Theorems

Id	If	Then	Description
T1	Compact	Countably compact	Asserted on page 21 of DOI 10.1007/978-1-4612-6290-9.
T2	Countably compact	Weakly countably compact	Let $A \subset X$ be countably infinite. If $A \subset X$ has no limit point then it must be closed. Moreover, for every…
T3	Sequentially compact	Countably compact	If $X$ is not countably compact then there is some countable open cover $\{U_n\}_{n \in \omega}$ of $X$ with no finite …
T4	Countably compact	Pseudocompact	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 countabl…
T5	Exhaustible by compacts	Hemicompact	See the proof of the theorem in Math StackExchange 4569500.
T6	Compact	Locally relatively compact	If a space is compact, the entire space is a compact open neighborhood of any of its points.
T7	Locally relatively compact	Weakly locally compact	Trivially, the closure of an open neighborhood of a point with compact closure is a compact neighbor…
T8	Exhaustible by compacts	Weakly locally compact	By definition.
T9	Compact	Exhaustible by compacts	Since compact spaces are $\sigma$-compact and locally compact.
T10	Extremally disconnected ∧ Locally Hausdorff	Sequentially discrete	Shown in the answer by Ulli to Math StackExchange 4751804. Stated for Hausdorff spaces, but the pro…
T11	Has a countable $k$-network	Has a countable network	Evident from the definitions, as a $k$-network is a network.
T12	Fully normal	Paracompact	Proved in Math StackExchange 4862626.
T13	Compact	Strongly paracompact	This holds as any finite cover is star-finite.
T14	Paracompact	Metacompact	This holds as any locally finite refinement is point finite.
T15	Paracompact	Countably paracompact	This holds as any countable open cover is an open cover and thus has a locally finite refinement.
T16	Metacompact	Countably metacompact	This holds as any countable open cover is an open cover and thus has a point finite refinement.
T17	Countably compact	Countably paracompact	This holds as any finite cover is locally finite.
T18	Countably paracompact	Countably metacompact	This holds as any locally finite refinement is point finite.
T19	$T_1$ ∧ Weakly countably compact	Countably compact	Proven on pages 19-20 of DOI 10.1007/978-1-4612-6290-9.
T20	Pseudometrizable	Has a $\sigma$-locally finite base	A portion of Theorem 40.3 (Nagata-Smirnov metrization theorem) of zbMATH 0951.54001. See Math StackE…
T21	Separable	Countable chain condition	If $\{x_n\}$ is a countable dense subset and $U_\alpha$ a pairwise disjoint collection of open sets, then choos…
T22	Anticompact ∧ Countable	Has a countable $k$-network	In an Anticompact space the collection of all singletons is a $k$-network, which is countable if the s…
T23	Regular ∧ Has a $\sigma$-locally finite base	Pseudometrizable	Theorem 40.3 (Nagata-Smirnov metrization theorem) of zbMATH 0951.54001, generalized to not assume $T…
T24	Meta-Lindelöf ∧ Separable	Lindelöf	See Math StackExchange 1095781 or Theorem 1 in https://dantopology.wordpress.com/2022/10/29/examples…
T25	Topological $n$-manifold	Locally $n$-Euclidean	By definition: see page 316 of zbMATH 0951.54001.
T26	$T_2$ ∧ Exhaustible by compacts	$T_4$	Asserted in Figure 7 of DOI 10.1007/978-1-4612-6290-9.
T27	Weakly locally compact ∧ $R_1$	Completely regular	By Theorem 19.3 in zbMATH 1052.54001 locally compact Hausdorff spaces are completely regular. The g…
T28	$T_2$ ∧ Countably compact ∧ First countable	$T_3$	Asserted in Figure 7 of DOI 10.1007/978-1-4612-6290-9.
T29	Has a countable network	Has a $\sigma$-locally finite network	Evident from the definitions, as a countable family of sets in $X$ is $\sigma$-locally finite.
T30	Regular ∧ Lindelöf	Strongly paracompact	Corollary 5.3.11 of zbMATH 0684.54001 (which uses "Lindelöf" to mean Lindelöf and $T_3$) shows that …
T31	Locally $n$-Euclidean ∧ $T_2$ ∧ Second countable	Topological $n$-manifold	By definition: see page 316 of zbMATH 0951.54001.
T32	$T_{2 \frac{1}{2}}$	$T_2$	Asserted on page 14 of DOI 10.1007/978-1-4612-6290-9 (where Comp. Haus means $T_{2.5}$).
T33	$T_3$	$T_{2 \frac{1}{2}}$	If $X$ is $T_3$, it is $T_2$. So given distinct points $a,b\in X$, there are disjoint open $O_a$ and $O_b$ contain…
T34	Has a $\sigma$-locally finite $k$-network	Has a $\sigma$-locally finite network	Evident from the definitions, as a $k$-network is a network.
T35	Completely regular	Regular	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 $f^{-1}((\frac{2}{3},1])$ a…
T36	Completely normal	Normal	Asserted in Figure 1 of DOI 10.1007/978-1-4612-6290-9 (where T_4 means normal and T_5 means complete…
T37	Normal ∧ $R_0$	Completely regular	See Math StackExchange 3990826.
T38	Ultraconnected	Path connected	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 $f:[0,1] \rightarrow X$
T39	Injectively path connected	Path connected	Evident from the definitions.
T40	Path connected	Connected	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 sam…
T41	Has a dispersion point	¬Empty	Immediate from the definitions.
T42	Discrete	$T_1$	Asserted on Figure 9 of DOI 10.1007/978-1-4612-6290-9.
T43	$T_1$ ∧ Scattered	Totally disconnected	If $X$ is $T_1$, any nontrivial connected subset is dense-in-itself. Thus if $X$ is scattered and $T_1$, it …
T44	Partition topology	Extremally disconnected	In a Partition topology space, every open set is closed.
T45	Extremally disconnected ∧ $T_2$	Totally separated	Given distinct points $x,y \in X$, take disjoint neighborhoods $U_x$ and $U_y$ of $x$ and $y$ repectively, by $T_2…
T46	Totally separated	Totally disconnected	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 $C = (C \cap U) \cup (C \cap V)$
T47	Totally disconnected	Totally path disconnected	Holds as path components are contained in components.
T48	Totally separated	Functionally Hausdorff	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 $f:X \rightarrow [0,1]$
T49	Totally path disconnected	$T_1$	See Math StackExchange 4580119.
T50	$T_2$ ∧ Separable	Cardinality $\leq 2^{\mathfrak c}$	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 $A=D \cap U_x$ f…
T51	Hyperconnected	Locally connected	Any open subset of a hyperconnected space is connected.
T52	Totally disconnected ∧ Has multiple points	¬Connected	Asserted on page 32 of DOI 10.1007/978-1-4612-6290-9; note that the singleton is ruled out.
T53	Lindelöf ∧ Weakly locally compact	Exhaustible by compacts	See Math StackExchange 4568032.
T54	Zero dimensional	Completely regular	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 funct…
T55	Cosmic	Has a countable network	Follows from the definition.
T56	Pseudometrizable	Fully normal	Asserted on page 34 of DOI 10.1007/978-1-4612-6290-9 for metrizable spaces. The result extends to p…
T57	Locally pseudometrizable	First countable	Take a pseudometrizable neighborhood. The balls of rational radius within this neighborhood form a l…
T58	Weakly locally compact	$k_1$-space	See Math StackExchange 919892.
T59	Sequential	$k_2$-space	As shown in Math StackExchange 2026072, Sequential spaces are $k_1$-space. The stronger conclusion …
T60	$k_1$-space ∧ $k_1$-Hausdorff	$k_3$-space	Immediate from the definitions, since $k_1$-Hausdorff means all Compact subspaces are $T_2$.
T61	Has a countable network ∧ $T_3$	Cosmic	Follows from the definition.
T62	Has a $\sigma$-locally finite base ∧ Weakly Lindelöf	Second countable	Every locally finite collection of nonempty open sets in a Weakly Lindelöf space is countable. (Proo…
T63	Locally injectively path connected	Locally path connected	Evident from the definitions.
T64	Locally path connected	Locally connected	Since any path connected set is connected.
T65	Weakly locally compact ∧ Metrizable	Completely metrizable	Proven as Theorem 2.3.30 of DOI 10.1007/b98956.
T66	LOTS	GO-space	By definition.
T67	Countable	Cardinality $\lt\mathfrak c$	Since $|\omega| < \mathfrak{c} = 2^{\omega}$.
T68	Cardinality $\lt\mathfrak c$	Cardinality $\leq\mathfrak c$	Immediate from the definitions.
T69	CGWH	$k_3$-space	See Lemma 1.4(c) in N. Strickland: The category of CGWH spaces. See also Math StackExchange 4303611…
T70	$k_3$-space	$T_1$	Consider a singleton $\{p\}\subseteq X$. Its intersection with every compact Hausdorff (hence $T_1$) subspace $K$ of…
T71	Hyperconnected	Strongly connected	As defined on page 223 of DOI 10.1007/978-1-4612-6290-9 all continuous functions from a hyperconnect…
T72	Ultraconnected	Collectionwise normal	In an Ultraconnected space, since any two nonempty closed sets intersect, any discrete family of clo…
T73	$T_0$ ∧ Zero dimensional	Totally separated	Asserted in Figure 9 (page 32) of DOI 10.1007/978-1-4612-6290-9.
T74	Countable	$\sigma$-compact	A countable set is a countable union of finite subsets and any finite set is compact.
T75	Injectively path connected ∧ Has multiple points	¬Cardinality $\lt\mathfrak c$	There is an injective path $f:[0,1]\to X$ joining two distinct points. Therefore $|X| \geq |[0,1]| = \mathfrak{c}$.
T76	Strongly connected	Pseudocompact	As defined on page 223 of DOI 10.1007/978-1-4612-6290-9 all continuous functions from a strongly con…
T77	Completely metrizable	Metrizable	Defined as such on page 37 of DOI 10.1007/978-1-4612-6290-9.
T78	Biconnected	Connected	Defined as such on page 33 of DOI 10.1007/978-1-4612-6290-9.
T79	Strongly connected	Connected	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 $f(U) = \{0\}$
T80	Functionally Hausdorff ∧ Has multiple points	¬Strongly connected	Choose distinct $x,y \in X$, and by Complete Hausdorff there is a continuous $f:X \rightarrow \mathbb{R}$ with $f(x)=0 \neq 1=f(y)$.
T81	Weakly locally compact ∧ KC	Locally relatively compact	Evident from the definitions.
T82	Locally Euclidean	Locally arc connected	Every point $x\in X$ has an open neighborhood $U$ homeomorphic to some $\mathbb R^n$. The open balls centered at $x$ wit…
T83	Locally injectively path connected ∧ ¬Discrete	¬Cardinality $\lt\mathfrak c$	By non-discreteness choose a non-isolated point of $X$. It has a Injectively path connected neighborho…
T84	$\sigma$-space	$T_3$	Follows from the definition.
T85	Discrete	Completely metrizable	The topology on a discrete space may be generated by the discrete metric $d(x,y)=1$ for all $x \neq y$. Then…
T86	Functionally Hausdorff	$T_{2 \frac{1}{2}}$	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 $f^{-1}([0,\frac{1}{3})$
T87	Ultraconnected	Strongly connected	Let $f:X \rightarrow \mathbb{R}$ be continuous. Since for all $a,b\in f(X)$, $f^{-1}(a)$ and $f^{-1}(b)$ are closed subsets of $X$ and thus …
T88	Path connected ∧ Has multiple points	¬Totally path disconnected	Follows from the definition on page 31 of DOI 10.1007/978-1-4612-6290-9 as the path connecting two d…
T89	Locally path connected ∧ ¬Discrete	¬Totally path disconnected	If a point of $X$ has a non-trivial path connected neighborhood, then $X$ is not totally path disconnect…
T90	Second countable	Has a $\sigma$-locally finite base	By the definition of a $\sigma$-locally finite base, see e.g. 23.9 of zbMATH 1052.54001. If $\{B_n:n<\omega\}$ is a …
T91	Hyperconnected ∧ Normal	Ultraconnected	If $X$ is not Ultraconnected, there are two disjoint nonempty closed sets $C$ and $D$. If $X$ is additionall…
T92	Has a dispersion point	Biconnected	The space $X$ is Connected by definition of Has a dispersion point. If $X=A\cup B$ with $A$ and $B$ disjoint an…
T93	Has a countable network	Hereditarily separable	Having a countable network is a hereditary property; and a space with a countable network is Separab…
T94	Injectively path connected ∧ Has multiple points	¬Biconnected	There is an injective path $f:[0,1]\to X$ joining two distinct points. Then, $f([0,1/3])$ and $f([2/3,1])$ ar…
T95	Connected ∧ Locally path connected	Path connected	If $X$ is locally path connected, then since the concatenation of two paths is again a path, path comp…
T96	Hyperconnected	Extremally disconnected	Any nonempty open set $U\subseteq X$ is dense in X, so its closure is the whole space, which is open.
