001}}.\n\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000137","property":"P000007","value":true,"uid":"","description":"\n\nLet $X$ be Bing's Discrete Extension Space and $Y$ Michael's Closed Subspace.\n\nAs the name suggests, $Y \\subset X$ is closed because $X \\setminus Y \\subset X \\setminus M$ and every point of $X \\setminus M$ is isolated. Since $X$ is $T_4$, so is $Y$.\n\nSee item #3 for space #143 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000137","property":"P000014","value":true,"uid":"","description":"\n\n$X$ is a subspace of {S136} and {S136|P14}.\n","refs":[]},{"space":"S000137","property":"P000015","value":false,"uid":"","description":"\n\nAsserted in the General Reference Chart for space #143\nin {{zb:0386.54001}}.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000137","property":"P000031","value":true,"uid":"","description":"\n\nLet $\\{U_\\alpha\\}$ be an open cover of $Y$. For each $x_r \\in M$, select $U_r$ with $x_r \\in U_r \\in \\{U_\\alpha\\}$. Let $V_r = \\{x \\in U_r\\ |\\ \\{r\\} \\in x\\} = U_r \\cap \\pi^{-1}(1)$. Then $\\{V_r\\} \\cup \\{\\{f\\}\\ |\\ f \\in F\\}$ is a point-finite open refinement of $\\{U_\\alpha\\}$.\n\nSee item #5 for space #143 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000137","property":"P000032","value":true,"uid":"","description":"\n\nSee item #6 for space #143 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000137","property":"P000059","value":true,"uid":"","description":"\n\nLet $S = \\{x\\in X : x(\\lambda) = 1\\text{ for finitely many }\\lambda\\in 2^{\\mathbb{R}}\\}$, then $|S| = |[2^{\\mathbb{R}}]^{<\\omega}| = |2^{\\mathbb{R}}| = 2^\\mathfrak{c}$ where $[A]^{<\\omega}$ denotes the set of all finite subsets of $A$. Also $|M| = \\mathfrak{c}$. Thus $|F| = |S\\setminus M| = 2^\\mathfrak{c}$, and so $|Y| = 2^\\mathfrak{c}$.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000137","property":"P000088","value":false,"uid":"","description":"\n\nAsserted in Example 13 of the Metrization Theory chapter\nin {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000137","property":"P000112","value":true,"uid":"","description":"\n\nShown in {{mo:497834}}.\n","refs":[{"name":"Answer to \"Is there a normal space with a $G_\\delta$ diagonal which is not submetrizable?\"","mo":497834}]},{"space":"S000137","property":"P000163","value":false,"uid":"","description":"\n\nLet $S = \\{x\\in X : x(\\lambda) = 1\\text{ for finitely many }\\lambda\\in 2^{\\mathbb{R}}\\}$, then $|S| = |[2^{\\mathbb{R}}]^{<\\omega}| = |2^{\\mathbb{R}}| = 2^\\mathfrak{c}$ where $[A]^{<\\omega}$ denotes the set of all finite subsets of $A$. Also $|M| = \\mathfrak{c}$. Thus $|F| = |S\\setminus M| = 2^\\mathfrak{c}$, and so $|Y| = 2^\\mathfrak{c} > \\mathfrak{c}$.","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000137","property":"P000227","value":true,"uid":"","description":"\n\n$M$ is a closed and discrete subset of cardinality $\\mathfrak c$.\n\nSee item #1 for space #143 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000138","property":"P000008","value":false,"uid":"","description":"\n\nSee {{mathse:5013612}}.\n","refs":[{"name":"Is Rudin's Dowker space hereditarily normal?","mathse":5013612}]},{"space":"S000138","property":"P000114","value":false,"uid":"","description":"\n\nNote that there are $\\aleph_2$ many $\\alpha < \\omega_2$ s.t. $\\omega < \\mathrm{cf}(\\alpha) < \\omega_2$. Thus, $(\\omega_1, \\alpha, \\omega_1, \\omega_1, \\cdots)$, where $\\alpha$ ranges over all $\\alpha < \\omega_2$ with $\\omega < \\mathrm{cf}(\\alpha) < \\omega_2$, are in $X$. Hence, $X$ has at least $\\aleph_2$ many elements.\n","refs":[]},{"space":"S000138","property":"P000127","value":true,"uid":"","description":"\n\nSee Sections II and III in {{zb:0224.54019}}.\n","refs":[{"name":"A normal space X for which X×I is not normal (M.E. Rudin)","zb":"0224.54019"}]},{"space":"S000138","property":"P000139","value":true,"uid":"","description":"\n\nThe function $f\\equiv\\omega_1$ is the only point in the open set\n$X\\cap\\prod_{n<\\omega} [0,\\omega_1+1)$, since the condition\n$\\omega<\\mathrm{cf}(g(n))$ implies $g(n)\\geq\\omega_1$.\n","refs":[]},{"space":"S000138","property":"P000147","value":true,"uid":"","description":"\n\nSee lemma 4 in {{zb:0224.54019}}.\n","refs":[{"name":"A normal space X for which X×I is not normal (M.E. Rudin)","zb":"0224.54019"}]},{"space":"S000138","property":"P000162","value":false,"uid":"","description":"\n\nSee IV.4 of {{zb:0224.54019}}.\n","refs":[{"name":"A normal space X for which X×I is not normal (M.E. Rudin)","zb":"0224.54019"}]},{"space":"S000138","property":"P000164","value":true,"uid":"","description":"\n\n$|X| \\leq \\aleph_\\omega^\\omega$ and $\\aleph_\\omega^\\omega$ is smaller than every measurable cardinal.\n","refs":[]},{"space":"S000138","property":"P000207","value":true,"uid":"","description":"\n\nBy theorem 2.2 of {{zb:0499.54016}},\nthis space is $2$-fully normal, and so {P207}.\n","refs":[{"name":"More on M. E. Rudin’s Dowker space (K. P. Hart)","zb":"0499.54016"}]},{"space":"S000139","property":"P000042","value":true,"uid":"","description":"\n\nA point $x\\ne\\infty$ has a local base of open neighborhoods, each equal to some arc of a circle around $x$.\nAnd the point $\\infty$ has a local base of open neighborhoods of the form $\\bigcup_{n\\in\\mathbb Z}U_n$, where each $U_n$ is an open arc around $\\infty$ in the n-th circle.\nThese neighborhoods are all {P37}.\n","refs":[]},{"space":"S000139","property":"P000063","value":false,"uid":"","description":"\n\n{S139} contains {S131} as a closed subspace, and {S131|P63}.\n","refs":[]},{"space":"S000139","property":"P000067","value":true,"uid":"","description":"\n\nSee {{mathse:4844916}}.\n","refs":[{"name":"Answer to \"Can a Fréchet-Urysohn hemicompact Hausdorff space fail to be locally compact?\"","mathse":4844916}]},{"space":"S000139","property":"P000068","value":false,"uid":"","description":"\n\nIt is easy to check being {P68} passes to closed subspaces. Since {S139}\ncontains a closed subspace homeomorphic to {S170} and {S170|P68}, the result follows.\n","refs":[]},{"space":"S000139","property":"P000080","value":true,"uid":"","description":"\n\nSee {{mathse:4844916}}.\n","refs":[{"name":"Answer to \"Can a Fréchet-Urysohn hemicompact Hausdorff space fail to be locally compact?\"","mathse":4844916}]},{"space":"S000139","property":"P000089","value":false,"uid":"","description":"\n\nDenote one of the circles by $S^1$ and let $p$ be the mutual intersection point of all the circles.\n\nConstruct a continuous map $f$ as the composition of the retraction of $X$ onto $S^1$ sending every point outside of $S^1$ to $p$, followed by a rotation of $S^1$ by a small angle.\nThe map $f$ has no fixed point.\n","refs":[]},{"space":"S000139","property":"P000111","value":true,"uid":"","description":"\n\nSee {{mathse:4844916}}.\n","refs":[{"name":"Answer to \"Can a Fréchet-Urysohn hemicompact Hausdorff space fail to be locally compact?\"","mathse":4844916}]},{"space":"S000139","property":"P000183","value":true,"uid":"","description":"\n\nEach of the circles (corresponding to an interval $[n,n+1]$, $n\\in\\mathbb Z$, with the endpoints identified) is a closed subspace of $X$ and {S170|P183}.\nAnd every compact subset of $X$ is contained in the union of a finite number of these circles.\n\nTherefore, taking a countable $k$-network from each of the (countably many) circles and forming their union gives a countable $k$-network for $X.$\n","refs":[]},{"space":"S000139","property":"P000200","value":false,"uid":"","description":"\n\nLet one of the circles be denoted by $S^1$. The map $X \\to S^1$ which is the identity on $S^1$ and which sends every other point to the mutual intersection point of the circles is a continuous retraction. Since {S170|P200}, it follows by Proposition 1.17 of {{zb:1044.55001}} that $X$ is not simply connected.\n\nSee Example 1.25 of {{zb:1044.55001}} for a comparison of $\\pi_1(X)$ with the fundamental group of {S201}.\n","refs":[{"name":"Algebraic Topology (Hatcher)","zb":"1044.55001"}]},{"space":"S000139","property":"P000204","value":true,"uid":"","description":"\n\nThe portion of each circle which is complementary to the mutual intersection point $p$ of the circles is an open subset of $X$. It follows that $X \\setminus \\{p\\}$ is disconnected. Since $X$ is connected, $p$ is a cut point.\n","refs":[]},{"space":"S000139","property":"P000205","value":false,"uid":"","description":"\n\nEvery point besides the mutual intersection point of the circles is a non-cut point.\n","refs":[]},{"space":"S000139","property":"P000206","value":true,"uid":"","description":"\n\nIf Player I picks $x_n= \\infty$ in every turn, clearly $\\infty \\in \\bigcap \\{U_n:n<\\omega\\}$.\nSo suppose for some $n$, $x_n\\neq \\infty$, then Player II can choose $V_n$ to be homeomorphic to {S25} and the assertion follows since {S25|P206}.\n","refs":[]},{"space":"S000139","property":"P000214","value":false,"uid":"","description":"\n\n$X$ contains {S131} as a subspace and {S131|P214}.\n","refs":[]},{"space":"S000139","property":"P000243","value":true,"uid":"","description":"\n\nThe set $X\\setminus\\{\\infty\\}$ is open and dense in $X$.\nIt is {P243}\nbecause it is homemorphic to a countable disjoint union of intervals $(0,1)\\cong\\mathbb R$\nand {S25|P243}.\n","refs":[]},{"space":"S000140","property":"P000023","value":false,"uid":"","description":"\n\nSee {{mathse:4854178}}.","refs":[{"name":"What are the compactness properties of $\\mathbb R$, extended by a point with co-countable open neighborhoods?","mathse":4854178}]},{"space":"S000140","property":"P000031","value":true,"uid":"","description":"\n\nSee {{mathse:4854178}}.","refs":[{"name":"What are the compactness properties of $\\mathbb R$, extended by a point with co-countable open neighborhoods?","mathse":4854178}]},{"space":"S000140","property":"P000032","value":false,"uid":"","description":"\n\nSee {{mathse:4854178}}.","refs":[{"name":"What are the compactness properties of $\\mathbb R$, extended by a point with co-countable open neighborhoods?","mathse":4854178}]},{"space":"S000140","property":"P000037","value":false,"uid":"","description":"\n\nIf $\\gamma\\colon [0,1]\\to X$ is a path joining any point in $\\mathbb R$ to $\\infty$, and WLOG $\\gamma^{-1}(\\{\\infty\\})=\\{1\\}$, then $\\gamma(1-\\frac{1}{n})$ is a sequence in $\\mathbb R$ converging to $\\infty$, contradicting that $\\mathbb R$ is sequentially closed in $X$, as explained in {{mathse:4850979}}.\n","refs":[{"name":"Answer to 'Radial/pseudoradial implies Fréchet-Urysohn/sequential for locally countable spaces'","mathse":4850979}]},{"space":"S000140","property":"P000039","value":false,"uid":"","description":"\n\n$(0,1)$ and $(1,2)$ are disjoint open subsets.\n","refs":[]},{"space":"S000140","property":"P000041","value":true,"uid":"","description":"\n\n$X$ is locally connected at each point in $\\mathbb R$ by local connectedness of $\\mathbb R$. Every neighborhood of $\\infty$ is connected, since all neighborhoods of $\\infty$ are dense, so there can be no separation of an open set containing $\\infty$. \n","refs":[]},{"space":"S000140","property":"P000046","value":false,"uid":"","description":"\n\n$\\mathbb R$ is a path connected subset of $X$.\n","refs":[]},{"space":"S000140","property":"P000060","value":true,"uid":"","description":"\n\nLet $f\\colon X\\to \\mathbb R$, with $f(\\infty)=p$. Then for each $n\\in \\mathbb N$, $f^{-1}((p-\\frac{1}{n},p+\\frac{1}{n}))$ contains a neighborhood of $\\infty$, hence is cocountable in $X$. It follows that the intersection of these neighborhoods, $f^{-1}(\\{p\\})$, is also co-countable. But then the image of the connected space $X$ under $f$ is countable and connected, hence a singleton, so that $f$ is constant.\n","refs":[{"name":"Answer to 'Radial/pseudoradial implies Fréchet-Urysohn/sequential for locally countable spaces'","mathse":4850979}]},{"space":"S000140","property":"P000064","value":true,"uid":"","description":"\n\nFollows from Theorem 1.15 of {{mr:0431104}} as its dense subspace $\\mathbb R$ is {P64}.\nSee also {{wikipedia:Baire_space}}.\n","refs":[{"name":"Baire spaces (R. C. Haworth; R. A McCoy)","mr":431104},{"name":"Baire space","wikipedia":"Baire_space"}]},{"space":"S000140","property":"P000065","value":true,"uid":"","description":"\n\nBy construction, $|X| = |\\mathbb R\\cup\\{\\infty\\}| = \\mathfrak{c}$.\n","refs":[]},{"space":"S000140","property":"P000073","value":true,"uid":"","description":"\n\nLet $S \\subseteq X$ be a nonempty irreducible ({P39}) set.\nThe subspace $\\mathbb R$ is open in $X$ and Hausdorff.\nIt follows that $S$ cannot contain more than one point from $\\mathbb R$, and thus $S$ is finite.\nOn the other hand, $X$ is {P2}.\nConsequently $S$ is discrete, which implies it must be a singleton.\n","refs":[]},{"space":"S000140","property":"P000084","value":false,"uid":"","description":"\n\n$\\infty$ has no Hausdorff neighborhood, since all of its neighborhoods are dense.\n\n","refs":[]},{"space":"S000140","property":"P000086","value":false,"uid":"","description":"\n\nEvery neighborhood of $\\infty$ is non-Hausdorff, which is not the case for other points in $X$.\n","refs":[]},{"space":"S000140","property":"P000089","value":true,"uid":"","description":"\n\nAs explained [here](https://topology.pi-base.org/spaces/S000140/properties/P000037), $\\infty$ cannot be joined by a path to any other point.\n\nThus if $f\\colon X\\to X$ is continuous, then either $f(\\mathbb R)=\\{\\infty\\}$, whereby $f(\\infty)$ must equal $\\infty$ in order for $f(X)$ to be connected, or $f(\\mathbb R)\\subseteq \\mathbb R$. In the latter case, we either again have $f(\\infty)=\\infty$, or we have $f(X)\\subseteq \\mathbb R$, which implies (since $X$ is {P60}) that $f$ is constant.\n\nWe conclude that either $\\infty$ is a fixed point, or $f$ is a constant map $f\\equiv p$, so that $p$ is a fixed point.\n","refs":[]},{"space":"S000140","property":"P000098","value":true,"uid":"","description":"\n\nSee {{mathse:4854178}}.","refs":[{"name":"What are the compactness properties of $\\mathbb R$, extended by a point with co-countable open neighborhoods?","mathse":4854178}]},{"space":"S000140","property":"P000101","value":true,"uid":"","description":"\n\nLet $r\\colon X\\to A$ be a retraction. If $r$ is constant, its image is closed, since singletons in $X$ are closed. Suppose then that $r$ is not constant. Since $X$ is {P60}, $A$ must contain $\\infty$. \n\nMoreover, $\\infty$ cannot be joined by a path to any other point, as explained [here](https://topology.pi-base.org/spaces/S000140/properties/P000037), so by continuity of $r$, we cannot have $\\infty\\in r(\\mathbb R)$, as otherwise we would have $r(\\mathbb R)=\\{\\infty\\}$, contradicting that $r$ is not constant.\n\nTherefore $r(\\mathbb R)\\subseteq \\mathbb R$, so $r$ restricts to a retraction of $\\mathbb R$. Then $A\\cap \\mathbb R$ is closed in $\\mathbb R$, since the {S25} {P101}. As $\\infty\\in A$, $A$ is closed in $X$.","refs":[]},{"space":"S000140","property":"P000131","value":true,"uid":"","description":"\n\n$X$ is the union of the {P131} spaces $\\mathbb R$ and $\\{\\infty\\}$.\n","refs":[]},{"space":"S000140","property":"P000143","value":false,"uid":"","description":"\n\nSee {{mathse:4854178}}.","refs":[{"name":"What are the compactness properties of $\\mathbb R$, extended by a point with co-countable open neighborhoods?","mathse":4854178}]},{"space":"S000140","property":"P000169","value":false,"uid":"","description":"\n\nSince every open neighborhood of $\\infty$ is dense, the only regular open neighborhood is $X$.