hcju/proofwikips huggingface dataset:
Finite Complement Space is Locally Connected
Let $T = \struct {S, \tau}$ be a finite complement topology on an infinite set $S$. Then $T$ is a locally connected space.
Mapping Induces Partition on Domain
Let $f: S \to T$ be a mapping. Let $F$ be defined as: :$F = \set {\map {f^{-1} } x: x \in T}$ where $\map {f^{-1} } x$ is the preimage of $x$. Then $F$ is a partition of $S$.
Countable Finite Complement Space is not Path-Connected
Let $T = \struct {S, \tau}$ be a countable finite complement topology. Then $T$ is not path-connected.
Countable Finite Complement Space is not Locally Path-Connected
Let $T = \struct {S, \tau}$ be a countable finite complement topology. Then $T$ is not locally path-connected.
Arc-Connectedness in Uncountable Finite Complement Space
Let $T = \struct {S, \tau}$ be a finite complement topology on a uncountable set $S$. If the Continuum Hypothesis is accepted as true, then: :$T$ is arc-connected :$T$ is locally arc-connected.
Countable Complement Topology is Topology
Let $T = \struct {S, \tau}$ be a countable complement space. Then $\tau$ is a topology on $T$.
Countable Complement Topology is Expansion of Finite Complement Topology
Let $T = \struct {S, \tau}$ be the countable complement topology on an infinite set $S$. Let $T' = \struct {S, \tau'}$ be the finite complement topology on the same infinite set $S$. Then $\tau$ is an expansion of $\tau'$.
Countable Complement Space is T1
Let $T = \struct {S, \tau}$ be a countable complement topology. Then $T$ is a $T_1$ (Fréchet) space.
Uncountable Subset of Countable Complement Space Intersects Open Sets
Let $T = \struct {S, \tau}$ be a countable complement topology on an uncountable set $S$. Let $H \subseteq S$ be an uncountable subset of $S$. Then the intersection of $H$ with any non-empty open set of $T$ is uncountable.
Countable Complement Space is not T2
Let $T = \struct {S, \tau}$ be a countable complement topology on an uncountable set $S$. Then $T$ is not a $T_2$ (Hausdorff) space.
Countable Complement Space is Irreducible
Let $T = \struct {S, \tau}$ be a countable complement topology on an uncountable set $S$. Then $T$ is an irreducible space.
Countable Complement Space is not T3, T4 or T5
Let $T = \struct {S, \tau}$ be a countable complement topology on an uncountable set $S$. Then $T$ is not a $T_3$ space, $T_4$ space or $T_5$ space.
Closed Unit Interval is not Countably Infinite Union of Disjoint Closed Sets
Let $\mathbb I = \closedint 0 1$ be the closed unit interval. Then $\mathbb I$ cannot be expressed as the union of a countably infinite set of pairwise disjoint closed sets.
Compact Sets in Countable Complement Space
Let $T = \struct {S, \tau}$ be a countable complement topology on an uncountable set $S$. Then the compact sets of $T$ are exactly the finite subsets of $S$.
Countable Complement Space is not Sigma-Compact
Let $T = \struct {S, \tau}$ be a countable complement topology on an uncountable set $S$. Then $T$ is not a $\sigma$-compact space.
Countable Complement Space is Lindelöf
Let $T = \struct {S, \tau}$ be a countable complement topology on an uncountable set $S$. Then $T$ is a Lindelöf space.
Countable Complement Space is not First-Countable
Let $T = \struct {S, \tau}$ be a countable complement topology on an uncountable set $S$. Then $T$ is not a first-countable space.
Limit Points of Countable Complement Space
Let $T = \struct {S, \tau}$ be a countable complement space. Let $H \subseteq S$ be an uncountable subset of $S$. Then every point of $S$ is a limit point of $H$. Let $H \subseteq S$ be an countable or finite subset of $S$. Then $H$ contains all its limit points.
Countable Complement Space is not Separable
Let $T = \struct {S, \tau}$ be a countable complement topology on an uncountable set $S$. Then $T$ is not a separable space.
Countable Complement Space Satisfies Countable Chain Condition
Let $T = \struct {S, \tau}$ be a countable complement topology on an uncountable set $S$. Then $T$ satisfies the countable chain condition.