T97	Extremally disconnected ∧ Connected	Hyperconnected	A nonempty open set $U\subseteq X$ has a nonempty clopen closure by Extremally disconnected. Since the space is…
T98	$T_4$	$T_1$	By definition as in 15.1 of zbMATH 1052.54001.
T99	$T_1$ ∧ Normal	$T_4$	By definition as in 15.1 of zbMATH 1052.54001.
T100	$T_5$	$T_1$	By definition as on page 12 of DOI 10.1007/978-1-4612-6290-9.
T101	$T_1$ ∧ Completely normal	$T_5$	By definition as on page 12 of DOI 10.1007/978-1-4612-6290-9.
T102	First countable	Well-based	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…
T103	Well-based	Radial	Mentioned in the Introduction section of DOI 10.1016/j.topol.2014.08.002.
T104	Fully $T_4$	$T_1$	By definition as on page 23 of DOI 10.1007/978-1-4612-6290-9.
T105	$T_1$ ∧ Fully normal	Fully $T_4$	By definition as on page 23 of DOI 10.1007/978-1-4612-6290-9.
T106	Lindelöf ∧ Countably compact	Compact	If $X$ is Lindelof, any open cover has a countable subcover, and if $X$ is countably compact, this subco…
T107	Countably compact ∧ Meta-Lindelöf	Compact	Theorem 1 and the following Remark of DOI 10.3792/pja/1195526497 show that in any topological space …
T108	Totally disconnected ∧ Locally connected	Discrete	If every point has a connected neighborhood and the only connected sets are single points, then ever…
T109	Ultraconnected ∧ $R_0$	Indiscrete	If $X$ is $R_0$ and $x,y\in X$ are topologically distinct, then $\operatorname{cl}\{x\}$ and $\operatorname{cl}\{y\}$ are disjoint closed subsets…
T110	Normal ∧ Pseudocompact	Weakly countably compact	Suppose $X$ is a normal space which is not weakly countably compact. There is a countably infinite cl…
T111	Biconnected ∧ Cardinality $\geq 4$	$T_0$	See Math StackExchange 4814029.
T112	$T_5$	$T_4$	If $A, B \subset X$ are disjoint closed sets, $cl(A) \cap B = A \cap cl(B) = \emptyset$.
T113	$T_4$	$T_{3 \frac{1}{2}}$	In a $T_1$ space points are closed. By Urysohn's Lemma, for any point $a$ and closed set $B$ disjoint fro…
T114	$T_{3 \frac{1}{2}}$	Functionally Hausdorff	Since $X$ is $T_1$ points of $X$ are closed, and since $X$ is Completely regular there is a function separ…
T115	$T_{3 \frac{1}{2}}$	$T_3$	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 $O_B = f^{-1}((\frac{1}{2},1])$
T116	$\aleph_0$-space	Has a countable $k$-network	Follows from the definition.
T117	Has a countable $k$-network ∧ $T_3$	$\aleph_0$-space	Follows from the definition.
T118	$T_2$	$T_1$	By definition, see page 11 of DOI 10.1007/978-1-4612-6290-9.
T119	$T_1$	$T_0$	By definition, see page 11 of DOI 10.1007/978-1-4612-6290-9.
T120	Embeddable in $\mathbb R$	GO-space	By definition, since $\mathbb R$ is a LOTS.
T121	Compact	$\sigma$-compact	By definition; see Figure 3 on page 21 of DOI 10.1007/978-1-4612-6290-9.
T122	$\sigma$-compact	Lindelöf	If $X = \cup_{n \in \omega} C_n$ with each $C_n$ compact and $\mathcal{U}$ is any open cover, then some finite subcollection $\mathcal{U}_n$ cove…
T123	Lindelöf ∧ Countably paracompact	Paracompact	If $X$ is Lindelof, any open cover has a countable subcover, and if $X$ is countably paracompact, this s…
T124	Lindelöf ∧ Countably metacompact	Metacompact	If $X$ is Lindelof, any open cover has a countable subcover, and if $X$ is countably metacompact, this s…
T125	GO-space ∧ Compact	LOTS	Let $\tau$ be the topology on $X$ and let $<$ be the order on $X$ in the definition of GO-space, with correspon…
T126	Cozero complemented	$T_{3 \frac{1}{2}}$	Required by the definition given in MR 2247908.
T127	Homogeneous ∧ Lindelöf ∧ Locally Hausdorff ∧ Baire	$T_2$	See Theorem 4.1 in DOI 10.1090/S0002-9939-07-09100-9.
T128	Lindelöf	Weakly Lindelöf	By the definition given in e.g. DOI 10.2307/1996310.
T129	Countable chain condition	Weakly Lindelöf	$wc(X)\leq c(X)$ where $wc(X)$ is the weak covering number of $X$ and $c(X)$ is the cellularity of $X$. $X$ has CCC …
T130	Has a $\sigma$-locally finite base	First countable	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 s…
T131	GO-space ∧ Connected	LOTS	The given topology on $X$ and the order topology induced by the order in the definition of GO-space co…
T132	Metrizable ∧ Strongly Choquet	Completely metrizable	Shown in zbMATH 0819.04002. See also Wikipedia Choquet_game.
T133	Completely metrizable	Čech complete	Theorem 4.3.26 of zbMATH 0684.54001.
T134	Baire ∧ ¬Empty	¬Meager	By their definitions, see 25.1 and 25.2 of zbMATH 1052.54001.
T135	Strongly Choquet	Baire	A consequence of Exercise 8.15 and the statement before Exercise 8.13 in zbMATH 0819.04002.
T136	Weakly locally compact ∧ Regular	Baire	See Theorem 34, p. 200 of MR 0370454, or Proposition 20.18, p. 538 of MR 1417259.
T137	Has a $\sigma$-locally finite base	Has a $\sigma$-locally finite $k$-network	Evident from the definitions, as a base for the topology is a $k$-network.
T138	Cardinality $=\mathfrak c$	¬Cardinality $\lt\mathfrak c$	See zbMATH 1052.54001 for a discussion on cardinalities.
T139	Cardinality $=\mathfrak c$	Cardinality $\leq\mathfrak c$	Immediate from the definitions.
T140	Menger	Lindelöf	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 o…
T141	$\sigma$-compact	Markov Menger	See the proof of Theorem 2.2 in DOI 10.1016/S0166-8641(96)00075-2, which we summarize here.
T142	Countable chain condition ∧ $T_{3 \frac{1}{2}}$	Cozero complemented	Given any cozero $U \subset X$, $X \setminus \overline U$ contains a maximal pairwise-disjoint collection of cozero sets $\mathcal{V}$. $\mathcal{V}$ is cou…
T143	Door	$T_0$	Given two distinct points $x$ and $y$, by the Door property either $\{x\}$ is open (and is a neighborhood of…
T144	Discrete	Door	Evident from the definitions.
T145	Door ∧ $T_2$	Scattered	See Math StackExchange 3789612.
T146	$T_3$	Regular	See 14.1 of zbMATH 1052.54001.
T147	$\sigma$-space	Has a $\sigma$-locally finite network	Follows from the definition.
T148	Regular ∧ $T_0$	$T_3$	$T_3$ is often defined as regular + $T_1$, but on page 12 of DOI 10.1007/978-1-4612-6290-9 the authors n…
T149	$T_{3 \frac{1}{2}}$	Completely regular	See e.g. zbMATH 1052.54001
T150	Has a $\sigma$-locally finite network ∧ $T_3$	$\sigma$-space	Follows from the definition.
T151	Completely regular ∧ $T_0$	$T_{3 \frac{1}{2}}$	$T_{3.5}$ is often defined as regular + $T_1$, but on page 14 of DOI 10.1007/978-1-4612-6290-9 the authors…
T152	$T_6$	$T_1$	By definition, see DOI 10.1007/978-1-4612-6290-9.
T153	$T_1$ ∧ Perfectly normal	$T_6$	By definition, see DOI 10.1007/978-1-4612-6290-9.
T154	$T_6$	$T_5$	See Figure 2 of DOI 10.1007/978-1-4612-6290-9.
T155	Regular	Semiregular	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 …
T156	Perfectly normal	Completely normal	Separation by a continuous function implies separation by open sets.
T157	Dowker	$T_4$	By definition.
T158	Dowker	¬Countably paracompact	By definition.
T159	$T_4$ ∧ ¬Countably paracompact	Dowker	By definition.
T160	Rothberger	Menger	By definition, see e.g. DOI 10.14712/1213-7243.2015.201.
T161	Strategic Menger	Menger	See Figure 3 of DOI 10.14712/1213-7243.2015.201.
T162	Markov Menger	2-Markov Menger	See Figure 3 of DOI 10.14712/1213-7243.2015.201.
T163	$\sigma$-compact	$\sigma$-relatively-compact	All compact subsets are relatively compact. See Math StackExchange 4702452.
T164	Regular ∧ $\sigma$-relatively-compact	$\sigma$-compact	The closure of a relatively-compact set in a Regular space is compact. See Proposition 4.4 in DOI 10…
T165	Markov Menger	$\sigma$-relatively-compact	Let $\sigma(\mathcal{U}, n)$ be a winning Markov strategy for $F$ in the Menger game, and let $\mathfrak{C}$ be the collection of all…
T166	$\sigma$-relatively-compact	Markov Menger	The second player may cover the nth relatively compact subset during the nth round of the game.
T167	Second countable ∧ Strategic Menger	Markov Menger	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…
T168	2-Markov Menger	Strategic Menger	See Figure 3 of DOI 10.14712/1213-7243.2015.201.
T169	Scattered	$T_0$	By definition on page 33 of DOI 10.1007/978-1-4612-6290-9, given two distinct points $x,y$, the subspa…
T170	$R_1$ ∧ Paracompact	Fully normal	Follows from Math StackExchange 4969398 ($R_1$ and Paracompact imply Regular), zbMATH 0684.54001 The…
T171	Locally Euclidean ∧ Has multiple points	¬Hyperconnected	If there is a point $x\in X$ that has an open neighborhood $U$ homeomorphic to $\mathbb R^n$ with $n\ge 1$, the open set $U$ …
T172	Locally Euclidean ∧ ¬Empty	Strongly Choquet	Player 2 can choose a neighborhood $V_0$ that is homeomorphic to some $\mathbb R^n$. Then the game is played on $\mathbb R^n$
T173	Locally Hausdorff	Sober	See Proposition 3.5 of MR 0702721.
T174	Sober	$T_0$	See page 124 of MR 1002193.
T175	Locally Euclidean ∧ Lindelöf	Second countable	See Math StackExchange 4416020.
T176	Spectral	Sober	By definition. See example 21, section 2.6 of MR 1077251.
T177	Pseudometrizable	Proximal	The entourage-picker in the proximal game may choose the metric entourage of radius $2^{-n}$ during round…
T178	Corson compact	Proximal	Real lines are Metrizable and therefore Proximal, and the $\Sigma$-products and closed subsets of Proximal …
T179	$T_2$ ∧ Compact ∧ Proximal	Corson compact	Main result of DOI 10.1016/j.topol.2014.05.010.
T180	$T_6$	Cozero complemented	As a set is closed if and only if it is cozero complemented.
T181	Metrizable	Locally metrizable	$X$ is the metrizable neighborhood for each point of $x$.
T182	$\aleph$-space	Has a $\sigma$-locally finite $k$-network	Follows from the definition.