\n","refs":[]},{"space":"S000140","property":"P000171","value":true,"uid":"","description":"\n\nSee {{mathse:4854178}}.","refs":[{"name":"What are the compactness properties of $\\mathbb R$, extended by a point with co-countable open neighborhoods?","mathse":4854178}]},{"space":"S000140","property":"P000173","value":true,"uid":"","description":"\n\nSee {{mathse:4850979}}.","refs":[{"name":"Answer to 'Radial/pseudoradial implies Fréchet-Urysohn/sequential for locally countable spaces'","mathse":4850979}]},{"space":"S000140","property":"P000180","value":true,"uid":"","description":"\n\nSee {{mathse:4850979}}.","refs":[{"name":"Answer to 'Radial/pseudoradial implies Fréchet-Urysohn/sequential for locally countable spaces'","mathse":4850979}]},{"space":"S000140","property":"P000183","value":true,"uid":"","description":"\n\nThe compact subsets of $X=\\mathbb R\\cup\\{\\infty\\}$ are precisely the sets of the form $K$ or $K\\cup\\{\\infty\\}$ with $K$ compact in $\\mathbb R$ (see {{mathse:4854178}}).\n\nLet $\\mathcal{N}$ be a countable $k$-network of $\\mathbb{R}$ with the Euclidean topology, for example a countable base for the topology. Then it is easy to check that the countable collection consisting of all the elements of $\\mathcal N$ and the singleton $\\{\\infty\\}$ is a $k$-network for $X$.\n\nSee {{mathse:4972337}}.\n","refs":[{"name":"What are the compactness properties of $\\mathbb{R}$, extended by a point with co-countable open neighborhoods?","mathse":4854178},{"name":"Pseudocharacter of $T_1$ spaces with a countable k-network","mathse":4972337}]},{"space":"S000140","property":"P000189","value":true,"uid":"","description":"\n\n$\\mathbb R$ is a dense subset of $X$ and {S25|P189}.\n","refs":[]},{"space":"S000140","property":"P000204","value":false,"uid":"","description":"\n\nThe point $\\infty$ is not a cut point, since $X \\setminus \\{\\infty\\}=\\mathbb R$ is connected.\nAnd if $x\\in\\mathbb R$, the intervals $(\\leftarrow, x)$ and\n$(x, \\to)$ in $\\mathbb{R}$ are connected,\nhence so are their closures $(\\leftarrow, x)\\cup\\{\\infty\\}$ and $(x, \\to)\\cup\\{\\infty\\}$.\nSince the two closures have a point in common, their union $X\\setminus\\{x\\}$ is also connected.\n","refs":[]},{"space":"S000140","property":"P000210","value":true,"uid":"","description":"\n\nSee {{mathse:5126685}}.\n","refs":[{"name":"Do any of the Arkhangel'skii $\\alpha_i$ properties hold for $\\mathbb{R}$ extended by a point with co-countable open neighborhoods?","mathse":5126685}]},{"space":"S000140","property":"P000219","value":false,"uid":"","description":"\n\n$X$ and its subspace $\\mathbb R$ have the same cardinality.\nBut they are not homeomorphic:\n{S25|P3} and {S140|P3}.\n","refs":[]},{"space":"S000140","property":"P000229","value":true,"uid":"","description":"\n\n{S140|P37} and $\\mathbb{R} \\subset X$ inherits its usual topology. Thus the path components of $X$ are $\\mathbb{R}$ and $\\{\\infty\\}$, with their usual topologies.\n\nThe result follows since {S25|P229} and {S162|P229}.\n","refs":[]},{"space":"S000140","property":"P000243","value":true,"uid":"","description":"\n\n{S25} is an open dense subspace of $X$\nand {S25|P243}.\n","refs":[]},{"space":"S000141","property":"P000016","value":true,"uid":"","description":"\n\n$X$ is the union of its two compact subspaces $\\omega_1+1$ ({S36}), and $1+\\omega^*$, homeomorphic to $\\omega+1$ ({S20}).\n","refs":[]},{"space":"S000141","property":"P000051","value":true,"uid":"","description":"\n\nLet $A\\subseteq X$ be nonempty. If $A$ meets $\\omega^*$, any point in their intersection is isolated in $X$ (and in $A$). Otherwise, $A\\subseteq\\omega_1+1$ and hence there is a point of $A$ isolated in $A$, since {S36|P51}.\n","refs":[]},{"space":"S000141","property":"P000061","value":false,"uid":"","description":"\n\nEvery element $x\\in\\omega^*$ has an immediate predecessor and an immediate successor in the order of $X$.\nSo the singleton $\\{x\\}$ is a clopen set in $X$, and hence a cozero set.\nThus the set $U=\\omega^*$ is a cozero set in $X$, as a countable union of cozero sets.\n\nSuppose $V=\\text{coz}(g)$, with $g:X\\to\\mathbb R$, is a cozero set disjoint from $U$.\nSince the point $p$ (between $\\omega_1$ and $\\omega^*$) is in $\\overline U\\setminus U$,\nnecessarily $g(p)=0$.\nOn the other hand, a continuous function $h:\\omega_1+1\\to\\mathbb R$ with\n$h(\\omega_1)=r\\in\\mathbb R$ has constant value $r$\non an interval $[x,\\omega_1]\\subseteq\\omega_1+1$ for some $x<\\omega_1$.\nIndeed, $f^{-1}(r)=\\bigcap_{n\\in\\mathbb N}f^{-1}((r-\\frac1n,r+\\frac1n))$ is a countable\nintersection of neighborhoods of $\\omega_1$, which is also a neighborhood of $\\omega_1$.\nIt follows that the function $g$ has value $0$ on an open interval $(x,p)$ of $X$ for some $xa_1>a_2>\\cdots$ converging to $0$ and given a point $q=\\langle x,0\\rangle$ on the $x$-axis, the set of points $\\langle x,a_n\\rangle$ ($n\\in\\omega$) on the vertical line above $q$ is closed in $X$, but also infinite and discrete, hence not compact. If $p$ is a point on the $x$-axis, every neighborhood of $p$ contains such a set (with some $q\\ne p$). So $p$ has no compact neighborhood.\n","refs":[]},{"space":"S000143","property":"P000027","value":false,"uid":"","description":"\n\nSee {{mo:160496}}.\n","refs":[{"name":"Answer to \"A question about cardinal functions\"","mo":160496}]},{"space":"S000143","property":"P000028","value":true,"uid":"","description":"\n\nEasy to check, using a decreasing sequence of balls for a point above the $x$-axis and a decreasing sequence of \"butterfly neighborhoods\" for a point on the $x$-axis.\n","refs":[{"name":"$\\aleph_0$-spaces (E. Michael)","zb":"0148.16701"}]},{"space":"S000143","property":"P000037","value":true,"uid":"","description":"\n\nSuppose $p$ is a point on the $x$-axis and consider a circle in $X$ that is tangent to the $x$-axis at the point $p$.\nBy examining \"butterfly neighborhoods\" of $p$, it is easy to check that a path starting at $p$ and following the circle will be continuous.\n\nOn the other hand, two points above the $x$-axis can be joined by a path following a straight line. By combining both types of path, one can construct a path between any two points of $X$.\n","refs":[]},{"space":"S000143","property":"P000042","value":true,"uid":"","description":"\n\nEach \"butterfly neighborhood\" of a point $p$ on the $x$-axis is {P37},\nusing the same argument used to show {S143|P37}.\nAnd for a point $q$ above the $x$-axis,\na small open ball centered at $q$ is also {P37}.\n","refs":[]},{"space":"S000143","property":"P000064","value":true,"uid":"","description":"\n\nThe complement of the $x$-axis is homeomorphic to {S176}\nand {S176|P64}.\nSo $X$ is {P64} since it contains a dense subspace that is {P64}.\n","refs":[{"name":"Baire space on Wikipedia","wikipedia":"Baire_space"}]},{"space":"S000143","property":"P000068","value":false,"uid":"","description":"\n\nIt is easy to check being {P68} passes to closed subspaces. Since {S170} embeds as a closed subspace of\n{S143} and {S170|P68}, the result follows.\n","refs":[]},{"space":"S000143","property":"P000089","value":false,"uid":"","description":"\n\nThe translation $\\langle x,y\\rangle\\mapsto\\langle x+1,y\\rangle$ is continuous and has no fixed point.\n","refs":[]},{"space":"S000143","property":"P000166","value":true,"uid":"","description":"\n\nThis topology is finer than the subspace topology induced from {S176}, which is separable and metrizable.\n","refs":[]},{"space":"S000143","property":"P000182","value":true,"uid":"","description":"\n\nThe subspaces consisting of the $x$-axis and its complement in $X$ are homeomorphic respectively to {S25} and {S176}.\nEach of them is {P27}, hence {P182}.\nThe union of a countable network from each of the subspaces is a countable network for $X$.\n\nSee {{mo:160496}}.\n","refs":[{"name":"Answer to \"A question about cardinal functions\"","mo":160496}]},{"space":"S000143","property":"P000200","value":false,"uid":"","description":"\n\nSee {{mathse:4998216}}.\n","refs":[{"name":"Is the butterfly space contractible?","mathse":4998216}]},{"space":"S000144","property":"P000001","value":true,"uid":"","description":"\n\nBy construction of the Alexandrov topology from a poset.\n","refs":[{"name":"Alexandrov topology on Wikipedia","wikipedia":"Alexandrov_topology"}]},{"space":"S000144","property":"P000078","value":true,"uid":"","description":"\n\nBy construction.\n","refs":[]},{"space":"S000144","property":"P000126","value":false,"uid":"","description":"\n\nThe singleton $\\{a\\}$ is neither open nor closed.\n","refs":[]},{"space":"S000144","property":"P000176","value":true,"uid":"","description":"\n\nBy construction.\n","refs":[]},{"space":"S000144","property":"P000196","value":false,"uid":"","description":"\n\nThe subset $\\{a,b\\}$ is not connected.\n","refs":[]},{"space":"S000144","property":"P000201","value":true,"uid":"","description":"\n\nThe maximum point $1$ belongs to every nonempty open set.\n","refs":[]},{"space":"S000144","property":"P000202","value":true,"uid":"","description":"\n\nThe minimum point $0$ belongs to every nonempty closed set.\n","refs":[]},{"space":"S000144","property":"P000204","value":false,"uid":"","description":"\n\nIf $0$ is not removed, then the space is {P202}, namely $0$, so it will be {P36}. If $1$ is not removed, then the space is {P201}, so it will be {P36}. Thus, removing at most one point cannot disconnect the space.\n","refs":[]},{"space":"S000145","property":"P000002","value":true,"uid":"","description":"\n\nThe free ultrafilter $\\mathscr U$ does not contain any singleton. Hence singletons are not open and every singleton is closed.\n","refs":[{"name":"Examples of connected door spaces","mathse":3088994}]},{"space":"S000145","property":"P000039","value":true,"uid":"","description":"\n\nAny two nonempty open sets belong to $\\mathscr U$, and hence have nonempty intersection by the definition of filter.\n","refs":[{"name":"Examples of connected door spaces","mathse":3088994}]},{"space":"S000145","property":"P000044","value":true,"uid":"","description":"\n\nLet $Y$ be a {P36} subset of $X$ with at least two points.\nIf $Y$ is closed in $X$, by the property of ultrafilters all subsets of $Y$ are closed, thus $Y$ is {P52}, which is not possible.\nThis implies that $Y$ is open, that is, every {P36} subset of $X$ must be open.\n\nNow, because $X$ is {P36}, $X$ cannot be partitioned into two nonempty open subsets.\nWe conclude that $X$ cannot be partitioned into two {P36} subsets, each with at least two points.\n","refs":[]},{"space":"S000145","property":"P000086","value":true,"uid":"","description":"\n\nLet $p, q \\in X$ be arbitrary distinct points.\nDefine $f(x) = \\begin{cases} q & \\text{if }x = p, \\\\ p & \\text{if }x = q, \\\\ x & \\text{otherwise}. \\end{cases}$\n\nThen $f(x)$ is involutary and continuous since for every open subset $U$ and every point $x \\in X$, both $U \\setminus \\{x\\} = U \\cap \\left( X \\setminus \\{x\\} \\right)$ and $U \\cup \\{x\\}$ are open.\n","refs":[]},{"space":"S000145","property":"P000089","value":true,"uid":"","description":"\n\nSee {{mathse:5021484}}.\n","refs":[{"name":"Does a free ultrafilter topology have the fixed point property?","mathse":5021484}]},{"space":"S000145","property":"P000101","value":false,"uid":"","description":"\n\nLet $p, q \\in X$ be arbitrary distinct points.\nDefine $f(x) = \\begin{cases} q & \\text{if }x = p, \\\\ x & \\text{otherwise}. \\end{cases}$\n\nThen $f(x)$ is continuous since for every open subset $U$ and every point $x \\in X$, both $U \\setminus \\{x\\} = U \\cap \\left( X \\setminus \\{x\\} \\right)$ and $U \\cup \\{x\\}$ are open.\n\nTherefore, $f$ is a retraction from $X$ to $X \\setminus \\{p\\}$, which is not closed.\n","refs":[]},{"space":"S000145","property":"P000126","value":true,"uid":"","description":"\n\nGiven a nonempty proper subset $A\\subseteq X$, either $A\\in\\mathscr U$ or $X\\setminus A\\in\\mathscr U$ by a property of ultrafilters. That is, $A$ is either open or closed.\n","refs":[{"name":"Examples of connected door spaces","mathse":3088994}]},{"space":"S000145","property":"P000138","value":false,"uid":"","description":"\n\nLet $A$ be an infinite closed proper subset of $X$, and take an arbitrary $p \\in X \\setminus A$.\nFor each $S \\subseteq A$, the function $f_S:X\\to X$ defined by\n$$f_S(x) = \\begin{cases} p & \\text{if }x\\in S, \\\\ x & \\text{otherwise}, \\end{cases}$$\nis continuous since for every nonempty open set $U$, its inverse image $f^{-1}(U)$ contains $U\\setminus A$, which is open and nonempty; hence $f^{-1}(U)$ is open.\n\nAnd there are continuum many such functions $f_S$.\n","refs":[]},{"space":"S000145","property":"P000181","value":true,"uid":"","description":"\n\nBy construction.\n","refs":[]},{"space":"S000146","property":"P000027","value":true,"uid":"","description":"\n\nSince {S146} is a subspace of {S30}, the result follows from {S30|P27}.\n","refs":[]},{"space":"S000146","property":"P000050","value":true,"uid":"","description":"\n\nThe product of {P50} spaces is {P50}, so this follows from {S27|P50}.\n","refs":[]},{"space":"S000146","property":"P000053","value":true,"uid":"","description":"\n\nSince {S146} is a subspace of {S30}, the result follows from {S30|P53}.\n","refs":[]},{"space":"S000146","property":"P000056","value":true,"uid":"","description":"\n\nLet $F_q = \\{f\\in\\mathbb Q^\\omega:f(0)=q\\}$, then $F_q$ is a closed set with empty interior, and $\\mathbb Q^\\omega = \\bigcup_{q \\in \\mathbb Q} F_q$.\n","refs":[]},{"space":"S000146","property":"P000065","value":true,"uid":"","description":"\n\n$\\left| \\mathbb Q^\\omega \\right| = \\mathfrak c$.\n","refs":[]},{"space":"S000146","property":"P000066","value":false,"uid":"","description":"\n\n{S28} is homeomorphic to $\\mathbb Z^\\omega$, which can be viewed as a closed subspace of {S146}.\nAnd {P66} is preserved by closed subspaces.\nBut {S28|P66}.\n","refs":[]},{"space":"S000146","property":"P000087","value":true,"uid":"","description":"\n\nThis space is an additive subgroup of {S30}, which has a group topology.\n","refs":[]},{"space":"S000147","property":"P000017","value":false,"uid":"","description":"\n\nShown in {{mathse:4744210}}.\n","refs":[{"name":"Why can't a Lusin set be sigma-compact?","mathse":4744210}]},{"space":"S000147","property":"P000046","value":true,"uid":"","description":"\n\nAssume there are $a,b\\in X$ with $a0$ contains a compact neighborhood of $x$ of the form $[y,\\rightarrow)$.\n","refs":[]},{"space":"S000151","property":"P000138","value":false,"uid":"","description":"\n\nFix $\\theta\\in[0,1]\\setminus\\mathbb Q$.\nDefine a map on $X$ by the formula:\n$f_\\theta(x)=\\begin{cases}\\frac{x}{2}&\\text{for }x<\\theta,\\\\\n\\frac{x+1}{2}&\\text{for }x>\\theta,\\end{cases}\\ x\\in X$.\n\nThe set $X\\setminus f_\\theta(X)=(\\frac{\\theta}2,\\frac{\\theta+1}2)\\cap\\mathbb Q$ is different for each irrational $\\theta$.\n\nContinuity of $f_\\theta$ can be verified on the canonical basis for the topology on $X$.\nThe map is strictly increasing and\n$f_\\theta^{-1}((x,\\rightarrow))=(f_\\theta^{-1}(x),\\rightarrow)$ for every $x\\in f_\\theta(X)$.\nFor $x\\in X\\setminus f_\\theta(X)$ the set\n$f_\\theta^{-1}((x,\\rightarrow))=(\\theta,\\rightarrow)$ is open as well.\n","refs":[]},{"space":"S000151","property":"P000139","value":false,"uid":"","description":"\n\nEvery neighborhood of a point $x<1$ contains points $y>x$.\n\nEvery neighborhood of $1$ contains points $y<1$.\n","refs":[]},{"space":"S000151","property":"P000181","value":true,"uid":"","description":"\n\nBy definition.