Closed Interval in Reals is Uncountable

Let $a, b$ be extended real numbers such that $a < b$. Then the closed interval $\set {x \in \R : a \le x \le b} \subseteq \R$ is uncountable.


Metric Space is T5

Let $M = \struct {A, d}$ be a metric space. Then $M$ is a $T_5$ space.


Metric Space is Perfectly T4

Let $M = \struct {A, d}$ be a metric space. Then $M$ is a perfectly $T_4$ space.


Metric Space fulfils all Separation Axioms

Let $M = \struct {A, d}$ be a metric space. Then $M$, considered as a topological space, fulfils all separation axioms: :$M$ is a $T_0$ (Kolmogorov) space :$M$ is a $T_1$ (Fréchet) space :$M$ is a $T_2$ (Hausdorff) space :$M$ is a semiregular space :$M$ is a $T_{2 \frac 1 2}$ (completely Hausdorff) space  :$M$ is a $T_3$ space  :$M$ is a regular space :$M$ is an Urysohn space :$M$ is a $T_{3 \frac 1 2}$ space :$M$ is a Tychonoff (completely regular) space :$M$ is a $T_4$ space :$M$ is a normal space :$M$ is a $T_5$ space  :$M$ is a completely normal space :$M$ is a perfectly $T_4$ space :$M$ is a perfectly normal space


Metric Space is Fully T4

Let $M = \struct {A, d}$ be a metric space. Then $M$ is a fully $T_4$ space.


Metric Space is Fully Normal

Let $M = \struct {A, d}$ be a metric space. Then $M$ is a fully normal space.


Metric Space is Paracompact

Let $M = \struct {A, d}$ be a metric space. Then $M$ is a paracompact space.


Metric Space is First-Countable

Let $M = \struct {A, d}$ be a metric space. Then $M$ is first-countable.


Metric Space is Separable iff Second-Countable

A metric space is separable {{iff}} it is second-countable.


Metric Space is Lindelöf iff Second-Countable

Let $M = \struct {A, d}$ be a metric space. Then $M$ is Lindelöf {{iff}} $M$ is second-countable.


Metric Space is Countably Compact iff Sequentially Compact

Let $M$ be a metric space. Then $M$ is countably compact {{iff}} $M$ is sequentially compact.


Metric Space is Weakly Countably Compact iff Countably Compact

Let $M = \struct {A, d}$ be a metric space. Then $M$ is weakly countably compact {{iff}} $M$ is countably compact.


Countably Compact Metric Space is Compact

Let $M = \struct {A, d}$ be a metric space. Let $M$ be countably compact. Then $M$ is compact.


Metric Space is Weakly Locally Compact iff Strongly Locally Compact

Let $M = \struct {A, d}$ be a metric space. Then $M$ is weakly locally compact {{iff}} $M$ is strongly locally compact.


Sequence of Implications of Metric Space Compactness Properties

Let $P_1$ and $P_2$ be compactness properties and let: :$P_1 \implies P_2$ mean: :If a metric space $M$ satsifies property $P_1$, then $M$ also satisfies property $P_2$. Then the following sequence of implications holds: {| |- | align="center" | Sequentially Compact || | align="center" | $\implies$ || | align="center" | Weakly $\sigma$-Locally Compact || | align="center" | $\implies$ || | align="center" | Weakly Locally Compact || |- | align="center" | $\Big\Updownarrow$ || | align="center" | || | align="center" | $\Big\Downarrow$ || | align="center" | || | align="center" | $\Big\Updownarrow$ || |- | align="center" | Countably Compact || | align="center" | || | align="center" | $\sigma$-Compact || | align="center" | || | align="center" | Strongly Locally Compact || |- | align="center" | $\Big\Updownarrow$ || | align="center" | || | align="center" | $\Big\Downarrow$ || |- | align="center" | Compact || | align="center" | || | align="center" | Separable || |- | align="center" | $\Big\Updownarrow$ || | align="center" | || | align="center" | $\Big\Updownarrow$ || |- | align="center" | Weakly Countably Compact || | align="center" | || | align="center" | Lindelöf || |- | align="center" | || | align="center" | || | align="center" | $\Big\Updownarrow$ || |- | align="center" | || | align="center" | || | align="center" | Second-Countable || |}


Extremally Disconnected Metric Space is Discrete

Let $M = \struct {A, d}$ be a metric space which is extremally disconnected. Then $M$ is the discrete topology.


Indicator is Well-Defined

Let $G$ be a finite group, and $a \in G$. Let $H$ be a subgroup of $G$. Then the indicator of $a$ in $H$ is well defined.


Compact Metric Space is Totally Bounded

Let $M = \struct {A, d}$ be a metric space which is compact. Then $M$ is totally bounded.


Subgroup Generated by Subgroup and Element

Let $G$ be a finite abelian group. Let $H$ be a proper subgroup of $G$. Let $a \in G \setminus H$. Let $n$ be the indicator of $a$ in $H$. Then: :$K = \left\{{x a^k: x \in H, \ 0 \le k < n}\right\}$ is a subgroup of $G$ such that $H \subseteq K$, and each element of $K$ has a unique representation in this form. Moreover: :$K = \left\langle{H, a}\right\rangle$ where $\left\langle{H, a}\right\rangle$ denotes the subgroup generated by $\left\{{H, a}\right\}$, and: :$\left\vert{K}\right\vert = n \left\vert{H}\right\vert$ where $\left\vert{K}\right\vert$ denotes the order of $K$.


Number of Characters on Finite Abelian Group

Let $G$ be a finite abelian group. Then the number of characters $G \to \C^\times$ is $\left\vert{G}\right\vert$.


Sequentially Compact Metric Space is Complete

Let $M = \struct {A, d}$ be a metric space which is sequentially compact. Then $M$ is complete.


Topological Completeness is Topological Property

Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces which are homeomorphic. Let $T_1$ be topologically complete. Then $T_2$ is also topologically complete.


Topological Completeness is Weakly Hereditary

Let $T = \struct {S, \tau}$ be a topological space which is topologically complete. Let $V \subseteq S$ be a closed subspace of $T$. Then $V$ is also topologically complete. That is, topological completeness is weakly hereditary.


Metric Space Compact iff Complete in All Equivalent Metrics

Let $M_1 = \struct {A, d_1}$ be a metric space. Then $M_1$ is compact {{iff}} $M_2 = \struct {A, d_2}$ is a complete metric space whenever $d_2$ is equivalent to $d_1$.


Urysohn's Metrization Theorem

Let $T = \struct {S, \tau}$ be a topological space which is regular and second-countable. Then $T$ is metrizable.


Metrizable iff Regular and has Sigma-Locally Finite Basis

Let $T = \struct {S, \tau}$ be a topological space. Then $T$ is metrizable {{iff}} $T$ is regular and has a $\sigma$-locally finite basis.


Uniformity iff Quasiuniformity has Symmetric Basis

Let $S$ be a set. Let $\UU$ be a quasiuniformity on $S$. Then $\UU$ is a uniformity {{iff}} $\UU$ has a symmetric filter basis.


Quasiuniformity Induces Topology

Let $\UU$ be a quasiuniformity on a set $S$. Let $\tau \subseteq \powerset S$ be a subset of the power set of $S$, created from $\UU$ by: :$\tau := \set {\map u x: u \in \UU, x \in S}$ where: :$\forall x \in S: \map u x = \set {y: \tuple {x, y} \in u}$ That is, where $\map u x$ is the image of $x$ under $u$, where $u$ is considered as a relation on $S$. Then $\tau$ is a topology on $S$. That is, the quasiuniform space $\struct {\struct {S, \UU}, \tau}$ is also the topological space $\struct {S, \tau}$.


Topological Space is Quasiuniformizable

Every topological space is quasiuniformizable.


T3 1/2 Space is Uniformizable

Let $T$ be a topological space. Then $T$ is uniformizable {{iff}} $T$ is a $T_{3 \frac 1 2}$ space.


Pseudometric Space generates Uniformity

Let $P = \struct {A, d}$ be a pseudometric space. Let $\UU$ be the set of sets defined as: :$\UU := \set {u \in A \times A: \exists \epsilon \in \R_{>0}: u_\epsilon \subseteq u}$ where: :$\R_{>0}$ is the set of strictly positive real numbers :$u_\epsilon$ is defined as: ::$u_\epsilon := \set {\tuple {x, y}: \map d {x, y} < \epsilon}$ Then $\UU$ is a uniformity on $A$ which generates a uniform space with the same topology as the topology induced by $d$.


Definition:Pseudometrizable Uniformity

Let $P = \struct {A, d}$ be a pseudometric space. Let $\UU$ be the uniformity on $A$ defined as: :$\UU := \set {u_\epsilon: \epsilon \in \R_{>0} }$ where: :$\R_{>0}$ is the set of strictly positive real numbers :$u_\epsilon$ is defined as: ::$u_\epsilon := \set {\tuple {x, y}: \map d {x, y} < \epsilon}$ Then $\UU$ is defined as '''pseudometrizable'''.


Metric Space generates Uniformity

Let $M = \struct {A, d}$ be a metric space. Let $\UU$ be the set of sets defined as: :$\UU := \set {u \in A \times A: \exists \epsilon \in \R_{>0}: u_\epsilon \subseteq u}$ where: :$\R_{>0}$ is the set of strictly positive real numbers :$u_\epsilon$ is defined as: ::$u_\epsilon := \set {\tuple {x, y}: \map d {x, y} < \epsilon}$ Then $\UU$ is a uniformity on $A$ which generates a uniform space with the same topology as the topology induced by $d$.


Discrete Topology is Topology

:$\tau$ is a topology on $S$.


Discrete Topology is Finest Topology

:$\tau$ is the finest topology on $S$.


Set in Discrete Topology is Clopen

:$\forall U \subseteq S: U$ is both closed and open in $\struct {S, \tau}$.


Topological Space is Discrete iff All Points are Isolated

:$\tau$ is the discrete topology on $S$ {{iff}} all points in $S$ are isolated points of $T$.


Point in Discrete Space is Adherent Point

Let $T = \struct {S, \tau}$ be a discrete topological space. Let $U \subseteq S$. Then $x$ is an adherent point of $U$ {{iff}} $x \in U$.