Countable Complement Space is Locally Connected
Let $T = \struct {S, \tau}$ be a countable complement topology on an uncountable set $S$. Then $T$ is a locally connected space.
Countable Complement Space is not Weakly Countably Compact
Let $T = \struct {S, \tau}$ be a countable complement topology on an uncountable set $S$. Then $T$ is not weakly countably compact.
Saturated Implies Universal
Let $\kappa$ be an infinite cardinal. Let $\mathcal{M}$ be a model of the $\mathcal{L}$-theory $T$. If $\mathcal{M}$ is $\kappa$-saturated, then it is $\kappa^+$-universal, where $\kappa^+$ is the successor cardinal of $\kappa$.
Double Pointed Countable Complement Topology fulfils no Separation Axioms
Let $T = \struct {S, \tau}$ be a countable complement topology on an uncountable set $S$. Let $T \times D$ be the double pointed topology on $T$. Then $T \times D$ is not a $T_0$ space, $T_1$ space, $T_2$ space, $T_3$ space, $T_4$ space or $T_5$ space.
Double Pointed Countable Complement Topology is Weakly Countably Compact
Let $T = \struct {S, \tau}$ be a countable complement topology on an uncountable set $S$. Let $T \times D$ be the double pointed topology on $T$. Then $T \times D$ is weakly countably compact.
Open Extension of Double Pointed Countable Complement Topology is T4 Space
Let $T = \struct {S, \tau_S}$ be a countable complement topology on an uncountable set $S$. Let $D = \struct {\set {0, 1}, \tau_D}$ be the indiscrete topology on two points. Let $T \times D$ be the double pointed topology on $T$. Let $\paren {T \times D}^*_{\bar p}$ be the open extension topology on $S \times \set {0, 1} \cup \set p$ where $p \notin S \times \set {0, 1}$. Then $\paren {T \times D}^*_{\bar p}$ is a $T_4$ space, and no other separation axioms are fulfilled. That is, $\paren {T \times D}^*_{\bar p}$ is not a $T_0$ space, $T_1$ space, $T_2$ space, $T_3$ space or $T_5$ space.
Saturated Models of same Cardinality are Isomorphic
Let $T$ be an $\LL$-theory. Let $\kappa$ be an infinite cardinal. If $\MM$ and $\NN$ are saturated models of $T$ and the cardinality of the universes of $\MM$ and $\NN$ are both $\kappa$, then $\MM$ and $\NN$ are isomorphic.
Infinite Ramsey's Theorem
Let $k, n \in \N$. For any set $S$, let $S^{\left({n}\right)}$ denote the set $\left\{ {\left\{ {s_1, \ldots, s_n}\right\}: \text{each } s_i \in S}\right\}$ of cardinality $n$ subsets of $S$. Let $X$ be an infinite set. Then: :for every partition $P$ of $X^{\left({n}\right)}$ into $k$ many components :there is an infinite subset $Y \subseteq X$ such that: :each member of $Y^{\left({n}\right)}$ is in the same component of $P$.
Compact Complement Topology is T1
Let $T = \struct {S, \tau}$ be a compact complement space. Then $T$ is a $T_1$ (Fréchet) space.
Compact Complement Topology is Irreducible
Let $T = \struct {\R, \tau}$ be the compact complement topology on $\R$. Then $T$ is an irreducible space.
Weierstrass-Casorati Theorem
Let $f$ be a holomorphic function defined on the open ball $\map B {a, r} \setminus \set a$. Let $f$ have an essential singularity at $a$. Then: :$\forall s < r: f \sqbrk {\map B {a, s} \setminus \set a}$ is a dense subset of $\C$. {{Disambiguate|Definition:Dense}}
Compact Complement Topology is Locally Connected
Let $T = \struct {\R, \tau}$ be the compact complement topology on $\R$. Then $T$ is a locally connected space.
Compact Complement Topology is Compact
Let $T = \struct {\R, \tau}$ be the compact complement topology on $\R$. Then $T$ is a compact space.