\n","refs":[]},{"space":"S000151","property":"P000192","value":false,"uid":"","description":"\n\nEvery set in a {P196} space is irreducible.\n\nGiven an irrational $\\theta\\in(0,1)$, the set $[0,\\theta)\\cap\\mathbb Q$ is closed and irreducible.\nBut it is not the closure $(\\leftarrow,x]$ of a point $x\\in X$.\n","refs":[]},{"space":"S000151","property":"P000196","value":true,"uid":"","description":"\n\nThe topology on $X$ is coarser than the one from {S150},\nand {S150|P196}.\n","refs":[]},{"space":"S000151","property":"P000201","value":true,"uid":"","description":"\n\nThe point $1$ is a generic point, as it belongs to every nonempty open set.\n","refs":[]},{"space":"S000151","property":"P000202","value":true,"uid":"","description":"\n\nThe point $0$ is a focal point, as it belongs to every nonempty closed set.\n","refs":[]},{"space":"S000152","property":"P000039","value":true,"uid":"","description":"\n\nSee part 2 of {{mathse:5007860}}.\n","refs":[{"name":"Answer to \"For a compact sober \"highly non-$T_1$\" space, how much \"highly connectedness\" is needed to imply it's a spectral space?\"","mathse":5007860}]},{"space":"S000152","property":"P000073","value":true,"uid":"","description":"\n\nSee part 2 of {{mathse:5007860}}.\n","refs":[{"name":"Answer to \"For a compact sober \"highly non-$T_1$\" space, how much \"highly connectedness\" is needed to imply it's a spectral space?\"","mathse":5007860}]},{"space":"S000152","property":"P000075","value":false,"uid":"","description":"\n\nSee part 2 of {{mathse:5007860}}.\n","refs":[{"name":"Answer to \"For a compact sober \"highly non-$T_1$\" space, how much \"highly connectedness\" is needed to imply it's a spectral space?\"","mathse":5007860}]},{"space":"S000152","property":"P000181","value":true,"uid":"","description":"\n\nBy construction.\n","refs":[]},{"space":"S000152","property":"P000202","value":true,"uid":"","description":"\n\nThe point $-1$ is a focal point of $X$ as it is a minimum of the poset.\n\nSee part 2 of {{mathse:5007860}}.\n","refs":[{"name":"Answer to \"For a compact sober \"highly non-$T_1$\" space, how much \"highly connectedness\" is needed to imply it's a spectral space?\"","mathse":5007860}]},{"space":"S000152","property":"P000204","value":false,"uid":"","description":"\n\nGiven $x\\in X$, there is a largest point $y\\in X\\setminus\\{x\\}$.\nEvery nonempty open set of the subspace $X\\setminus\\{x\\}$ contains $y$,\nand therefore $X\\setminus\\{x\\}$ is {P39} and thus {P36}\n[(Explore)](https://topology.pi-base.org/spaces?q=Hyperconnected+%2B+%7EConnected).\n","refs":[]},{"space":"S000152","property":"P000226","value":true,"uid":"","description":"\n\nAssume $O_1 \\supsetneq O_2 \\supsetneq \\dots$ is an infinite decreasing sequence of open sets. Since $\\{-1,0_a,0_b\\}$ is an initial segment of the partial order, we must then have $\\{-1,0_a,0_b\\} \\cap O_n = \\emptyset$ for all $n\\geq 4$. Therefore $O_n$ must be finite for $n>4$, which leads to a contradiction.\n","refs":[]},{"space":"S000153","property":"P000015","value":false,"uid":"","description":"\n\n{P15} is a hereditary property. The space $X$ contains as a subspace a copy of {S35}, which is not {P15}.\n","refs":[{"name":"Long line on Wikipedia","wikipedia":"Long_line_(topology)"}]},{"space":"S000153","property":"P000019","value":false,"uid":"","description":"\n\nThe sequence $(1/n)_n$ has no accumulation point in $X$.\n","refs":[{"name":"Long line on Wikipedia","wikipedia":"Long_line_(topology)"}]},{"space":"S000153","property":"P000061","value":true,"uid":"","description":"\n\nSee the \"Related spaces\" section at the end of {{mathse:5106636}}.\n","refs":[{"name":"Answer to \"Is the long ray cozero complemented?\"","mathse":5106636}]},{"space":"S000153","property":"P000062","value":false,"uid":"","description":"\n\nFor $\\alpha \\in \\omega_1$, let $A_\\alpha = \\{x\\in X: x<(\\alpha,0)\\}$. The open cover $\\{A_\\alpha\\ |\\ \\alpha \\in \\omega_1\\}$ has no countable subcollection dense in\nthe space.\n\nSee also item #1 for space #45 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000153","property":"P000065","value":true,"uid":"","description":"\n\n$X$ has the same cardinality as {S38}.\n","refs":[{"name":"Long line on Wikipedia","wikipedia":"Long_line_(topology)"}]},{"space":"S000153","property":"P000076","value":true,"uid":"","description":"\n\nIf $f$ is the map from the proof that {S149|P76}, then $f$ restricted to {S153} is a closed embedding into the $\\Sigma$-product $(0, \\infty)\\times \\Sigma_{0 < \\alpha < \\omega_1} \\mathbb{R}$, which by the same argument is {P76}. \n","refs":[]},{"space":"S000153","property":"P000083","value":false,"uid":"","description":"\n\n{S38} is homeomorphic to the closed subspace $[\\langle1,0\\rangle,\\infty) \\subseteq X$ and {S38|P83}.\n","refs":[]},{"space":"S000153","property":"P000086","value":true,"uid":"","description":"\n\nSee {{mathse:1641524}}. Also, {{mo:314665}}.\n","refs":[{"name":"Homogeneous space minus a point","mathse":1641524},{"name":"Answer to \"Is every uncountable, homogeneous connected T2-space isomorphic to a subspace of $\\mathbb R^\\omega$?\"","mo":314665}]},{"space":"S000153","property":"P000089","value":false,"uid":"","description":"\n\nThe subspace $(\\langle 0, 0 \\rangle, \\langle 1, 0 \\rangle]$, which is homeomorphic to {S210}, is a retract of $X$ via\n$$\\langle \\alpha,x \\rangle \\mapsto \\begin{cases}\\langle \\alpha,x \\rangle,\\quad \\alpha=0\\\\\n\\langle1,0\\rangle,\\quad\\text{otherwise}\n\\end{cases}$$\nand {S210|P89}.\n","refs":[]},{"space":"S000153","property":"P000133","value":true,"uid":"","description":"\n\n$X$ is an order-convex subset of {S38}, which is {P133}.\n","refs":[{"name":"Long line on Wikipedia","wikipedia":"Long_line_(topology)"}]},{"space":"S000153","property":"P000155","value":true,"uid":"","description":"\n\nEvery point has a neighborhood isomorphic to the open interval $(0,1)$, because any non-degenerate interval $[p,q]$ in $X$ is homeomorphic to the interval $[0,1]\\subseteq\\mathbb R$.\n\nSee item #6 for space #45 in {{zb:0386.54001}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000153","property":"P000162","value":false,"uid":"","description":"\n\nThe subspace $X\\subseteq Y$ of {S153} given by $X = \\{(x, 0) : 0 < x < \\omega_1\\}$ is a closed copy of $\\omega_1$ in $Y$. If $Y$ were realcompact, then its closed subspace $X$ would be realcompact (see theorem 8.10 in {{zb:1380.46022}}). But $\\omega_1$ is a pseudocompact non-compact Tychonoff space, so is not realcompact, see {T388}.\n","refs":[{"name":"Rings of Continuous Functions (Gillman and Jerison)","zb":"1380.46022"}]},{"space":"S000153","property":"P000186","value":true,"uid":"","description":"\n\n$X$ is a subspace of {S149} and {S149|P186}.\n","refs":[]},{"space":"S000153","property":"P000198","value":true,"uid":"","description":"\n\n$X$ is the union of the subspaces $\\{0\\} \\times (0, 1)$ and $\\left[ 1, \\omega_1 \\right) \\times [0, 1)$,\nwhich are homeomorphic to {S25} and {S38}, respectively.\nEach has countable extent, as {S25|P198} and {S38|P198}.\n","refs":[]},{"space":"S000153","property":"P000199","value":false,"uid":"","description":"\n\nSimilar to the proof that {S38|P199}.\n","refs":[{"name":"Prove the long line is not contractible.","mathse":1282097}]},{"space":"S000153","property":"P000200","value":true,"uid":"","description":"\n\nSimilar to the proof that {S38|P200}.\n","refs":[]},{"space":"S000154","property":"P000061","value":false,"uid":"","description":"\n\nNote that in any normal space the cozero sets are exactly the open $F_\\sigma$-sets (see Corollary 1.5.13 of {{zb:0684.54001}}), hence in $X$ since {S154|P13}.\n\nSuppose that $X$ is cozero complemented and let $U\\subseteq \\mathbb{R}$ be any countably infinite set. Then $U$ is an open $F_\\sigma$-set, so it follows that $U$ is a cozero set. Find a cozero set $V$ disjoint from $U$ such that $U\\cup V$ is dense in $X$. Since $X\\setminus V$ is infinite we have $\\infty\\notin V$. Since $V$ is a cozero set, write $V = \\bigcup_{n=1}^\\infty F_n$ where $F_n$ are closed. Then $\\infty\\notin F_n$ so $F_n$ is finite. It follows that $V$ is countable. But then $X = \\overline{U\\cup V} = U\\cup V\\cup \\{\\infty\\}$ is countable, contradiction.\n","refs":[{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"space":"S000154","property":"P000065","value":true,"uid":"","description":"\n\nBy construction.\n","refs":[]},{"space":"S000154","property":"P000091","value":true,"uid":"","description":"\n\nIt follows from Proposition 3.1 of {{zb:0232.46019}} that the one-point compactification of the disjoint union of {P91} spaces is {P91},\nand {S154} is the one-point compactification of {S3},\nwhich is the disjoint union of copies of {S162}.\n\nSee also {{mathse:5071093}}.\n","refs":[{"name":"Weakly compact sets - their topological properties and the Banach spaces they generate (J. Lindenstrauss)","zb":"0232.46019"},{"name":"Is the one-point compactification of an uncountable discrete space Eberlein compact?","mathse":5071093}]},{"space":"S000154","property":"P000120","value":false,"uid":"","description":"\n\nSuppose an open neighbourhood $U$ of $\\infty$ is orderable. Assume that $(\\leftarrow,\\infty)_U$ is uncountable. Pick a point $x_0\\in(\\leftarrow,\\infty)_U$. It is isolated, hence it has a successor $x_1$. Applying this argument inductively we construct an increasing sequence $(x_n)_{n=0}^\\infty\\subset (\\leftarrow,\\infty)_U$. Note that for each $n<\\omega$ the interval $(\\leftarrow,x_n]_U$ is finite. Since $(\\leftarrow,\\infty)_U$ is uncountable, there exists $y\\in \\bigcap_{n=0}^\\infty (x_n,\\infty)_U$ and the open interval $(y,\\rightarrow)_U$ is a not cofinite neighbourhood of $\\infty$. A contradiction.\n","refs":[]},{"space":"S000154","property":"P000151","value":true,"uid":"","description":"\n\nIn the Rothberger game $\\mathsf{G}_1(\\mathcal{O}_X, \\mathcal{O}_X)$,\nin the first round let Player 2 choose $U_1\\in \\mathcal{U}_1$ to be any open set containing $\\infty$.\nThe set $X\\setminus U_1$ is finite, so in subsequent rounds\nlet Player 2 choose $U_n$ so that they cover $X\\setminus U_1$.\nAfter a finite number of rounds we'll obtain that $\\{U_1, ..., U_n\\}$ is a cover of $X$.\n","refs":[]},{"space":"S000154","property":"P000203","value":true,"uid":"","description":"\n\nAll points except $\\infty$ are isolated.\n","refs":[]},{"space":"S000154","property":"P000210","value":true,"uid":"","description":"\n\nLet $x \\in X$.\nIf $x = \\infty$, all neighborhoods of $x$ are cofinite, so we can follow the same argument as in {T800}.\nOtherwise $x$ is isolated, so there cannot be a countably infinite set converging to $x$. Hence, the assertion {P210} is trivially true.\n","refs":[]},{"space":"S000155","property":"P000072","value":true,"uid":"","description":"\n\nIn definition 3.7 of {{doi:10.14712/1213-7243.2015.201}} a set-theoretic statement $\\mathcal{A}(\\kappa)$ is introduced, and in theorem 3.8 it's shown that $\\mathcal{A}(\\omega_1)$ holds. In theorem 5.6 it's shown that $\\mathcal{A}(\\kappa)$ implies that $L(\\kappa)$ is robustly Menger, where $L(\\kappa)$ is the one-point Lindelöfication of discrete space of size $\\kappa$. In this case $L(\\omega_1)$ is {S155}. From theorem 5.5 it follows that robustly Menger implies 2-Markov Menger, so {S155|P72}.\n","refs":[{"name":"Applications of limited information strategies in Menger's game","doi":"10.14712/1213-7243.2015.201"}]},{"space":"S000155","property":"P000114","value":true,"uid":"","description":"\n\nBy construction.\n","refs":[]},{"space":"S000155","property":"P000133","value":true,"uid":"","description":"\n\nSee Proposition 2 in {{mathse:4833761}}, where $X$ is $F_{\\omega_1,\\omega_1}$.\n","refs":[{"name":"LOTS and radial properties for generalized Fort/Fortissimo spaces","mathse":4833761}]},{"space":"S000155","property":"P000147","value":true,"uid":"","description":"\n\nThe point $p$ is a P-point, since its neighborhoods are the cocountable subsets of $X$ containing the point and a countable intersection of them is still such a neighborhood.\nAnd every isolated point is also a P-point.\n","refs":[]},{"space":"S000155","property":"P000151","value":true,"uid":"","description":"\n\nSee {{mathse:4727833}}.\n","refs":[{"name":"What kinds of selection principles hold for Fortissimo space?","mathse":4727833}]},{"space":"S000155","property":"P000174","value":true,"uid":"","description":"\n\nConsider the ordered set $Y=((\\omega_1\\times\\mathbb Z)\\cup\\{\\infty\\},<)$,\nwhere the product $\\omega_1\\times\\mathbb Z$ has the lexicographic order\nand the element $\\infty$ is larger than all the others.\nThe space $X$ is homeomorphic to $Y$ with its corresponding order topology\n(see the proof of Proposition 2 in {{mathse:4833761}}).\n\n$X$ is well-based at the point $\\infty$, since the intervals $(\\alpha,\\infty]$\nfor $\\alpha<\\infty$ form a neighborhood base totally ordered by inclusion.\nAnd $X$ is trivially well-based at each of the other (isolated) points.\n","refs":[{"name":"LOTS and radial properties for generalized Fort/Fortissimo spaces","mathse":4833761}]},{"space":"S000155","property":"P000203","value":true,"uid":"","description":"\n\nAll points except $p$ are isolated.\n","refs":[]},{"space":"S000156","property":"P000050","value":true,"uid":"","description":"\n\nBasic open sets in a convergent sequence are clopen; this passes to {S156}.\n","refs":[{"name":"What separation properties are satisfied by the Arens space?","mathse":4673615}]},{"space":"S000156","property":"P000051","value":true,"uid":"","description":"\n\nLet $A$ be a nonempty subset of the Arens space $Y$. Either $A$ contains a point isolated in $Y$, or $A$ is contained in a subspace of $Y$ homeomorphic to {S20} which is {P51}. In both cases $A$ contains a point isolated in $A$.\n","refs":[{"name":"What separation properties are satisfied by the Arens space?","mathse":4673615}]},{"space":"S000156","property":"P000057","value":true,"uid":"","description":"\n\nBy construction.\n","refs":[{"name":"Note on convergence in topology.","mr":37500}]},{"space":"S000156","property":"P000080","value":false,"uid":"","description":"\n\nThe Arens space contains {S23} as a subspace which is not {P80}. Since {P80} is a hereditary property, the Arens space does not satisfy the property either.\n\nSee .\n","refs":[{"name":"A note on the Arens' space and sequential fan.","mr":1485766}]},{"space":"S000156","property":"P000104","value":true,"uid":"","description":"\n\nSee {{mathse:5012106}}.\n","refs":[{"name":"Is the Arens space symmetrizable?","mathse":5012106}]},{"space":"S000156","property":"P000111","value":true,"uid":"","description":"\n\nSee {{mathse:4672783}}.\n","refs":[{"name":"Is the Arens space hemicompact but not locally compact?","mathse":4672783}]},{"space":"S000156","property":"P000183","value":true,"uid":"","description":"\n\nSee the last part of {{mathse:4673444}}.\n","refs":[{"name":"Answer to \"Is the Arens space hemicompact but not locally compact?\"","mathse":4673444}]},{"space":"S000156","property":"P000243","value":true,"uid":"","description":"\n\nThe set $\\omega\\times\\omega$ is open dense in $X$ and discrete.\nSo the singletons $\\{x\\}$ for $x\\in\\omega\\times\\omega$ form a countable $\\pi$-base for $X$.\n","refs":[]},{"space":"S000157","property":"P000002","value":true,"uid":"","description":"\n\nBy inspection, every singleton is closed.\n","refs":[]},{"space":"S000157","property":"P000059","value":true,"uid":"","description":"\n\n$\\aleph_1\\le\\mathfrak c$.