Interior Equals Closure of Subset of Discrete Space

Let $A \subseteq S$. Then: :$A = A^\circ = A^-$ where: :$A^\circ$ is the interior of $A$ :$A^-$ is the closure of $A$.


Mapping from Discrete Space is Continuous

Let $T_1 = \struct {S_1, \tau_1}$ be the discrete topological space on $S_1$. Let $T_2 = \struct {S_2, \tau_2}$ be any other topological space. Let $\phi: S_1 \to S_2$ be a mapping. Then $\phi$ is continuous.


Exponential Function is Superfunction

The function $f : \C \to \C$, defined as :$f \left({ z }\right) = c^z$ is a superfunction for any complex number $c$.


Standard Discrete Metric induces Discrete Topology

Let $M = \struct {A, d}$ be the (standard) discrete metric space on $A$. Then $d$ induces the discrete topology on $A$. Thus the discrete topology is metrizable.


Discrete Space satisfies all Separation Properties

Let $T = \struct {S, \powerset S}$ be the discrete topological space on $S$. Then $T$ fulfils all separation axioms: :$T$ is a $T_0$ (Kolmogorov) space :$T$ is a $T_1$ (Fréchet) space :$T$ is a $T_2$ (Hausdorff) space :$T$ is a semiregular space :$T$ is a $T_{2 \frac 1 2}$ (completely Hausdorff) space  :$T$ is a $T_3$ space  :$T$ is a regular space :$T$ is an Urysohn space :$T$ is a $T_{3 \frac 1 2}$ space :$T$ is a Tychonoff (completely regular) space :$T$ is a $T_4$ space :$T$ is a normal space :$T$ is a $T_5$ space  :$T$ is a completely normal space :$T$ is a perfectly $T_4$ space :$T$ is a perfectly normal space


Point in Discrete Space is Neighborhood

Let $T = \struct {S, \tau}$ be a discrete topological space. Let $x \in S$. Then $\set x$ is a neighborhood of $x$ in $T$.


Discrete Space is Strongly Locally Compact

Let $T = \struct {S, \tau}$ be a discrete topological space. Then $T$ is strongly locally compact.


Discrete Space is First-Countable

Let $T = \struct {S, \tau}$ be a discrete topological space. Then $T$ is first-countable.


Discrete Space has Open Locally Finite Cover

Let $T = \struct {S, \tau}$ be a discrete topological space. Consider the set $\CC$ of all singleton subsets of $S$: :$\CC := \set {\set x: x \in S}$ Then $\CC$ is an open cover of $T$ which is locally finite. This cover is the finest cover on $S$. That is, if $\VV$ is a cover of $T$, then $\CC$ is a refinement of $\VV$.


Discrete Space is Paracompact

Let $T = \struct {S, \tau}$ be a discrete topological space. Then $T$ is paracompact.


Singleton Set in Discrete Space is Compact

Let $T = \struct {S, \tau}$ be a topological space where $\tau$ is the discrete topology on $S$. Let $x \in S$. Then $\set x$ is compact.


Countable Discrete Space is Sigma-Compact

Let $T = \struct {S, \tau}$ be a countable discrete topological space. Then $T$ is $\sigma$-compact.


Countable Discrete Space is Lindelöf

Let $T = \struct {S, \tau}$ be a countable discrete topological space. Then $T$ is a Lindelöf space.


Basis for Discrete Topology

Let $S$ be a set. Let $\tau$ be the discrete topology on $S$. Let $\BB$ be the set of all singleton subsets of $S$: :$\BB := \set {\set x: x \in S}$. Then $\BB$ is a basis for $T$.


Countable Discrete Space is Second-Countable

Let $T = \struct {S, \tau}$ be a countable discrete topological space. Then $T$ is second-countable.


Countable Discrete Space is Separable

Let $T = \struct {S, \tau}$ be a countable discrete topological space. Then $T$ is separable.


Uncountable Discrete Space is not Separable

Let $T = \struct {S, \tau}$ be an uncountable discrete topological space. Then $T$ is not separable.


Uncountable Discrete Space is not Second-Countable

Let $T = \struct {S, \tau}$ be an uncountable discrete topological space. Then $T$ is not second-countable.


Uncountable Discrete Space is not Lindelöf

Let $T = \struct {S, \tau}$ be an uncountable discrete topological space. Then $T$ is not a Lindelöf space.


Uncountable Discrete Space is not Sigma-Compact

Let $T = \struct {S, \tau}$ be an uncountable discrete topological space. Then $T$ is not $\sigma$-compact.


Discrete Space is Compact iff Finite

Let $T = \left({S, \tau}\right)$ be a topological space where $\tau$ is the discrete topology on $S$. Then $T$ is compact {{iff}} $S$ is a finite set, thereby making $\tau$ the finite discrete topology on $S$.


Finite Space is Sequentially Compact

Let $T = \struct {S, \tau}$ be a topological space where $S$ is a finite set. Then $T$ is sequentially compact.


Discrete Space is Fully Normal

Let $T = \struct {S, \tau}$ be a discrete topological space. Then $T$ is fully normal.


Discrete Space is Fully T4

Let $T = \struct {S, \tau}$ be a discrete topological space. Then $T$ is fully $T_4$.


Finite Topological Space is Compact

Let $T = \struct {S, \tau}$ be a topological space where $S$ is a finite set. Then $T$ is compact.


Finite Space is Second-Countable

Let $T = \struct {S, \tau}$ be a topological space where $S$ is a finite set. Then $T$ is a second-countable space.


Finite Discrete Space satisfies all Compactness Properties

Let $T = \struct {S, \tau}$ be a finite discrete topological space. Then $T$ satisfies the following compactness properties: :$T$ is compact. :$T$ is Sequentially Compact. :$T$ is Countably Compact. :$T$ is Weakly Countably Compact. :$T$ is a Lindelöf Space :$T$ is Pseudocompact. :$T$ is $\sigma$-Compact. :$T$ is Locally Compact. :$T$ is Weakly Locally Compact. :$T$ is Strongly Locally Compact. :$T$ is $\sigma$-Locally Compact. :$T$ is Weakly $\sigma$-Locally Compact. :$T$ is Fully Normal. :$T$ is Fully $T_4$. :$T$ is Paracompact. :$T$ is Countably Paracompact. :$T$ is Metacompact. :$T$ is Countably Metacompact.


Finite Space Satisfies All Compactness Properties

Let $T = \struct {S, \tau}$ be a topological space where $S$ is a finite set. Then $T$ satisfies the following compactness properties: : $T$ is compact. : $T$ is sequentially compact. : $T$ is countably compact. : $T$ is weakly countably compact. : $T$ is a Lindelöf space. : $T$ is pseudocompact. : $T$ is $\sigma$-compact. : $T$ is strongly locally compact. : $T$ is $\sigma$-locally compact. : $T$ is weakly $\sigma$-locally compact. : $T$ is locally compact. : $T$ is weakly locally compact. : $T$ is paracompact. : $T$ is countably paracompact. : $T$ is metacompact. : $T$ is countably metacompact.


Discrete Space is not Dense-In-Itself

Let $T = \struct {S, \tau}$ be a discrete topological space. Then $T$ is not dense-in-itself.


Non-Trivial Discrete Space is not Connected

Let $T = \struct {S, \tau}$ be a non-trivial discrete topological space. Then $T$ is not connected. Thus also:


Discrete Space is Locally Path-Connected

Let $T = \struct {S, \tau}$ be a discrete topological space. Then $T$ is locally path-connected.


Point is Path-Connected to Itself

Let $T = \left({S, \tau}\right)$ be a topological space. Let $a \in S$. Then $a$ is path-connected to itself.


Discrete Uniformity is Uniformity

Let $S$ be a set. Let $\UU$ be the discrete uniformity on $S$ Then $\UU$ is indeed a uniformity.


Indiscrete Topology is Topology

:$\tau$ is a topology on $S$.


Indiscrete Topology is Coarsest Topology

:$\tau$ is the coarsest topology on $S$.


Open and Closed Sets in Indiscrete Topology

Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space. Let $H \subseteq S$.


Subset of Indiscrete Space is Compact and Sequentially Compact


Limit Points of Indiscrete Space

Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space consisting of at least two points. Let $H$ be a subset of $T$ such that $H \ne \O$. Then every point of $T$ is a limit point of $H$.


Sequence in Indiscrete Space converges to Every Point

Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space. Let $\sequence {s_n}$ be a sequence in $T$. Then $\sequence {s_n}$ converges to every point of $S$.


Empty Set is Nowhere Dense

Let $T = \struct {S, \tau}$ be a topological space. Then the empty set $\O$ is nowhere dense in $T$.


Interior of Open Set

Let $T = \left({S, \tau}\right)$ be a topological space. Let $U \subseteq T$ be open in $T$. Then: : $U^\circ = U \iff U \in \tau$ where $U^\circ$ is the interior of $U$. That is, a subset of $S$ is open in $T$ {{iff}} it equals its interior.


Interior of Subset of Indiscrete Space

Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space. Let $H \subset S$ (that is, let $H$ be a proper subset of $T$). Then: :$H^\circ = H^{\circ -} = H^{\circ - \circ} = \O$ where: :$H^\circ$ denotes the interior of $H$ :$H^-$ denotes the closure of $H$.


Closure of Subset of Indiscrete Space

Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space. Let $\O \subsetneq H \subseteq S$ (that is, let $H$ be a non-empty subset of $T$). Then: :$H^- = H^{- \circ} = H^{- \circ -} = S$ where: :$H^\circ$ denotes the interior of $H$ :$H^-$ denotes the closure of $H$.


Boundary of Subset of Indiscrete Space

Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space. Let $\O \subsetneq H \subsetneq S$ (that is, let $H$ be a non-empty proper subset of $T$). Then: :$\partial H = S$ where $\partial H$ denotes the boundary of $H$. If $H = \O$ or $H = S$ then $\partial H = \O$.


Boundary of Boundary of Subset of Indiscrete Space

Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space. Let $H \subseteq S$. Then: :$\map \partial {\partial H} = \O$ where $\partial H$ denotes the boundary of $H$.


Indiscrete Space is Separable

Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space such that $S$ has more than one element. Then $T$ is separable.


Mapping to Indiscrete Space is Continuous

Let $T_1 = \struct {S_1, \tau_1}$ be any topological space. Let $T_2 = \struct {S_2, \tau_2}$ be the indiscrete topological space on $S_2$. Let $\phi: S_1 \to S_2$ be a mapping. Then $\phi$ is continuous.