Countable Local Basis in Compact Complement Topology
Let $T = \struct {\R, \tau}$ be the compact complement topology on $\R$. Let $p \in \R$. Then sets of the form: :$\openint {-\infty} {-n} \cup \openint {p - \dfrac 1 n} {p + \dfrac 1 n} \cup \openint n \infty$ form a countable local basis for $p$.
Compact Complement Topology is First-Countable
Let $T = \struct {\R, \tau}$ be the compact complement topology on $\R$. Then $T$ is a first-countable space.
Compact Complement Topology is Second-Countable
Let $T = \struct {\R, \tau}$ be the compact complement topology on $\R$. Then $T$ is a second-countable space.
Compact Complement Topology is Coarser than Euclidean Topology
Let $T = \struct {\R, \tau}$ be the compact complement topology on $\R$. Then $\tau$ is coarser than the usual (Euclidean) topology in $\R$.
Fort Topology is Topology
Let $T = \struct {S, \tau_p}$ be a Fort space. Then $\tau_p$ is a topology on $T$.
Fort Space is Excluded Point Space with Finite Complement Space
Let $T = \struct {S, \tau_p}$ be a Fort space on an infinite set $S$. Then $\tau_p$ is the minimal topology that is generated by the excluded point topology and the finite complement topology.
Fort Space is T1
Let $T = \struct {S, \tau_p}$ be a Fort space on an infinite set $S$. Then $T$ is a $T_1$ (Fréchet) space.
Fort Space is T5
Let $T = \struct {S, \tau_p}$ be a Fort space on an infinite set $S$. Then $T$ is a $T_5$ space.
Fort Space is Completely Normal
Let $T = \struct {S, \tau_p}$ be a Fort space on an infinite set $S$. Then $T$ is a completely normal space. Consequently, $T$ satisfies all weaker separation axioms.
Exchange Principle
Let $D$ be a strongly minimal set in $\mathcal M$. Let $A$ be a subset of $D$. Let $b, c \in D$. If $b$ is algebraic over $A \cup \left\{ {c}\right\}$ but not over $A$, then $c$ is algebraic over $A \cup \left\{ {b}\right\}$.
Uncountable Fort Space is not Perfectly Normal
Let $T = \struct {S, \tau_p}$ be a Fort space on an uncountable set $S$. Then $T$ is not a perfectly normal space.
Countable Fort Space is Perfectly Normal
Let $T = \struct {S, \tau_p}$ be a Fort space on a countably infinite set $S$. Then $T$ is a perfectly normal space.
Fort Space is Compact
Let $T = \struct {S, \tau_p}$ be a Fort space on an infinite set $S$. Then $T$ is a compact space.
Boubaker's Theorem/Proof of Uniqueness
Let $\left({R, +, \circ}\right)$ be a commutative ring. Let $\left({D, +, \circ}\right)$ be an integral subdomain of $R$ whose zero is $0_D$ and whose unity is $1_D$. Let $X \in R$ be transcendental over $D$. Let $D \left[{X}\right]$ be the ring of polynomial forms in $X$ over $D$. Consider the following properties: {{begin-eqn}} {{eqn | n = 1       | l = \sum_{k \mathop = 1}^N {p_n \left({0}\right)}       | r = -2N }} {{eqn | n = 2       | l = \sum_{k \mathop = 1}^N {p_n \left({\alpha_k}\right)}       | r = 0 }} {{eqn | n = 3       | l = \left.{\sum_{k \mathop = 1}^N \frac {\mathrm d p_x \left({x}\right)} {\mathrm d x} }\right\vert_{x \mathop = 0}       | r = 0 }} {{eqn | n = 4       | l = \left.{\sum_{k \mathop = 1}^N \frac {\mathrm d {p_n}^2 \left({x}\right)} {\mathrm d x^2} }\right\vert_{x \mathop = 0}       | r = \frac 8 3 N \left({N^2 - 1}\right) }} {{end-eqn}} where, for a given positive integer $n$, $p_n \in D \left[{X}\right]$ is a non-null polynomial such that $p_n$ has $N$ roots $\alpha_k$ in $F$. Then the subsequence $\left \langle {B_{4 n} \left({x}\right)}\right \rangle$ of the Boubaker polynomials is the unique polynomial sequence of $D \left[{X}\right]$ which verifies simultaneously the four properties $(1) - (4)$.