\nHence $\\aleph_2=\\aleph_1^+\\le\\mathfrak c^+\\le 2^{\\mathfrak c}$.\n","refs":[]},{"space":"S000157","property":"P000114","value":false,"uid":"","description":"\n\n$|X|=\\aleph_2\\ne\\aleph_1$.\n","refs":[]},{"space":"S000157","property":"P000120","value":false,"uid":"","description":"\n\nEvery neighbourhood of $p$ is homeomorphic to $X$ and {S157|P154}, hence not {P133}.\n","refs":[]},{"space":"S000157","property":"P000147","value":true,"uid":"","description":"\n\nThe point $p$ is a P-point, since its neighborhoods are the cocountable subsets of $X$ containing the point and a countable intersection of them is still such a neighborhood.\nAnd every isolated point is also a P-point.\n","refs":[]},{"space":"S000157","property":"P000151","value":true,"uid":"","description":"\n\nSee {{mathse:4727833}}.\n","refs":[{"name":"What kinds of selection principles hold for Fortissimo space?","mathse":4727833}]},{"space":"S000157","property":"P000154","value":false,"uid":"","description":"\n\nSee Proposition 4 in {{mathse:4833761}}, where $X$ is $F_{\\omega_2,\\omega_1}$.\n","refs":[{"name":"LOTS and radial properties for generalized Fort/Fortissimo spaces","mathse":4833761}]},{"space":"S000157","property":"P000172","value":true,"uid":"","description":"\n\nSee Proposition 1 in {{mathse:4833761}}, where $X$ is $F_{\\omega_2,\\omega_1}$.\n","refs":[{"name":"LOTS and radial properties for generalized Fort/Fortissimo spaces","mathse":4833761}]},{"space":"S000157","property":"P000174","value":false,"uid":"","description":"\n\nSuppose by contradiction that $X$ is well-based at the point $p$.\nSo there is a chain (totally ordered by inclusion) of neighborhoods of $p$ forming a local base at $p$.\nTheir complements form a chain $\\mathscr C$ of countable subsets of $X\\setminus\\{p\\}$ whose union is $X\\setminus\\{p\\}$.\nBut this is impossible since, as shown for example in {{mathse:342091}}, $|\\bigcup\\mathscr C|\\le\\aleph_1<\\aleph_2$.\n","refs":[{"name":"Why does the union of a chain of countable sets have cardinality at most $\\aleph_1$?","mathse":342091}]},{"space":"S000157","property":"P000203","value":true,"uid":"","description":"\n\nAll points except $p$ are isolated.\n","refs":[]},{"space":"S000158","property":"P000016","value":true,"uid":"","description":"\n\nShown in section 17.2 of {{zb:1052.54001}}.\n","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"space":"S000158","property":"P000095","value":true,"uid":"","description":"\n\nEvery interval $[a,b]$ with $a \\mathfrak{c}$. See {S38|P65} and {S103|P59}.\n","refs":[]},{"space":"S000173","property":"P000172","value":false,"uid":"","description":"\n\n$X$ contains {S103} as a subspace and {S103|P172}.\n","refs":[]},{"space":"S000173","property":"P000199","value":false,"uid":"","description":"\n\nA product is contractible only if its factors are contractible. See {{mathse:4996405}}. Since {S38|P199}, it follows that $X$ is not contractible.\n","refs":[{"name":"Answer to \"An arbitrary product of contractible spaces is contractible\"","mathse":4996405}]},{"space":"S000173","property":"P000200","value":true,"uid":"","description":"\n\nA product is {P37} if its factors are.\nIn addition, the fundamental group factors through products, so the same is true of simply-connectedness; see Proposition 1.12 of {{zb: \"1044.55001\"}}.\nSince {S38|P200} and {S103|P200}, it follows that $X$ is simply connected.\n","refs":[{"name":"Algebraic Topology (Hatcher)","zb":"1044.55001"}]},{"space":"S000173","property":"P000214","value":false,"uid":"","description":"\nSee {{mathse:5106079}}.\n","refs":[{"name":"Do the Arkhangel'skii $\\alpha_i$ properties hold for uncountable products?","mathse":5106079}]},{"space":"S000173","property":"P000215","value":false,"uid":"","description":"\n\n$X$ contains {S38} as a subspace and {S38|P162}.\n","refs":[]},{"space":"S000174","property":"P000030","value":false,"uid":"","description":"\n\nProposition 2 of {{doi:10.4064/FM-97-2-53-55}}.\n","refs":[{"name":"A perfectly normal locally metrizable non-paracompact space","doi":"10.4064/FM-97-2-53-55"}]},{"space":"S000174","property":"P000067","value":true,"uid":"","description":"\n\nProposition 1 of {{doi:10.4064/FM-97-2-53-55}}.\n","refs":[{"name":"A perfectly normal locally metrizable non-paracompact space","doi":"10.4064/FM-97-2-53-55"}]},{"space":"S000174","property":"P000082","value":true,"uid":"","description":"\n\nBy construction in {{doi:10.4064/FM-97-2-53-55}}.\n","refs":[{"name":"A perfectly normal locally metrizable non-paracompact space","doi":"10.4064/FM-97-2-53-55"}]},{"space":"S000174","property":"P000088","value":true,"uid":"","description":"\n\nRemark 2 of {{doi:10.4064/FM-97-2-53-55}}.\n","refs":[{"name":"A perfectly normal locally metrizable non-paracompact space","doi":"10.4064/FM-97-2-53-55"}]},{"space":"S000174","property":"P000112","value":true,"uid":"","description":"\n\nBy construction, the topology on $X$ is finer than the product topology on $\\omega_1^\\omega$, where $\\omega_1$ is equipped with the discrete topology. The said product topology on $\\omega_1^\\omega$ is generated by the Baire metric,\n\n$$d(\\langle \\alpha_i \\rangle, \\langle \\beta_i \\rangle) = \\frac{1}{i}$$\n\nwhere $i$ is the first coordinate s.t. $x_i \\neq y_i$. See {S102}.\n","refs":[{"name":"A perfectly normal locally metrizable non-paracompact space","doi":"10.4064/FM-97-2-53-55"}]},{"space":"S000174","property":"P000163","value":true,"uid":"","description":"\n\n$|X| = |\\omega_1^\\omega| \\leq (2^\\omega)^\\omega = 2^\\omega = \\mathfrak{c}$.\n","refs":[]},{"space":"S000175","property":"P000010","value":false,"uid":"","description":"\n\nSee {{mathse:5091633}}. See also {{mathse:3394284}}.\n","refs":[{"name":"Answer to \"Separation Properties of the Radial Plane Topology\"","mathse":5091633},{"name":"Answer to \"Topology Between the Radial Topology and Standard One on R^2\"","mathse":3394284}]},{"space":"S000175","property":"P000022","value":false,"uid":"","description":"\n\nProjection onto the $x$ coordinate is an unbounded continuous function.\n","refs":[]},{"space":"S000175","property":"P000026","value":true,"uid":"","description":"\n\n$\\mathbb Q^2$ is a dense subset.\n","refs":[{"name":"Problem 5468 of The American Mathematical Monthly","doi":"10.2307/2315929"}]},{"space":"S000175","property":"P000042","value":true,"uid":"","description":"\n\nLet $U$ be an open set in $X$.\nFix $x_0\\in U$ and define $V$ as the set of all such $x\\in U$\nthat there exists a finite sequence $(x_k)_{k=1}^n$ of points in $U$\nsuch that $x=x_n$ and for every $k=1,\\ldots,n$ the points $x_k$ and $x_{k-1}$ belong\nto an open segment contained in $U$.\nIt is easy to check that $V$ is path connected and is radially open.\nTherefore for any $x_0\\in U$ there exists a path connected\nneighbourhood $V\\subset U$.\n","refs":[]},{"space":"S000175","property":"P000065","value":true,"uid":"","description":"\n\nBy construction.\n","refs":[{"name":"Problem 5468 of The American Mathematical Monthly","doi":"10.2307/2315929"}]},{"space":"S000175","property":"P000079","value":true,"uid":"","description":"\n\nFollows as the space is the quotient of a topological sum of first-countable spaces (Euclidean lines).\n","refs":[{"name":"Problem 5468 of The American Mathematical Monthly","doi":"10.2307/2315929"}]},{"space":"S000175","property":"P000086","value":true,"uid":"","description":"\n\nGiven points $p,q\\in\\mathbb R^2$, the function $f(x)=x-p+q$ is a bijection mapping $p$ to $q$. Since the function is linear, it preserves lines passing through points and therefore is a homeomorphism.\n","refs":[]},{"space":"S000175","property":"P000089","value":false,"uid":"","description":"\n\nEvery translation is a continuous self-homeomorphism.\n","refs":[]},{"space":"S000175","property":"P000095","value":true,"uid":"","description":"\n\nBy construction every line segment is homeomorphic to {S158}.\n","refs":[]},{"space":"S000175","property":"P000166","value":true,"uid":"","description":"\n\nFollows as the topology is finer than the Euclidean topology.\n","refs":[{"name":"Problem 5468 of The American Mathematical Monthly","doi":"10.2307/2315929"}]},{"space":"S000175","property":"P000172","value":false,"uid":"","description":"\n\nShown in {{mathse:4784524}}: $\\mathbb Q^2$ is dense in {S175},\nbut there exists a point of {S175} which is not the limit of any\nsequence (and therefore any transfinite sequence) from $\\mathbb Q^2$.\n\nSee also {{mathse:3634961}} and {{doi:10.2307/2315929}}.\n","refs":[{"name":"Is the radial plane radial?","mathse":4784524},{"name":"Problem 5468 of The American Mathematical Monthly","doi":"10.2307/2315929"},{"name":"Answer to Radially open set topology is separable?","mathse":3634961}]},{"space":"S000175","property":"P000204","value":false,"uid":"","description":"\n\nSince every segment is homeomorphic to {S158},\nin the space with a removed point\nthere still exists a path (union of at most two segments)\nbetween an arbitrary pair of points.\n","refs":[]},{"space":"S000175","property":"P000227","value":true,"uid":"","description":"\n\nContains a closed discrete subspace $\\{(x,y):x^2+y^2=1\\}$ of cardinality $\\mathfrak c$.\n","refs":[{"name":"Problem 5468 of The American Mathematical Monthly","doi":"10.2307/2315929"}]},{"space":"S000176","property":"P000022","value":false,"uid":"","description":"\n\nSee {{zb:1052.54001}}.\n","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"space":"S000176","property":"P000027","value":true,"uid":"","description":"\n\nSee {{zb:1052.54001}}.\n","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"space":"S000176","property":"P000053","value":true,"uid":"","description":"\n\nSee {{zb:1052.54001}}.\n","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"space":"S000176","property":"P000097","value":false,"uid":"","description":"\n\nThe space contains a subspace homeomorphic to {S170}, and {S170|P97}.\n","refs":[]},{"space":"S000176","property":"P000120","value":false,"uid":"","description":"\n\nIt contains {S170} as a non-open subspace,\nwhich is a contradiction with Lemma 3.5 in {{zb:1458.54023}}.\n","refs":[{"name":"Locally ordered topological spaces (P. Pikul)","zb":"1458.54023"}]},{"space":"S000176","property":"P000123","value":true,"uid":"","description":"\n\nSee {{zb:1052.54001}}.\n","refs":[{"name":"General Topology (Willard)","zb":"1052.54001"}]},{"space":"S000176","property":"P000204","value":false,"uid":"","description":"\n\n{S176} remains {P37} with the removal of any point.\n","refs":[]},{"space":"S000176","property":"P000238","value":true,"uid":"","description":"\n\nThe standard addition and scalar multiplication on $\\mathbb R^2$ are continuous, and together satisfy the axioms of a real vector space.\n","refs":[]},{"space":"S000177","property":"P000003","value":true,"uid":"","description":"\n\nSee Example 3.1 of {{doi:10.1016/0016-660X(72)90026-8}}.\n","refs":[{"name":"A topological view of P-spaces (A. Misra)","doi":"10.1016/0016-660X(72)90026-8"}]},{"space":"S000177","property":"P000004","value":false,"uid":"","description":"\n\nSee Example 3.1 of {{doi:10.1016/0016-660X(72)90026-8}}.\n","refs":[{"name":"A topological view of P-spaces (A. Misra)","doi":"10.1016/0016-660X(72)90026-8"}]},{"space":"S000177","property":"P000010","value":true,"uid":"","description":"\n\nSee Example 3.1 of {{doi:10.1016/0016-660X(72)90026-8}}.\n","refs":[{"name":"A topological view of P-spaces (A. Misra)","doi":"10.1016/0016-660X(72)90026-8"}]},{"space":"S000177","property":"P000051","value":true,"uid":"","description":"\n\nEvery point $a_{\\alpha\\beta}$ and $b_{\\alpha\\beta}$ is isolated in $X$. And any set not containing these points is a subset of $\\{a,b\\}\\cup\\{c_\\gamma:\\gamma<\\omega_1\\}$, which is a discrete subspace of $X$.\n","refs":[{"name":"A topological view of P-spaces (A. Misra)","doi":"10.1016/0016-660X(72)90026-8"}]},{"space":"S000177","property":"P000114","value":true,"uid":"","description":"\n\nBy construction.\n","refs":[{"name":"A topological view of P-spaces (A. Misra)","doi":"10.1016/0016-660X(72)90026-8"}]},{"space":"S000177","property":"P000147","value":true,"uid":"","description":"\n\nSee Example 3.1 of {{doi:10.1016/0016-660X(72)90026-8}}.\n","refs":[{"name":"A topological view of P-spaces (A. Misra)","doi":"10.1016/0016-660X(72)90026-8"}]},{"space":"S000178","property":"P000004","value":true,"uid":"","description":"\n\nSee Example 3.1 of {{doi:10.1016/0016-660X(72)90026-8}}.\n","refs":[{"name":"A topological view of P-spaces (A. Misra)","doi":"10.1016/0016-660X(72)90026-8"}]},{"space":"S000178","property":"P000010","value":false,"uid":"","description":"\n\nSee Example 3.1 of {{doi:10.1016/0016-660X(72)90026-8}}.\n","refs":[{"name":"A topological view of P-spaces (A. Misra)","doi":"10.1016/0016-660X(72)90026-8"}]},{"space":"S000178","property":"P000048","value":true,"uid":"","description":"\n\nIf $x, y$ are two distinct points of $X$, then either one of them is of the form $a_{\\alpha\\beta}$ so the clopen set $U = \\{a_{\\alpha\\beta}\\}$ separates them, or none of them is of the form $a_{\\alpha\\beta}$, but one is of the form $c_{\\gamma}$ so that the clopen set $U = \\{c_{\\gamma}\\}\\cup\\{a_{\\gamma\\beta}:\\beta < \\omega_1\\}$ separates them.","refs":[]},{"space":"S000178","property":"P000051","value":true,"uid":"","description":"\n\n{S177} is {P51}, which is a property preserved by subspaces.\n","refs":[{"name":"A topological view of P-spaces (A. Misra)","doi":"10.1016/0016-660X(72)90026-8"}]},{"space":"S000178","property":"P000114","value":true,"uid":"","description":"\n\nBy construction.\n","refs":[{"name":"A topological view of P-spaces (A. Misra)","doi":"10.1016/0016-660X(72)90026-8"}]},{"space":"S000178","property":"P000147","value":true,"uid":"","description":"\n\n{S177} is {P147}, which is a property preserved by subspaces.\n","refs":[{"name":"A topological view of P-spaces (A. Misra)","doi":"10.1016/0016-660X(72)90026-8"}]},{"space":"S000179","property":"P000005","value":true,"uid":"","description":"\n\nBy construction.","refs":[{"name":"A Lindelöf group with non-Lindelöf square","doi":"10.1016/j.aim.2017.11.021"}]},{"space":"S000179","property":"P000026","value":false,"uid":"","description":"\n\nSee Lemma 10 of {{doi:10.1016/j.aim.2017.11.021}}.","refs":[{"name":"A Lindelöf group with non-Lindelöf square","doi":"10.1016/j.aim.2017.11.021"}]},{"space":"S000179","property":"P000087","value":true,"uid":"","description":"\n\nBy construction.","refs":[{"name":"A Lindelöf group with non-Lindelöf square","doi":"10.1016/j.aim.2017.11.021"}]},{"space":"S000179","property":"P000114","value":true,"uid":"","description":"\n\nBy construction, the underlying set is a group generated by $\\aleph_1$ elements, so the space has cardinality $\\aleph_1$ as well.","refs":[{"name":"A Lindelöf group with non-Lindelöf square","doi":"10.1016/j.aim.2017.11.021"}]},{"space":"S000179","property":"P000131","value":true,"uid":"","description":"\n\nSee Lemma 10 of {{doi:10.1016/j.aim.2017.11.021}}.","refs":[{"name":"A Lindelöf group with non-Lindelöf square","doi":"10.1016/j.aim.2017.11.021"}]},{"space":"S000179","property":"P000149","value":false,"uid":"","description":"\n\nSee the argument beginning at the bottom of page 227 of {{doi:10.1016/j.aim.2017.11.021}}. There, they prove that the constructed group $grp(\\mathscr L)$ has the property that $grp(\\mathscr L)^2$ is not {P18}, hence it is not {P18} in all of its finite powers.","refs":[{"name":"A Lindelöf group with non-Lindelöf square","doi":"10.1016/j.aim.2017.11.021"}]},{"space":"S000180","property":"P000006","value":true,"uid":"","description":"\n\nSee construction in {{doi:10.1112/blms/8.3.239}}.