Indiscrete Space is Path-Connected

Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space. Then $T$ is path-connected.


Indiscrete Space is Connected

Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space. Then $T$ is connected.


Indiscrete Space is Arc-Connected iff Uncountable

Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space. Then $T$ is arc-connected {{iff}} $S$ is an uncountable set.


Indiscrete Space is Irreducible

Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space. Then $T$ is irreducible.


Indiscrete Space is Ultraconnected

Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space. Then $T$ is ultraconnected.


Indiscrete Non-Singleton Space is not T0

Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space which has more than one element. Then $T$ is not a $T_0$ (Kolmogorov) space.


Indiscrete Space is T5

Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space. Then $T$ is a $T_5$ space.


Indiscrete Space is T3

Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space . Then $T$ is a $T_3$ space.


Indiscrete Space is Pseudometrizable

Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space. Then $T$ is pseudometrizable.


Partition Topology is Topology

Let $S$ be a set. Let $\PP$ be a partition of $S$. Let $\tau$ be the set of subsets of $S$ defined as: :$a \in \tau \iff a$ is the union of sets of $\PP$ Then $\tau$ is a topology on $S$.


Basis for Partition Topology

Let $S$ be a set. Let $\PP$ be a partition of $S$. Let $\tau$ be the partition topology on $S$ defined as: :$a \in \tau \iff a$ is the union of sets of $\mathcal P$ Then $\PP$ forms a basis of $\tau$.


Open Set in Partition Topology is also Closed

Let $T = \struct {S, \tau}$ be a topological space. Then $T$ is a partition space {{iff}}: :$\forall U \subseteq S: U \in \tau \iff \relcomp S U \in \tau$ That is, a topological space is a partition space {{iff}} all open sets are closed, and all closed sets are also open.


Quotient Topology of Partition Topology is Discrete Space

Let $\PP$ be a partition of a set $S$. Let $T = \struct {S, \tau}$ be the partition space formed from $\PP$. Let $S / \PP$ be the quotient set of $S$ by $\PP$. Then the quotient topology $\tau_{S / \PP}$ is a discrete topology.


Partition of Singletons yields Discrete Topology

Let $S$ be a set which is non-empty. Let $\PP$ be the (trivial) partition of singletons on $S$: :$\PP = \set {\set x: x \in S}$ Then the partition topology on $\PP$ is the discrete topology.


Partition Topology is not T0

Let $S$ be a set and let $\PP$ be a partition on $S$ which is not the (trivial) partition of singletons. Let $T = \struct {S, \tau}$ be the partition space whose basis is $\PP$. Then $T$ is not a $T_0$ (Kolmogorov) space.


Partition Topology is T5

Let $S$ be a set and let $\PP$ be a partition on $S$ which is not the (trivial) partition of singletons. Let $T = \struct {S, \tau}$ be the partition space whose basis is $\PP$. Then $T$ is a $T_5$ space.


Partition Topology is T3 1/2

Let $S$ be a set and let $\PP$ be a partition on $S$. Let $T = \struct {S, \tau}$ be the partition space whose basis is $\PP$. Then $T$ is a $T_{3 \frac 1 2}$ space.


Odd-Even Topology is Second-Countable

Let $T = \struct {\Z_{>0}, \tau}$ be a topological space where $\tau$ is the odd-even topology on the strictly positive integers $\Z_{>0}$. Then $T$ is second-countable.


Odd-Even Topology is Weakly Countably Compact

Let $T = \struct {\Z_{>0}, \tau}$ be a topological space where $\tau$ is the odd-even topology on the strictly positive integers $\Z_{>0}$. Then $T$ is weakly countably compact.


Odd-Even Topology is not Countably Compact

Let $T = \struct {\Z_{>0}, \tau}$ be a topological space where $\tau$ is the odd-even topology on the strictly positive integers $\Z_{>0}$. Then $T$ is not countably compact.


Countable Discrete Space is not Weakly Countably Compact

Let $T = \struct {S, \tau}$ be a countable discrete space. Then $T$ is not weakly countably compact.


Weak Countable Compactness is not Preserved under Continuous Maps

Let $T_A = \struct {S_A, \tau_A}$ be a topological space which is weakly countably compact. Let $T_B = \struct {S_B, \tau_B}$ be another topological space. Let $\phi: T_A \to T_B$ be a continuous mapping. Then $T_B$ is not necessarily weakly countably compact.


Deleted Integer Topology is not Countably Compact

Let $S = \R_{\ge 0} \setminus \Z$, and let $\tau$ be the deleted integer topology on $S$. Then the topological space $T = \struct {S, \tau}$ is not countably compact.


Deleted Integer Topology is Second-Countable

Let $S = \R_{\ge 0} \setminus \Z$. Let $\tau$ be the deleted integer topology on $S$. Then the topological space $T = \struct {S, \tau}$ is second-countable.


Deleted Integer Topology is Weakly Countably Compact

Let $S = \R_{\ge 0} \setminus \Z$. Let $\tau$ be the deleted integer topology on $S$. Then the topological space $T = \struct {S, \tau}$ is weakly countably compact.


Pseudometric induces Topology

Let $S \ne \O$ be a non-empty set. Consider a pseudometric space $\struct {S, d}$ where $d: S \times S \to \R_{\ge 0}$ is a pseudometric. Then $\struct {S, d}$ gives rise to a topological space $\struct {S, \tau_d}$ whose topology $\tau_d$ is '''defined''' (or '''induced''') by $d$.


Open Sets in Pseudometric Space

Let $P = \left({A, d}\right)$ be a pseudometric space. Then $\varnothing$ and $A$ are both open in $P$.


Partition Space is Pseudometrizable

Let $T = \struct {S, \tau}$ be a partition space. Then $T$ is pseudometrizable.


Double Pointed Topology is not T0

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 $T$ is not a $T_0$ (Kolmogorov) space.


Double Pointed Discrete Real Number Space is Weakly Countably Compact

Let $T_\R = \struct {\R, \tau_\R}$ be the (uncountable) discrete space on the set of real numbers. Let $T_D = \struct {D, \tau_D}$ be the indiscrete topology on the doubleton $D = \set {a, b}$. Let $T = T_\R \times T_D$ be the double pointed (uncountable) discrete space which is the product space of $T_\R$ and $T_D$. Then $T$ is weakly countably compact.


Double Pointed Discrete Real Number Space is not Lindelöf

Let $T_\R = \struct {\R, \tau_\R}$ be the (uncountable) discrete space on the set of real numbers. Let $T_D = \struct {D, \tau_D}$ be the indiscrete topology on the doubleton $D = \set {a, b}$. Let $T = T_\R \times T_D$ be the double pointed (uncountable) discrete space which is the product space of $T_\R$ and $T_D$. Then $T$ is '''not''' a Lindelöf space.


Particular Point Topology is Topology

Let $T = \struct {S, \tau_p}$ be a particular point space. Then $\tau_p$ is a topology on $S$, and $T$ is a topological space.


Limit Points in Particular Point Space

Let $T = \struct {S, \tau_p}$ be a particular point space. Let $x \in S$ such that $x \ne p$. Then $x$ is a limit point of $p$.


Closure of Open Set of Particular Point Space

Let $T = \struct {S, \tau_p}$ be a particular point space. Let $U \in \tau_p$ be open in $T$ such that $U \ne \O$. Then: :$U^- = S$ where $U^-$ denotes the closure of $U$.


Interior of Closed Set of Particular Point Space

Let $T = \struct {S, \tau_p}$ be a particular point space. Let $V \subseteq S$ be closed in $T$ such that $V \ne S$. Then: :$V^\circ = \O$ where $V^\circ$ denotes the interior of $V$.


Point in Particular Point Space is not Omega-Accumulation Point

Let $T = \struct {S, \tau_p}$ be a particular point space. Let $x \in S$ such that $x \ne p$. Let $H \subseteq S$ such that $p \in H$. Then $x$ is not an $\omega$-accumulation point of $H$.


Particular Point Space is T0

Let $T = \struct {S, \tau_p}$ be a particular point space. Then $T$ is a $T_0$ (Kolmogorov) space.


Non-Trivial Particular Point Topology is not T1

Let $T = \struct {S, \tau_p}$ be a particular point space such that $S$ is not a singleton. Then $T$ is not a $T_1$ (Fréchet) space.


Particular Point Topology with Three Points is not T4

Let $T = \struct {S, \tau_p}$ be a particular point space such that $S$ is not a singleton or a doubleton. That is, such that $S$ has more than two distinct elements. Then $T$ is not a $T_4$ space.


Compact Space in Particular Point Space

Let $T = \struct {S, \tau_p}$ be a particular point space. Then $\set p$ is compact in $T$.


Finite Multiplicative Subgroup of Field is Cyclic

Let $\left({F, +, \times}\right)$ be a field. Let $\left({F^*, \times}\right)$ denote the multiplicative group of $F$. Let $C$ be a finite subgroup of $\left({F^*, \times}\right)$. Then $C$ is cyclic.


Induced Homomorphism of Polynomial Forms

Let $R$ and $S$ be commutative rings with unity. Let $\phi : R \to S$ be a ring homomorphism. Let $R \left[{X}\right]$ and $S \left[{X}\right]$ be the rings of polynomial forms over $R$ and $S$ respectively in the indeterminate $X$. Then the map $\overline{\phi}: R \left[{X}\right] \to S \left[{X}\right]$ given by: :$\overline{\phi} \left({a_0 + a_1 X + \cdots + a_n X^n}\right) = \phi \left({a_0}\right) + \phi \left({a_1}\right) X + \cdots + \phi \left({a_n}\right) X^n$ is a ring homomorphism.


Structure of Simple Algebraic Field Extension

Let $F / K$ be a field extension. Let $\alpha \in F$ be algebraic over $K$. Let $\mu_\alpha$ be the minimal polynomial of $\alpha$ over $K$. Let $K \left[{\alpha}\right]$ (resp. $K \left({\alpha}\right)$) be the subring (resp. subfield) of $F$ generated by $K \cup \left\{ {\alpha}\right\}$. Then: :$K \left[{\alpha}\right] = K \left({\alpha}\right) \simeq K \left[{X}\right] / \left\langle{\mu_\alpha}\right\rangle$ where $\left\langle{\mu_\alpha}\right\rangle$ is the ideal of the ring of polynomial functions generated by $\mu_\alpha$. {{explain|Link to Generator needs refining}} Moreover: :$n := \left[{K \left({\alpha}\right) : K}\right] = \deg \mu_\alpha$ and: :$1, \alpha, \dotsc, \alpha^{n - 1}$ is a basis of $K \left({\alpha}\right)$ over $K$.