Generating Function for Boubaker Polynomials
The Boubaker polynomials, defined as:    :$B_n \left({x}\right) = \begin{cases} 1 & : n = 0 \\ x & : n = 1 \\ x^2 + 2 & : n = 2 \\ x B_{n-1} \left({x}\right) - B_{n-2} \left({x}\right) & : n > 2 \end{cases}$ have as an ordinary generating function: : $\displaystyle f_{B_n, \operatorname{ORD}} \left({x, t}\right) = \sum_{n \mathop = 0}^{\infty} B_n \left({x}\right) t^n = \frac {1 + 3 t^2} {1 + t \left({t - x}\right)}$
Exponential Generating Function for Boubaker Polynomials
The Boubaker polynomials, defined as:            :$B_n \left({x}\right) = \begin{cases} 1 & : n = 0 \\ x & : n = 1 \\ x^2+2 & : n = 2 \\ x B_{n-1} \left({x}\right) - B_{n-2} \left({x}\right) & : n > 2 \end{cases}$ have as an exponential generating function: : $\displaystyle f_{B_n, \operatorname{EXP}} \left({x, t}\right) = \sum_{n=0}^\infty B_n \left({x}\right) \frac{t^n}{n!} = 4 x e^{t \frac x 2} \frac {\sin \left({t \sqrt {1 - \left({\frac x 2}\right)^2}}\right)} {\sqrt {1 - \left({\frac x 2}\right)^2}} - 2 e^{t \frac x 2} \cos \left({t \sqrt {1 - \left({\frac x 2}\right)^2}}\right) - 3$
Clopen Points in Fort Space
Let $T = \struct {S, \tau_p}$ be a Fort space on an infinite set $S$. Let $q \in S: q \ne p$. Then $\set q$ is both open and closed in $T$. $\set p$ itself, on the other hand, is closed but not open.
Fort Space is Totally Separated
Let $T = \struct {S, \tau_p}$ be a Fort space on an infinite set $S$. Then $T$ is a totally separated space.
Fort Space is Zero Dimensional
Let $T = \struct {S, \tau_p}$ be a Fort space on an infinite set $S$. Then $T$ is a zero dimensional space.
Formula and its Negation Cannot Both Cause Forking
Let $T$ be a complete $\mathcal{L}$-theory. Let $\mathfrak{C}$ be a monster model for $T$. Let $A\subseteq B$ be subsets of the universe of $\mathfrak{C}$. Let $\pi(\bar x)$ be an $n$-type over $B$. If $\pi$ does not fork over $A$, then for any formula $\phi(\bar x, \bar b)$, either $\pi \cup \{\phi\}$ or $\pi \cup \{\neg\phi\}$ does not fork over $A$.
Forking is Local
Let $T$ be a complete $\mathcal{L}$-theory. Let $\mathfrak{C}$ be a monster model for $T$. Let $A\subseteq B$ be subsets of the universe of $\mathfrak{C}$. Let $\pi(\bar x)$ be an $n$-type over $B$. $\pi$ forks over $A$ if and only if a finite subset of $\pi$ forks over $A$.
Subset of Finite Set is Finite
Let $X$ be a finite set. If $Y$ is a subset of $X$, then $Y$ is also finite.
Statements Equivalent to Non-Dividing Type
Let $T$ be a complete $\mathcal L$-theory. Let $\mathfrak C$ be a monster model for $T$. Let $A$ be a subset of the universe of $\mathfrak C$. The following are equivalent: :$(1): \quad$ $\operatorname{tp}(\bar c / A,\bar{b})$ does not divide over $A$. :$(2): \quad$ For every $\{\bar b_i : i \in I\}$ containing $\bar b$ which is order indiscernible over $A$, there is some $\bar c'$ with $\operatorname{tp}(\bar c' / A,\bar b) = \operatorname{tp}(\bar c / A,\bar b)$ such that $\{\bar b_i : i \in I\}$ is order indiscernible over $A,\bar c'$. :$(3): \quad$ For every $\{\bar b_i : i \in I\}$ containing $\bar b$ which is order indiscernible  over $A$, there is some $A,\bar b$-automorphism $f$ such that $\{f(\bar b_i) : i \in I\}$ is order indiscernible over $A,\bar c$.