\n","refs":[{"name":"A Scattered Space that is not Zero-Dimensional (R. C. Solomon)","doi":"10.1112/blms/8.3.239"}]},{"space":"S000180","property":"P000050","value":false,"uid":"","description":"\n\nSee construction in {{doi:10.1112/blms/8.3.239}}.\n","refs":[{"name":"A Scattered Space that is not Zero-Dimensional (R. C. Solomon)","doi":"10.1112/blms/8.3.239"}]},{"space":"S000180","property":"P000051","value":true,"uid":"","description":"\n\nSee construction in {{doi:10.1112/blms/8.3.239}}.\n","refs":[{"name":"A Scattered Space that is not Zero-Dimensional (R. C. Solomon)","doi":"10.1112/blms/8.3.239"}]},{"space":"S000180","property":"P000065","value":true,"uid":"","description":"\n\nThe space is constructed as a set of cardinality $\\mathfrak c$ to which points $x_{rs}$ are added for all $r\\in (0,1), s\\in F_r$ where $F_r$ is a particular set of sequences of reals, which each have cardinality at most $\\mathfrak{c}^{\\aleph_0}=\\mathfrak{c}$.\n","refs":[{"name":"A Scattered Space that is not Zero-Dimensional (R. C. Solomon)","doi":"10.1112/blms/8.3.239"}]},{"space":"S000180","property":"P000197","value":false,"uid":"","description":"\n\nBy construction, the subspace $S$ is a discrete subspace of cardinality $2^{\\aleph_0}$.\n","refs":[{"name":"A Scattered Space that is not Zero-Dimensional (R. C. Solomon)","doi":"10.1112/blms/8.3.239"}]},{"space":"S000181","property":"P000006","value":true,"uid":"","description":"\n\nAs a subspace of the countable product of {S36}; {P6} is productive and hereditary.","refs":[{"name":"A space which is 𝜎-compact but neither hemicompact nor second countable","mathse":4736734}]},{"space":"S000181","property":"P000017","value":true,"uid":"","description":"\n\nSee {{mathse:4736734}}.","refs":[{"name":"Vietoris topology on spaces dominated by second countable ones","doi":"10.1515/math-2015-0018"},{"name":"Answer to \"A space which is 𝜎-compact but neither hemicompact nor second countable\"","mathse":4736740}]},{"space":"S000181","property":"P000028","value":false,"uid":"","description":"\n\nHas a non-{P28} subspace\n{S36}.\n","refs":[{"name":"Answer to \"A space which is 𝜎-compact but neither hemicompact nor second countable\"","mathse":4736740}]},{"space":"S000181","property":"P000036","value":false,"uid":"","description":"\n\n$X\\cong (\\omega_1 + 1) \\times X$ and {S36|P36}.\n","refs":[]},{"space":"S000181","property":"P000081","value":false,"uid":"","description":"\n\n$X$ contains {S36} as a subspace and {S36|P81}.\n","refs":[]},{"space":"S000181","property":"P000103","value":false,"uid":"","description":"\n\n$X$ contains {S36} as a subspace and {S36|P103}.\n","refs":[]},{"space":"S000181","property":"P000111","value":false,"uid":"","description":"\n\nSee {{mathse:4736734}}.\n","refs":[{"name":"Answer to \"A space which is 𝜎-compact but neither hemicompact nor second countable\"","mathse":4736740},{"name":"Domination by second countable spaces and Lindelöf Σ-property","doi":"10.1016/j.topol.2010.10.014"}]},{"space":"S000181","property":"P000114","value":true,"uid":"","description":"\n\nTo choose a function $f:\\omega\\to[0,\\omega_1]$ with finite support, there are countably many ways\nto choose a finite subset $A$ of $\\omega$ as the support of $f$ and for each such $A$ there are\n$\\aleph_1$ way to define a function $A\\to(0,\\omega_1]$. So $|X|=\\aleph_0\\cdot\\aleph_1=\\aleph_1$.\n","refs":[{"name":"Answer to \"A space which is 𝜎-compact but neither hemicompact nor second countable\"","mathse":4736740}]},{"space":"S000181","property":"P000126","value":false,"uid":"","description":"\n\nThis space contains {S36} as a subspace, and {S36|P126}.\nTherefore {S181} is not {P126}.\n","refs":[]},{"space":"S000181","property":"P000167","value":false,"uid":"","description":"\n\n$X$ contains {S35} as a subspace and {S35|P167}.\n","refs":[]},{"space":"S000181","property":"P000191","value":false,"uid":"","description":"\n\n$X$ contains {S36} as a subspace and {S36|P191}.\n","refs":[]},{"space":"S000181","property":"P000197","value":false,"uid":"","description":"\n\n$X$ contains {S35} as a subspace and {S35|P197}.\n","refs":[]},{"space":"S000181","property":"P000210","value":true,"uid":"","description":"\n\n$X$ is a subspace of a countable product of {P210} spaces,\nsince {S36|P210}.\n","refs":[]},{"space":"S000181","property":"P000215","value":false,"uid":"","description":"\n\n$X$ contains {S35} as a subspace and {S35|P162}.\n","refs":[]},{"space":"S000182","property":"P000056","value":true,"uid":"","description":"\n\nSee {{mathse:3947741}}.\n","refs":[{"name":"Answer to \"Meager topological spaces of uncountable size\"","mathse":3947741}]},{"space":"S000182","property":"P000065","value":true,"uid":"","description":"\n\n$|X| = \\mathfrak{c}\\cdot \\aleph_0 = \\mathfrak{c}$","refs":[]},{"space":"S000182","property":"P000087","value":true,"uid":"","description":"\n\nThe result follows from {S3|P87}\nand {S27|P87},\nand the fact that a product of topological groups is a topological group.\n","refs":[]},{"space":"S000182","property":"P000093","value":true,"uid":"","description":"\n\nDisjoint union of countable spaces.","refs":[{"name":"Meager topological spaces of uncountable size","mathse":3198478}]},{"space":"S000182","property":"P000133","value":true,"uid":"","description":"\n\n$X$ is homeomorphic to $\\mathbb R\\times\\mathbb Q$ with order topology induced by the lexicographic order.\n","refs":[]},{"space":"S000182","property":"P000146","value":true,"uid":"","description":"\n\nIt is a consequence of the fact that {S27|P146}.\n\n","refs":[]},{"space":"S000182","property":"P000166","value":true,"uid":"","description":"\n\nIdentifying $X$ with the set $\\mathbb R\\times\\mathbb Q$ with $\\mathbb R$ discrete,\none can observe that the topology induced from {S176} is coarser.\nThe Euclidean topology on the set is separable.\n","refs":[]},{"space":"S000182","property":"P000227","value":true,"uid":"","description":"\n\nThe subspace $D \\times \\{0\\}$ is closed discrete with cardinality $\\mathfrak c$.\n","refs":[]},{"space":"S000183","property":"P000039","value":true,"uid":"","description":"\n\nAll nonempty open sets are co-countable, and thus intersect.\n","refs":[{"name":"Must US extremally disconnected spaces be sequentially discrete?","mo":452903}]},{"space":"S000183","property":"P000065","value":true,"uid":"","description":"\n\nBy construction, as $|X| = |Y \\cup Z| = \\mathfrak c + \\aleph_0 = \\mathfrak c$.\n","refs":[]},{"space":"S000183","property":"P000081","value":false,"uid":"","description":"\n\nThe closure of $Y$ is the whole space, but no countable set in $Y$ has\na limit point in $Z$ as countable sets in $Y$ are closed in $X$.\n","refs":[{"name":"Must US extremally disconnected spaces be sequentially discrete?","mo":452903}]},{"space":"S000183","property":"P000103","value":true,"uid":"","description":"\n\nConsider a countably compact subset $A\\subseteq X$. It must have finite intersection with the {S17}\nsubspace $Y$, as otherwise it would contain an infinite closed discrete set. So $A\\cap Y$ is closed in $X$.\n\nSince the {S20} subspace $Z$ is closed in $X$, the intersection $A\\cap Z$ is closed in $A$, hence countably compact. Since $Z$ is {P103}, $A\\cap Z$ is then closed in $Z$, and therefore closed in $X$.\n\nThen $A$ is closed in $X$ as the union of two closed sets.\n","refs":[{"name":"Must US extremally disconnected spaces be sequentially discrete?","mo":452903}]},{"space":"S000183","property":"P000167","value":false,"uid":"","description":"\n\n$Z$ is a copy of a nontrivial converging sequence.\n","refs":[{"name":"Must US extremally disconnected spaces be sequentially discrete?","mo":452903}]},{"space":"S000184","property":"P000001","value":false,"uid":"","description":"\n\nContains the non-{P1} subspace {S4}.\n","refs":[]},{"space":"S000184","property":"P000036","value":false,"uid":"","description":"\n\nBy construction as $X$ is the disjoint union of open\nsets $\\{0,1\\}$ and $\\{2,3\\}$.\n","refs":[]},{"space":"S000184","property":"P000078","value":true,"uid":"","description":"\n\nBy construction.\n","refs":[]},{"space":"S000184","property":"P000086","value":true,"uid":"","description":"\n\nIt is the product of two {P86} spaces; see {S1|P86} and {S4|P86}.\n","refs":[]},{"space":"S000184","property":"P000176","value":true,"uid":"","description":"\n\nBy construction.\n","refs":[]},{"space":"S000184","property":"P000185","value":true,"uid":"","description":"\n\nThe basis $\\{\\{0,1\\},\\{2,3\\}\\}$ is a partition.\n","refs":[]},{"space":"S000185","property":"P000010","value":true,"uid":"","description":"\n\nThe collection \n$\\{((\\omega\\setminus \\{n\\})\\times\\omega)\\cup\\{\\infty_x\\}:n<\\omega\\}\n\\cup\n\\{(\\omega\\times(\\omega\\setminus \\{0\\}))\\cup\\{\\infty_y\\}:n<\\omega\\}\n\\cup\n\\{\\{(n,m)\\}:n,m<\\omega\\}$ is a basis of regular open sets.\n","refs":[]},{"space":"S000185","property":"P000016","value":true,"uid":"","description":"\n\nGiven open sets containing $\\infty_x,\\infty_y$, only finitely-many\npoints remain uncovered.\n","refs":[]},{"space":"S000185","property":"P000047","value":true,"uid":"","description":"\n\nSince points of $\\omega^2$ are isolated and closed, the only connected subsets\nthat intersect $\\omega^2$ are singletons.\nThen as $\\{\\infty_x,\\infty_y\\}$ is disconnected, the result follows.\n","refs":[]},{"space":"S000185","property":"P000051","value":true,"uid":"","description":"\n\nIf a non-empty subset fails to contain an isolated point from $\\omega^2$,\nthen it is contained in the discrete subspace $\\{\\infty_x,\\infty_y\\}$.\n","refs":[]},{"space":"S000185","property":"P000057","value":true,"uid":"","description":"\n\nThe space is countable by definition.\n","refs":[]},{"space":"S000185","property":"P000061","value":true,"uid":"","description":"\n\nSingletons $\\{a\\}$ with $a\\in\\omega^2$ are clopen sets, hence cozero sets.\nSo every subset of $\\omega^2$ is a cozero set, as a countable union of cozero sets.\n\nThus, given a cozero set $A$, the set $B := X \\setminus (A \\cup \\{\\infty_1,\\infty_2\\})$ is a cozero set disjoint from $A$, and $A \\cup B$ is dense in $X$ since it contains $\\omega^2$.\n","refs":[]},{"space":"S000185","property":"P000082","value":true,"uid":"","description":"\n\nNeighborhoods of $\\infty_x$ and $\\infty_y$ missing the other\nare homeomorphic to a copy of the open metric fan\n\n$\\{(0,0)\\}\\cup\\left\\{\\left(\\frac{\\cos(1/m)}{n},\\frac{\\sin(1/m)}{n}\\right)\n:0","refs":[]},{"space":"S000188","property":"P000175","value":true,"uid":"","description":"\n\nBy definition.\n","refs":[]},{"space":"S000188","property":"P000176","value":false,"uid":"","description":"\n\nBy definition.\n","refs":[]},{"space":"S000188","property":"P000203","value":true,"uid":"","description":"\n$\\{0\\}$ and $\\{1\\}$ are open, and $\\{2\\}$ is not.\n","refs":[]},{"space":"S000189","property":"P000052","value":true,"uid":"","description":"\n\nAll subsets of this space are open by definition.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"}]},{"space":"S000189","property":"P000175","value":true,"uid":"","description":"\n\nBy definition.\n","refs":[]},{"space":"S000189","property":"P000176","value":false,"uid":"","description":"\n\nBy definition.\n","refs":[]},{"space":"S000190","property":"P000129","value":true,"uid":"","description":"\n By definition.\n","refs":[]},{"space":"S000190","property":"P000175","value":true,"uid":"","description":"\n\nBy construction\n","refs":[]},{"space":"S000190","property":"P000176","value":false,"uid":"","description":"\n\nBy construction\n","refs":[]},{"space":"S000191","property":"P000014","value":false,"uid":"","description":"\n\nThe {S87} is a subspace which is not {P13}.\n","refs":[]},{"space":"S000191","property":"P000022","value":false,"uid":"","description":"\n\nThe set $\\{\\langle n,0\\rangle:n\\in\\omega\\}$ is clopen in $X$ and infinite discrete, hence not {P22}.\n","refs":[]},{"space":"S000191","property":"P000023","value":false,"uid":"","description":"\n\nFor example, the point $p=\\langle\\omega_1,0\\rangle$ has no compact neighborhood. Every neighborhood of $p$ contains a set of the form $A=\\{\\langle \\alpha+n,0\\rangle : n\\in\\omega\\}$ for some $\\alpha<\\omega_1$. This set is closed in $X$ but not compact (since infinite discrete).\n","refs":[]},{"space":"S000191","property":"P000029","value":false,"uid":"","description":"\n\nThere are uncountably many isolated points.\n","refs":[]},{"space":"S000191","property":"P000049","value":false,"uid":"","description":"\n\nThe open sets $\\{\\langle 0,2n\\rangle: n\\in\\omega\\}$ and $\\{\\langle 0,2n+1\\rangle: n\\in\\omega\\}$ are disjoint, but their closures are not.\n","refs":[]},{"space":"S000191","property":"P000050","value":true,"uid":"","description":"\n\nThe space is the product of two {P50} spaces $Y$ and $Z$. For $Y=[0,\\omega_1]$ every neighborhood of $\\omega_1$ is clopen in $Y$, and for every $y\\in Y\\setminus\\{\\omega_1\\}$ the singleton $\\{y\\}$ is clopen in $Y$. The space $Z=[0,\\omega]$ is {S20}, which is {P50}.\n","refs":[]},{"space":"S000191","property":"P000051","value":true,"uid":"","description":"\n\nThe space $X$ has a finer topology than {S78}, which is {P51}.\n","refs":[]},{"space":"S000191","property":"P000081","value":false,"uid":"","description":"\n\nThe space $X$ contains as a subspace a copy of a \"Fortissimo\" space $W=[0,\\omega_1]$ (with $\\omega_1$ having cocountable neighborhoods and all other points isolated). The point $\\omega_1\\in W$ is in the closure of $[0,\\omega_1)$, but every countable subset of $[0,\\omega_1)$ is closed in $W$. So $W$ is not {P81}, and neither is $X$ since {P81} is a hereditary property.\n","refs":[]},{"space":"S000191","property":"P000114","value":true,"uid":"","description":"\n\nBy construction.\n","refs":[]},{"space":"S000191","property":"P000151","value":true,"uid":"","description":"\n\nAs the space $Y=[0,\\omega_1]$ is some type of Fortissimo space, it is {P151} as shown in {{mathse:4727833}}.\n\nThe space $X=Y\\times [0,\\omega]$ is a countable union of copies of $Y$ (the horizontal slices), and a countable union of {P151} subspaces is {P151} (by a suitable parallel meshing of the strategies for the various subspaces).\n","refs":[{"name":"What kinds of selection principles hold for Fortissimo space?","mathse":4727833}]},{"space":"S000191","property":"P000167","value":false,"uid":"","description":"\n\nThe sequence $(0, n)$ converges to $(0, \\omega)$.\n","refs":[]},{"space":"S000191","property":"P000172","value":false,"uid":"","description":"\n\nSee {{mathse:4821179}}.\n","refs":[{"name":"Radial and pseudoradial properties for Dieudonné plank and variant","mathse":4821179}]},{"space":"S000191","property":"P000173","value":true,"uid":"","description":"\n\nSee {{mathse:4821179}}.\n","refs":[{"name":"Radial and pseudoradial properties for Dieudonné plank and variant","mathse":4821179}]},{"space":"S000191","property":"P000210","value":true,"uid":"","description":"\n\n{S155|P210} and {S20|P210},\nhence so is their product.\n","refs":[]},{"space":"S000192","property":"P000016","value":true,"uid":"","description":"\n\nGiven any open cover, take two elements of the cover containing\n$\\infty_E,\\infty_D$; these cover all but a closed and bounded, and thus compact,\nsubset of $[0,\\infty)$.\n","refs":[]},{"space":"S000192","property":"P000027","value":true,"uid":"","description":"\n\nWitnessed by the basis of open sets consisting of\n$(p,q)$, $\\{\\infty_E\\}\\cup(p,\\infty)\\setminus E$, and\n$\\{\\infty_D\\}\\cup(p,\\infty)\\setminus D$ for $p,q\\in\\mathbb Q$.