hcju/proofwikips huggingface dataset:
Separable Metric Space is Second-Countable
Let $M = \struct {A, d}$ be a metric space. Let $M$ be separable. Then $M$ is second-countable.
Lipschitz Equivalence is Equivalence Relation
Let $A$ be a set. Let $\mathcal D$ be the set of all metrics on $A$. Let $\sim$ be the relation on $\mathcal D$ defined as: :$\forall d_1, d_2 \in \mathcal D: d_1 \sim d_2 \iff d_1$ is Lipschitz equivalent to $d_2$ Then $\sim$ is an equivalence relation.
Sequentially Compact Metric Space is Separable
A sequentially compact metric space is separable.
Sequentially Compact Metric Space is Second-Countable
A sequentially compact metric space is second-countable.
Lipschitz Equivalent Metrics are Topologically Equivalent
Let $M_1 = \struct {A, d_1}$ and $M_2 = \struct {A, d_2}$ be metric spaces on the same underlying set $A$. Let $d_1$ and $d_2$ be Lipschitz equivalent. Then $d_1$ and $d_2$ are topologically equivalent.
Compact Metric Space is Complete
A compact metric space is complete.
Pullback as Equalizer
Let $\mathbf C$ be a metacategory with binary products. Suppose that the following diagram commutes (possibly except for the square): ::$\begin{xy}\xymatrix@+1em{  P   \ar@/^/[rrd]^*+{p_1}   \ar@/_/[ddr]_*+{p_2}   \ar[rd]^*+{e} \\ &  A \times B   \ar[r]_*+{\pi_1}   \ar[d]^*+{\pi_2} &  A   \ar[d]^*+{f} \\ &  B   \ar[r]_*+{g} &  C }\end{xy}$ where $A \times B$ is the binary product of $A$ and $B$, and $\pi_1, \pi_2$ are the associated projections. Then the border of the diagram, i.e.: ::$\begin{xy}\xymatrix{  P   \ar[r]^*+{p_1}   \ar[d]_*+{p_2} &  A   \ar[d]^*+{f} \\  B   \ar[r]_*+{g} &  C }\end{xy}$ is a pullback of $f$ and $g$ iff $e: P \to A \times B$ is the equalizer of $f \circ \pi_1$ and $g \circ \pi_2$.
Minkowski's Inequality for Sums/Index 2
Let $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n \ge 0$ be non-negative real numbers. Then: :$\displaystyle \paren {\sum_{k \mathop = 1}^n \paren {a_k + b_k}^2}^{1/2} \le \paren {\sum_{k \mathop = 1}^n a_k^2}^{1/2} + \paren {\sum_{k \mathop = 1}^n b_k^2}^{1/2}$
Banach-Steinhaus Theorem
Let $X$ be a Banach space. Let $Y$ be a normed vector space with norm $\norm {\,\cdot\,}_Y$. Let $\family {T_\alpha: X \to Y}_{\alpha \mathop \in A}$ be an $A$-indexed family of bounded linear transformations from $X$ to $Y$. Suppose that: :$\displaystyle \forall x \in X: \sup_{\alpha \mathop \in A} \norm {T_\alpha x}_Y < \infty$ Then: :$\displaystyle \sup_{\alpha \mathop \in A} \norm {T_\alpha} < \infty$ where $\norm {T_\alpha}$ denotes the norm of the linear transformation $T_\alpha$. {{explain|We may need a definitive explanation for the intuitively understood $< \infty$.}}
Minkowski's Inequality for Sums/Index Greater than 1
Let $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n \in \R_{\ge 0}$ be non-negative real numbers. Let $p \in \R$ be a real number such that $p > 1$. Then: :$\displaystyle \left({\sum_{k \mathop = 1}^n \left({a_k + b_k}\right)^p}\right)^{1/p} \le \left({\sum_{k \mathop = 1}^n a_k^p}\right)^{1/p} + \left({\sum_{k \mathop = 1}^n b_k^p}\right)^{1/p}$
Minkowski's Inequality for Sums/Index Less than 1
Let $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n \in \R_{\ge 0}$ be non-negative real numbers. Let $p \in \R$ be a real number. If $p < 0$, then we require that $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n$ be strictly positive. If $p < 1$, $p \ne 0$, then: :$\displaystyle \left({\sum_{k \mathop = 1}^n \left({a_k + b_k}\right)^p}\right)^{1/p} \ge \left({\sum_{k \mathop = 1}^n a_k^p}\right)^{1/p} + \left({\sum_{k \mathop = 1}^n b_k^p}\right)^{1/p}$
Monotone Real Function is Darboux Integrable
Let $\closedint a b$ be a closed real interval, where $a < b$. Let $f: \closedint a b \to \R$ be a monotone real function. Then $f$ is Darboux integrable over $\closedint a b$.
Topologies Not Always Comparable by Coarseness
Let $S$ be a set with at least $2$ elements. Let $\mathbb T$ be the set of all topologies on $S$. For two topologies $\tau_a, \tau_b \in \mathbb T$, let $\tau_a \le \tau_b$ denote that $\tau_a$ is coarser than $\tau_b$. Then there exist $\tau_1, \tau_2 \in \mathbb T$ such that neither: :$\tau_1 \le \tau_2$ nor: :$\tau_2 \le \tau_1$ That is, there are always topologies on $S$ which are non-comparable.