Fortissimo Topology is Topology
Let $T = \struct {S, \tau_p}$ be a Fortissimo space. Then $\tau_p$ is a topology on $T$.
Fortissimo Space is T1
Let $T = \struct {S, \tau_p}$ be a Fortissimo space. Then $T$ is a $T_1$ (Fréchet) space.
Fortissimo Space is Excluded Point Space with Countable Complement Space
Let $T = \struct {S, \tau_p}$ be a Fortissimo space. Then $\tau_p$ is the minimal topology that is generated by the excluded point topology and the countable complement topology.
Fortissimo Space is T5
Let $T = \struct {S, \tau_p}$ be a Fortissimo space on an infinite set $S$. Then $T$ is a $T_5$ space.
Fortissimo Space is Lindelöf
Let $T = \struct {S, \tau_p}$ be a Fortissimo space. Then $T$ is a Lindelöf space.
Fortissimo Space is not Sigma-Compact
Let $T = \struct {S, \tau}$ be a Fortissimo space. Then $T$ is not a $\sigma$-compact space.
Fortissimo Space is not Separable
Let $T = \struct {S, \tau_p}$ be the fortissimo space on an uncountable set $S$. Then $T$ is not a separable space.
Fortissimo Space is not First-Countable
Let $T = \struct {S, \tau_p}$ be a Fortissimo space. Then $T$ is not a first-countable space.
Modified Fort Topology is Topology
Let $T = \struct {S, \tau_{a, b} }$  be a modified Fort space. Then $\tau_{a, b}$ is a topology on $S$.
Subset of Naturals is Finite iff Bounded
Let $X$ be a subset of the natural numbers $\N$. Then $X$ is finite  it is bounded.
Arens-Fort Space is Expansion of Countable Fort Space
Let $T = \struct {S, \tau}$ be the Arens-Fort space, where $S = \Z_{\ge 0} \times \Z_{\ge 0}$. Let $T_p = \struct {S, \tau_p}$ be the Fort space on $S$ where $p = \left({0, 0}\right)$. Then $\tau$ is an expansion of $\tau_p$. Furthermore, $S$ is countably infinite, so $T_p$ is a countable Fort space.
Arens-Fort Space is T1
Let $T = \struct {S, \tau}$ be the Arens-Fort space. Then $T$ is a $T_1$ (Fréchet) space.
Arens-Fort Space is Completely Hausdorff
Let $T = \struct {S, \tau}$ be the Arens-Fort space. Then $T$ is a $T_{2 \frac 1 2}$ (completely Hausdorff) space.
Arens-Fort Space is T5
Let $T = \struct {S, \tau}$ be the Arens-Fort space. Then $T$ is a $T_5$ space.
Arens-Fort Space is Completely Normal
Let $T = \struct {S, \tau}$ be the Arens-Fort space. Then $T$ is a completely normal space. Consequently, $T$ satisfies all weaker separation axioms.
Arens-Fort Space is not First-Countable
Let $T = \struct {S, \tau}$ be the Arens-Fort space. Then $T$ is not a first-countable space.
Countable Space is Separable
Let $T = \struct {S, \tau}$ be a topological space where $S$ is a countable set. Then $T$ is a separable space.
Arens-Fort Space is Separable
Let $T = \struct {S, \tau}$ be the Arens-Fort space. Then $T$ is a separable space.
Arens-Fort Space is Sigma-Compact
Let $T = \left({S, \tau}\right)$ be the Arens-Fort space. Then $T$ is a $\sigma$-compact space.
Arens-Fort Space is Lindelöf
Let $T = \struct {S, \tau}$ be the Arens-Fort space. Then $T$ is a Lindelöf space.
Arens-Fort Space is not Weakly Locally Compact
Let $T = \struct {S, \tau}$ be the Arens-Fort space. Then $T$ is not a weakly locally compact space.
Arens-Fort Space is not Compact
Let $T = \struct {S, \tau}$ be the Arens-Fort space. Then $T$ is not a compact space.
Arens-Fort Space is not Countably Compact
Let $T = \struct {S, \tau}$ be the Arens-Fort space. Then $T$ is not a countably compact space.