\n","refs":[]},{"space":"S000192","property":"P000036","value":true,"uid":"","description":"\n\nFollows as $[0,\\infty)$ is a connected dense subset.\n","refs":[{"name":"Connectedness of a certain space with closed retracts but non-unique sequential limits","mathse":4828298}]},{"space":"S000192","property":"P000037","value":false,"uid":"","description":"\n\nSee {{mathse:4828298}}.\n","refs":[{"name":"Connectedness of a certain space with closed retracts but non-unique sequential limits","mathse":4828298}]},{"space":"S000192","property":"P000041","value":false,"uid":"","description":"\n\nSee {{mathse:4828298}}.\n","refs":[{"name":"Connectedness of a certain space with closed retracts but non-unique sequential limits","mathse":4828298}]},{"space":"S000192","property":"P000044","value":false,"uid":"","description":"\n\nSee {{mathse:4828298}}.\n","refs":[{"name":"Connectedness of a certain space with closed retracts but non-unique sequential limits","mathse":4828298}]},{"space":"S000192","property":"P000046","value":false,"uid":"","description":"\n\n$X$ contains {S158} as a subspace and {S158|P46}.\n","refs":[]},{"space":"S000192","property":"P000060","value":false,"uid":"","description":"\n\nSee {{mathse:4828298}}.\n","refs":[{"name":"Connectedness of a certain space with closed retracts but non-unique sequential limits","mathse":4828298}]},{"space":"S000192","property":"P000061","value":true,"uid":"","description":"\n\nAny continuous function $f$ from $X$ to a Hausdorff space $Y$ takes the same value\nat $\\infty_E$ and $\\infty_D$, and thus factors through the quotient space $X/\\{\\infty_E,\\infty_D\\}$\nthat identifies the two infinity points.\nOne can check that the quotient space is homeomorphic to the interval $[0,+\\infty]$ of the\nextended real line, and hence homeomorphic to {S158}.\nSince this space is {P6}, it is the Tychonoff reflection of $X$,\nwith the quotient map $\\varphi$ as reflection map.\n\nThe map $\\varphi$ is open at each point $x$ of the dense subset $[0,\\infty)\\subseteq X$\n(i.e., sends neighborhoods of $x$ to neighborhoods of $f(x)$).\nSince {S158|P61},\nthe result then follows from the Remark after Proposition 2 in {{mathse:5108293}}.\n","refs":[{"name":"Cozero complemented property for a space and its Tychonoff reflection","mathse":5108293}]},{"space":"S000192","property":"P000064","value":true,"uid":"","description":"\n\nFollows from Theorem 1.15 of {{mr:0431104}} as its dense subspace $[0,\\infty)$ is {P64}.\nSee also {{wikipedia:Baire_space}}.\n","refs":[{"name":"Baire spaces (R. C. Haworth; R. A McCoy)","mr":431104},{"name":"Baire space","wikipedia":"Baire_space"}]},{"space":"S000192","property":"P000065","value":true,"uid":"","description":"\n\nBy construction, $|X| = |[0,\\infty)\\cup\\{\\infty_E,\\infty_D\\}| = \\mathfrak{c}$.\n","refs":[]},{"space":"S000192","property":"P000068","value":false,"uid":"","description":"\n\nIt is easy to check being {P68} passes to coarser topologies. Since the topology on the space is finer than\n{S65} and {S65|P68}, the result follows.\n","refs":[]},{"space":"S000192","property":"P000099","value":false,"uid":"","description":"\n\nThe sequence $a_n=n+0.5$ converges to both $\\infty_E$ and $\\infty_D$.\n","refs":[{"name":"Answer to 'A \"simple\" space with closed retracts but non-unique sequential limits'","mo":460346}]},{"space":"S000192","property":"P000101","value":true,"uid":"","description":"\n\nShown in {{mo:460346}}.\n","refs":[{"name":"Answer to 'A \"simple\" space with closed retracts but non-unique sequential limits'","mo":460346}]},{"space":"S000192","property":"P000120","value":true,"uid":"","description":"\n\nThe subspaces $[0,\\infty)$, $([0,\\infty)\\setminus D)\\cup\\{\\infty_D\\}$ and $([0,\\infty)\\setminus E)\\cup\\{\\infty_E\\}$ are open and {P133}\nwith the natural order on $\\mathbb R$ ($\\infty_\\ast$ is a maximal element).\n","refs":[]},{"space":"S000192","property":"P000169","value":false,"uid":"","description":"\n\nEvery regular open neighborhood of $\\infty_E$ contains $\\infty_D$, since $\\infty_D$ is in the interior\nof the closure of every neighborhood of $\\infty_E$.\n","refs":[]},{"space":"S000193","property":"P000129","value":true,"uid":"","description":"\n\nBy definition.\n","refs":[]},{"space":"S000193","property":"P000181","value":true,"uid":"","description":"\n\nBy construction.\n","refs":[]},{"space":"S000194","property":"P000065","value":true,"uid":"","description":"\n\nBy construction.\n","refs":[]},{"space":"S000194","property":"P000129","value":true,"uid":"","description":"\n\nBy definition.\n","refs":[]},{"space":"S000195","property":"P000002","value":true,"uid":"","description":"\n\nBy construction finite sets are closed.\n","refs":[{"name":"Answer to \"Is there a first countable, $T_1$, weakly Lindelof, sequentially compact space which is not also compact?\"","mathse":4874321}]},{"space":"S000195","property":"P000016","value":false,"uid":"","description":"\n\n$\\{\\alpha:\\alpha<\\omega_1\\}$ is an open cover with no finite subcover.\n","refs":[{"name":"Answer to \"Is there a first countable, $T_1$, weakly Lindelof, sequentially compact space which is not also compact?\"","mathse":4874321}]},{"space":"S000195","property":"P000020","value":true,"uid":"","description":"\n\nSince $\\omega_1$ is well-ordered, every sequence has a non-decreasing subsequence.\nAs open sets are initial segments with only finitely-many elements removed, this subsequence\nconverges to its supremum.\n","refs":[{"name":"Answer to \"Is there a first countable, $T_1$, weakly Lindelof, sequentially compact space which is not also compact?\"","mathse":4874321}]},{"space":"S000195","property":"P000026","value":true,"uid":"","description":"\n\n$\\omega$ is a countable dense subset.\n","refs":[{"name":"Answer to \"Is there a first countable, $T_1$, weakly Lindelof, sequentially compact space which is not also compact?\"","mathse":4874321}]},{"space":"S000195","property":"P000028","value":true,"uid":"","description":"\n\n$\\{(\\alpha+1)\\setminus F:F\\in[\\alpha]^{<\\aleph_0}\\}$ is a countable base at\neach $\\alpha<\\omega_1$.\n","refs":[{"name":"Answer to \"Is there a first countable, $T_1$, weakly Lindelof, sequentially compact space which is not also compact?\"","mathse":4874321}]},{"space":"S000195","property":"P000051","value":true,"uid":"","description":"\nSince $\\omega_1$ is well-ordered, a nonempty set $Y\\subseteq\\omega_1$ has a least element $\\alpha$.\nThen $\\alpha$ is isolated in $Y$ since $[0,\\alpha+1)\\cap Y=\\{\\alpha\\}$.\n","refs":[]},{"space":"S000195","property":"P000099","value":false,"uid":"","description":"\n\nThe sequence $a_n=n$ converges to every $\\alpha\\in\\omega_1\\setminus\\omega$.\n","refs":[]},{"space":"S000195","property":"P000114","value":true,"uid":"","description":"\n\nBy construction.\n","refs":[]},{"space":"S000196","property":"P000008","value":true,"uid":"","description":"\n\nShown at {{mathse:5017774}}.\n","refs":[{"name":"Why is the \"long circle\" completely normal ($T_5$)?","mathse":5017774}]},{"space":"S000196","property":"P000016","value":true,"uid":"","description":"\n\nAsserted in {{mathse:1499512}}.\nFollows from {S39|P16}.\n","refs":[{"name":"Answer to \"Path-connected and locally connected space that is not locally path-connected\"","mathse":1499512}]},{"space":"S000196","property":"P000037","value":true,"uid":"","description":"\n\nAsserted in {{mathse:1499512}}.\nFollows from {S38|P37}.\n","refs":[{"name":"Answer to \"Path-connected and locally connected space that is not locally path-connected\"","mathse":1499512}]},{"space":"S000196","property":"P000041","value":true,"uid":"","description":"\n\nAsserted in {{mathse:1499512}}.\n","refs":[{"name":"Answer to \"Path-connected and locally connected space that is not locally path-connected\"","mathse":1499512}]},{"space":"S000196","property":"P000042","value":false,"uid":"","description":"\n\nAsserted in {{mathse:1499512}}.\nFollows from {S39|P42}.\n","refs":[{"name":"Answer to \"Path-connected and locally connected space that is not locally path-connected\"","mathse":1499512}]},{"space":"S000196","property":"P000061","value":false,"uid":"","description":"\n\nLet $L=(\\omega_1\\times[0,1))\\cup\\{(\\omega_1,0)\\}$ be the {S39}\nwith minimum element $m=(0,0)$ and maximum element $M=(\\omega_1,0)$.\nThe long circle is the quotient space $X=L/\\{m,M\\}$ obtained by identifying $m$ and $M$.\nLet $q:L\\to X$ denote the quotient map, and let $p=q(m)=q(M)$ be the identified point.\n\nFix an element $z$ with $m0$ for $m\\in U$,\ndefine $\\mu(\\left<\\alpha,\\beta\\right>,U)=\\{\\left<\\alpha,\\beta\\right>\\}$ if $\\beta<\\omega_1$;\nand define $\\mu(\\left<\\alpha,\\omega_1\\right>,U)$\nto be any \"rectangle\" $(\\gamma,\\alpha]\\times(\\delta,\\omega_1]$ contained in $U$.\nIt is easy to check that the function $\\mu$ satisfies Definition 2 of {P109}.\n","refs":[{"name":"Strong collectionwise normality and M. E. Rudin’s Dowker space (Hart)","doi":"10.1090/S0002-9939-1981-0630058-4"}]},{"space":"S000207","property":"P000114","value":true,"uid":"","description":"\n\nSince $|\\omega_1|=|\\omega_1 + 1| = \\aleph_1$.\n","refs":[]},{"space":"S000207","property":"P000167","value":false,"uid":"","description":"\n\nThe subspace $\\omega_1\\times\\{\\omega_1\\}\\subseteq X$ is homeomorphic to {S35}\nand {S35|P167}.\n","refs":[]},{"space":"S000207","property":"P000207","value":false,"uid":"","description":"\n\nShown by Cohen in {{zb:0046.16403}}. See {{mathse:5004796}} for details in English.\n","refs":[{"name":"Sur une problème de M. Dieudonné (Cohen)","zb":"0046.16403"},{"name":"Answer to \"Is every $T_4$ topological space divisible?\"","mathse":5004796}]},{"space":"S000207","property":"P000217","value":true,"uid":"","description":"\n\nSee {{mathse:5121137}}.\n","refs":[{"name":"Cohen's modification of $\\omega_1\\times(\\omega_1+1)$ is strongly zero-dimensional","mathse":5121137}]},{"space":"S000208","property":"P000003","value":true,"uid":"","description":"\n\n$X$ is a subspace of $\\prod_n (\\omega_{n+1}+1)$ with box topology, which is {P3}.\n","refs":[]},{"space":"S000208","property":"P000008","value":false,"uid":"","description":"\n\n$X$ contains {S138} and {S138|P8}.\n","refs":[]},{"space":"S000208","property":"P000114","value":false,"uid":"","description":"\n\n$X$ contains {S138}, and {S138|P114} and {S138|P57}.\n","refs":[]},{"space":"S000208","property":"P000146","value":true,"uid":"","description":"\n\nSee section IV.4 of {{zb:0224.54019}}.\n\nSee also [K.P. Hart's notes](https://fa.ewi.tudelft.nl/~hart/37/onderwijs/old-courses/settop/rudin.pdf).\n","refs":[{"name":"A normal space X for which X×I is not normal (M.E. Rudin)","zb":"0224.54019"}]},{"space":"S000208","property":"P000147","value":true,"uid":"","description":"\n\nThe proof is the same as for {S138}. See Lemma 4 in {{zb:0224.54019}}.\n","refs":[{"name":"A normal space X for which X×I is not normal (M.E. Rudin)","zb":"0224.54019"}]},{"space":"S000208","property":"P000164","value":true,"uid":"","description":"\n\n$|X| \\leq \\aleph_\\omega^\\omega$ and $\\aleph_\\omega^\\omega$ is smaller than every measurable cardinal.\n","refs":[]},{"space":"S000209","property":"P000016","value":true,"uid":"","description":"\n\n$X$ is homeomorphic to the Alexandroff one-point compactification of {S83}.\n","refs":[{"name":"Alexandroff extension on Wikipedia","wikipedia":"Alexandroff_extension"}]},{"space":"S000209","property":"P000038","value":true,"uid":"","description":"\n\nThe map $[0, 2\\pi] \\to X$ defined by $t \\mapsto \\langle t, 1 \\rangle$ if $t < 2\\pi$ and $2\\pi \\mapsto \\langle 2\\pi, 2\\rangle$ is injective and continuous. It is clear by restricting this map to sub-intervals and reparameterizing the results that every pair of points is connected by an injective path.\n","refs":[]},{"space":"S000209","property":"P000060","value":false,"uid":"","description":"\n\nMap the origin $0_1$ to $0_2$, leaving the other points fixed, then compose with a projection of the circle to the $x$-axis. The result is a continuous non-constant map to the reals.\n","refs":[]},{"space":"S000209","property":"P000061","value":true,"uid":"","description":"\n\nThe Tychonoff reflection of $X$ is {S170},\nwith the reflection map $\\varphi:X\\to\\mathbb S^1$ sending the two \"origins\" $0_1,0_2$ to $0$ and every other point to itself.\n\nSince {S170|P61}\nand the reflection map $\\varphi$ is open,\nthe result follows from Proposition 2 in {{mathse:5108293}}.\n","refs":[{"name":"Cozero complemented property for a space and its Tychonoff reflection","mathse":5108293}]},{"space":"S000209","property":"P000068","value":false,"uid":"","description":"\n\nSimilar to {S83|P68}.\n","refs":[]},{"space":"S000209","property":"P000089","value":false,"uid":"","description":"\n\nMap the origin $0_1$ to $0_2$, leaving the other points fixed, then compose with a small rotation of the circle. The result is a continuous map without fixed point.\n","refs":[]},{"space":"S000209","property":"P000095","value":false,"uid":"","description":"\n\nUsing similar arguments as for {S83|P38},\none can show that any injective path $f:[0,1]\\to X$ from one origin to the other\nmust start at one origin, go in one direction around the circle without ever backtracking,\nand end up at the other origin.\nSo $f$ is necessarily surjective.\nBut its image, namely $X$, is not homeomorphic to $[0,1]$,\nfor example because {S209|P3}.\n","refs":[]},{"space":"S000209","property":"P000101","value":false,"uid":"","description":"\n\nSame argument as {S83|P101}.\n","refs":[]},{"space":"S000209","property":"P000110","value":true,"uid":"","description":"\n\nWrite a point in $S^1$ as its angle $\\theta$ and let $(\\theta, \\phi)$ be a smallest arc from $\\theta$ to $\\phi$. Let the two origins of $X$ be written as $\\theta = 0_1$ and $\\theta = 0_2$.\n\nFor each integer $n \\ge 1$, let $\\mathscr U_n$ be the open cover of $X$ consisting of all open arcs of length $\\frac{1}{n}$ which do not contain either of the origins, plus open arcs of the form $((-\\frac{1}{n}, \\frac{1}{n}) \\setminus \\{0\\}) \\cup \\{0_i\\}$ where $i = 1$ or $2$. These open covers form a development for $X$.\n","refs":[]},{"space":"S000209","property":"P000155","value":true,"uid":"","description":"\n\nEach point is contained in an open set homeomorphic to $S^1$, namely $X\\setminus\\{0_1\\}$ or $X\\setminus\\{0_2\\}$, and {S170|P155}.\n","refs":[]},{"space":"S000209","property":"P000169","value":false,"uid":"","description":"\n\nSame argument as {S83|P169}.\n","refs":[]},{"space":"S000209","property":"P000200","value":false,"uid":"","description":"\n\nThe map sending the origins to $0_1$ and fixing all other points is a retraction onto {S170}. Since {S170|P200}, it follows by Proposition 1.17 of {{zb:1044.55001}} that $X$ is not simply connected.\n","refs":[{"name":"Algebraic Topology (Hatcher)","zb":"1044.55001"}]},{"space":"S000209","property":"P000204","value":false,"uid":"","description":"\n\nFor any $p \\in X$, $X \\backslash \\{p\\}$ is either homeomorphic to {S83} or {S170}.\n","refs":[]},{"space":"S000209","property":"P000216","value":true,"uid":"","description":"\n\nIt is enough to show that each open set in $X$ is {P30}.\nThe whole space is {P30} since {S209|P16}.\nAnd a proper open subset of $X$ is a disjoint union of open sets,\neach homeomorphic to either {S170}, {S25} or {S83}.\nObserve each of {S170|P30}, {S25|P30} and {S83|P30}.\nSince the {P30} property is preserved by taking disjoint unions, the result follows.