Subset of Particular Point Space is either Open or Closed

Let $T = \left({S, \tau_p}\right)$ be a particular point space. Let $H \subseteq S$ be any subset of $T$. Then $H$ is either open or closed in $T$. The only sets which are both closed and open in $T$ are $S$ and $\varnothing$.


Particular Point Space is Weakly Locally Compact

Let $T = \struct {S, \tau_p}$ be a particular point space. Then $T$ is weakly locally compact.


Infinite Particular Point Space is not Compact

Let $T = \struct {S, \tau_p}$ be an infinite particular point space. Then $T$ is not compact.


Uncountable Particular Point Space is not Lindelöf

Let $T = \struct {S, \tau_p}$ be an uncountable particular point space. Then $T$ is not a Lindelöf space.


Particular Point Space is Separable

Let $T = \struct {S, \tau_p}$ be a particular point space. Then $T$ is separable.


Field Adjoined Set

Let $F$ be a field. Let $S \subseteq F$ be a subset of $F$. Let $K \le F$ be a subfield of $F$. The subring $K \sqbrk S$ of $F$ generated by $K \cup S$ is the set of all finite linear combinations of powers of elements of $S$ with coefficients in $K$. The subfield $\map K S$ of $F$ generated by $K \cup S$ is the set of all $x y^{-1} \in F$ with $a, b \in K \sqbrk S$, $b \ne 0$. $\map K S$ is isomorphic to the field of quotients $Q$ of $K \sqbrk S$.


Separability in Uncountable Particular Point Space

Let $T = \struct {S, \tau_p}$ be an uncountable particular point space. Let $H = S \setminus \set p$ where $\setminus$ denotes set difference. Then $H$ is not separable.


Uncountable Particular Point Space is not Second-Countable

Let $T = \struct {S, \tau_p}$ be an uncountable particular point space. Then $T$ is not second-countable.


Particular Point Space is First-Countable

Let $T = \struct {S, \tau_p}$ be a particular point space. Then $T$ is first-countable.


Homeomorphic Non-Comparable Particular Point Topologies

Let $S$ be a set with at least two elements. Let $p, q \in S: p \ne q$. Let $\tau_p$ and $\tau_q$ be the particular point topologies on $S$ by $p$ and $q$ respectively. Then the topological spaces $T_p = \struct {S, \tau_p}$ and $T_q = \struct {S, \tau_q}$ are homeomorphic. However, $\tau_p$ and $\tau_q$ are not comparable.


Universal Property for Simple Field Extensions

Let $F / K$ be a field extension. Let $\alpha \in F$ be an algebraic over $K$. Let $\mu_\alpha$ be the minimal polynomial over $\alpha$ over $K$. Let $\psi : K \left({\alpha}\right) \to L$ be a homomorphism. Let $\phi = \psi \restriction_K$. Let $\overline \phi: K \left[{X}\right] \to L \left[{X}\right]$ be the Induced Homomorphism of Polynomial Forms. Then $\psi\left({\alpha}\right)$ is a root of $\overline \phi \left({\mu_\alpha}\right)$ in $L$. Conversely let $L$ be a field. Let $\phi: K \to L$ be a homomorphism. Let $\beta$ be a root of $\overline \phi  \left({\mu_\alpha}\right)$ in $L$. Then there exists a unique field homomorphism $\psi : K \left({\alpha}\right) \to L$ extending $\phi$ that sends $\alpha$ to $\beta$.


Finite Field Extension is Algebraic

Let $L / K$ be a finite field extension. Then $L / K$ is algebraic.


Finitely Generated Algebraic Extension is Finite

Let $L/K$ be a field extension and $\alpha_1,\ldots,\alpha_n \in L$ algebraic over $K$. Then $K \left({\alpha_1, \ldots, \alpha_n}\right) / K$ is a finite field extension.


Maximal Algebraic Extension is Subfield

Let $L/K$ be a extension of fields. Let $K^a$ be the maximal algebraic extension of $K$ contained in $L$. Then $K^a$ is a subfield of $L$.


Closed Set in Particular Point Space has no Limit Points

Let $T = \struct {S, \tau_p}$ be a particular point space. Let $H \subsetneq S$ be closed in $T$. Then $H$ has no limit points.


Particular Point Space is Scattered

Let $T = \struct {S, \tau_p}$ be a particular point space. Then $T$ is scattered.


Particular Point Space is Irreducible

Let $T = \left({S, \tau_p}\right)$ be a particular point space. Then $T$ is irreducible.


Particular Point Space is not Ultraconnected

Let $T = \struct {S, \tau_p}$ be a particular point space with at least three points. Then $T$ is not ultraconnected.


Dispersion Point in Particular Point Space

Let $T = \struct {S, \tau_p}$ be a particular point space. Then $p$ is dispersion point of $T$.


Infinite Particular Point Space is not Weakly Countably Compact

Let $T = \struct {S, \tau_p}$ be an infinite particular point space. Then $T$ is not weakly countably compact.


Fundamental Theorem of Symmetric Polynomials

Let $K$ be a field. Let $f$ a symmetric polynomial over $K$. Then $f$ can be written uniquely as a polynomial in the elementary symmetric polynomials.


Particular Point Space is Pseudocompact

Let $T = \struct {S, \tau_p}$ be a particular point space. Then $T$ is pseudocompact.


Particular Point Space is Path-Connected

Let $T = \struct {S, \tau_p}$ be a particular point space. Then $T$ is path-connected.


Particular Point Space is not Arc-Connected

Let $T = \struct {S, \tau_p}$ be a particular point space. Then $T$ is not arc-connected.


Basis for Particular Point Space

Let $T = \left({S, \tau_p}\right)$ be a particular point space. Consider the set $\mathcal B$ defined as: :$\mathcal B = \left\{{\left\{{x, p}\right\}: x \in S}\right\} \cup \left\{{p}\right\}$ Then $B$ is a basis for $S$.


Particular Point Space is Non-Meager

Let $T = \left({S, \tau_p}\right)$ be a particular point space. Then $T$ is non-meager.


Infinite Particular Point Space is not Countably Metacompact

Let $T = \struct {S, \tau_p}$ be an infinite particular point space. Then $T$ is not countably metacompact.


Exportation and Self-Conditional

:$p \land q \implies r \dashv \vdash \paren {p \implies q} \implies \paren {p \implies r}$


Łoś-Vaught Test

Let $T$ be a satisfiable $\mathcal L$-theory with no finite models. Let $T$ be $\kappa$-categorical for some infinite cardinal $\kappa \ge \left|{\mathcal L}\right|$. Then $T$ is complete.


Sierpiński Space is Irreducible

Let $T = \struct {\set {0, 1}, \tau_0}$ be a Sierpiński space. Then $T$ is irreducible.


Sierpiński Space is Path-Connected

Let $T = \struct {\set {0, 1}, \tau_0}$ be a Sierpiński space. Then $T$ is path-connected.


Sierpiński Space is not Arc-Connected

Let $T = \struct {\set {0, 1}, \tau_0}$ be a Sierpiński space. Then $T$ is not arc-connected.


Sierpiński Space is T5

Let $T = \struct {\set {0, 1}, \tau_0}$ be a Sierpiński space. Then $T$ is a $T_5$ space.


Sierpiński Space is T4

Let $T = \struct {\set {0, 1}, \tau_0}$ be a Sierpiński space. Then $T$ is a $T_4$ space.


Upward Löwenheim-Skolem Theorem

{{Disambiguate|Definition:Model|I suspect model of a first-order theory $\LL$, which is more specific than what is linked to now}} Let $T$ be an $\LL$-theory with an infinite model. Then for each infinite cardinal $\kappa \ge \card \LL$, there exists a model of $T$ with cardinality $\kappa$.


Compactness Theorem

Let $\LL$ be the language of predicate logic. Let $T$ be a set of $\LL$-sentences. Then $T$ is satisfiable {{iff}} $T$ is finitely satisfiable.


Łoś's Theorem

Let $\LL$ be a language. Let $I$ be an infinite set. Let $\UU$ be an ultrafilter on $I$. Let $\map \phi {v_1, \ldots, v_n}$ be an $\LL$-formula. Let $\MM$ be the ultraproduct: :$\displaystyle \paren {\prod_{i \mathop \in I} \MM_i} / \UU$ where each $\MM_i$ is an $\LL$-structure. Then, for all $m_1 = \paren {m_{1, i} }_\UU, \dots, m_n = \paren {m_{n, i} }_\UU$ in $\MM$: :$\MM \models \map \phi {m_1, \ldots, m_n}$ {{iff}}: :the set $\set {i \in I: \MM_i \models \map \phi {m_{1, i}, \ldots, m_{n, i} } }$ is in $\UU$. In particular, for all $\LL$-sentences $\phi$, we have that: :$\MM \models \phi$ {{iff}} $\set {i \in I: \MM_i \models \phi}$ is in $\UU$.


Closed Extension Topology is Topology

Let $T = \struct {S, \tau}$ be a topological space. Let $\tau^*_p$ be the closed extension topology of $\tau$. Then $\tau^*_p$ is a topology on $S^*_p = S \cup \set p$.


Quantifier Free Formula is Preserved by Superstructure

Let $\MM, \NN$ be $\LL$-structures such that  $\MM$ is a substructure of $\NN$. {{Disambiguate|Definition:Structure}} Let $\map \phi {\bar x}$ be a quantifier-free $\LL$-formula, and let $\bar a \in\MM$. Then $\MM \models \map \phi {\bar a}$ {{iff}} $\NN \models \map \phi {\bar a}$.


Algebraically Closed Field is Infinite

Let $F$ be an algebraically closed field. Then $F$ is infinite.


Field of Uncountable Cardinality K has Transcendence Degree K

Let $F$ be a field of uncountable cardinality $\kappa$. Then $F$ has transcendence degree $\kappa$ over its prime field.