T183	First countable	Fréchet Urysohn	Fix a sequence of neighborhoods and pick a point from each of these neighborhoods.
T184	Fréchet Urysohn	Sequential	Asserted in the introduction to MR 0687569.
T185	Sequential	Countably tight	Asserted in the introduction to MR 0687569.
T186	Locally countable	Countably tight	As shown in Math StackExchange 3809662.
T187	Finite	Countable	See zbMATH 1052.54001 for a discussion on cardinalities.
T188	Sequentially compact ∧ Sequentially discrete	Finite	In an infinite Sequentially discrete space, a sequence with distinct terms has no convergent subsequ…
T189	Finite	Second countable	Space is finite implies the topology is finite, hence countable. Thus the topology itself is a count…
T190	Cardinality $=\aleph_1$	Cardinality $\leq\mathfrak c$	Evident from the definitions.
T191	Cardinality $=\aleph_1$	¬Countable	Evident from the definitions.
T192	$k_2$-space ∧ $k_2$-Hausdorff	CGWH	See Proposition 11.4 in C. Rezk, "Compactly generated spaces" or Proposition 2.14 in N. Strickland, …
T193	$T_2$	Locally Hausdorff	Follows directly from their definitions.
T194	CGWH	KC	See Math StackExchange 1072014.
T195	Locally Hausdorff	$T_1$	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…
T196	Weakly locally compact ∧ $T_2$	Čech complete	Theorem 3.3.9 of zbMATH 0684.54001 asserts that any locally compact Hausdorff space is an open subse…
T197	Has a $\sigma$-locally finite $k$-network ∧ $T_3$	$\aleph$-space	Follows from the definition.
T198	Finite	Noetherian	Finite spaces are compact, and finiteness is a hereditary property.
T199	Polish	Separable	By definition.
T200	Polish	Completely metrizable	By definition.
T201	Separable ∧ Completely metrizable	Polish	By definition.
T202	Locally pseudometrizable ∧ $R_1$ ∧ Paracompact	Pseudometrizable	If $X$ is Locally pseudometrizable, $R_1$ and Paracompact, its Kolmogorov quotient is Locally metrizab…
T203	GO-space ∧ Locally connected	Weakly locally compact	Given a Locally connected GO-space $X$, there is a base for the topology consisting of Connected open …
T204	Discrete	Homogeneous	Evident from the definitions as all self-bijections are homeomorphisms.
T205	Radial	Pseudoradial	Follows directly from the definitions: given a radially closed set $A$ and $a\in cl(A)$, by Radial there ex…
T206	Fréchet Urysohn	Radial	Evident from the definitions, as ordinary sequences are transfinite sequences.
T207	Sequential	Pseudoradial	Evident from the definitions, as ordinary sequences are transfinite sequences.
T208	Indiscrete ∧ Has multiple points	¬Has an isolated point	Evident from the definitions.
T209	Has an isolated point ∧ Homogeneous	Discrete	Let $a$ be an isolated point of $X$. Let $x\in X$; there is an homeomorphism of $X$ onto itself taking $a$ to $x$, …
T210	Locally countable ∧ Pseudoradial	Sequential	Given a transfinite sequence with values in a set $A$ and converging to a point $p\in X$, first replace it …
T211	Countably tight ∧ Radial	Fréchet Urysohn	Established in Math StackExchange 4850979.
T212	Countable ∧ First countable	Second countable	Evident from the definitions: the countable union of the countable point-bases is a countable basis.…
T213	Collectionwise normal	Normal	Evident from the definitions as two closed disjoint subsets form a discrete family.
T214	Fully normal	Strongly collectionwise normal	Shown in Math StackExchange 5005987 using techniques from zbMATH 0078.14803.
T215	Corson compact	Fréchet Urysohn	Proven in U.120 of DOI 10.1007/978-3-319-16092-4.
T216	GO-space	Radial	The result for LOTS is stated as evident in the first sentence of the proof of Satz 1 in DOI 10.4064…
T217	Countably tight ∧ Well-based	First countable	See Math StackExchange 4856518.
T218	Discrete	Locally finite	The set of singletons is a basis of finite sets.
T219	Metrizable ∧ Compact	Eberlein compact	In DOI 10.2140/pjm.1977.72.487 Eberlein compacts are characterized as compact spaces with $\sigma$-point-fi…
T220	Eberlein compact	Corson compact	Noted on page 494 of DOI 10.2140/pjm.1977.72.487.
T221	Countable sets are discrete	$T_1$	Immediate from the definitions.
T222	Countable sets are discrete	Anticompact	In a Countable sets are discrete space every countable subspace is discrete (as each of its subsets …
T223	Countably compact ∧ Sequential	Sequentially compact	Theorem 1.20 of DOI 10.14288/1.0080490. Refer to zbMATH 0684.54001 Thm 3.10.31 for a proof assuming …
T224	Weakly locally compact ∧ Regular	Locally relatively compact	By Theorem 17 in chapter 5, p. 146, of MR 0370454 every point in a weakly locally compact regular sp…
T225	First countable ∧ P-space	Alexandrov	Given a point $x\in X$, the intersection of a countable local base at $x$ is contained in every neighborhoo…
T226	US	$T_1$	Let $x, y \in X$ with $x \neq y$. Since $X$ is US, the constant sequence $x_n := x$ converges to $x$ but not to $y$. Thus t…
T227	KC	Weak Hausdorff	Continuous images of compact are always compact, and thus closed given KC.
T228	$T_2$	$k_1$-Hausdorff	Follows as $T_2$ is hereditary.
T229	Weak Hausdorff	$k_2$-Hausdorff	See Proposition 11.2 of https://ncatlab.org/nlab/files/Rezk_CompactlyGeneratedSpaces.pdf.
T230	First countable ∧ US	$T_2$	See Math StackExchange 1369459.
T231	Embeddable in $\mathbb R$	Embeddable into Euclidean space	By definition.
T232	Totally path disconnected ∧ Embeddable in $\mathbb R$	Zero dimensional	See Math StackExchange 3824535.
T233	Connected ∧ Embeddable in $\mathbb R$	Locally path connected	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 $a,b\in A$ i…
T234	Strongly KC	KC	Follows from the definitions.
T235	Sequential ∧ US	Strongly KC	Shown in Lemma 3.10 of DOI 10.1007/s10587-009-0022-6.
T236	GO-space ∧ Countably tight	First countable	Shown by KP Hart at MathOverflow 312803. To see this, suppose $X$ is a GO-space for some linear order …
T237	Hemicompact	$\sigma$-compact	17I.1 of zbMATH 1052.54001
T238	Countable	Locally countable	Every open set is countable in a countable space.
T239	Compact ∧ Connected ∧ Locally connected ∧ Metrizable	Injectively path connected	This is Theorem 31.2 in zbMATH 1052.54001.
T240	Path connected ∧ Weak Hausdorff	Arc connected	It is a classical result that Path connected $T_2$ spaces are Arc connected. This is shown for examp…
T241	$\aleph_0$-space	Cosmic	Evident from the definitions, as a $k$-network is a network.
T242	Cosmic	$\sigma$-space	Evident from the definitions, as a countable family of sets is $\sigma$-locally finite.
T243	$\aleph_0$-space	$\aleph$-space	Evident from the definitions, as a countable family of sets is $\sigma$-locally finite.
T244	$\aleph$-space	$\sigma$-space	Evident from the definitions, as a $k$-network is a network.
T245	Locally compact	Weakly locally compact	Having a local base of compact neighborhoods implies at least one compact neighborhood. See Wikiped…
T246	Weakly locally compact ∧ Regular	Locally compact	By Theorem 17 in chapter 5, p. 146, of MR 0370454 every point in a weakly locally compact regular sp…
T247	Discrete ∧ Indiscrete	¬Has multiple points	The only spaces that are both discrete and indiscrete are the empty and singleton spaces.
T248	¬Has multiple points	Discrete	The spaces with less than two points are the empty space and the singleton space, which are both dis…
T249	¬Has multiple points	Indiscrete	The spaces with less than two points are the empty space and the singleton space, which are both ind…
T250	¬Finite	Has multiple points	Trivially from the definitions.
T251	Indiscrete	Compact	All open covers are finite to begin with.
T252	Partition topology	Pseudometrizable	The pseudometric where $d(x,y)=0$ if the points $x$ and $y$ lie in the same element of the partition and $d(x,y)=1$
T253	Has multiple points ∧ $T_0$	¬Indiscrete	Take two distinct points in $X$. Since $X$ is $T_0$, there is an open set containing one of the points an…
T254	Hereditarily Lindelöf	Lindelöf	Evident from the definitions.
T255	Lindelöf ∧ $G_\delta$ space	Hereditarily Lindelöf	If $X$ is Lindelof and a $G_\delta$ space, every open set in $X$ is an $F_\sigma$ and hence is also Lindelof. This use…
T256	Perfectly normal	$G_\delta$ space	Follows from the definition of Perfectly normal.
T257	Normal ∧ $G_\delta$ space	Perfectly normal	Follows from the definition of Perfectly normal.
T258	Regular ∧ Hereditarily Lindelöf	Perfectly normal	Shown in Math StackExchange 322506.
T259	Countable	Has a countable network	Evident, as the collection of all singletons in $X$ is a network.
T260	Has a countable network	Hereditarily Lindelöf	Having a countable network is a hereditary property; and a space with a countable network is Lindelö…
T261	Countable ∧ $R_0$	$G_\delta$ space	Every open subset of $X$ is an $F_\sigma$, as it is the countable union of the closures of its singletons.
T262	$R_1$ ∧ Hyperconnected	Indiscrete	If two points in a $R_1$ space are topologically distinguishable, they have disjoint open neighborho…
T263	Hereditarily Lindelöf ∧ Scattered	Countable	See Math StackExchange 4955301.
T264	Metrizable	Pseudometrizable	Evident from the definitions.
T265	Pseudometrizable ∧ $T_0$	Metrizable	If two distinct points $x$ and $y$ satisfy $d(x,y)=0$, they are not topologically distinguishable.
T266	Finite	Locally finite	All open sets in a finite space are finite by definition.
T267	Alexandrov ∧ $T_1$	Discrete	Stated on p. 18 of MR 1711071.
T268	Pseudometrizable	Perfectly normal	Every closed subset $A$ of $X$ is the zero-set of a real-valued continuous function, namely, $f(x)=d(x,A)$
T269	Weakly locally compact ∧ $T_2$ ∧ Totally disconnected	Zero dimensional	See Math StackExchange 11423 or Theorem 6.2.9 in zbMATH 0684.54001 (where Totally disconnected is ca…
T270	Second countable	First countable	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 a…
T271	Second countable	Has a countable $k$-network	Follows from the definitions, as a base for the topology is a $k$-network.
T272	Second countable	$k$-Lindelöf	See bof's answer at Math StackExchange 4727903.
T273	GO-space	$T_5$	For a LOTS space, see items #3-6 of example #39 in DOI 10.1007/978-1-4612-6290-9_6 or Math StackExch…
T274	Discrete	LOTS	Since discrete spaces are homeomorphic precisely when they have the same cardinality, it suffices to…
T275	LOTS ∧ Connected ∧ Separable	Second countable	Let $X$ be a connected linearly ordered topological space with countable dense subset $Q$. Then the coll…
T276	Hereditarily Lindelöf	Has countable spread	Follows as Lindelöf discrete spaces are countable.
T277	GO-space ∧ Countable chain condition	Hereditarily Lindelöf	For the result with the stronger hypothesis of LOTS in place of GO-space, see Theorem 2.2 in DOI 10.…
T278	GO-space ∧ Countable chain condition	First countable	For the result with the stronger hypothesis of LOTS in place of GO-space, Exercise 3.12.4(a) in zbMA…
T279	Hemicompact ∧ First countable	Weakly locally compact	See Math StackExchange 2919068.
T280	Locally countable ∧ $T_1$	Totally path disconnected	Let $I$ be the interval $[0,1]$ and consider a path $f:I\to X$. Suppose first that $X$ is Countable and $T_1$. …
T281	$T_2$	$R_1$	Follows directly from the definitions, and stated in Wikipedia Hausdorff_space.