Lindelöf T3 Space is Paracompact
Let $T = \struct {S, \tau}$ be a $T_3$ space which is also Lindelöf. Then $T$ is paracompact.
Arens-Fort Space is not Connected
Let $T = \struct {S, \tau}$ be the Arens-Fort space. Then $T$ is not a connected space.
Arens-Fort Space is not Locally Connected
Let $T = \struct {S, \tau}$ be the Arens-Fort space. Then $T$ is not a locally connected space.
Clopen Points in Arens-Fort Space
Let $T = \struct {S, \tau}$ be the Arens-Fort space. Let $q \in S: q \ne \tuple {0, 0}$. Then $\set q$ is both open and closed in $T$. $\set {\tuple {0, 0} }$ itself is closed, but not open.
Neighborhood of Origin of Arens-Fort Space is Closed
Let $T = \struct {S, \tau}$ be the Arens-Fort space. Every neighborhood of $\tuple {0, 0}$ is closed in $T$.
Arens-Fort Space is Zero Dimensional
Let $T = \struct {S, \tau}$ denote the Arens-Fort space. Then $T$ is zero dimensional.
Arens-Fort Space is Totally Separated
Let $T = \struct {S, \tau}$ be the Arens-Fort space. Then $T$ is totally separated.
Isolated Points in Arens-Fort Space
Let $T = \struct {S, \tau}$ be the Arens-Fort space. Let $q \in S: q \ne \tuple {0, 0}$. Then $q$ is an isolated point of $T$.
Point in Topological Space is Open iff Isolated
Let $T = \left({S, \tau}\right)$ be a topological space. Let $x \in S$. Then $\left\{{x}\right\}$ is open in $T$  $x$ is an isolated point of $T$.
Arens-Fort Space is Scattered
Let $T = \struct {S, \tau}$ be the Arens-Fort space. Then $T$ is a scattered space.
Fortissimo Space is not Metrizable
Let $T = \struct {S, \tau_p}$ be a Fortissimo space. Then $\tau_p$ is not a metrizable topology.
Fortissimo Space is Paracompact
Let $T = \struct {S, \tau_p}$ be a Fortissimo space. Then $T$ is a paracompact space.
Clopen Points in Modified Fort Space
Let $T = \struct {S, \tau_{a, b} }$  be a modified Fort space. Then all points in $S \setminus \set {a, b}$ are both open and closed in $T$. $a$ and $b$ themselves are not open in $T$, but they are closed in $T$.
Modified Fort Space is Compact
Let $T = \struct {S, \tau_{a, b} }$ be a modified Fort space. Then $T$ is a compact space.
Modified Fort Space is T1
Let $T = \struct {S, \tau_{a, b} }$ be a modified Fort space. Then $T$ is a $T_1$ (Fréchet) space.
Modified Fort Space is not T2
Let $T = \struct {S, \tau_{a, b} }$ be a modified Fort space. Then $T$ is not a $T_2$ (Hausdorff) space.
Modified Fort Space is not T3, T4 or T5
Let $T = \struct {S, \tau_{a, b} }$ be a modified Fort space. Then $T$ is not a $T_3$ space, $T_4$ space or $T_5$ space.
Basis for Euclidean Topology on Real Number Line
Let $\R$ be the set of real numbers. Let $\BB$ be the set of subsets of $\R$ defined as: :$\BB = \set {\openint a b: a, b \in \R}$ That is, $\BB$ is the set of all open real intervals of $\R$: :$\openint a b := \set {x \in \R: a < x < b}$ Then $\BB$ forms a basis for the Euclidean topology on $\R$.