Theory of Algebraically Closed Fields of Characteristic p is Complete

Let $p$ be either $0$ or a prime number. Let $ACF_p$ be the theory of algebraically closed fields of characteristic $p$ in the language $\mathcal L_r = \left\{ {0, 1, +, -, \cdot}\right\}$ for rings, where: :$0, 1$ are constants and: :$+, -, \cdot$ are binary functions.  Then $ACF_p$ is complete.


Lefschetz Principle (First-Order)

Let $\phi$ be a sentence in the language $\LL_r = \set {0, 1, +, -, \cdot}$ for rings, where $0, 1$ are constants and $+, -, \cdot$ are binary functions. {{TFAE}} :$(1): \quad \phi$ is true in every algebraically closed field of characteristic $0$. :$(2): \quad \phi$ is true in some algebraically closed field of characteristic $0$. :$(3): \quad \phi$ is true in algebraically closed fields of characteristic $p$ for arbitrarily large primes $p$. :$(4): \quad \phi$ is true in algebraically closed fields of characteristic $p$ for sufficiently large primes $p$. Note in particular that since $\C$ is an algebraically closed field of characteristic $0$, these are equivalent to $\phi$ being true in the field $\C$.


CNF Satisfiability Problem is NP-Complete

The conjunctive normal form boolean satisfiability problem (CNF SAT) is NP-complete.


Closed Sets of Closed Extension Topology

Let $T = \struct {S, \tau}$ be a topological space. Let $T^*_p = \struct {S^*_p, \tau^*_p}$ be the closed extension space of $T$. Then the closed sets of $T^*_p$ (apart from $S^*_p$) are the closed sets of $T$. This explains why $\tau^*_p$ is called the closed extension topology of $\tau$.


Particular Point Topology is Closed Extension Topology of Discrete Topology

Let $S$ be a set and let $p \in S$. Let $\tau_p$ be the particular point topology on $S$. Let $T = \struct {S \setminus \set p, \vartheta}$ be the discrete topological space on $S \setminus \set p$. Then $T^* = \struct {S, \tau_p}$ is a closed extension space of $T$.


Closed Extension Topology is not T1

Let $T = \struct {S, \tau}$ be a topological space. Let $T^*_p = \struct {S^*_p, \tau^*_p}$ be the closed extension space of $T$. Then $T^*_p$ is not a $T_1$ (Fréchet) space.


Algebraic Numbers are Countable

The set $\Bbb A$ of algebraic numbers is countable.


Limit Points in Closed Extension Space

Let $T = \struct {S, \tau}$ be a topological space. Let $T^*_p = \struct {S^*_p, \tau^*_p}$ be the closed extension space of $T$. Let $x \in S$. Then $x$ is a limit point of $p$.


Closure of Open Set of Closed Extension Space

Let $T = \struct {S, \tau}$ be a topological space. Let $T^*_p = \struct {S^*_p, \tau^*_p}$ be the closed extension space of $T$. Then $U^- = S^*_p$ where $U^-$ denotes the closure of $U$ in $T^*_p$.


Closed Extension Space is Irreducible

Let $T = \struct {S, \tau}$ be a topological space. Let $T^*_p = \struct {S^*_p, \tau^*_p}$ be the closed extension space of $T$. Then $T^*_p$ is irreducible.


Polynomial over Field has Finitely Many Roots

Let $F$ be a field. Let $F \left[{x}\right]$ be the ring of polynomial functions in the indeterminate $x$. If $p \in F \left[{x}\right]$ be non-null, then $p$ has finitely many roots in $F$.


Set of Polynomials over Infinite Set has Same Cardinality

Let $S$ be a set of infinite cardinality $\kappa$. Let $S \left[{x}\right]$ be the set of polynomial forms over $S$ in the indeterminate $x$. Then $S \left[{x}\right]$ has cardinality $\kappa$.


Excluded Point Topology is Topology

Let $T = \left({S, \tau_{\bar p}}\right)$ be an excluded point space. Then $\tau_{\bar p}$ is a topology on $S$, and $T$ is a topological space.


Excluded Point Topology is Open Extension Topology of Discrete Topology

Let $S$ be a set and let $p \in S$. Let $\tau_{\bar p}$ be the excluded point topology on $S$. Let $T = \struct {S \setminus \set p, \tau_D}$ be the discrete topological space on $S \setminus \set p$. Then $T^* = \struct {S, \tau_{\bar p} }$ is an open extension space of $T$.


Product of Countable Discrete Space with Sierpiński Space is Paracompact

Let $T_X = \struct {S, \tau}$ be a countable discrete space. Let $T_Y = \struct {\set {0, 1}, \tau_0}$ be a Sierpiński space. Let $T_X \times T_Y$ be the product space of $T_X$ and $T_Y$. Then $T_X \times T_Y$ is paracompact.


Reduction of Explicit ODE to First Order System

Let $\map {x^{\paren n} } t = \map F {t, x, x', \ldots, x^{\paren {n - 1} } }$, $\map x {t_0} = x_0$ be an explicit ODE with $x \in \R^m$. Let there exist $I \subseteq \R$ such that there exists a unique particular solution: :$x: I \to \R^m$ to this ODE. Then there exists a system of first order ODEs: :$y' = \map {\tilde F} {t, y}$ with $y = \tuple {y_1, \ldots, y_{m n} }^T \in \R^{m n}$ such that: :$\tuple {\map {y_1} t, \ldots, \map {y_m} t} = \map x t$ for all $t \in I$ and $\map y {t_0} = x_0$.


Open Continuous Image of Paracompact Space is not always Countably Metacompact

Let $T_A = \struct {X_A, \tau_A}$ be a topological space which is paracompact. Let $T_B = \struct {X_B, \tau_B}$ be another topological space. Let $\phi: T_A \to T_B$ be a mapping which is both continuous and open. Then it is not necessarily even the case that $T_B$ is countably metacompact, let alone paracompact.


Fixed Point Formulation of Explicit ODE

Let $x' = \map f {t, x}$ with $\map x {t_0} = x_0$ be an explicit ODE of dimension $n$. For $a, b \in \R$, let $\mathcal X = \map {\mathcal C} {\closedint a b; \R^n}$ be the space of continuous functions on the closed interval $\closedint a b$. Let $T: \mathcal X \to \mathcal X$ be the map defined by: :$\displaystyle \map {\paren {T x} } t = x_0 + \int_{t_0}^t \map f {s, \map x s} \rd s$ {{explain|Definition of $\paren {T x}$}} Then a fixed point of $T$ in $\mathcal X$ is a solution to the above ODE.


Cardinality of Infinite Union of Infinite Sets

Let $\kappa$ be an infinite cardinal. Let $X_i$ be sets of cardinality at most $\kappa$ indexed by a set $I$ of cardinality at most $\kappa$. Then their union $\displaystyle \bigcup_{i \mathop \in I} X_i$ has cardinality at most $\kappa$. Furthermore, if at least one of the $X_i$ is size $\kappa$, then the union has cardinality $\kappa$.


Well-Ordering Theorem

Every set is well-orderable.


Well-Ordering Theorem is Equivalent to Axiom of Choice

The Well-Ordering Theorem holds {{iff}} the Axiom of Choice holds. That is, every set can be well-ordered {{iff}} every collection of sets has a choice function.


Tychonoff Topology is Coarsest Topology such that Projections are Continuous

Let $\mathbb X = \family {\struct {X_i, \tau_i} }_{i \mathop \in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set. Let $X$ be the cartesian product of $\mathbb X$: :$\displaystyle X := \prod_{i \mathop \in I} X_i$ Let $\tau$ be the Tychonoff topology on $X$. For each $i \in I$, let $\pr_i : X \to X_i$ be the corresponding projection which maps each ordered tuple in $X$ to the corresponding element in $X_i$: :$\forall x \in X: \map {\pr_i} x = x_i$ Then $\tau$ is the coarsest topology on $X$ such that all the $\pr_i$ are continuous.


Condition for Closed Extension Space to be T0 Space

Let $T = \struct {S, \tau}$ be a topological space. Let $T^*_p = \struct {S^*_p, \tau^*_p}$ be the closed extension space of $T$. Then $T^*_p$ is a $T_0$ (Kolmogorov) space {{iff}} $T$ is.


Excluded Point Space is T0

Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space. Then $T$ is a $T_0$ (Kolmogorov) space.


Open Extension Topology is not T1

Let $T = \struct {S, \tau}$ be a topological space. Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the open extension space of $T$. Then $T^*_{\bar p}$ is not a $T_1$ (Fréchet) space.


Extreme Value Theorem

Let $X$ be a compact metric space and $Y$ a normed vector space. Let $f: X \to Y$ be a continuous mapping. Then $f$ is bounded, and there exist $x, y \in X$ such that: :$\forall z \in X: \norm {\map f x} \le \norm {\map f z} \le \norm {\map f y}$ where $\norm {\map f x}$ denotes the norm of $\map f x$. Moreover, $\norm f$ attains its minimum and maximum.


Continuous Functions on Compact Space form Banach Space

Let $X$ be a compact Hausdorff space. Let $Y$ be a Banach space. Let $\CC = \CC \struct {X; Y}$ be the set of all continuous mappings $X \to Y$. {{explain|Work out what convention ProofWiki has already evolved for $\CC \struct {X; Y}$.}} Let $\norm {\,\cdot\,}_\infty$ be the supremum norm on $\CC$. Then $\struct {\CC, \norm {\,\cdot\,} }$ is a Banach space.


Supremum Norm is Norm

Let $S$ be a set. Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space over $K \in \set {\R, \C}$. Let $\BB$ be the set of bounded mappings $S \to X$. Let $\norm {\, \cdot \,}_\infty$ be the supremum norm on $\BB$. Then $\norm {\, \cdot \,}_\infty$ is a norm on $\BB$. {{MissingLinks|Add a link that establishes that $\BB$ is a vector space}}


One-Step Vector Subspace Test

Let $V$ be a vector space over a division ring $K$. Let $U \subseteq V$ be a non-empty subset of $V$ such that: :$\forall u, v \in U: \forall \lambda \in K: u + \lambda v \in U$ Then $U$ is a subspace of $V$.


Ax-Grothendieck Theorem

Let $f: \C^n \to \C^n$ be a polynomial map. Let $f$ be injective. Then $f$ is surjective.


Condition for Open Extension Space to be T0 Space

Let $T = \struct {S, \tau}$ be a topological space. Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the open extension space of $T$. Then $T^*_{\bar p}$ is a $T_0$ (Kolmogorov) space {{iff}} $T$ is.