T282	Regular	$R_1$	Follows directly from the definitions, and stated in Wikipedia Hausdorff_space.
T283	$R_1$ ∧ $T_0$	$T_2$	Follows directly from the definitions, and stated in Wikipedia Hausdorff_space.
T284	Alexandrov	Locally compact	See Theorem 5 in https://arxiv.org/abs/0708.2136. The smallest (open) neighborhood $U$ of a point $x$ i…
T285	Alexandrov	First countable	The smallest neighborhood of a point in an Alexandrov space forms a local base with a single element…
T286	$R_1$	$R_0$	Follows directly from the definitions, and stated in Wikipedia Separation_axiom.
T287	$T_1$	$R_0$	Follows directly from the definitions, and stated in Wikipedia T1_space.
T288	$R_0$ ∧ $T_0$	$T_1$	Follows directly from the definitions. See Wikipedia T1_space.
T289	$G_\delta$ space	$R_0$	See Math StackExchange 4547643.
T290	Finite	Baire	A finite space has only finitely many open sets, and the intersection of two open dense sets is an o…
T291	Anticompact ∧ Countable	Hemicompact	Let $X=\omega$ be a countable space such that every compact subset is finite. Then $X$ is hemicompact: let $K_n=\{0,\dots,n\}$
T292	Anticompact ∧ $k_1$-space	Locally finite	An Anticompact $k_1$-space has a topology generated by its finite subsets, i.e. it is Alexandrov. Th…
T293	Locally finite	Anticompact	Any compact subset is covered by finitely many finite sets and is therefore finite.
T294	GO-space ∧ Separable	Hereditarily separable	For the result with the stronger hypothesis of LOTS in place of GO-space, see Theorem 3.3 in DOI 10.…
T295	Has multiple points	¬Empty	A space with at least two points is not empty.
T296	Indiscrete	Hereditarily connected	The two open sets $\emptyset$ and $X$ form a chain under inclusion.
T297	Countably-many continuous self-maps	Countable	Each constant map is a continuous map.
T298	Countably-many continuous self-maps ∧ Regular	Finite	See an anonymous comment in MathOverflow 418619. This extends the result of Math StackExchange 42311…
T299	Finite	Countably-many continuous self-maps	A Finite space only has finitely-many self-maps, continuous or not.
T300	Cardinality $\lt\mathfrak c$ ∧ Completely regular	Zero dimensional	Shown in Math StackExchange 4528918 for countable spaces. The same argument works for any cardinali…
T301	Countably-many continuous self-maps ∧ Alexandrov	Finite	See Math StackExchange 4231158 and Math StackExchange 4575225.
T302	Countable ∧ Compact ∧ $R_1$	Pseudometrizable	It is shown in Math StackExchange 3705764 that countable compact Hausdorff spaces are metrizable. T…
T303	Anticompact ∧ Compact	Finite	Follows directly from the definitions.
T304	Anticompact ∧ $\sigma$-compact	Countable	Follows directly from the definitions.
T305	Sequentially discrete	Totally path disconnected	See Math StackExchange 4882280.
T306	Scattered ∧ ¬Empty	Has an isolated point	The space itself is non-empty and thus contains an isolated point.
T307	Locally finite ∧ $T_0$	Scattered	First assume $X$ is nonempty and show it has an isolated point. Take a nonempty finite open set $U\subseteq X$ t…
T308	$R_0$ ∧ Has an isolated point ∧ Has multiple points	¬Connected	By Has an isolated point there is a singleton $\{x\}$ that is open. By $R_0$ the open set $\{x\}$ contains $\overline{\{x\}}$
T309	Connected ∧ Cardinality $\lt\mathfrak c$	Strongly connected	The continuous image of a Connected and Cardinality $\lt\mathfrak c$ space is Connected and Cardinal…
T310	$T_2$	Has closed retracts	See Math StackExchange 805274.
T311	Has closed retracts	$T_1$	See MathOverflow 191016.
T312	Compact ∧ KC	Has closed retracts	Asserted in MathOverflow 434451; let $R$ be a retract, then $R$ is the continuous image of the Compact s…
T313	Hyperconnected	Countable chain condition	In a hyperconnected space, no two nonempty open sets can be disjoint. So the space satisfies Counta…
T314	Has an isolated point	¬Meager	A space containing an isolated point cannot be meager, because no set containing the isolated point …
T315	Empty	Meager	The empty space is nowhere dense in itself, hence is a meager space.
T316	Alexandrov	Locally path connected	See Math StackExchange 2965227.
T317	Scattered	Baire	In a scattered space every nonempty subset, in particular every nonempty open set, has an isolated p…
T318	Anticompact ∧ $T_1$	$k_1$-Hausdorff	Every Compact subset is Finite and $T_1$, and thus Discrete and $T_2$.
T319	Countable ∧ ¬Has an isolated point ∧ $T_1$	Meager	As $X$ is $T_1$ and without isolated point, every singleton is closed and with empty interior, and thus …
T320	Ultraconnected	$\sigma$-connected	By definition, an ultraconnected space cannot be partitioned into disjoint closed sets.
T321	$k_{\omega,1}$-space	Hemicompact	Shown in Math StackExchange 4585825.
T322	$k_{\omega,1}$-space ∧ $T_1$ ∧ First countable	Weakly locally compact	See Math StackExchange 4585825.
T323	$k_{\omega,1}$-space	$k_1$-space	Let $K_n$ witness $k_{\omega,1}$-space, and let $C$ have closed intersection with every Compact subspace…
T324	$k_3$-space	$k_2$-space	See Math StackExchange 4646084.
T325	$k_2$-space	$k_1$-space	See Math StackExchange 4646084.
T326	Pseudometrizable	Locally pseudometrizable	The entire space is the desired neighborhood.
T327	Locally metrizable	Locally pseudometrizable	Every metric is a pseudometric.
T328	Locally metrizable	Locally Hausdorff	Around every point there is a neighborhood that is Metrizable, hence $T_2$.
T329	Locally Euclidean	Locally metrizable	Every point has a neighborhood that is homeomorphic to an open subset of some $\mathbb R^n$, hence Metrizable. …
T330	Locally pseudometrizable	$R_0$	Around every point there is a neighborhood that is Pseudometrizable, hence $R_0$. And a space that is…
T331	Locally pseudometrizable ∧ $T_0$	Locally metrizable	Every point has a neighborhood that is Pseudometrizable and $T_0$, hence Metrizable via (Pseudometri…
T332	Locally Euclidean	Locally compact	Every point has a neighborhood that is homeomorphic to $\mathbb R^n$ for some non-negative integer $n$. Since $\mathbb R^n$ …
T333	Topological $n$-manifold	$T_2$	By definition: see page 316 of zbMATH 0951.54001.
T334	Has a countable network ∧ $T_0$	Cardinality $\leq\mathfrak c$	See page 127 in zbMATH 0684.54001.
T335	$T_4$	Normal	By definition as in 15.1 of zbMATH 1052.54001.
T336	$T_5$	Completely normal	By definition as on page 12 of DOI 10.1007/978-1-4612-6290-9.
T337	Fully $T_4$	Fully normal	By definition as on page 23 of DOI 10.1007/978-1-4612-6290-9.
T338	$T_6$	Perfectly normal	By definition, see DOI 10.1007/978-1-4612-6290-9.
T339	Spectral	Compact	By definition. See example 21, section 2.6 of MR 1077251.
T340	Topological $n$-manifold	Second countable	By definition: see page 316 of zbMATH 0951.54001.
T341	Countable	Markov Rothberger	Suppose $X$ is countable and let $\{ F_n : n \in \omega \}$ be an enumeration of the non-empty finite subsets of $X$. In …
T342	Ultraparacompact	Strongly paracompact	Any partition is star-finite.
T343	Strongly paracompact	Paracompact	Any star-finite collection of open sets is locally finite.
T344	Lindelöf ∧ Zero dimensional	Ultraparacompact	See Proposition 4 of DOI 10.48550/arXiv.1306.6086.
T345	Has a group topology	Completely regular	For each open neighborhood $U$ of the identity, $D_U=\{(x,y):xy^{-1}\in U\}$ is a basic entourage forming a unif…
T346	Has a group topology	¬Empty	A group cannot be empty.
T347	Has a group topology	Homogeneous	Every element can be sent to every other element by some left-translation, which is a homeomorphism.…
T348	Has a group topology ∧ First countable	Pseudometrizable	This is essentially the Birkhoff-Kakutani theorem (https://terrytao.wordpress.com/2011/05/17/the-bir…
T349	Indiscrete	Homogeneous	All bijections of the space onto itself are continuous.
T350	Alexandrov	P-space	Follows immediately from definitions.
T351	Regular ∧ P-space	Zero dimensional	Shown in the proof of Proposition 3.2 in DOI 10.1016/0016-660X(72)90026-8.
T352	Has a countable $k$-network	Has a $\sigma$-locally finite $k$-network	Evident from the definitions, as a countable family of sets in $X$ is $\sigma$-locally finite.
T353	Strategic Menger	$\omega$-Menger	Evident from the definitions.
T354	Markov Rothberger	Strategically Rothberger	Evident from the definitions.
T355	Strategically Rothberger	$\omega$-Rothberger	Evident from the definitions.
T356	$\omega$-Menger	$\omega$-Lindelöf	Evident from the definitions and similar to the proof of Menger ⇒ Lindelöf.
T357	Markov Rothberger	Markov Menger	Evident from the definitions and similar to the comment that every Rothberger space is Menger in DOI…
T358	Strategically Rothberger	Strategic Menger	Evident from the definitions and similar to the comment that every Rothberger space is Menger in DOI…
T359	$\omega$-Rothberger	$\omega$-Menger	Evident from the definitions and similar to the comment that every Rothberger space is Menger in DOI…
T360	$\omega$-Lindelöf	Lindelöf	Evident from the definitions.
T361	$\omega$-Rothberger	Rothberger	Evident from the definitions.
T362	$\omega$-Menger	Menger	Evident from the definitions.
T363	$T_1$ ∧ Markov Rothberger	Countable	Proved for $T_{3 \frac{1}{2}}$ spaces as one of the implications in Theorem 17 of DOI 10.1016/j.topo…
T364	$k$-Lindelöf	$\omega$-Lindelöf	See the proof provided at Math StackExchange 4717687.
T365	Markov $k$-Rothberger	Strategically $k$-Rothberger	Evident from the definitions.
T366	Strategically $k$-Rothberger	$k$-Rothberger	Evident from the definitions.
T367	Markov $k$-Menger	Strategically $k$-Menger	Evident from the definitions.
T368	Strategically $k$-Menger	$k$-Menger	Evident from the definitions.
T369	$k$-Menger	$k$-Lindelöf	Evident from the definitions and similar to the proof of Menger ⇒ Lindelöf.
T370	$k$-Rothberger	$k$-Menger	Evident from the definitions and similar to the comment that every Rothberger space is Menger in DOI…
T371	Strategically $k$-Rothberger	Strategically $k$-Menger	Evident from the definitions and similar to the comment that every Rothberger space is Menger in DOI…
T372	Markov $k$-Rothberger	Markov $k$-Menger	Evident from the definitions and similar to the comment that every Rothberger space is Menger in DOI…
T373	Hemicompact	Markov $k$-Rothberger	See Theorem 3.22 of MR 3991109.
T374	$T_1$ ∧ First countable ∧ $k$-Menger	Hemicompact	See Proposition 5 of DOI 10.1016/j.topol.2005.07.015. In that proof, one can take $O_n(K)$ to be $X \setminus \{ x_n(K) \}$
T375	$k$-Menger	$\omega$-Menger	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 $k$-…
T376	Has a $\sigma$-locally finite network ∧ Lindelöf	Has a countable network	Every $\sigma$-locally finite collection of sets in a Lindelöf space is countable.
T377	Anticompact ∧ $\omega$-Lindelöf	$k$-Lindelöf	Evident from the definitions since all compact sets are finite.
T378	Anticompact ∧ $\omega$-Menger	$k$-Menger	Evident from the definitions since all compact sets are finite.
T379	Anticompact ∧ Strategically Rothberger	Strategically $k$-Rothberger	Evident from the definitions since all compact sets are finite.