\n","refs":[]},{"space":"S000210","property":"P000089","value":false,"uid":"","description":"\n \n The map $x\\mapsto (x+1)/2$ is continuous from $X$ to itself and without fixed point.\n ","refs":[]},{"space":"S000210","property":"P000095","value":true,"uid":"","description":"\n\nEach interval $[a,b]$ with $a\\ne\\min X$,\none can find an increasing sequence of points $x^{(n)}\\in X$ converging to $a$\n(i.e., cofinal in $(\\leftarrow,a)$) as follows.\n\nIf the values $a_0,a_1,\\dots$ are non-zero infinitely many times,\ndefine\n$$x^{(n)}=\\left\\in(\\leftarrow,a).$$\nThe sequence $(x^{(n)})_n$ is non-decreasing and by removing repeated entries\none obtains an increasing subsequence converging to $a$.\n\nOtherwise, $a=\\left$ with $a_m$ the last non-zero value.\nTake an increasing sequence $b_1\\in(\\leftarrow,a).$$\nThe sequence $(x^{(n)})_n$ is increasing and converges to $a$.\n\nSimilarly, given a point $a\\ne\\max X$,\none can find a decreasing sequence of points $y^{(n)}\\in X$ converging to $a$.\n\nSo for $a\\notin\\{\\min X,\\max X\\}$, the countably many intervals\n$(x^{(n)},y^{(n)})$ form a local base at $a$. And for $a\\in\\{\\min X,\\max X\\}$\none can use the intervals $[a,y^{(n)})$ or $(x^{(n)},a]$.\n","refs":[]},{"space":"S000212","property":"P000036","value":true,"uid":"","description":"\n\nSee {{mathse:4453440}}.\n","refs":[{"name":"Answer to \"Is there a nontrivial LOTS that is connected and totally path disconnected?\"","mathse":4453440}]},{"space":"S000212","property":"P000046","value":true,"uid":"","description":"\n\nSee {{mathse:4453440}}.\n","refs":[{"name":"Answer to \"Is there a nontrivial LOTS that is connected and totally path disconnected?\"","mathse":4453440}]},{"space":"S000212","property":"P000061","value":false,"uid":"","description":"\n\n(similar to {S41|P61})\n\nSince {S212|P28},\nevery open interval $(z,w)\\subseteq X$ is an $F_\\sigma$ set.\nFor $a\\in[0,1]$, let $I_a$ denote the open interval\nfrom $\\left$ to $\\left$, which is therefore an $F_\\sigma$ set.\nThe set $U=\\bigcup\\{I_a:a\\in[0,1]\\cap\\mathbb Q\\}$ is an open $F_\\sigma$ set,\nhence a cozero set since {S212|P13}\n(see Corollary 1.5.13 in {{zb:0684.54001}}).\n\nSuppose by contradiction $V$ is a cozero set disjoint from $U$ with $U\\cup V$ dense in $X$.\nNecessarily, $V\\cap I_a\\ne\\varnothing$ for $a\\in[0,1]\\setminus\\mathbb Q$.\nAnd $V\\cap\\overline I_a=\\varnothing$ for $a\\in[0,1]\\cap\\mathbb Q$.\nSo if $\\pi:X\\to[0,1]$ is the projection onto the $0$-th coordinate,\n$\\pi(V)=[0,1]\\setminus\\mathbb Q$.\nSince every cozero set is an $F_\\sigma$ set, $V=\\bigcup_n F_n$ for some closed sets $F_n$,\nand $\\pi(V)=\\bigcup_n\\pi(F_n)$.\nSo there is at least one $F_n$ with $\\pi(F_n)$ infinite; this $\\pi(F_n)$ contains an accumulation point $r \\in [0, 1]$.\nThen either $u=\\left\\in F_n\\subseteq V$ or $v=\\left\\in F_n\\subseteq V$.\nBut any neighborhood of a point like $u$ or $v$ contains $I_a$ for some $a\\in\\mathbb Q$, so $V\\cap U\\ne\\varnothing$.\nThis contradiction shows that $X$ is not {P61}.\n","refs":[{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"space":"S000212","property":"P000076","value":false,"uid":"","description":"\n\n$X$ contains the closed subspace\n$\\bigl( (0, 1] \\times \\{0\\}^\\omega \\bigr) \\cup \\bigl( [0, 1) \\times \\{1\\}^\\omega \\bigr)$,\nwhich is homeomorphic to {S93}\n(by the map sending $\\left$ to $\\left$).\n\n{S93|P76}, so neither is $X$\nsince the {P76} property is hereditary with respect to closed sets.\n","refs":[]},{"space":"S000212","property":"P000133","value":true,"uid":"","description":"\n\nBy definition.\n","refs":[]},{"space":"S000213","property":"P000001","value":true,"uid":"","description":"\n\nThe topology is the Alexandrov topology for a poset (instead of a general preorder).\n","refs":[]},{"space":"S000213","property":"P000010","value":false,"uid":"","description":"\n\nThe smallest open neighborhood of $0$ is $V=\\{0, 1, 3\\}$,\nwhich is dense in $X$ and thus not a regular open set.\n","refs":[]},{"space":"S000213","property":"P000013","value":false,"uid":"","description":"\n\nThe closed sets $\\{0\\}$ and $\\{2\\}$ cannot be separated by disjoint open sets.\n","refs":[]},{"space":"S000213","property":"P000036","value":true,"uid":"","description":"\n\nBy inspection, there are no nontrivial clopen sets.\n","refs":[]},{"space":"S000213","property":"P000078","value":true,"uid":"","description":"\n\nBy construction.\n","refs":[]},{"space":"S000213","property":"P000089","value":false,"uid":"","description":"\n\n$f(x) = (x + 2) \\bmod 4$ defines a continuous map $X\\to X$ without fixed points.\n","refs":[]},{"space":"S000213","property":"P000176","value":true,"uid":"","description":"\n\nBy construction.\n","refs":[]},{"space":"S000213","property":"P000200","value":false,"uid":"","description":"\n\n{S51} $K$ is a covering space of $X$ with covering map $p \\colon K \\to X$, $x \\mapsto x \\bmod 4$.\n\nThis covering map has non-singleton fibers and {S51|P37}, hence $X$ is not {P200} (see for example Theorem 54.4 in {{zb:0951.54001}}).\n","refs":[{"name":"Topology (Munkres)","zb":"0951.54001"}]},{"space":"S000213","property":"P000204","value":false,"uid":"","description":"\n\n$X \\setminus \\{p\\}$ for $p\\in X$ is homeomorphic to\neither {S7}\nor {S11}.\nAnd both of them are {P36}.\n","refs":[]},{"space":"S000214","property":"P000058","value":false,"uid":"","description":"\n\nThe cardinality equals $\\aleph_0^{\\aleph_1}=2^{\\aleph_1}\\geq 2^{\\aleph_0}$.\n","refs":[]},{"space":"S000214","property":"P000059","value":true,"uid":"","description":"\n\nThe cardinality equals $\\aleph_0^{\\aleph_1}=2^{\\aleph_1}\\leq 2^\\mathfrak{c}$.\n","refs":[]},{"space":"S000214","property":"P000087","value":true,"uid":"","description":"\n\nSee {{mo:500191}}.\n","refs":[{"name":"Answer to: \"Find a topological group which is $T_5$ but not $T_6$\"","mo":500191}]},{"space":"S000214","property":"P000114","value":false,"uid":"","description":"\n\nThe cardinality equals $\\aleph_0^{\\aleph_1}=2^{\\aleph_1}$.\n","refs":[]},{"space":"S000214","property":"P000133","value":true,"uid":"","description":"\n\nBy definition.\n","refs":[]},{"space":"S000214","property":"P000174","value":true,"uid":"","description":"\n\nThe family $\\{ (-e_\\lambda,\\,e_\\lambda) : \\lambda<\\omega_1 \\}$, where\n$$e_\\lambda(\\alpha)=\\begin{cases}0&\\text{ for }\\alpha<\\lambda\\\\\n1&\\text{ for }\\alpha\\geq \\lambda\n\\end{cases}\\quad (\\alpha<\\omega_1)$$\nis a well-ordered base of neighbourhoods for the constant $0$ element\n(we do not need to consider other points since {S214|P86}).\nIndeed, if $f>0$ and $\\alpha<\\omega_1$ is the first position\n$f(\\alpha)>0$, then $f> e_{\\alpha+1}>0$. The same can be done for\nthe lower bound of an interval around $0$ (with $-e_{\\alpha+1}$).\n","refs":[]},{"space":"S000214","property":"P000191","value":false,"uid":"","description":"\n\nSee {{mo:500191}}.\n","refs":[{"name":"Answer to: \"Find a topological group which is $T_5$ but not $T_6$\"","mo":500191}]},{"space":"S000214","property":"P000206","value":true,"uid":"","description":"\n\nObserve that the family $\\{E_\\lambda(x):x\\in X,\\ \\lambda<\\omega_1\\}$, where\n$$E_\\lambda(x):=(x-e_\\lambda,\\,x+e_\\lambda),\n\\quad e_\\lambda(\\alpha)=\\begin{cases}0&\\text{ for }\\alpha<\\lambda\\\\\n1&\\text{ for }\\alpha\\in[\\lambda,\\omega_1)\n\\end{cases}$$\nis a base for the order topology on $X$\n(see {S214|P174}).\n\nThe second player can pick sets\n$V_n=E_{\\alpha_n}(x_n)$ with $\\alpha_n$ greater than the first indices\non which $x_n$ differs from $x_{n-1}+e_{\\alpha_{n-1}}$ and $x_{n-1}-e_{\\alpha_{n-1}}$ (then $\\overline{V_n}\\subset V_{n-1}$).\nNecessarily $\\alpha_n>\\alpha_{n-1}$.\nDefine $f\\in X$ as follows:\n$$f(\\alpha)=\\begin{cases}x_n(\\alpha)&\\text{ if }\\alpha\\in[\\alpha_{n-1},\\alpha_n)\\\\\n0&\\text{ for }\\alpha\\in[\\gamma,\\omega_1)\n\\end{cases}$$\nwith $\\alpha_{-1}:=0$ and $\\gamma:=\\sup\\{\\alpha_n:n\\in\\omega\\}$.\n\nNote that since $x_n\\in E_{\\alpha_{n-1}}(x_{n-1})$ we have $f(\\alpha)=x_n(\\alpha)$ for all $\\alpha<\\alpha_n$. It can be readily verified, that\n$fx_n-e_{\\alpha_n}$, for each $n<\\omega$,\nhence $f\\in\\bigcap_{n<\\omega} V_n$.\n","refs":[]},{"space":"S000214","property":"P000216","value":true,"uid":"","description":"\n\nSince $\\mathbb Z\\cong$ {S2|P30},\nthe lexicographic product $\\mathbb Z^{\\omega_1}$ is {P30}\nby Theorem 4.2.2 in {{zb:0282.54017}} (accessible [here](https://ir.cwi.nl/pub/12753)).\nBy Theorem 2.4.7 therein and the fact that {S214|P86}, it suffices to prove that $X\\setminus\\{\\mathbf 0\\}$\nis {P30}, where $\\mathbf 0\\in X$ denotes the element with all coordinates $0$.\n\nObserve that $X\\setminus\\{\\mathbf0\\}$ is the disjoint union of the open order-convex sets\n$U_{\\lambda,n}=\\{0\\}^{\\lambda}\\times\\{n\\}\\times\\mathbb Z^{(\\lambda,\\omega_1)}$ for $\\lambda<\\omega_1$ and $n\\in\\mathbb Z\\setminus\\{0\\}$.\nEach of these subspaces is order-isomorphic and homeomorphic to $X$, hence {P30}.\nConsequently, so is $X\\setminus\\{\\mathbf0\\}$.\n","refs":[{"name":"Metrizability in generalized ordered spaces (M.J.Faber, 1974)","zb":"0282.54017"}]},{"space":"S000214","property":"P000227","value":true,"uid":"","description":"\n\nThe subset $\\{f\\in X: f(\\lambda)=0\\text{ for }\\lambda\\in[\\omega,\\omega_1)\\}$\nis closed discrete and of cardinality continuum.\n","refs":[]},{"space":"S000216","property":"P000006","value":true,"uid":"","description":"\n\n$X$ is contained in {S108} and {S108|P6}.\n","refs":[]},{"space":"S000216","property":"P000026","value":true,"uid":"","description":"\n\n$\\mathbb{N}\\subseteq X$ is countable and dense.\n","refs":[]},{"space":"S000216","property":"P000049","value":true,"uid":"","description":"\n\n$X$ contains {S2} and is therefore a dense subspace of {S108} and {S108|P49}.\n","refs":[]},{"space":"S000216","property":"P000065","value":true,"uid":"","description":"\n\n$X$ is in bijection with $\\mathbb{R}$.\n","refs":[]},{"space":"S000216","property":"P000112","value":true,"uid":"","description":"\n\nExtend $\\varphi$ to $X$ so that if $E = E_r = \\{\\varphi^{-1}(s_n) : n\\in\\omega\\}$ and $s_n\\to r$ then $\\varphi(p_E) = r$. If $\\varphi(p_E)\\in U$ where $U\\subseteq \\mathbb{R}$ is open, find $N$ such that $s_n\\in U$ for $n\\geq N$. Since {S216|P49}, $V =\\overline{E}\\setminus\\varphi^{-1}(\\{s_1, s_2, ..., s_N\\})$ is an open neighbourhood of $p_E$ and $\\varphi(V)\\subseteq U$. So $\\varphi:X\\to\\mathbb{R}$ is a continuous injection, hence $X$ is submetrizable.\n","refs":[]},{"space":"S000216","property":"P000227","value":true,"uid":"","description":"\n\n$D$ is a closed discrete subspace of $X$ of size $\\mathfrak{c}$.\n","refs":[]},{"space":"S000216","property":"P000243","value":true,"uid":"","description":"\n\n$X$ is a dense subspace of {S108} and {S108|P243}.\n","refs":[]},{"space":"S000217","property":"P000001","value":true,"uid":"","description":"\n\nGiven $\\alpha<\\beta$ in $X$, the open set $[0,\\alpha]$ contains $\\alpha$ and not $\\beta$.\n","refs":[]},{"space":"S000217","property":"P000093","value":true,"uid":"","description":"\n\nEach basic open set of the form $[0,\\alpha]$ for $\\alpha\\in\\omega_1$ is countable.\n","refs":[]},{"space":"S000217","property":"P000114","value":true,"uid":"","description":"\n\nBy construction.\n","refs":[]},{"space":"S000217","property":"P000196","value":true,"uid":"","description":"\n\nBy construction.\n","refs":[]},{"space":"S000217","property":"P000219","value":true,"uid":"","description":"\n\nLet $Y\\subseteq X$ with $|Y|=|X|$.\nUsing well-orderedness and the fact that $\\omega_1$ is an initial ordinal,\none can show that there is a unique order isomorphism $f:Y\\to X$.\nSince $X$ is Alexandrov, so is $Y$ with its subspace topology $\\tau_Y$.\nThe smallest $\\tau_Y$-neighborhood of a point $y\\in Y$ is\n$$(\\leftarrow,y]_X\\cap Y = (\\leftarrow,y]_Y,$$\nwhich coincides with the smallest neighborhood of $y$ in the left ray topology of $Y$.\nThis shows that $f$ is a homeomorphism.\n","refs":[]},{"space":"S000217","property":"P000226","value":true,"uid":"","description":"\n\nSince the ordinal numbers are well-ordered,\nevery nonempty collection of open sets has a minimal element.\n","refs":[]},{"space":"S000218","property":"P000006","value":true,"uid":"","description":"\n\n{S35|P6} and {S36|P6},\nhence so is their product.\n","refs":[]},{"space":"S000218","property":"P000013","value":false,"uid":"","description":"\n\nThe closed sets $H=\\omega_1\\times\\{\\omega_1\\}$\nand $K=\\{\\langle\\alpha,\\alpha\\rangle:\\alpha<\\omega_1\\}$\nare disjoint and cannot be separated by disjoint open sets.\n\nSee {{mathse:1080057}}.\n","refs":[{"name":"$[0,\\omega_1]\\times[0,\\omega_1)$ is not $T_4$","mathse":1080057}]},{"space":"S000218","property":"P000020","value":true,"uid":"","description":"\n\n{S35|P20} and {S36|P20},\nhence so is their product.\n","refs":[]},{"space":"S000218","property":"P000023","value":true,"uid":"","description":"\n\n{S35|P23} and {S36|P23},\nhence so is their product.\n","refs":[]},{"space":"S000218","property":"P000051","value":true,"uid":"","description":"\n\n{S35|P51} and {S36|P51},\nhence so is their product.\n","refs":[]},{"space":"S000218","property":"P000062","value":false,"uid":"","description":"\n\nThe clopen subset $\\omega_1\\times\\{0\\}\\subseteq X$ is homeomorphic to {S35}\nand {S35|P62}.\n","refs":[]},{"space":"S000218","property":"P000081","value":false,"uid":"","description":"\n\nThe subspace $\\{0\\}\\times(\\omega_1+1)\\subseteq X$ is homeomorphic to {S36}\nand {S36|P81}.\n","refs":[]},{"space":"S000218","property":"P000103","value":false,"uid":"","description":"\n\n$X$ contains {S36} as a subspace and {S36|P103}.\n","refs":[]},{"space":"S000218","property":"P000114","value":true,"uid":"","description":"\n\nSince $\\aleph_1\\cdot\\aleph_1=\\aleph_1$.\n","refs":[]},{"space":"S000218","property":"P000172","value":false,"uid":"","description":"\n\nThe subspace $[0,\\omega]\\times[0,\\omega_1]\\subseteq X$ is homeomorphic to {S78}\nand {S78|P172}.\n","refs":[]},{"space":"S000218","property":"P000210","value":true,"uid":"","description":"\n\n{S35|P210} and {S36|P210},\nhence so is their product.\n","refs":[]},{"space":"S000219","property":"P000052","value":true,"uid":"","description":"\n\nBy definition.\n","refs":[]},{"space":"S000219","property":"P000059","value":true,"uid":"","description":"\n\nBy definition $|X| = 2 ^ \\mathfrak{c}$","refs":[]},{"space":"S000219","property":"P000163","value":false,"uid":"","description":"\n\nBy definition $|X| = 2 ^ \\mathfrak{c} > \\mathfrak{c}$.\n","refs":[]},{"space":"S000220","property":"P000001","value":true,"uid":"","description":"\n\nGiven $\\alpha<\\beta$ in $X$, $[\\beta,\\rightarrow)$ is a neighborhood of $\\beta$ missing $\\alpha$.\n","refs":[]},{"space":"S000220","property":"P000026","value":false,"uid":"","description":"\n\n{S221} has a coarser topology than $X$\nand {S221|P26}.\n","refs":[]},{"space":"S000220","property":"P000038","value":false,"uid":"","description":"\n\n{S221} has a coarser topology than $X$, but {S221|P38}.\n","refs":[]},{"space":"S000220","property":"P000089","value":false,"uid":"","description":"\n\nThe function $f:X\\to X,\\alpha\\mapsto\\alpha+1$ is continuous and has no fixed point.\n","refs":[]},{"space":"S000220","property":"P000090","value":true,"uid":"","description":"\n\nA point $\\alpha\\in X$ has $[\\alpha,\\rightarrow)$ as smallest neighborhood.\n","refs":[]},{"space":"S000220","property":"P000114","value":true,"uid":"","description":"\n\nBy definition.