Non-Trivial Excluded Point Topology is not T1

Let $T = \struct {S, \tau_{\bar p} }$ be a excluded point space such that $S$ is not a singleton. Then $T$ is not a $T_1$ (Fréchet) space.


Limit Points in Open Extension Space

Let $T = \struct {S, \tau}$ be a topological space. Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the open extension space of $T$. Let $x \in S$. Then $p$ is a limit point of $x$.


Limit Points in Excluded Point Space

Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space. Let $x \in S$ such that $x \ne p$. Then $p$ is the only limit point of $x$. Similarly, let $U \subseteq S$. Then $p$ is the only limit point of $U$.


Excluded Point Space is T5

Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space. Then $T$ is a $T_5$ space.


Excluded Point Space is Compact

Let $T = \left({S, \tau_{\bar p}}\right)$ be an excluded point space. Then $T$ is a compact space.


Open Extension Space is Compact

Let $T = \struct {S, \tau}$ be a topological space. Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the open extension space of $T$. Then $T^*_{\bar p}$ is a compact space.


Open Extension Space is Connected

Let $T = \struct {S, \tau}$ be a topological space. Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the open extension space of $T$. Then $T^*_{\bar p}$ is a connected space.


Excluded Point Space is Connected

Let $T = \left({S, \tau_{\bar p}}\right)$ be an excluded point space. Then $T^*_{\bar p}$ is a connected space.


Open Extension Space is Ultraconnected

Let $T = \struct {S, \tau}$ be a topological space. Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the open extension space of $T$. Then $T^*_{\bar p}$ is ultraconnected.


Excluded Point Space is Ultraconnected

Let $T = \left({S, \tau_{\bar p}}\right)$ be an excluded point space. Then $T$ is ultraconnected.


Excluded Point Space is not Irreducible

Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space with at least three points. Then $T^*_{\bar p}$ is not irreducible.


Excluded Point Space is Path-Connected

Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space. Then $T^*_{\bar p}$ is path-connected.


Open Extension Space is Path-Connected

Let $T = \struct {S, \tau}$ be a topological space. Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the open extension space of $T$. Then $T^*_{\bar p}$ is path-connected.


Excluded Point Space is not Arc-Connected

Let $S$ be a set with at least two distinct elements. Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space. Then $T$ is not arc-connected.


Basis for Excluded Point Space

Let $T = \left({S, \tau_{\bar p}}\right)$ be an excluded point space. Consider the set $\mathcal B$ defined as: :$\mathcal B = \left\{{\left\{{x}\right\}: x \in S \setminus \left\{{p}\right\}}\right\} \cup \left\{{S}\right\}$ Then $B$ is a basis for $S$.


Partition Topology is Zero Dimensional

Let $T = \struct {S, \tau}$ be a partition space. Then $T$ is zero dimensional.


Discrete Space is Zero Dimensional

Let $T = \struct {S, \tau}$ be a discrete space. Then $T$ is zero dimensional.


Discrete Space is Scattered

Let $T = \struct {S, \tau}$ be a topological space where $\tau$ is the discrete topology on $S$. Then $T$ is a scattered space.


Discrete Space is Extremally Disconnected

Let $T = \struct {S, \tau}$ be a topological space where $\tau$ is the discrete topology on $S$. Then $T$ is extremally disconnected.


Totally Separated Space is Totally Disconnected

Let $T = \struct {S, \tau}$ be a topological space which is totally separated. Then $T$ is totally disconnected.


Open Extension Topology is not Perfectly T4

Let $T = \struct {S, \tau}$ be a topological space. Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the open extension space of $T$. Then $T^*_{\bar p}$ is not a perfectly $T_4$ space.


Excluded Point Space is not Perfectly T4

Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space. Then $T$ is not a perfectly $T_4$ space.


Type is Realized in some Elementary Extension

Let $\mathcal M$ be an $\mathcal L$-structure. Let $A$ be a subset of the universe of $\mathcal M$. Let $p$ be an $n$-type over $A$. There exists an elementary extension of $\mathcal M$ which realizes $p$.


Subset of Excluded Point Space is not Dense-in-itself

Let $T = \struct {S, \tau_{\bar p} }$ be a excluded point space such that $S$ is not a singleton. Let $H \subseteq S$. Then $H$ is not dense-in-itself.


Excluded Point Space is Scattered

Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space. Then $T$ is a scattered space.


Dispersion Point of Excluded Point Space

Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space. Then $p$ is a dispersion point of $T$.


Excluded Point Space is First-Countable

Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space. Then $T$ is first-countable.


Countable Excluded Point Space is Second-Countable

Let $T = \struct {S, \tau_{\bar p} }$ be a countable excluded point space. Then $T$ is a second-countable space.


Uncountable Excluded Point Space is not Second-Countable

Let $T = \struct {S, \tau_{\bar p} }$ be an uncountable excluded point space. Then $T$ is not second-countable.


Fractional Sobolev Embedding Theorem

Let $S'$ denote the space of tempered distributions. Let $\mathcal F : S' \to S'$ denote the Fourier transform. For each $s \in \R$ and $p \in [1, \infty]$ let $W^{s,p} \left({\R^n}\right) = \left\{{u \in S': \langle \xi \rangle^s \hat{u} \in L^p \left({\R^n}\right)}\right\}$ where $\displaystyle \left \langle{\xi}\right \rangle = \left({1 + \left|{\xi}\right|^2}\right)^\frac 1 2$. Then : 1. If $s > t$ then $W^{s,p} \left({\R^n}\right)$ embeds continuously into $W^{t,q}\left({\R^n}\right)$ where $q$ is given by $\displaystyle \frac 1 q = \frac 1 p - \frac {s-t} n$. {{stub}} {{namedfor|Sergei Lvovich Sobolev}} Category:Fourier Analysis 33i3o3ouq6duaqzehxwbx7ff7dp5v5c


Countable Excluded Point Space is Separable

Let $T = \struct {S, \tau_{\bar p} }$ be a countable excluded point space. Then $T$ is a separable space.


Uncountable Excluded Point Space is not Separable

Let $T = \struct {S, \tau_{\bar p} }$ be an uncountable excluded point space. Then $T$ is not separable.


Excluded Set Topology is Topology

Let $T = \struct {S, \tau_{\bar H} }$ be an excluded set space. Then $\tau_{\bar H}$ is a topology on $S$, and $T$ is a topological space.


Either-Or Topology is Topology

Let $T = \struct {S, \tau}$ be the either-or space. Then $\tau$ is a topology on $T$.


Closed Sets of Either-Or Topology

Let $T = \struct {S, \tau}$ be the either-or space. Then the closed sets of $T$ are: :$\O$ :$S$ :$\set {-1}$ :$\set 1$ :$\set {-1, 1}$ :Any subset of $\closedint {-1} 1$ containing $\set 0$ as a subset.


Type Space is Compact

Let $\mathcal M$ be an $\mathcal L$-structure, and let $A$ be a subset of the universe of  $\mathcal M$. The type space $S_n^{\mathcal M}(A)$ of $n$-types over $A$ is compact.


Omitting Types Theorem

Let $\mathcal L$ be the language of predicate logic with a countable signature. Let $T$ be an $\mathcal L$-theory. Let $\left\{{p_i: i \in \N}\right\}$ be a countable set of non-isolated $n$-types of $T$. There is a countable $\mathcal L$-structure $\mathcal M$ such that $\mathcal M \models T$ and $\mathcal M$ omits each $p_i$.


Vinogradov's Theorem

Let $\Lambda$ be the von Mangoldt function. For $N \in \Z$, let: :$\displaystyle \map R N = \sum_{n_1 + n_2 + n_3 \mathop = N} \map \Lambda {n_1} \, \map \Lambda {n_2} \, \map \Lambda {n_3}$ be a weighted count of the number of representations of $N$ as a sum of three prime powers. Let $\SS$ be the arithmetic function: :$\displaystyle \map \SS N = \prod_{p \mathop \nmid N} \paren {1 + \frac 1 {\paren {p - 1}^3} } \prod_{p \mathop \divides N} \paren {1 - \frac 1 {\paren {p - 1}^2} }$ where: :$p$ ranges over the primes :$p \nmid N$ denotes that $p$ is not a divisor of $N$ :$p \divides N$ denotes that $p$ is a divisor of $N$. Then for any $A > 0$ and sufficiently large odd integers $N$: :$\map R N = \dfrac 1 2 \map \SS N N^2 + \map \OO {\dfrac {N^2} {\paren {\log N}^A} }$ where $\OO$ denotes big-O notation.


Vinogradov's Theorem/Major Arcs

Let $B \in \R_{>0}$. Then: :$\displaystyle \int_\mathcal M \map F \alpha^3 \map e {-N \alpha} \rd \alpha = \frac {N^2} 2 \map {\mathcal S} N + \map {\mathcal O} {\frac {N^2} {\paren {\ln N}^{B/2} } }$ where the implied constant depends only on $B$.


Exponential Dominates Polynomial

Let $\exp$ denote the real exponential function. For any fixed $k \in \N$ and $\alpha > 0$ there exists $N \in \N$ such that $x^k < \exp \left({\alpha x}\right)$ for all real $x > N$. {{expand|it seems that the proof would apply to $x^k, x \in \R$ without any changes to the proof. Changes made as suggested Nov. 2016.}}


Power Dominates Logarithm

Let $\epsilon \in \R_{>0}$. Let $B \in \N$ be arbitrary. Then there exists $N \in \N$ such that: :$\forall n > N: \left({\ln n}\right)^B < n^\epsilon$


Condition for Open Extension Space to be T5 Space

Let $T = \struct {S, \tau}$ be a topological space. Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the open extension space of $T$. Then $T^*_{\bar p}$ is a $T_5$ space {{iff}} $T$ is a $T_5$ space.


Open Extension Topology is T4

Let $T = \struct {S, \tau}$ be a topological space. Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the open extension space of $T$. Then $T^*_{\bar p}$ is a $T_4$ space.


Open Extension Topology is not T3

Let $T = \struct {S, \tau}$ be a topological space. Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the open extension space of $T$. Then $T^*_{\bar p}$ is not a $T_3$ space.


Excluded Point Topology is not T3

Let $T = \struct {S, \tau_{\bar p} }$ be a excluded point space. Then $T$ is not a $T_3$ space.