T380	Anticompact ∧ $\omega$-Rothberger	$k$-Rothberger	Evident from the definitions since all compact sets are finite.
T381	$T_1$ ∧ Markov $k$-Rothberger	Hemicompact	Proved for $T_{3 \frac{1}{2}}$ spaces as one of the implications of Theorem 3.22 in MR 3991109. Show…
T382	$T_4$ ∧ Metacompact ∧ Cardinality less than every measurable cardinal	Realcompact	See DOI 10.4153/CJM-1972-081-9 corollary 2.
T383	Cardinality $\leq 2^{\mathfrak c}$	Cardinality less than every measurable cardinal	See Theorem 12.5 in DOI 10.1007/978-1-4615-7819-2: in ZFC a measurable cardinal must be strongly ina…
T384	$T_3$ ∧ Lindelöf	Realcompact	See Theorem 3.8.2 of zbMATH 0684.54001 (where the Lindelöf property assumes $T_3$).
T385	Realcompact	$T_{3 \frac{1}{2}}$	$T_{3 \frac{1}{2}}$ is hereditary, and Realcompact spaces are [closed] subsets of $T_{3 \frac{1}{2}}…
T386	Pseudocompact ∧ Realcompact	Compact	Take the space $H\subseteq \mathbb R^\kappa$ (by Realcompact); its projection $H_\alpha\subseteq\mathbb R$ for each factor $\alpha<\kappa$ must be bounded (by P…
T387	Has a $\sigma$-locally finite $k$-network ∧ Lindelöf	Has a countable $k$-network	Every $\sigma$-locally finite collection of sets in a Lindelöf space is countable.
T388	$G_\delta$ space	Countably metacompact	We will use Ishikawa's characterization of Countably metacompact.
T389	Countably metacompact ∧ Normal	Countably paracompact	See theorem 2 of DOI 10.4153/CJM-1951-026-2.
T390	Cardinality $\leq\mathfrak c$	Cardinality $\leq 2^{\mathfrak c}$	Since for any cardinal $\kappa$, $\kappa < 2^{\kappa}$.
T391	¬Cardinality $\lt\mathfrak c$ ∧ ¬Cardinality $=\mathfrak c$	¬Cardinality $\leq\mathfrak c$	Direct from the definitions.
T392	Regular ∧ Markov $k$-Menger	Hemicompact	See MathOverflow 450531.
T393	Extremally disconnected ∧ Semiregular	Zero dimensional	If a space is Semiregular, it has a basis $\mathcal B$ of sets $O\in\mathcal B$ with $\operatorname{int}\operatorname{cl}O=O$. In a Extremally disconnected …
T394	Metrizable ∧ Realcompact	Cardinality less than every measurable cardinal	See MathOverflow 469593. In particular this answer by K. P. Hart.
T395	P-space ∧ Lindelöf ∧ $T_2$	Normal	See Math StackExchange 4744652.
T396	P-space ∧ Functionally Hausdorff	Totally separated	See Corollary 5.2 in DOI 10.1016/0016-660x(72)90026-8 (where "totally disconnected" is used with the…
T397	P-space ∧ Countably compact ∧ $T_1$	Finite	See proposition 4.1 a) in DOI 10.1016/0016-660x(72)90026-8.
T398	P-space ∧ $T_3$ ∧ Pseudocompact	Finite	See proposition 4.1 e) in DOI 10.1016/0016-660x(72)90026-8 and Math StackExchange 4744697.
T399	P-space ∧ $T_1$ ∧ Weakly locally compact	Discrete	See proposition 4.1 d) in DOI 10.1016/0016-660x(72)90026-8.
T400	Connected ∧ Basically disconnected	Strongly connected	The case of the empty space is obvious. Assume $X$ is a nonempty, connected and basically disconnected…
T401	Normal	Pseudonormal	Immediate from the definitions.
T402	$T_1$ ∧ Pseudonormal	Regular	Let $H$ be a closed set and $x$ be a point not in $H$. The singleton $\{x\}$ is countable and closed by $T_1$…
T403	Regular ∧ P-space	Pseudonormal	Shown in Math StackExchange 4744732.
T404	Sequential ∧ US ∧ Density $\leq\mathfrak c$	Cardinality $\leq\mathfrak c$	Proved in Math StackExchange 4850951 with a generalization of the ideas in Theorem 4.4 of MR 0776620…
T405	$T_2$ ∧ Lindelöf ∧ First countable	Cardinality $\leq\mathfrak c$	See Theorem 4.5 of MR 0776620.
T406	Pseudonormal ∧ Countable	Normal	Immediate from the definitions.
T407	Metrizable	Submetrizable	Immediate from the definition.
T408	Submetrizable	Functionally Hausdorff	Given two points $x,y$, there exists a function $f:X\to[0,1]$ continuous in the coarser Metrizable topolog…
T409	Separable ∧ Submetrizable	Has a coarser separable metrizable topology	Suppose $D$ is a countable dense set in $X$. If $\tau$ is a coarser metrizable topology on $X$, the set $D$ is s…
T410	Has a coarser separable metrizable topology	Submetrizable	Immediate from the definition.
T411	Discrete ∧ Cardinality $\leq\mathfrak c$	Has a coarser separable metrizable topology	Embed the space as a subset of $\mathbb R$, then Euclidean Real Numbers witnesses Has a coarser separable metr…
T412	Has a coarser separable metrizable topology	Cardinality $\leq\mathfrak c$	Suppose $X$ has a coarser topology that is Separable and Metrizable, hence also $T_2$ and First counta…
T413	Anticompact ∧ $T_1$	Sequentially discrete	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 fi…
T414	Sequentially discrete	US	Suppose the sequence $(x_n)$ converges to a point $x$. Then the sequence $(x_1,x,x_2,x,\dots)$ that alternate…
T415	P-space ∧ $T_1$	Countable sets are discrete	If singletons are closed and countable unions of closed sets are closed, then every countable set is…
T416	Sequential ∧ Sequentially discrete	Discrete	If $X$ is sequential, its topology is equal to its sequential coreflection. So if the sequential core…
T417	Separable ∧ Countable sets are discrete	Countable	The space $X$ contains a countable dense subset, which is closed by Countable sets are discrete, hence…
T418	Countably tight ∧ Countable sets are discrete	Discrete	Let $A$ be an arbitrary subset of $X$. By Countably tight, every point $p\in\overline A$ is in the closure of a count…
T419	Semi-Hausdorff	$T_1$	Immediate from the definitions.
T420	$T_2$	Semi-Hausdorff	Assume $x,y$ are separated by the disjoint open sets $U,V$. Let $U_{r}=int(cl(U))$ be regular open. It follo…
T421	$T_1$ ∧ Semiregular	Semi-Hausdorff	Given distinct points $x,y$, let $U$ be an open neighborhood of $x$ missing $y$ by $T_1$. By Semiregular, $U$ …
T422	Semi-Hausdorff ∧ Has multiple points	¬Hyperconnected	Let $x,y$ be distinct points with a regular open set $U$ containing $x$ and missing $y$. Then $U$ is not dense…
T423	Compact ∧ $k_1$-Hausdorff	$T_2$	Immediate from the definitions.
T424	$k_1$-Hausdorff	KC	A characterization of $k_1$-Hausdorff in DOI 10.1007/BF02194829 is that all Compact subsets are clos…
T425	$k_2$-Hausdorff	US	See Math StackExchange 4760309.
T426	$T_1$ ∧ Has a dispersion point	Totally path disconnected	See Math StackExchange 4807248.
T427	Exhaustible by compacts ∧ KC	Paracompact	See Math StackExchange 4810248 for a proof.
T428	Cardinality $\geq 3$	Has multiple points	By definition.
T429	Connected ∧ ¬Cardinality $\geq 3$ ∧ ¬Empty	Has a dispersion point	By definition, since nonempty connected spaces where Cardinality $\geq 3$ fails (exactly The Singlet…
T430	Cardinality $\geq 4$	Cardinality $\geq 3$	By definition.
T431	¬Finite	Cardinality $\geq 4$	By definition.
T432	Biconnected ∧ Alexandrov ∧ Cardinality $\geq 4$	Has a dispersion point	Explored in Math StackExchange 4812062; shown a few different ways assuming Finite, and generalized …
T433	Connected ∧ ¬Cardinality $\geq 4$	Biconnected	Biconnected holds vacuously as $X$ is Connected and every pair of subsets, each with at least two poin…
T434	Hemicompact ∧ $k_1$-space	$k_{\omega,1}$-space	Shown in Math StackExchange 4585825
T435	Locally Hausdorff ∧ Locally compact	$k_3$-space	Shown in Math StackExchange 4848174
T436	Locally Hausdorff ∧ $k_2$-space	$k_3$-space	Shown in Math StackExchange 4848174
T437	Discrete	Locally $n$-Euclidean	Every $x\in X$ has the singleton $\{x\}$ as a neighborhood homeomorphic to $\mathbb R^0$.
T438	Locally $n$-Euclidean ∧ Has an isolated point	Discrete	A point is isolated iff it has a neighborhood homeomorphic to $\mathbb R^0$. So if this holds for one point, it…
T439	Compact ∧ Countable	Sequentially compact	Shown in Math StackExchange 4851117
T440	Hereditarily separable	Separable	Immediate from the definition.
T441	Hereditarily separable	Countably tight	Suppose $A\subseteq X$ and $p\in\overline A$. Since $X$ is Hereditarily separable, there is a countable dense subset $D\subseteq A$, wher…
T442	$\sigma$-compact ∧ KC	Metacompact	Let $X=\bigcup_{n<\omega}K_n$ witness $\sigma$-compact and take an open cover $\mathcal U$. Let $\mathcal U_n$ be a finite refinement of …
T443	Fixed point property	Connected	If $X$ is not connected, let $p\in U$, $q\in V$, where $X= U\cup V$ is a separation by disjoint nonempty open sets $U$ an…
T444	Has a group topology ∧ Has multiple points	¬Fixed point property	If $g\in X$ is not the identity, then $x\mapsto g\cdot x$ is a continuous self-map with no fixed point.
T445	Compact ∧ Connected ∧ LOTS ∧ ¬Empty	Fixed point property	Let $f\colon X\to X$ be continuous. For $x_0\in X$, if $f(x_0)>x_0$, then let $U,V$ be disjoint open neighborhoods with…
T446	Fixed point property	¬Empty	The empty set has one (empty) self-map, which has no fixed point.
T447	Fixed point property	$T_0$	Suppose $X$ is not $T_0$, with two topologically indistinguishable points $a$ and $b$. Then the map $f:X\to X$ …
T448	Indiscrete	Partition topology	The basis $\{X\}$ is a partition of $X$.
T449	Has a dispersion point ∧ Cardinality $\geq 3$	$T_0$	See Math StackExchange 4895754.
T450	Indiscrete	Second countable	A space with only finitely many open sets must by definition have a countable basis.
T451	Indiscrete ∧ ¬Cardinality $\lt\mathfrak c$	Locally injectively path connected	Any function into an indiscrete space is continuous, and cardinality $\ge \mathfrak c$ permits injectivity from $[0,1]$
T452	Has a cut point	Connected	By definition.
T453	Normal ∧ Paracompact	Fully normal	Proved in Math StackExchange 4862626.
T454	Countably infinite	Countable	By definition.
T455	Countably infinite	¬Finite	By definition.
T456	Countable ∧ ¬Finite	Countably infinite	By definition.
T457	Corson compact	Compact	By definition.
T458	Corson compact	$T_2$	Follows as Corson compact spaces embed within a $T_2$
T459	Embeddable into Euclidean space	Second countable	Second countable is a hereditary property, and each $\mathbb R^n$ is Second countable as a countable product of…
T460	Embeddable into Euclidean space	Metrizable	Metrizable is a hereditary property, by simply restricting the domain of any metric to a given subse…
T461	Topological $n$-manifold	Embeddable into Euclidean space	See section 50, Exercise 7 on page 316 of zbMATH 0951.54001.
T462	Separable ∧ Metrizable ∧ Zero dimensional	Embeddable in $\mathbb R$	See Theorem 1.3.17 in zbMATH 0401.54029.