Real Number Line satisfies all Separation Axioms
Let $\struct {\R, \tau_d}$ be the the real number line with the usual (Euclidean) topology. Then $\struct {\R, \tau_d}$ fulfils all separation axioms: :$\struct {\R, \tau_d}$ is a $T_0$ (Kolmogorov) space :$\struct {\R, \tau_d}$ is a $T_1$ (Fréchet) space :$\struct {\R, \tau_d}$ is a $T_2$ (Hausdorff) space :$\struct {\R, \tau_d}$ is a semiregular space :$\struct {\R, \tau_d}$ is a $T_{2 \frac 1 2}$ (completely Hausdorff) space  :$\struct {\R, \tau_d}$ is a $T_3$ space  :$\struct {\R, \tau_d}$ is a regular space :$\struct {\R, \tau_d}$ is an Urysohn space :$\struct {\R, \tau_d}$ is a $T_{3 \frac 1 2}$ space :$\struct {\R, \tau_d}$ is a Tychonoff (completely regular) space :$\struct {\R, \tau_d}$ is a $T_4$ space :$\struct {\R, \tau_d}$ is a normal space :$\struct {\R, \tau_d}$ is a $T_5$ space  :$\struct {\R, \tau_d}$ is a completely normal space :$\struct {\R, \tau_d}$ is a perfectly $T_4$ space :$\struct {\R, \tau_d}$ is a perfectly normal space
Elementary Amalgamation Theorem
Let $\MM$ and $\NN$ be $\LL$-structures. Let $B$ be a subset of the universe of $\MM$ such that there is a partial elementary embedding $f: B \to \NN$. There is an elementary extension $\AA$ of $\MM$ and an elementary embedding $g: \NN \to \AA$ such that $\map g {\map f b} = b$ for all $b \in B$. $\array { & & \AA& & \\ &\preceq & & \nwarrow \exists g & & \!\!\!\!\!\!\!\!\!\!\!\!\!: g f = \operatorname {id}_B \\ \MM & & & &\!\!\!\!\!\!\!\! \NN \\ & \supseteq & & \nearrow f \\ & & B& & \\ }$
Baire Space iff Open Sets are Non-Meager
Let $T = \struct {S, \tau}$ be a topological space. Then $T$ is a Baire space  every non-empty open set of $T$ is non-meager in $T$.
Extension Realizing All Types
Let $\MM$ be an $\LL$-structure. Let $M$ be its universe. There is an elementary extension $\NN$ of $\MM$ such that every type over $M$ (relative to $\MM$) is realized in $\NN$.
Baire Space is Non-Meager
Let $T = \struct {S, \tau}$ be a Baire space (in the context of topology). Then $T$ is non-meager in $T$.
Separation Axioms on Double Pointed Topology/T3 Axiom
Let $T_1 = \struct {S, \tau_S}$ be a topological space. Let $D = \struct {A, \set {\O, A} }$ be the indiscrete topology on an arbitrary doubleton $A = \set {a, b}$. Let $T = \struct {T_1 \times D, \tau}$ be the double pointed topological space on $T_1$. Then $T \times D$ is a $T_3$ space  $T$ is also a $T_3$ space.
Separation Axioms on Double Pointed Topology/T4 Axiom
Let $T_1 = \struct {S, \tau_S}$ be a topological space. Let $D = \struct {A, \set {\O, A} }$ be the indiscrete topology on an arbitrary doubleton $A = \set {a, b}$. Let $T = \struct {T_1 \times D, \tau}$ be the double pointed topological space on $T_1$. Then $T \times D$ is a $T_4$ space  $T$ is also a $T_4$ space.
Separation Axioms on Double Pointed Topology/T5 Axiom
Let $T_1 = \struct {S, \tau_S}$ be a topological space. Let $D = \struct {A, \set {\O, A} }$ be the indiscrete topology on an arbitrary doubleton $A = \set {a, b}$. Let $T = \struct {T_1 \times D, \tau}$ be the double pointed topological space on $T_1$. Then $T \times D$ is a $T_5$ space  $T$ is also a $T_5$ space.
Separation Axioms on Double Pointed Topology
Let $T = \struct {S, \tau_S}$ be a topological space. Let $D = \struct {\set {a, b}, \tau_D}$ be the indiscrete topology on two points. Let $T \times D$ be the double pointed topology on $T$. Then: :$T \times D$ is not a $T_0$ (Kolmogorov) space, a $T_1$ (Fréchet) space, a $T_2$ (Hausdorff) space or a $T_{2 \frac 1 2}$ (completely Hausdorff) space. :$T \times D$ is a $T_3$ space, a $T_{3 \frac 1 2}$ space, a $T_4$ space or a $T_5$ space  $T$ is.