Condition for Open Extension Space to be Separable

Let $T = \struct {S, \tau}$ be a topological space. Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the open extension space of $T$. Then $T^*_{\bar p}$ is a separable space {{iff}} $T$ is.


Finitely Satisfiable Theory has Maximal Finitely Satisfiable Extension

Let $T$ be a finitely satisfiable $\mathcal L$-theory. Then there exists a finitely satisfiable $\mathcal L$-theory $T'$ which contains $T$ as a subset such that: :for all $\mathcal L$-sentences $\phi$, either $\phi \in T'$ or $\neg \phi \in T'$.


Condition for Open Extension Space to be Second-Countable

Let $T = \struct {S, \tau}$ be a topological space. Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the open extension space of $T$. Then $T^*_{\bar p}$ is a second-countable space {{iff}} $T$ is.


Limit Points of Either-Or Topology

Let $T = \struct {S, \tau}$ be the either-or space. Let $H \subseteq S$ be any subset of $S$. Then no element of $S$ can be a limit point of $H$ except $0$.


Either-Or Topology is T5

Let $T = \struct {S, \tau}$ be the either-or space. Then $T$ is a $T_5$ space.


Either-Or Topology is Lindelöf

Let $T = \struct {S, \tau}$ be the either-or space. Then $T$ is a Lindelöf space.


Subspace of Either-Or Space less Zero is not Lindelöf

Let $T = \struct {S, \tau}$ be the either-or space. Let $H = S \setminus \set 0$ be the set $S$ without zero. Then the topological subspace $T_H = \struct {H, \tau_H}$ is not a Lindelöf space.


Basis for Either-Or Topology

Let $T = \left({S, \tau}\right)$ be the either-or space. Let $\mathcal B$ be the set: :$\mathcal B := \left\{{\left\{{x}\right\}: x \in S, x \ne 0}\right\} \cup \left\{{ \left({-1 \,.\,.\, 1}\right) }\right\}$ ... that is, the set of all singleton subsets of $S$ less $\left\{{0}\right\}$ and including the open real interval $\left({-1 \,.\,.\, 1}\right)$. Then $\mathcal B$ is a basis for $T$.


Either-Or Topology is Locally Path-Connected

Let $T = \struct {S, \tau}$ be the either-or space. Then $T$ is a locally path-connected space.


Limit Points of Infinite Subset of Finite Complement Space

Let $T = \struct {S, \tau}$ be a finite complement space. Let $H \subseteq S$ be an infinite subset of $S$. Then every point of $S$ is a limit point of $H$.


Finite Complement Topology is Separable

Let $T = \struct {S, \tau}$ be a finite complement topology on an infinite set $S$. Then $T$ is a separable space.


Closure of Infinite Subset of Finite Complement Space

Let $T = \struct {S, \tau}$ be a finite complement space. Let $H \subseteq S$ be an infinite subset of $S$. Then $H^- = S$ where $H^-$ is the closure of $S$.


Subspace of Finite Complement Topology is Compact

Let $T = \struct {S, \tau}$ be a finite complement topology on an infinite set $S$. Then every topological subspace of $T$, including $T$ itself, is a compact space.


Kluyver's Formula for Ramanujan's Sum

Let $q \in \N_{>0}$. Let $n \in \N$. Let $\map {c_q} n$ be Ramanujan's sum. Let $\mu$ denote the Möbius function. Then: :$\displaystyle \map {c_q} n = \sum_{d \mathop \divides \gcd \set {q, n} } d \map \mu {\frac q d}$ where $\divides$ denotes divisibility.


Condition for Complex Root of Unity to be Primitive

Let $n, k \in \N$. Then $\alpha_k = \map \exp {\dfrac {2 \pi i k} n}$ is a primitive $n$th root of unity {{iff}} $\gcd \set {n, k} = 1$.


Relation of Boubaker Polynomials to Chebyshev Polynomials

The Boubaker polynomials are related to Chebyshev polynomials through the equations: : $B_n \left({2x}\right) = \dfrac {4x}n \dfrac{\mathrm d}{\mathrm d x} T_n \left({x}\right) - 2 T_n \left({x}\right)$ : $B_n \left({2x}\right) = -2 T_n \left({x}\right) + 4 x U_{n-1} \left({x}\right)$ where: : $T_n$ denotes the Chebyshev polynomials of the first kind : $U_n$ denotes the Chebyshev polynomials of the second kind.


Relation of Boubaker Polynomials to Fermat Polynomials

The Boubaker polynomials are related to Fermat polynomials by: : $B_n \left({x}\right) = \dfrac 1 {\left({\sqrt 2}\right)^n} F_n \left({\dfrac {2 \sqrt 2 x} 3}\right) + \dfrac 3 {\left({\sqrt 2}\right)^{n-2}} F_{n-2} \left({\dfrac {2 \sqrt 2 x} 3}\right): \quad n = 0, 1, 2, \ldots$


F-Sigma and G-Delta Subsets of Uncountable Finite Complement Space

Let $T = \struct {S, \tau}$ be a finite complement topology on an uncountable set $S$. Then countably infinite subsets of $S$ are $F_\sigma$ sets and are neither open nor closed sets. Their relative complements in $S$ are $G_\delta$ sets, and are also neither open nor closed sets.


Skolem's Paradox

Let $\mathcal L$ be a countable first-order language. Let $T$ be an $\mathcal L$-theory which axiomatizes some version of set theory (for example, ZFC). There is a countable model of $T$.


Uncountable Finite Complement Space is not First-Countable

Let $T = \struct {S, \tau}$ be a finite complement topology on an uncountable set $S$. Then $T$ is not first-countable.


Separable Space need not be First-Countable

Let $T = \struct {S, \tau}$ be a topological space which is separable. Then $T$ does not necessarily have to be first-countable.


Asymptotic Growth of Euler Phi Function

Let $\phi$ be the Euler $\phi$ function. For any $\epsilon > 0$ and sufficiently large $n$: :$n^{1 - \epsilon} < \phi \left({n}\right) < n$


Finite Complement Space is T1

Let $T = \struct {S, \tau}$ be a finite complement topology on any set $S$ which contains at least two points. Then $T$ is a $T_1$ (Fréchet) space.


Finite Complement Space is Irreducible

Let $T = \struct {S, \tau}$ be a finite complement topology on an infinite set $S$. Then $T$ is irreducible.


Infinite Subset of Finite Complement Space Intersects Open Sets

Let $T = \struct {S, \tau}$ be a finite complement topology on an infinite set $S$. Let $H \subseteq S$ be an infinite subset of $S$. Then the intersection of $H$ with any non-empty open set of $T$ is infinite.


Irreducible Hausdorff Space is Singleton

Let $T = \left({S, \tau}\right)$ be a topological space which is irreducible and Hausdorff. Then $T$ contains only one point.


Finite Complement Space is not T2

Let $T = \struct {S, \tau}$ be a finite complement topology on an infinite set $S$. Then $T$ is not a $T_2$ (Hausdorff) space.


Finite Complement Space is not T3, T4 or T5

Let $T = \struct {S, \tau}$ be a finite complement topology on an infinite set $S$. Then $T$ is not a $T_3$ space, $T_4$ space or $T_5$ space.


Double Pointed Finite Complement Topology is Compact

Let $T = \struct {S, \tau}$ be a finite complement topology on an infinite set $S$. Let $T \times D$ be the double pointed topology on $T$. Then $T \times D$ is compact.


Double Pointed Finite Complement Topology fulfils no Separation Axioms

Let $T = \struct {S, \tau}$ be a finite complement topology on an infinite 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.


Vaughan's Identity

Let $\Lambda$ be von Mangoldt's function. Let $\mu$ be the Möbius function. Then for $y, z \ge 1$ and $n > z$: :$\displaystyle \map \Lambda n = \sum_{\substack {d \mathop \divides n \\ d \mathop \le y}} \map \mu d \, \map \ln {\frac n d} - \mathop {\sum \sum}_{\substack {d c \mathop \divides n \\ d \mathop \le y, \, c \mathop \le z}} \map \mu d \, \map \Lambda c + \mathop {\sum \sum}_{\substack {d c \mathop \divides n \\ d \mathop > y, \, c \mathop > z}} \map \mu d \, \map \Lambda c$ where $\divides$ denotes divisibility.


Sum Over Divisors of von Mangoldt is Logarithm

Let $\Lambda$ be von Mangoldt's function. Then for $n \geq 1$ :$\displaystyle \sum_{d | n} \Lambda(d) = \log n$


Multiplicative Function that Converges to Zero on Prime Powers

Let $f$ be a multiplicative function such that: :$\displaystyle \lim_{p^k \mathop \to \infty} \map f {p^k} = 0$ where $p^k$ runs though ''all'' prime powers. Then: :$\displaystyle \lim_{n \mathop \to \infty} \map f n = 0$ where $n$ runs through the integers.


Finite T1 Space is Discrete

Let $S$ be a finite set. Let $T = \left({S, \tau}\right)$ be a $T_1$ (Fréchet) space. Then $\tau$ is the discrete topology on $S$.


Finite Complement Topology is Minimal T1 Topology

Let $T = \struct {S, \tau}$ be a finite complement space. Let $\tau'$ be a topology on $S$ such that $T' = \struct {S, \tau'}$ is a $T_1$ (Fréchet) space. Then $\tau$ is comparable with $\tau'$ such that $\tau$ is coarser than $\tau'$. That is, of all the topologies on $S$ fulfilling the $T_1$ separation axiom, the finite complement space is the smallest. Thus the finite complement topology is known as the '''minimal $T_1$ topology''' on any given set.


Finite Complement Space is Connected

Let $T = \struct {S, \tau}$ be a finite complement topology on an infinite set $S$. Then $T$ is a connected space.


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: : $\displaystyle f_{B_n, \operatorname{EXP}} \left({x, t}\right) = \sum_{n=0}^\infty B_n \left({x}\right) \frac{t^n}{n!} = 4 x e^{t \frac x 2} \frac {\sin \left({t \sqrt {1 - \left({\frac x 2}\right)^2}}\right)} {\sqrt {1 - \left({\frac x 2}\right)^2}} - 2 e^{t \frac x 2} \cos \left({t \sqrt {1 - \left({\frac x 2}\right)^2}}\right) - 3$ {{Proofread}}


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 {{iff}} 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 $T$ {{iff}} $x$ 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 {{iff}} 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 {{iff}} $T$ 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 {{iff}} $T$ 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 {{iff}} $T$ 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 {{iff}} $T$ is.