T463	GO-space ∧ Second countable	Embeddable in $\mathbb R$	For the result with the stronger hypothesis of LOTS in place of GO-space, see Math StackExchange 491…
T464	Ultraparacompact ∧ $R_0$	Zero dimensional	In a $R_0$ space, the closure of each point is contained in all of its open neighborhoods. Then for …
T465	GO-space ∧ Totally disconnected	Zero dimensional	See Theorem 5.1 in zbMATH 0905.54021, where a GO-space is called a "line".
T466	Alexandrov ∧ $R_0$	Partition topology	Each open set is a union of closed sets by $R_0$, hence is a closed set by Alexandrov.
T467	Partition topology	Alexandrov	The smallest open neighborhood containing each point is the element of the partition that it lies in…
T468	Partition topology ∧ Connected	Indiscrete	If a Partition topology space is Connected, then the partition generating it only has a single eleme…
T469	Partition topology	Ultraparacompact	The partition generating a Partition topology space is a refinement of any open cover.
T470	Partition topology ∧ Homogeneous ∧ ¬Empty	Has a group topology	The partition elements of a Homogeneous Partition topology space must all have the same cardinality,…
T471	Has a group topology ∧ W-space	Embeds in a topological $W$-group	Immediate from the definitions.
T472	Embeddable into Euclidean space	Embeds in a topological $W$-group	Follows as $\mathbb R^n$ is Has a group topology and First countable (and therefore W-space).
T473	First countable	W-space	Follows from Theorem 3.2 on page 342 of DOI 10.1016/0016-660X(76)90024-6. Given a decreasing local b…
T474	W-space	Fréchet Urysohn	Asserted on page 340 of DOI 10.1016/0016-660X(76)90024-6. Given $x\in\overline A$, play the $W$ game at $x$ using Play…
T475	Proximal	W-space	Shown in Lemma 6 of DOI 10.1016/j.topol.2014.06.014. Note that while the article assumes spaces to b…
T476	$T_2$ ∧ Compact ∧ Embeds in a topological $W$-group	Corson compact	Shown in Theorem 3.11 of DOI 10.1016/j.jmaa.2023.127992.
T477	Corson compact	Embeds in a topological $W$-group	By definition as any $\Sigma$ product of reals Has a group topology and is W-space.
T478	Embeds in a topological $W$-group	Completely regular	Follows as Completely regular is hereditary and Has a group topology ⇒ Completely regular.
T479	Embeds in a topological $W$-group	W-space	Follows as W-space is hereditary.
T480	Compact ∧ Connected ∧ $T_2$	Continuum	By definition.
T481	Continuum	$T_2$	By definition.
T482	Continuum	Compact	By definition.
T483	Continuum	Connected	By definition.
T484	$\sigma$-connected	Connected	By definition.
T485	Continuum	$\sigma$-connected	See Theorem 6.1.27 of zbMATH 0684.54001 or Math StackExchange 6314.
T486	Weakly locally compact ∧ Connected ∧ Locally connected ∧ $T_2$	$\sigma$-connected	Asserted in DOI 10.4153/CMB-1973-069-1. Two different proofs are given in Math StackExchange 4906575…
T487	Countable ∧ $R_0$ ∧ ¬Indiscrete	¬$\sigma$-connected	The closures of the singletons are pairwise disjoint and partition $X$ into at least two and at most c…
T488	GO-space ∧ Countably compact	Sequentially compact	Every sequence in a totally ordered set has a monotone subsequence (see for example Math StackExchan…
T489	Ordinal space	LOTS	By definition.
T490	Ordinal space	Weakly locally compact	In an ordinal space $\alpha$, every $\beta\in\alpha$ has $[0,\beta]$ as a compact neighborhood.
T491	Ordinal space	Scattered	Given any nonempty subset $Y$ of an ordinal space $\alpha$, $\min Y$, the least element of $Y$, is an isolated poin…
T492	Ordinal space	Well-based	In an ordinal space $\alpha$, every $\beta\in\alpha$ has $\{(\gamma,\beta]\}_{\gamma<\beta}$ as a neighborhood base of $\beta$ well-ordered by reverse…
T493	Countable ∧ Discrete	Ordinal space	The space is homeomorphic to a finite ordinal space or $\omega$.
T494	Ordinal space ∧ Sequentially discrete	Discrete	Every ordinal number $\alpha\ge\omega+1$ is not sequentially discrete because $0,1,2,\cdots$ converges to $\omega\in\alpha$.
T495	Ordinal space ∧ $G_\delta$ space	Countable	Given an ordinal $\alpha>\omega_1$, $\{\omega_1\}$ is a closed subset of $\alpha$ that is not $G_\delta$. To see this, if $\{(\alpha_n,\omega_1]:n\in\omega\}$
T496	Has a $\sigma$-locally finite network ∧ Regular	$G_\delta$ space	See Math StackExchange 4944702.
T497	Locally compact ∧ KC	$T_{3 \frac{1}{2}}$	If every point has a local base of compact neighborhoods and every compact set is closed, every poin…
T498	Hyperconnected ∧ Locally relatively compact	Compact	Assuming $X$ is nonempty, let $x$ be a point of $X$. By Locally relatively compact, $x$ has an open neighbo…
T499	Locally countable ∧ Lindelöf	Countable	Cover $X$ with countable open sets, then take a countable subcover by Lindelöf. The union is countable…
T500	Has points $G_\delta$	$T_1$	If $X$ is Has points $G_\delta$, every point is an intersection of open sets, which is one of the char…
T501	First countable ∧ $T_1$	Has points $G_\delta$	Suppose $X$ is First countable and $T_1$ and let $x\in X$. Take a countable local base of open neighborhoo…
T502	$G_\delta$ space ∧ $T_1$	Has points $G_\delta$	Clear from the definitions, as points are closed in a $T_1$ space.
T503	Weakly locally compact ∧ $T_2$ ∧ Has points $G_\delta$	First countable	See Math StackExchange 240480.
T504	$k_{\omega,3}$-space	$k_{\omega,1}$-space	Immediate from the definitions.
T505	$k_{\omega,1}$-space ∧ $k_1$-Hausdorff	$k_{\omega,3}$-space	Immediate from the definitions, since $k_1$-Hausdorff means all Compact subspaces are $T_2$.
T506	$k_{\omega,3}$-space	$k_3$-space	Let $K_n$ witness $k_{\omega,3}$-space, and let $C$ have closed intersection with every Compact $T_2$ su…
T507	$k_{\omega,3}$-space	$T_4$	See Math StackExchange 4952092 and page 113 of zbMATH 0416.54027 (available here).
T508	Regular ∧ Lindelöf ∧ Scattered ∧ Has points $G_\delta$	Countable	See Math StackExchange 4954574. Also stated (with a typo) as Corollary 2.5 in DOI 10.1017/S144678870…
T509	Functionally Hausdorff ∧ Has a countable network	Submetrizable	See Corollary in answer to MathOverflow 280359.
T510	P-space ∧ Has points $G_\delta$	Discrete	Each point is open, since it is a countable intersection of open sets in a P-space space.
T511	Sober	Quasi-sober	Follows directly from the definitions.
T512	Quasi-sober ∧ $T_0$	Sober	Every nonempty irreducible closed subset of $X$ is the closure of a point, and that point is unique si…
T513	Locally finite	Quasi-sober	See Math StackExchange 5017256.
T514	Quasi-sober ∧ Hyperconnected ∧ $R_0$	Indiscrete	If $X$ is Quasi-sober, Hyperconnected, and nonempty, it has a generic point $x$. One of the characteriza…
T515	$G_\delta$ space ∧ P-space	Partition topology	If $X$ is $G_\delta$ space and P-space, then every closed subset of $X$ is open.
T516	$\aleph_0$-space ∧ First countable	Metrizable	See result (B) in zbMATH 0148.16701 (https://www.jstor.org/stable/24901448).
T517	$\aleph_0$-space ∧ Weakly locally compact	Metrizable	See result (C) in zbMATH 0148.16701 (https://www.jstor.org/stable/24901448).
T518	Well-based ∧ Has points $G_\delta$	First countable	See Math StackExchange 4963514.
T519	$R_1$	Quasi-sober	The Kolmogorov quotient of a $R_1$ space is $T_2$, which in turn implies Sober (Explore). Thus, the …
T520	Submetrizable	Has a $G_\delta$-diagonal	Suppose the topology of $X$ contains a coarser Metrizable topology induced by a metric $d$. For each $n\ge 1$
T521	Locally pseudometrizable	Quasi-sober	The Kolmogorov quotient of a Locally pseudometrizable space is Locally metrizable, which in turn imp…
T522	LOTS ∧ Connected	Locally connected	Let $X$ be a Connected LOTS, then $X$ is Dedekind-complete and has a dense ordering (see Math StackExcha…
T523	LOTS ∧ Path connected	Locally path connected	See discussion at Math StackExchange 4965472.
T524	Completely metrizable ∧ ¬Has an isolated point ∧ ¬Empty	¬Cardinality $\lt\mathfrak c$	See MathOverflow 22830, or Eric Wofsey's answer to Math StackExchange 2304575.
T525	Separable ∧ Corson compact	Metrizable	Asserted at the top of page 372 of DOI 10.1090/S0002-9939-1987-0884482-0.
T526	Locally path connected ∧ Locally Hausdorff	Locally arc connected	An arbitrary neighborhood $V$ of a point $x$ contains a $T_2$ neighborhood $W$ of $x$ by Locally Hausdorff. …
T527	Spectral	Locally compact	The compact open sets form a (global) basis and therefore also can comprise local bases.
T528	Noetherian ∧ Sober	Spectral	It immediately follows by the definitions that Spectral is equivalent to Sober for Noetherian spaces…
T529	Compact ∧ $T_2$ ∧ Totally disconnected	Stone space	By definition.
T530	Spectral ∧ $T_1$	Stone space	See Lemma 5.23.8 at the Stacks project.
T531	Stone space	Spectral	See Theorem 4.2 in MR 0861951.
T532	Stone space	Totally separated	See Theorem 4.2 in {mr:0861951}.
T533	Has a countable network ∧ Weakly locally compact ∧ $T_2$	Second countable	See Math StackExchange 2500413 or Theorem 3.3.5 in zbMATH 0684.54001.
T534	Countably compact ∧ Submetrizable	Metrizable	Let $f \colon X \to Y$ be a continuous bijection from a Countably compact space $X$ to a Metrizable space $Y$.
T535	Locally $n$-Euclidean	Locally Euclidean	By definition.
T536	Locally Euclidean ∧ Connected	Locally $n$-Euclidean	If a point $x$ has at the same time an open neighborhood homeomorphic to $\mathbb R^n$ and an open neighborhood h…
T537	Locally Euclidean ∧ Homogeneous	Locally $n$-Euclidean	If one point has a neighborhood homeomorphic to $\mathbb R^n$ for some $n$, then all points do with the same $n$ vi…
T538	Pseudoradial ∧ Sequentially discrete	P-space	See Math StackExchange 4975331.
T539	GO-space ∧ Extremally disconnected	Discrete	See Math StackExchange 4913523. The answer proves the result for LOTS but the proof generalizes easi…
T540	$T_2$ ∧ Hereditarily Lindelöf	Has points $G_\delta$	See Math StackExchange 4972410.
T541	Path connected	$\sigma$-connected	See Math StackExchange 4975686.
T542	Shrinking	Normal	The argument in zbMATH 0712.54016 for this result goes as follows. Suppose $E$ and $F$ are disjoint clos…
T543	Shrinking	Countably paracompact	This implication appears in the diagram on page 191 of zbMATH 0712.54016 and is mentioned in passing…
T544	Metacompact	Submetacompact	This is evident from the definitions. The single point-finite open refinement guaranteed by metacomp…
T545	Submetacompact ∧ Normal	Shrinking	See Theorem 6.2 of zbMATH 0712.54016.
T546	Hereditarily connected	Hyperconnected	The open sets are totally ordered by set inclusion and thus no two nonempty open sets are disjoint.