\n","refs":[]},{"space":"S000220","property":"P000196","value":true,"uid":"","description":"\n\nThe collection of open sets is totally ordered by inclusion.\n","refs":[]},{"space":"S000220","property":"P000208","value":true,"uid":"","description":"\n\nEvery nonempty set $A\\subseteq X$ has a minimum element $\\alpha$ by well-orderedness.\nAs a subspace, $A$ is {P202} with $\\alpha$ as a focal point;\nhence $A$ is {P16}.\n","refs":[]},{"space":"S000220","property":"P000219","value":true,"uid":"","description":"\n\nSee the first paragraph of section 6 in {{zb:1286.54032}},\nwhere the topology on $X$ is called the \"upper topology\" on $\\omega_1$.\n","refs":[{"name":"The Toronto Problem (W. R. Brian)","zb":"1286.54032"}]},{"space":"S000221","property":"P000026","value":false,"uid":"","description":"\n\nAny countable set $D\\subseteq X$ is bounded in $\\omega_1$\nand hence disjoint from some open set $(\\alpha,\\rightarrow)$.\nTherefore $D$ is not dense in $X$.\n","refs":[]},{"space":"S000221","property":"P000028","value":true,"uid":"","description":"\n\nEach point $\\alpha\\ne 0$ has the countable collection $\\{(\\beta,\\rightarrow) : \\beta<\\alpha\\}$\nas a local base. The point $0$ has $X$ as unique neighborhood.\n","refs":[]},{"space":"S000221","property":"P000038","value":false,"uid":"","description":"\n\nLet $f:[0,1]\\to X$ be a path in $X$.\nTake a countable dense subset $E$ of $[0,1]$.\nIts image $f(E)$ is countable and thus bounded in $X$, say by some $u\\in\\omega_1$.\nSo $f(E)$ is contained in the closed set $[0,u]$.\nAnd by density of $E$ in $[0,1]$, $f([0,1])$ is also contained in $[0,u]$.\nSince $[0,u]$ is countable, $f$ cannot be injective.\n","refs":[]},{"space":"S000221","property":"P000089","value":true,"uid":"","description":"\n\nThe specialization preorder on $X$ corresponds to the natural order of ordinals\n($\\alpha\\in\\overline{\\{\\beta\\}}\\iff\\alpha\\le\\beta$).\nSince continuous functions preserve the specialization preorder,\nany continuous $f:X\\to X$ is monotone non-decreasing\n($\\alpha\\le\\beta\\implies f(\\alpha)\\le f(\\beta)$).\n\nNow let $\\alpha_0=0$ and $\\alpha_n=f(\\alpha_{n-1})$ for $n=1,2,\\dots$.\nThis defines a non-decreasing sequence $\\alpha_0\\le \\alpha_1\\le \\cdots$ in $\\omega_1$.\nLet $\\alpha_\\omega:=\\sup_n \\alpha_n\\in X$.\nFor each $n$ we have $\\alpha_n\\le\\alpha_\\omega$, and hence $\\alpha_{n+1}=f(\\alpha_n)\\le f(\\alpha_\\omega)$.\nConsequently, $\\alpha_\\omega\\le f(\\alpha_\\omega)$.\nOn the other hand, $\\alpha_\\omega$ belongs to the closure of $\\{\\alpha_n:n\\in\\omega\\}$.\nBy continuity, $f(\\alpha_\\omega)$ belongs to the closure of $\\{\\alpha_{n+1}:n\\in\\omega\\}$.\nSo $f(\\alpha_\\omega)$ cannot be strictly larger than $\\alpha_\\omega$.\nThis proves that $f(\\alpha_\\omega)=\\alpha_\\omega$.\n","refs":[]},{"space":"S000221","property":"P000090","value":false,"uid":"","description":"\n\nThe intersection of all the neighborhoods of the point $\\omega \\in X$ is $[\\omega,\\rightarrow)$, which is not open.\n","refs":[]},{"space":"S000221","property":"P000114","value":true,"uid":"","description":"\n\nBy definition.\n","refs":[]},{"space":"S000221","property":"P000196","value":true,"uid":"","description":"\n\n$X$ has a coarser topology than {S220}\nand {S220|P196}.\n","refs":[]},{"space":"S000221","property":"P000208","value":true,"uid":"","description":"\n\n$X$ has a coarser topology than {S220}\nand {S220|P208}.\n","refs":[]},{"space":"S000221","property":"P000219","value":false,"uid":"","description":"\n\nLet $Y$ denote the subspace of successor cardinals in $X$, that is,\n$Y=\\{\\beta+1:\\beta\\in\\omega_1\\}$. Note that $|Y|=|X|$.\n\nBecause $\\omega_1$ is an initial ordinal, $(Y,\\le)$ is order-isomorphic to $(\\omega_1,\\le)$.\nAnd since an element $\\alpha=\\beta+1\\in Y$ has $[\\alpha,\\to)=(\\beta,\\to)$ as smallest neighborhood in $X$,\nit has $[\\alpha,\\to)\\cap Y$ as smallest neighborhood in $Y$.\nThis shows that the subspace topology on $Y$ is homeomorphic to {S220}.\n\nBut {S220} is not homeomorphic to {S221}, for example because only the first one is {P90}.\n","refs":[]},{"space":"S000222","property":"P000013","value":false,"uid":"","description":"\n\nSee item #6 for space #103 in {{zb:0386.54001}}.\nSee also {{mathse:4171891}}.\n","refs":[{"name":"Counterexamples in Topology","zb":"0386.54001"},{"name":"Proof that uncountable products of $\\mathbb Z$ are not normal","mathse":4171891}]},{"space":"S000222","property":"P000029","value":true,"uid":"","description":"\n\nSee Corollary 2.3.18 of {{zb:0684.54001}}\nand the discussion at {{mo:459262}}.\n","refs":[{"name":"Countable chain condition in topology","mo":459262},{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"space":"S000222","property":"P000050","value":true,"uid":"","description":"\n\nIt follows from the fact that\n{S2|P50}.\n\n","refs":[]},{"space":"S000222","property":"P000059","value":false,"uid":"","description":"\n\nThe space has cardinality $\\omega^{2^\\mathfrak{c}} \\geq 2^{2^\\mathfrak{c}} > 2^\\mathfrak{c}$.\n","refs":[]},{"space":"S000222","property":"P000081","value":false,"uid":"","description":"\n\nThe space contains a subspace homeomorphic to the Cantor cube $\\{0,1\\}^{\\aleph_1}$\nwhich, according to Theorem 6.2.16 in {{zb:0684.54001}}, contains\na copy of {S36}, since {S36|P50},\nis {P2} and has weight $\\aleph_1$.\nAnd {S36|P81}.\n\n\n","refs":[{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"space":"S000222","property":"P000087","value":true,"uid":"","description":"\n\nThe space is a product of spaces satisfying the property:\n{S2|P87}.\n","refs":[]},{"space":"S000222","property":"P000103","value":false,"uid":"","description":"\n\nThe space contains a subspace homeomorphic to the Cantor cube $\\{0,1\\}^{\\aleph_1}$\nwhich, according to Theorem 6.2.16 in {{zb:0684.54001}}, contains\na copy of {S36}, since {S36|P50},\nis {P2} and has weight $\\aleph_1$.\nAnd {S36|P103}.\n\n\n","refs":[{"name":"General Topology (Engelking, 1989)","zb":"0684.54001"}]},{"space":"S000222","property":"P000140","value":false,"uid":"","description":"\n\nSee {{mathse:4644785}}.\n","refs":[{"name":"The product of uncountably many copies of the real line is not a k-space","mathse":4644785}]},{"space":"S000222","property":"P000162","value":true,"uid":"","description":"\n\nThe product $\\mathbb Z^{2^\\mathfrak c}\\cong X$ is a closed subset\nof $\\mathbb R^{2^\\mathfrak c}$.\n","refs":[]},{"space":"S000222","property":"P000164","value":true,"uid":"","description":"\n\n$|X| = \\omega^{2^\\mathfrak{c}}$ is smaller than every measurable cardinal.\n","refs":[]},{"space":"S000222","property":"P000172","value":false,"uid":"","description":"\n\nThe proof is the same as for\n{S103|P172}.\n\n\n","refs":[]},{"space":"S000222","property":"P000191","value":false,"uid":"","description":"\n\n{S101}\ncan be embedded as a subspace of {S222}\nand {S101|P191}.\n","refs":[]},{"space":"S000222","property":"P000197","value":false,"uid":"","description":"\n\nThe set of characteristic functions of the singletons,\n$A=\\{ f\\in X: f(2^\\mathfrak{c})=\\{0,1\\},\\ |f^{-1}(1)|=1\\}$ is uncountable and discrete.\n","refs":[]},{"space":"S000222","property":"P000206","value":true,"uid":"","description":"\n\nThe second player can always pick sets of the form\n$V_n=\\{f\\in X: f(i)=x_n(i) \\text{ for }i\\in F_n\\}$,\nwhere $x_n\\in X$ and $F_n\\subset 2^\\mathfrak{c}$ is finite,\nfor such sets form a basis for the product topology.\n\nSince $V_{n+1}\\subseteq V_n$, we have $F_{n+1}\\supseteq F_n$.\nOn the countable set $F=\\bigcup_{n=0}^\\infty F_n$ define\n$n(i):=\\min\\{n<\\omega: i\\in F_n\\}$.\nThen any point $f\\in X$ satisfying $f(i)=x_{n(i)}(i)$,\nfor all $i\\in F$, belongs to $\\bigcap_{n<\\omega} V_n$.\n","refs":[]},{"space":"S000222","property":"P000209","value":true,"uid":"","description":"\n\nFollows from {S2|P209}\nand {S2|P3},\nand the Hewitt-Marczewski-Pondiczery theorem (See {{zb:0060.39602}} or \n[PlanetMath]())\nwhich shows that {P209} is preserved by\npowers up to $2^{\\mathfrak c}$ for {P3} spaces.\n","refs":[{"name":"A remark on density characters (Hewitt, 1946)","zb":"0060.39602"}]},{"space":"S000222","property":"P000214","value":false,"uid":"","description":"\nSee {{mathse:5106079}}.\n","refs":[{"name":"Do the Arkhangel'skii $\\alpha_i$ properties hold for uncountable products?","mathse":5106079}]},{"space":"S000222","property":"P000227","value":true,"uid":"","description":"\n\n$X$ contains a closed subspace homeomorphic to {S101}, and {S101|P227}.\n","refs":[]},{"space":"S000223","property":"P000013","value":false,"uid":"","description":"\n\nSee the remark after Proposition 4 of {{zb:0632.04005}}.\n$X$ is shown to be non-normal by applying Jones' lemma ({T836}) to a certain closed subspace $F\\subseteq X$.\n","refs":[{"name":"On completely Ramsey sets (S. Plewik)","zb":"0632.04005"}]},{"space":"S000223","property":"P000028","value":true,"uid":"","description":"\n\n$\\{[x \\cap n, x]: n < \\omega\\}$ is a countable local base around $x$, for any $x \\in [\\omega]^\\omega$: if $s, A$ are so that $x \\in [s, A]$, that is $s \\subseteq x \\subseteq A \\cup s$ and $\\max(s) < \\min(x \\setminus s)$, then $s = x \\cap (\\max(s) + 1)$ and $[s, x] \\subseteq [s, A]$. \n","refs":[]},{"space":"S000223","property":"P000029","value":false,"uid":"","description":"\n\nFor $x: \\omega \\to 2$, let $A_x = \\left\\{\\sum_{n = 0}^m 2^n x(n): m < \\omega\\right\\}$. Then, when $x \\neq y$, $A_x \\cap A_y$ is finite, and so $\\{[\\emptyset, A_x]: x \\in 2^\\omega\\}$ is an uncountable family of pairwise disjoint nonempty open sets.\n","refs":[]},{"space":"S000223","property":"P000050","value":true,"uid":"","description":"\n\nEach basic open set $[s, A]$ with $s$ finite and $A$ infinite is clopen.\nIndeed, suppose $x\\notin[s,A]$.\nIf $x\\cap[0,\\max(s)]\\ne s$, let $t=x\\cap[0,\\max(s)]$.\nOtherwise, necessarily $s\\subseteq x$ and $\\max(s)<\\min(x\\setminus s)$, and therefore $x\\not\\subseteq(s\\cup A)$, in which case let $t=x\\cap[0,\\min(x\\setminus(s\\cup A)]$.\nIn both cases, $x \\in [t, x]$ and $[t, x] \\cap [s, A] = \\emptyset$.\n","refs":[]},{"space":"S000223","property":"P000064","value":true,"uid":"","description":"\n\nEvery meager set in $X$ is nowhere dense, which is a strong form of being Baire.\nSee Corollary 8 in {{zb:0292.02054}} () or Lemma 26.27(ii) in {{zb:1007.03002}}.\n","refs":[{"name":"A new proof that analytic sets are Ramsey (E. Ellentuck)","zb":"0292.02054"},{"name":"Set theory (T. Jech, 2003)","zb":"1007.03002"}]},{"space":"S000223","property":"P000065","value":true,"uid":"","description":"\n\nBy construction.\n","refs":[]},{"space":"S000223","property":"P000093","value":false,"uid":"","description":"\n\nEvery non-empty open set has cardinality $\\mathfrak{c}$ by construction.\n","refs":[]},{"space":"S000223","property":"P000139","value":false,"uid":"","description":"\n\nEvery nonempty open set has cardinality $\\mathfrak{c}$.\n","refs":[]},{"space":"S000223","property":"P000166","value":true,"uid":"","description":"\n\nWhen one only considers open sets $[s, A]$ where $A = \\omega$, i.e. $\\{x \\in [\\omega]^\\omega: s \\subseteq x \\land \\max(s) < \\min(x \\setminus s)\\}$, then, the resulting topology is strictly coarser than the Ellentuck topology and homeomorphic to {S28}. Namely, conceiving {S28} as the set of infinite sequences of natural numbers, the map sending $\\{a_n: n < \\omega\\}$, where $a_n < a_m$ for $n < m$, to $f: \\omega \\to \\omega$ given by $f(0) = a_0$ and $f(n+1) = a_{n+1} - (a_n + 1)$, is a homeomorphism.\n","refs":[]},{"space":"S000225","property":"P000016","value":false,"uid":"","description":"\n\n$X$ is not a bounded subset of $\\mathbb R^2$.\n","refs":[]},{"space":"S000225","property":"P000089","value":false,"uid":"","description":"\n\nThe map $(x,y)\\mapsto (x+1,y)$ has no fixed point.\n","refs":[]},{"space":"S000225","property":"P000120","value":false,"uid":"","description":"\n\n$X$ contains an open subspace homeomorphic to {S176}\nand {S176|P120}.\n","refs":[]},{"space":"S000225","property":"P000122","value":false,"uid":"","description":"\n\nThe point $(0,0)\\in X$ does not have a neighborhood homeomorphic to some $\\mathbb R^n$.\n","refs":[]},{"space":"S000225","property":"P000184","value":true,"uid":"","description":"\n\nBy definition.\n","refs":[]},{"space":"S000225","property":"P000199","value":true,"uid":"","description":"\n\nSince $X$ is a convex subset of $\\mathbb R^2$,\nit can be deformation retracted to a point using a straight-line homotopy.\n","refs":[]},{"space":"S000225","property":"P000204","value":false,"uid":"","description":"\n\n$X$ is the product of two connected spaces, each with at least two points.\nSuch a product minus a point is always connected\n(see {{mathse:214317}}).\n","refs":[{"name":"$(X\\times Y)\\setminus (A\\times B)$ is connected for $A,B$ proper subsets of connected $X,Y$","mathse":214317}]},{"space":"S000225","property":"P000236","value":true,"uid":"","description":"\n\n$X$ satisfies {P236} with $n=2$.\n","refs":[]},{"space":"S000226","property":"P000002","value":false,"uid":"","description":"\n\n{S10} embeds into $X$ and {S10|P2}.\n","refs":[]},{"space":"S000226","property":"P000018","value":true,"uid":"","description":"\n\nLet $\\mathscr U$ be an open cover of $X$.\nThere is a countable $\\mathscr V\\subseteq\\mathscr U$ with $Y\\subseteq\\bigcup\\mathscr V$\nbecause {S155|P18}.\nSince every neighborhood of $p$ contains $Z$, $\\mathscr V$ is also a cover of $X$.\n","refs":[]},{"space":"S000226","property":"P000021","value":false,"uid":"","description":"\n\n$Y$ is a closed subspace of $X$\nand {S155|P21}.\n","refs":[]},{"space":"S000226","property":"P000114","value":true,"uid":"","description":"\n\nBy construction.\n","refs":[]},{"space":"S000226","property":"P000136","value":true,"uid":"","description":"\n\nThe subspace $Y$ is closed in $X$\nand anticompact ({S155|P136}).\nSo every compact subset of $X$ intersects $Y$ in a finite set, and is itself finite.\n","refs":[]},{"space":"S000226","property":"P000147","value":true,"uid":"","description":"\n\nThe neighborhoods of $p$ in $X$ are the sets $V\\cup Z$ with $V$ neighborhood of $p$ in $Y$.\nSince {S155|P147},\n$p$ is also a P-point in $X$.\nAnd every other (isolated) point is trivially a P-point.\n","refs":[]},{"space":"S000226","property":"P000174","value":true,"uid":"","description":"\n\nThe neighborhoods of $p$ in $X$ are the sets $V\\cup Z$ with $V$ neighborhood of $p$ in $Y$.\nSince {S155|P174},\nthere is a neighborhood base $\\mathscr N$ for $p$ in $Y$ that is totally ordered by inclusion.\nThen $\\{V\\cup Z:V\\in\\mathscr N\\}$ is a totally ordered neighborhood base for $p$ in $X$.\nAnd $X$ is trivially well-based at each of the other (isolated) points.\n","refs":[]},{"space":"S000226","property":"P000203","value":true,"uid":"","description":"\n\nAll points except $p$ are isolated.\n","refs":[]},{"space":"S000226","property":"P000229","value":true,"uid":"","description":"\n\nThe path components of $X$ are $Z$ and the singletons $\\{x\\}$ with $x\\in Y\\setminus\\{p\\}$\n(because {S10|P37} and\nevery $\\{x\\}$ with $x\\in Y\\setminus\\{p\\}$ is clopen in $X$).\n\nAnd since {S10|P229},\nevery path component of $X$ is {P229}.\n","refs":[]},{"space":"S000226","property":"P000234","value":false,"uid":"","description":"\n\nEvery neighborhood $V$ of $p$ contains a point $x\\in Y\\setminus\\{p\\}$.\nHence $V$ is not connected since $\\{x\\}$ is clopen in $X$.\n","refs":[]}],"version":{"ref":"main","sha":"b0894a88d775fb8a5ec315857ac884c11545d432"}}

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