Separation Axioms on Double Pointed Topology/T3.5 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 $T_{3 \frac 1 2}$ space {{iff}} $T$ is also a $T_{3 \frac 1 2}$ space.


Square of Vandermonde Matrix

The square of the Vandermonde matrix of order $n$: : $\mathbf V = \begin{bmatrix}   x_1 & x_2 & \cdots & x_n \\   x_1^2 & x_2^2 & \cdots & x_n^2 \\ \vdots & \vdots & \ddots & \vdots \\   x_1^n & x_2^n & \cdots & x_n^n \end{bmatrix}$ is symmetrical in $x_1, \ldots, x_n$. {{questionable|The case $n {{=}} 2$ left me clueless to what could possibly be intended here; only $\mathbf {V V}^T$ is trivially seen symmetric in the $x_n$, but this can hardly be called a square}}


Open and Closed Sets in Multiple Pointed Topology

Let $T = \left({S, \tau}\right)$ be a topological space. Let $A$ be a finite set whose cardinality is greater than $1$. Let $D = \left({A, \left\{{\varnothing, A}\right\}}\right)$ be the indiscrete space on $A$. Let $T \times D$ be a multiple pointed topological space generated from $T$ and $D$. Let $H \subseteq T$. Then: : $H \times A$ is open in $T \times D$ {{iff}} $H$ is open in $T$ : $H \times A$ is closed in $T \times D$ {{iff}} $H$ is closed in $T$ : $H \times \varnothing$ is both open and closed in $T \times D$ Let $\varnothing \subset H_A \subset A$, that is, let $H_A$ be a proper subset of $A$. Then $H \times H_A$ is neither open nor closed in $T \times D$.


Basis for Box Topology

Let $\left \langle {\left({S_i, \tau_i}\right)} \right \rangle_{i \mathop \in I}$ be an $I$-indexed family of topological spaces. Let $S$ be the cartesian product of $\left \langle {S_i} \right \rangle_{i \mathop \in I}$. That is: :$\displaystyle S := \prod_{i \mathop \in I} S_i$ Define: :$\displaystyle \mathcal B := \left\{{\prod_{i \mathop \in I} U_i: \forall i \in I: U_i \in \tau_i}\right\}$ Then $\mathcal B$ is a synthetic basis on $S$.


Space with Open Point is Non-Meager

Let $T = \struct {S, \tau}$ be a topological space. Let $x \in S$ be an open point. Then $T$ is a non-meager space.


Continuous Mapping on Finite Union of Closed Sets

Let $T = \struct {X, \tau}$ and $S = \struct {Y,\sigma}$ be topological spaces. For all $i \in \set {1, 2, \ldots, n}$, let $C_i$ be closed in $T$. Let $f: X \to Y$ be a mapping such that the restriction $f \restriction_{C_i}$ is continuous for all $i$. Then $f$ is continuous on $C = \displaystyle \bigcup_{i \mathop = 1}^n C_i$, that is, $f \restriction_C$ is continuous. If $\family {C_i}$ is infinite, the result does not necessarily hold.


Equivalence of Definitions of T2 Space

{{TFAE|def = T2 Space|view = $T_2$ (Hausdorff) space}} Let $T = \struct {S, \tau}$ be a topological space.


Equivalence of Definitions of T3 Space

{{TFAE|def = T3 Space|view = $T_3$ space}} Let $T = \struct {S, \tau}$ be a topological space.


Big Implies Saturated

Let $\mathcal M$ be an $\mathcal L$-structure. Let $\kappa$ be a cardinal. If $\mathcal M$ is $\kappa$-big, then it is $\kappa$-saturated.


Limit Points in T1 Space

Let $T = \struct {S, \tau}$ be a topological space which satisfies the $T_1$ (Fréchet) axiom. Let $H \subset S$ be any subset of $S$. Let $x \in H$. Then $x$ is a limit point of $H$ {{iff}} every neighborhood of $x$ contains infinitely many points of $H$.


Singleton Point is Isolated

Let $T = \left({S, \tau}\right)$ be a topological space. Let $x \in S$. Then $x$ is an isolated point of the singleton set $\left\{{x}\right\}$, but not necessarily an isolated point of $T$.


Tietze Extension Theorem

Let $T = \struct {S, \tau}$ be a topological space which is normal. Let $A \subseteq S$ be a closed set in $T$. Let $f: A \to \R$ be a continuous mapping from $A \subseteq S$ to the real number line under the usual (Euclidean) topology. Then there exists a continuous extension $g: S \to \R$, i.e. such that: :$\forall s \in A: \map f s = \map g s$


Hermitian Operators have Real Eigenvalues

Hermitian operations have real eigenvalues.


Hausdorff Space iff Diagonal Set on Product is Closed

Let $T = \struct{S, \tau}$ be a topological space. Let $\Delta_S$ be the diagonal set on $S$: :$\Delta_S = \set{\tuple{x, x} \in S \times S: x \in S}$ where $S \times S$ is the Cartesian product of $S$ with itself. Let $T^2 = \struct{S \times S, \TT}$ be the product space with Tychonoff topology $\TT$. Then $T$ is a $T_2$ (Hausdorff) space iff $\Delta_S$ is closed in $T^2$.


T0 Property is Hereditary

Let $T = \struct {S, \tau}$ be a topological space which is a $T_0$ (Kolmogorov) space. Let $T_H = \struct {H, \tau_H}$, where $\O \subset H \subseteq S$, be a subspace of $T$. Then $T_H$ is a $T_0$ (Kolmogorov) space.


T1 Property is Hereditary

Let $T = \struct {S, \tau}$ be a topological space which is a $T_1$ (Fréchet) space. Let $T_H = \struct {H, \tau_H}$, where $\varnothing \subset H \subseteq S$, be a subspace of $T$. Then $T_H$ is a $T_1$ (Fréchet) space.


T2 Property is Hereditary

Let $T = \struct {S, \tau}$ be a topological space which is a $T_2$ (Hausdorff) space. Let $T_H = \struct {H, \tau_H}$, where $\O \subset H \subseteq S$, be a subspace of $T$. Then $T_H$ is a $T_2$ (Hausdorff) space. That is, the property of being a $T_2$ (Hausdorff) space is hereditary.


Completely Hausdorff Property is Hereditary

Let $T = \struct {S, \tau}$ be a topological space which is a $T_{2 \frac 1 2}$ (completely Hausdorff) space. Let $T_H = \struct {H, \tau_H}$, where $\O \subset H \subseteq S$, be a subspace of $T$. Then $T_H$ is a $T_{2 \frac 1 2}$ (completely Hausdorff) space.


Closure in Subspace

Let $T = \struct {S, \tau}$ be a topological space. Let $T_H = \struct {H, \tau_H}$, where $\O \subset H \subseteq S$, be a subspace of $T$. Let $U \subseteq H$ and let $\map {\cl_T} U$ be the closure of $U$ in $T$. Then $\map {\cl_T} U \cap H$ is the closure of $U$ in $T_H$. That is: :$\map {\cl_T} U \cap H = \map {\cl_H} U$ where $\map {\cl_H} U$ denotes the closure of $U$ in $T_H$.


T3 Property is Hereditary

Let $T = \struct {S, \tau}$ be a topological space which is a $T_3$ space. Let $T_H = \struct {H, \tau_H}$, where $\O \subset H \subseteq S$, be a subspace of $T$. Then $T_H$ is a $T_3$ space.


T3 1/2 Property is Hereditary

Let $T = \struct {S, \tau}$ be a topological space which is a $T_{3 \frac 1 2}$ space. Let $T_H = \struct {H, \tau_H}$, where $\O \subset H \subseteq S$, be a subspace of $T$. Then $T_H$ is a $T_{3 \frac 1 2}$ space.


T5 Property is Hereditary

Let $T = \struct {S, \tau}$ be a topological space which is a $T_5$ space. Let $T_H = \struct {H, \tau_H}$, where $\O \subset H \subseteq S$, be a subspace of $T$. Then $T_H$ is a $T_5$ space.


T5 Space iff Every Subspace is T4

Let $T = \struct {S, \tau}$ be a topological space. Then $T$ is a $T_5$ space {{iff}} every subspace of $T$ is a $T_4$ space.


Product Space is T0 iff Factor Spaces are T0

Let $T_A = \struct {S_A, \tau_A}$ and $T_B = \struct {S_B, \tau_B}$ be topological spaces. Let $T = T_A \times T_B$ be the product space formed from $T_A$ and $T_B$. Then $T$ is a $T_0$ (Kolmogorov) space {{iff}} $T_A$ and $T_B$ are themselves both $T_0$ (Kolmogorov) spaces.


Product Space is T1 iff Factor Spaces are T1

Let $\mathbb S = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an indexed family of topological spaces for $\alpha$ in some indexing set $I$. Let $\displaystyle T = \struct {S, \tau} = \displaystyle \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\mathbb S$. Then $T$ is a $T_1$ (Fréchet) space {{iff}} each of $\struct {S_\alpha, \tau_\alpha}$ is a $T_1$ (Fréchet) space.


Gödel's Incompleteness Theorems/First

Let $T$ be the set of theorems of some recursive set of sentences in the language of arithmetic such that $T$ contains minimal arithmetic. $T$ cannot be both consistent and complete.


Undecidability Theorem

Let $T$ be the set of theorems of some consistent theory in the language of arithmetic which contains minimal arithmetic $Q$. $T$ is not recursive.


Set of Gödel Numbers of Arithmetic Theorems Not Definable in Arithmetic

Let $T$ be the set of theorems of some consistent theory in the language of arithmetic which contains minimal arithmetic. The set of Gödel numbers of the theorems of $T$ is not definable in $T$.


Diagonal Lemma

Let $T$ be the set of theorems of some theory in the language of arithmetic which contains minimal arithmetic. For any formula $\map B y$ in the language of arithmetic, there is a sentence $G$ such that :$T \vdash G \leftrightarrow \map B {\hat G}$ where $\hat G$ is the Gödel number of $G$ (more accurately, it is the term in the language of arithmetic obtained by applying the function symbol $s$ to $0$ this many times). {{Disambiguate|Definition:Logical Formula}}