T547	Hereditarily connected	Completely normal	Any Hereditarily connected space is trivially Normal (as there are no disjoint nonempty closed sets)…
T548	Hyperconnected ∧ Completely normal	Hereditarily connected	See condition (10) of theorem 23 at DOI 10.5186/aasfm.1977.0321 (reading $T_5$ as Completely normal).
T549	Hereditarily connected	Well-based	The topology is totally ordered by inclusion, so the set of open neighborhoods of any point is as we…
T550	Shrinking ∧ Hyperconnected	Ultraparacompact	In a Hyperconnected space $X$, the closure of every nonempty open set is $X$, so any open cover that adm…
T551	Hereditarily connected ∧ Locally countable ∧ ¬Countable	Cardinality $=\aleph_1$	By Locally countable choose a countable open neighborhood of each point. So $X$ can be covered by a f…
T552	Has a point with a unique neighborhood ∧ Homogeneous	Indiscrete	If one point in a homogeneous space has $X$ as its only neighborhood, then the same is true of every p…
T553	Hereditarily connected	Locally path connected	Every subset of a Hereditarily connected space is Ultraconnected, and Ultraconnected ⇒ Path connecte…
T554	Hereditarily connected ∧ Compact ∧ Sober ∧ Alexandrov	Spectral	Shown in Proposition 1.6.7 of DOI 10.1017/9781316543870.
T555	Hyperconnected ∧ Ultraconnected ∧ Cardinality $\geq 4$	¬Biconnected	See Math StackExchange 4987074.
T556	Ultraparacompact ∧ Connected ∧ ¬Empty	Has a point with a unique neighborhood	Any clopen partition of a Connected space $X$ must include $X$, so to admit clopen refinements every ope…
T557	$T_0$ ∧ Alexandrov ∧ Second countable	Countable	In an Alexandrov space, the smallest basis is the set of smallest neighborhoods of points. When it i…
T558	Has a cut point	Cardinality $\geq 3$	By definition, since spaces that are not Connected have at least two points.
T559	Has countable extent ∧ Discrete	Countable	Follows from the definitions.
T560	Hereditarily separable	Has countable spread	Follows as separable discrete spaces are countable.
T561	Has countable spread	Has countable extent	Follows from the definitions.
T562	Lindelöf	Has countable extent	Closed sets in a Lindelöf space are Lindelöf, and Lindelöf discrete spaces are countable.
T563	Has countable spread	Countable chain condition	See Math StackExchange 4988266.
T564	Locally finite	Locally countable	By definition, as finite sets are countable.
T565	Locally finite	Alexandrov	Given a finite neighborhood of a point, only finitely many intersections with open sets are needed t…
T566	Hereditarily connected	Has countable spread	The Discrete subspaces of a hereditarily connected space contain at most one point.
T567	Hereditarily connected ∧ Locally finite	Countable	By Locally finite choose a finite open neighborhood of each point. So $X$ can be covered by a family o…
T568	Door ∧ Hyperconnected	Anticompact	Suppose by contradiction that $A$ is an infinite compact subset of $X$. Write $A=B\cup C$ with $B,C$ infinite a…
T569	Weakly countably compact	Has countable extent	Immediate from the definitions.
T570	Proximal	Completely regular	See Definition 1.7 of DOI 10.1016/j.topol.2014.05.010: "A topological space is proximal iff it admit…
T571	Almost discrete	¬Discrete	By definition: all points in a discrete space are isolated, but an almost discrete space has exactly…
T572	Almost discrete	Door	Sets that contain the non-isolated point are closed and sets that don't are open.
T573	Almost discrete	Scattered	For a nonempty $Y\subseteq X$, either $|Y|=1$ and it trivially contains an isolated point, or $Y$ must contain a po…
T574	Almost discrete	Ultraparacompact	Any open cover must include an open neighborhood $U$ of the non-isolated point. The complement of $U$ ca…
T575	Almost discrete	Hereditarily collectionwise normal	Let $X$ be almost discrete. Every subspace of $X$ is either almost discrete or discrete. Thus every sub…
T576	Door ∧ Ultraconnected ∧ Has multiple points	Almost discrete	Per Corollary 2 of zbMATH 0646.54028 (https://www.jstor.org/stable/20489255).
T577	Door ∧ $T_2$ ∧ ¬Discrete	Almost discrete	Discussed after the proof of the main theorem of MR 923909 (https://www.jstor.org/stable/20489255).
T578	Almost discrete ∧ Connected	Has a dispersion point	Removing the non-isolated point results in a Discrete space.
T579	Has a dispersion point ∧ Cardinality $\geq 3$	Has a cut point	$X$ is connected since it Has a dispersion point. Also, (Totally disconnected ∧ Has multiple points) ⇒…
T580	W-space ∧ Countable	First countable	Shown in Corollary 3.4 of DOI 10.1016/0016-660X(76)90024-6.
T581	W-space ∧ Regular ∧ Separable	First countable	Shown in Theorem 3.6 of DOI 10.1016/0016-660X(76)90024-6.
T582	W-space ∧ $T_2$ ∧ Has countable spread	Has points $G_\delta$	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 1…
T583	Contractible	Simply connected	It follows that $X$ is Path connected by Math StackExchange 715720. Since $X$ is contractible, every map…
T584	Contractible	¬Empty	The empty space is not homotopy equivalent to a one-point space.
T585	Hereditarily connected ∧ ¬Empty	Contractible	See Math StackExchange 5000498.
T586	Metacompact	Meta-Lindelöf	Evident from the definitions.
T587	Lindelöf	Meta-Lindelöf	Evident from the definitions, as a countable cover is point-countable.
T588	Connected ∧ LOTS ∧ Cardinality $\geq 3$	Has a cut point	$X$ is connected by assumption. And on the other hand, given three points $a < b < c$ in $X$, $X\setminus\{b\} = (-\infty, b) \cup (b, \infty)$
T589	Indiscrete	Simply connected	Every function from some topological space to an indiscrete space $X$ is continuous. Therefore $X$ is Pa…
T590	Simply connected	Path connected	By definition.
T591	Has a generic point	Contractible	By assumption, there exists a generic point $p \in X$. We argue that $F : X \times [0, 1] \to X$, defined by
T592	Has a generic point	Separable	If $p$ is a generic point, then $\{p\}$ is a countable dense set.
T593	Has a generic point	Hyperconnected	By assumption, there exists a generic point $p \in X$. Since every nonempty open subset contains $p$ and $\overline{\{p\}} = X$
T594	Quasi-sober ∧ Hyperconnected ∧ ¬Empty	Has a generic point	By definition.
T595	Has a generic point ∧ $R_0$	Indiscrete	Suppose $X$ has a generic point $p$. This means that $\overline{\{p\}} = X$. By definition of $R_0$, it follows that $\overline{\{p\}}$
T596	Has a generic point ∧ Homogeneous	Indiscrete	If $X$ is homogeneous and has a generic point, then every point is a generic point. It immediately fol…
T597	Hyperconnected ∧ Has an isolated point	Has a generic point	In a hyperconnected space, every open set is dense, so an open singleton contains a generic point.
T598	Has a point with a unique neighborhood	Ultraconnected	Every nonempty closed set must contain any point whose only neighborhood is $X$.
T599	Has a point with a unique neighborhood	Ultraparacompact	If the only neighborhood of a point is $X$, any open cover must include $X$ and therefore admits a refin…
T600	Has a point with a unique neighborhood	Sequentially compact	Any sequence converges to all points whose only neighborhood is $X$.
T601	Has a point with a unique neighborhood	Contractible	By assumption, there exists a point $p \in X$ whose only neighborhood is $X$. We argue that $F : X \times [0, 1] \to X$, def…
T602	Path connected ∧ Has a dispersion point ∧ ¬Has a generic point	Has a point with a unique neighborhood	Shown in David Gao's answer to Math StackExchange 4993007.
T603	Separable	Density $\leq\mathfrak c$	Follows as every countable set has cardinality $\leq\mathfrak c$.
T604	Cardinality $\leq\mathfrak c$	Density $\leq\mathfrak c$	Follows as every space is a dense subset of itself.
T605	Almost discrete	Sober	See Math StackExchange 5012490.
T606	Almost discrete ∧ Compact	Spectral	See Math StackExchange 5012490.
T607	Almost discrete ∧ $T_1$	Strongly KC	Suppose $X$ satisfies the hypotheses, and let $p$ be the non-isolated point of $X$.
T608	Totally disconnected	Sober	Suppose that $C$ is a Hyperconnected non-empty closed subset of $X$. As $X$ is Totally disconnected, $C$ is …
T609	Has a generic point ∧ $T_0$ ∧ Alexandrov	Has an isolated point	See Math StackExchange 4994238.
T610	Čech complete	$T_{3 \frac{1}{2}}$	By the definition of Čech complete.
T611	Embeddable in $\mathbb R$ ∧ Connected ∧ ¬Empty	Contractible	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 $x\in X$
T612	Alexandrov ∧ Lindelöf	Markov Rothberger	For an Alexandrov space, the set of smallest neighborhoods of points is an open cover. By Lindelöf, …
T613	Has a point with a unique neighborhood	Markov Rothberger	A singleton whose only neighborhood is the entire space is a countable set as needed for Markov Roth…
T614	Radial ∧ Has points $G_\delta$	Fréchet Urysohn	See Math StackExchange 4993377.
T615	Developable ∧ $T_0$	Semimetrizable	See Math StackExchange 5041858.
T616	Semimetrizable	First countable	The balls $B_d(x,1/n)$ for $n=1,2,\dots$ form a neighborhood base at the point $x$.
T617	Has a generic point	Strongly Choquet	Regardless of how Player 2 plays, the intersection of $U_n$ will always contain any generic point of t…
T618	Ultraconnected ∧ Has a cut point	Has a closed point	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…
T619	Hyperconnected ∧ Has a cut point	Has an isolated point	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 h…
T620	Hereditarily connected	¬Has a cut point	Clear from the definitions, since removing a point does not disconnect the space.
T621	Has a closed point	¬Empty	By definition.
T622	Ultraconnected ∧ Shrinking ∧ ¬Empty	Has a point with a unique neighborhood	Assume to the contrary that $X$ is not Has a point with a unique neighborhood. The intersection of $\overline{\{x\}}$
T623	Čech complete	$k_1$-space	See Theorem 3.9.5 of zbMATH 0684.54001.
T624	Semimetrizable	Symmetrizable	Immediate from definitions.
T625	Symmetrizable	$T_1$	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.
T626	Symmetrizable	Sequential	Let $A \subseteq X$ be sequentially closed and $x \in X \setminus A$. It suffices to show $d(x,A)>0$. Assume to the contrary that …
T627	Symmetrizable ∧ Has countable extent	Hereditarily Lindelöf	See Theorem 1.2.5 in DOI 10.2991/978-94-6239-216-8. Note that while the book assumes $T_2$, the proo…
T628	Symmetrizable ∧ Fréchet Urysohn ∧ US	Semimetrizable	See the third part of Math StackExchange 5016336.
T629	GO-space	Monotonically normal	See Corollary 5.6 in zbMATH 0269.54009. Also here for the case of a LOTS.
T630	$T_1$ ∧ ¬Empty	Has a closed point	Every singleton subset of a $T_1$ space is closed.
T631	Has a cut point ∧ ¬Has a closed point	Has an isolated point	See Theorem 3.2 of DOI 10.1090/S0002-9939-99-04839-X.
T632	Ultraconnected ∧ Has a closed point	Has a point with a unique neighborhood	Since $X$ is ultraconnected, a closed singleton must be a subset of every nonempty closed set. So a cl…
T633	Homogeneous ∧ Has a closed point	$T_1$	Homeomorphisms preserve closed sets.
T634	Almost discrete	Has a closed point	Every singleton subset except for one is open. The union of the open singletons is therefore the com…
T635	Cut point space	Has a cut point	By definition.
T636	Has a cut point ∧ Homogeneous	Cut point space	Cut points are preserved under homeomorphism, so every point is a cut point.
T637	Cut point space	$T_0$	Removing one of a pair of topologically indistinguishable points can't change connectivity, so any c…
T638	Cut point space	¬Compact	Corollary 3.10 in DOI 10.1090/s0002-9939-99-04839-x.
T639	Cut point space	Has a closed point	By Theorem 3.7 in DOI 10.1090/s0002-9939-99-04839-x, a cut point space has infinitely many closed po…
T640	Para-Lindelöf ∧ Weakly Lindelöf	Lindelöf	See Math StackExchange 5005198.
T641	Locally relatively compact ∧ $T_0$ ∧ ¬Empty	Has a closed point	Since the space $X$ is nonempty and Locally relatively compact, it has a nonempty closed Compact subse…
T642	¬Empty ∧ Čech complete ∧ ¬Has an isolated point	¬Rothberger	See Math StackExchange 5004551.
T643	$T_1$ ∧ First countable ∧ Almost discrete	Completely metrizable	Shown in Math StackExchange 4987443.
T644	Almost discrete	Strongly Choquet	Shown in Math StackExchange 4987327.
T645	Čech complete ∧ ¬Empty	Strongly Choquet	For a $T_2$ Compact space, Player 2 may win by choosing $V_n$ with $\overline{V_n}\subseteq U_n$.
T646	Strongly Choquet	¬Empty	By definition.
T647	Locally finite ∧ ¬Empty	Strongly Choquet	During round 0, Player 2 chooses any finite neighborhood $V_0$ of $x_0$ that is contained in $U_0$. Then $\{U_n\}_{n \geq 1}$
T648	Strongly collectionwise normal	Collectionwise normal	Shown in zbMATH 0046.16403; see also Math StackExchange 345476.
T649	Door ∧ ¬Empty	Has a closed point	If a Door space has no closed points, then every singleton is open, which implies that the space is …
T650	Noetherian	Compact	By definition.
T651	Noetherian	Locally connected	See Lemma 5.9.6 from the Stacks project, which makes a stronger claim.
T652	Noetherian	Locally compact	For each $x \in X$, the set of open neighborhoods of $x$ is a local basis of compact neighborhoods, since ev…
T653	Paracompact	Para-Lindelöf	Immediate from definitions.
T654	Lindelöf	Para-Lindelöf	Immediate from definitions.
T655	Para-Lindelöf	Meta-Lindelöf	Immediate from definitions.
T656	Ultraconnected	Strongly collectionwise normal	$X^2$ is the only neighborhood of the diagonal, and $X^2\circ X^2=X^2$.
T657	Noetherian	Hereditarily Lindelöf	By definition, because a compact space is Lindelöf.
T658	Compact ∧ Partition topology	Noetherian	The ascending chain condition on open sets holds since there are only finitely many open sets.
T659	Noetherian ∧ $R_1$	Partition topology	First observe that a Noetherian $T_2$ space is Discrete, since every subset is compact, hence closed…
T660	Almost discrete ∧ Noetherian	Finite	Let $p \in X$ be the only non-isolated point. The subspace $X \setminus \{p\}$ is Discrete and Noetherian, hence Finite …
T661	Has countable spread ∧ Normal	Collectionwise normal	Given a discrete family $(F_i)_{i \in I}$ of nonempty closed sets in $X$, take a point $x_i\in F_i$ for each $i$. Th…
T662	Has countable extent ∧ $T_4$	Collectionwise normal	Given a discrete family $(F_i)_{i \in I}$ of nonempty closed sets in $X$, take a point $x_i\in F_i$ for each $i$. Ev…
T663	Has countable spread ∧ Completely normal	Hereditarily collectionwise normal	Has countable spread is a hereditary property and so is Completely normal. Thus, the result follows …
T664	Monotonically normal	Countably paracompact	See Theorems 1.2 and 2.3 of Chapter 17 of DOI 10.1016/C2009-0-12309-7.
T665	Hereditarily collectionwise normal	Collectionwise normal	By definition.
T666	Hereditarily collectionwise normal	Completely normal	By Collectionwise normal ⇒ Normal, every subspace of $X$ is Normal.
T667	Monotonically normal	$T_5$	If $X$ is Monotonically normal, it is Normal from part (i) of Definition (1), and is also $T_1$, hence…
T668	Monotonically normal	Hereditarily collectionwise normal	See Theorem 3.1 in zbMATH 0269.54009.
T669	Metrizable	Monotonically normal	Let $(X,d)$ be a metric space. Write $B(x,\epsilon)$ for the open ball centered at $x$ with radius $\epsilon$.
T670	Countably paracompact ∧ Hyperconnected	Countably compact	Let $\mathscr U$ be a countable open cover of $X$. Let $\mathscr V$ be a locally finite open refinement of $\mathscr U$ covering $X$. Giv…
T671	Almost discrete ∧ ¬Semiregular	Extremally disconnected	By Math StackExchange 5016854, an almost discrete space that is not semiregular must be the disjoint…
T672	Almost discrete ∧ ¬Semiregular	Locally finite	By Math StackExchange 5016854, an almost discrete space that is not semiregular must be the disjoint…
T673	Basically disconnected ∧ $T_{3 \frac{1}{2}}$	Cozero complemented	Suppose $X$ satisfies the hypotheses. Then $X$ is $T_{3 \frac{1}{2}}$. And if $U$ is a cozero set, its cl…
T674	Almost discrete ∧ $T_1$	Monotonically normal	$X$ is $T_1$.
T675	Hereditarily connected ∧ Compact ∧ Sober ∧ Cardinality $\lt\mathfrak c$	Spectral	See part 4 of Math StackExchange 5007860.
T676	Has a point with a unique neighborhood ∧ Locally injectively path connected	Injectively path connected	$X$ has a point $p$ whose only neighborhood is $X$. As $X$ is Locally injectively path connected, $p$ has a In…
T677	Locally finite	Has a $\sigma$-locally finite network	Let $\mathcal{N} = \{\{x\}:x\in X\}$. Then $\mathcal{N}$ is a locally finite network for $X$.
T678	Anticompact ∧ Has a $\sigma$-locally finite network	Has a $\sigma$-locally finite $k$-network	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 f…
T679	Hyperconnected ∧ ¬Has a point with a unique neighborhood ∧ Has a closed point	¬Pseudonormal	Let $p\in X$ be a closed point and find open $U$ with $p\in U$ and $U\neq X$. If $X$ were Pseudonormal then there would…
T680	Weakly locally compact ∧ $R_1$ ∧ ¬Empty	Strongly Choquet	A space $X$ is Strongly Choquet if and only if the Kolmogorov quotient of $X$ is Strongly Choquet.
T681	Ultraparacompact	Fully normal	Let $\mathcal V$ be an open cover of $X$. Since $X$ is Ultraparacompact, $\mathcal V$ has an open refinement $\mathcal U$ that is a part…
T682	Noetherian ∧ Has a generic point ∧ $T_0$	Fixed point property	See Math StackExchange 5014330.
T683	Almost discrete ∧ Sequential	Fréchet Urysohn	See Math StackExchange 5016263.
T684	Locally $n$-Euclidean ∧ Connected ∧ $T_2$	Homogeneous	See Math StackExchange 5015469. Note that the proof does not rely on Second countable.
T685	Topological $n$-manifold ∧ Compact ∧ Has multiple points	¬Contractible	Let $M$ be a Compact Contractible $n$-manifold. Since Contractible spaces are Connected
T686	Metacompact ∧ Collectionwise normal ∧ Regular	Paracompact	This is the Michael-Nagami theorem.
T687	Noetherian ∧ Has a countable $k$-network	Second countable	If $X$ is Noetherian, then each pseudobase necessarily contains every open set. If, in addition, $X$ is …
T688	$\sigma$-compact ∧ Homogeneous ∧ KC ∧ ¬Meager	Weakly locally compact	Suppose $X$ satisfies the hypotheses. By $\sigma$-compact and KC, $X$ is a countable union of compact cl…
T689	Locally countable ∧ Locally pseudometrizable ∧ Locally connected	Partition topology	By the known result Countable ∧ Pseudometrizable ∧ Connected ⇒ Indiscrete
T690	KC ∧ Hereditarily Lindelöf	Strongly KC	Let $A \subseteq X$ be a Countably compact subset. It must be Lindelöf because $X$ is Hereditarily Lindelöf.
T691	$T_2$ ∧ Extremally disconnected ∧ P-space ∧ Cardinality less than every measurable cardinal	Discrete	See Math StackExchange 5022632.
T692	Connected ∧ Strongly paracompact	Lindelöf	Follows from the fact that every star-finite open cover of a connected space is countable. See MathO…
T693	Extremally disconnected	Basically disconnected	Evident, as every cozero set is an open set.
T694	Basically disconnected ∧ Perfectly normal	Extremally disconnected	Follows from the definitions since in a Perfectly normal space open sets are cozero sets.
T695	P-space	Basically disconnected	Every cozero set is open and an $F_\sigma$ set, which is closed in a P-space. Hence every cozero set is clo…
T696	Strongly connected	Basically disconnected	If $X$ is Strongly connected, the only cozero sets are $\emptyset$ and $X$, which are clopen.
T697	Basically disconnected ∧ Completely regular	Zero dimensional	Suppose $X$ satisfies the hypotheses. Given an open neighborhood $U$ of a point $p$, there is a continuous…
T698	Weakly locally compact ∧ Para-Lindelöf ∧ Connected	Lindelöf	Call a space weakly locally Lindelöf if every point has a neighborhood that is Lindelöf.
T699	Semimetrizable	$G_\delta$ space	Let $A \subseteq X$ be a closed subset.
T700	Door ∧ ¬Anticompact	Almost discrete	See Math StackExchange 4995169.
T701	Door ∧ Connected ∧ ¬Almost discrete	Hyperconnected	According to Theorem 1 in zbMATH 1400.39025, there are three types of connected door spaces:
T702	Locally injectively path connected ∧ ¬Discrete	¬Biconnected	Suppose $X$ is Locally injectively path connected and not Discrete. There is at least one point with a…
T703	Arc connected	Injectively path connected	Evident from the definitions.
T704	Locally arc connected	Locally injectively path connected	Evident from the definitions.
T705	Arc connected	$T_1$	Given distinct points $a,b\in X$, there is an arc $f:[0,1]\to X$ joining $a$ to $b$. The image $A=f([0,1])$ is homeo…
T706	Locally arc connected	$T_1$	Each point $x\in X$ has an open neighborhood $U$ that is Arc connected. By Arc connected ⇒ $T_1$, $U$ is $T_1$.…
T707	Proximal	Collectionwise normal	See Theorem 10 of DOI 10.1016/j.topol.2014.06.014.
T708	Proximal	Countably paracompact	See Corollary 4 of DOI 10.1016/j.topol.2014.06.014.
T709	Path connected ∧ US	Injectively path connected	See Math StackExchange 4862260.
T710	Developable	First countable	Evident from the definitions.
T711	Developable	$R_0$	See Lemma in Math StackExchange 5041859.
T712	Pseudometrizable	Developable	Letting $\mathscr U_n$ be the collection of all open balls of radius $1/n$ gives a development for $X$.
T713	Collectionwise normal ∧ Developable	Pseudometrizable	This is essentially one of Bing's metrization theorems.
T714	Lindelöf ∧ Developable	Second countable	See Math StackExchange 148565.
T715	Moore space	Developable	By definition.
T716	Moore space	$T_3$	By definition.
T717	Developable ∧ $T_3$	Moore space	By definition.
T718	Countable ∧ $T_1$	Has a $G_\delta$-diagonal	Any set in a Countable $T_1$ space is a $G_\delta$ set, since its complement is an $F_\sigma$ set as a countable u…
T719	Developable ∧ $T_0$	Has a $G_\delta$-diagonal	Suppose the sequence of open covers $\mathscr U_1,\mathscr U_2,\dots$ is a development for $X$. For each $x\in X$, the collection …
T720	Has a $G_\delta$-diagonal	Has points $G_\delta$	Suppose the diagonal $\Delta$ is a $G_\delta$ set in $X^2$. Intersecting with $\{x\}\times X$ shows that $\{x\}$ is a $G_\delta$ set in $X$.…
T721	LOTS ∧ Has a $G_\delta$-diagonal	Metrizable	See Theorem 2.3 in zbMATH 0555.54015.
T722	LOTS ∧ Has multiple points	¬Biconnected	Suppose $X$ satisfies the hypotheses. If $X$ is not Connected, it is not Biconnected. Otherwise, since $X$