Pullback of Subset Inclusion
Denote with $\mathbf{Set}$ the category of sets. Let $A, B$ be sets, and let $f: A \to B$ be a mapping. Let $V \subseteq B$ be a subset of $B$. Denote with $i: V \to B$ the inclusion mapping. Let $f^{-1} \left({V}\right) \subseteq A$ be the preimage of $V$ under $f$. Denote with $j: f^{-1} \left({V}\right) \to A$ the inclusion mapping. Denote with $\bar f = f \restriction_{f^{-1} \left({V}\right)}$ the restriction of $f$ to $f^{-1} \left({V}\right)$. Then: ::$\begin{xy}\xymatrix{  f^{-1} \left({V}\right)   \ar[r]^*+{\bar f}   \ar[d]_*+{j} &  V   \ar[d]^*+{i} \\  A   \ar[r]_*+{f} &  B }\end{xy}$ is a pullback diagram in $\mathbf{Set}$.
Poincaré Plane is Abstract Geometry
The Poincaré plane $\struct {\H, L_H}$ is an abstract geometry.
Pullback Lemma
Let $\mathbf C$ be a metacategory. Suppose that the following is a commutative diagram in $\mathbf C$: ::$\begin{xy}\xymatrix@+0.5em{  F   \ar[d]_*+{h''}   \ar[r]^*+{f'} &  E   \ar[d]^*+{h'}   \ar[r]^*+{g'} &  D   \ar[d]^*+{h} \\  A   \ar[r]_*+{f} &  B   \ar[r]_*+{g} &  C }\end{xy}$ Suppose furthermore that the right square is a pullback. Then the left square is a pullback iff the outer rectangle is.
Carathéodory's Theorem (Convex Analysis)
Let $E \subset \R^\ell$. Let $\mathbb x \in \map {\operatorname{co} } E$. {{explain|Definition of $\map {\operatorname{co} } E$}} Then $\mathbf x$ is a convex combination of affinely independent points of $E$.  In particular, $\mathbf x $ is a convex combination of at most $\ell + 1$ points of $E$.
Limit of Sine of X over X/Corollary
:$\displaystyle \lim_{x \mathop \to 0} \frac x {\sin x} = 1$
Dicyclic Group is Group/Corollary
The quaternion group $Q_4$ is a non-abelian group.
Integral of Constant/Definite
:$\displaystyle \int_a^b c \ \ \mathrm dx = c \left({b-a}\right)$.
Primitive of Constant
:$\ds \int c \rd x = c x + C$ where $C$ is an arbitrary constant.
Unbounded Set of Real Numbers is not Compact
Let $\R$ be the set of real numbers considered as a Euclidean space. Let $S \subseteq \R$ be unbounded in $\R$. Then $S$ is not a compact subspace of $\R$.
Continuous Mapping is Sequentially Continuous
Let $X$ and $Y$ be topological spaces. Let $x \in X$. Let $f: X \to Y$ be a mapping that is continuous at $x$. Then $f$ is sequentially continuous at $x$.
Sequential Continuity is Equivalent to Continuity in Metric Space
Let $\left({X, d}\right)$ and $\left({Y, e}\right)$ be metric spaces. Let $f: X \to Y$ be a mapping. Let $x \in X$. Then $f$ is continuous at $x$  $f$ is sequentially continuous at $x$.
Pullback of Commutative Triangle
Let $\mathbf C$ be a metacategory. Suppose that the following is a commutative diagram in $\mathbf C$: ::$\begin{xy}\xymatrix@+1em@L+2px{  A'   \ar[rr]^*{h_\alpha}   \ar[dd]_*{\alpha'} & &  A   \ar[rd]^*{\gamma}   \ar[dd]^(.4)*{\alpha} \\ &  B'   \ar[ld]^*{\beta'}   \ar[rr] |{\hole} ^(.3)*{h_\beta} & &  B   \ar[ld]^*{\beta} \\  C'   \ar[rr]_*{h} & &  C }\end{xy}$ and that the two squares in it are pullback diagrams. Then there is a unique morphism $\gamma': A' \to B'$ making the following commute: ::$\begin{xy}\xymatrix@+1em@L+2px{  A'   \ar[rr]^*{h_\alpha}   \ar[dd]_*{\alpha'}   \ar@{-->}[rd]^*{\gamma'} & &  A   \ar[rd]^*{\gamma}   \ar[dd]^(.4)*{\alpha} \\ &  B'   \ar[ld]^*{\beta'}   \ar[rr] |{\hole} ^(.3)*{h_\beta} & &  B   \ar[ld]^*{\beta} \\  C'   \ar[rr]_*{h} & &  C }\end{xy}$ Furthermore, $\gamma'$ makes the following a pullback: ::$\begin{xy}\xymatrix@+.5em@L+2px{  A'   \ar[r]^*{h_\alpha}   \ar[d]_*{\gamma'} &  A   \ar[d]^*{\gamma} \\  B'   \ar[r]_*{h_\beta} &  B }\end{xy}$
Associative Commutative Idempotent Operation is Distributive over Itself
Let $\left({S, \circ}\right)$ be an algebraic structure, such that: :$(1): \quad \circ$ is associative :$(2): \quad \circ$ is commutative :$(3): \quad \circ$ is idempotent. Then $\circ$ is distributive over itself. That is: :$\forall a, b, c \in S: \left({a \circ b}\right) \circ \left({a \circ c}\right) = a \circ b \circ c = \left({a \circ c}\right) \circ \left({b \circ c}\right)$
Intersection Distributes over Intersection
Set intersection is distributive over itself: :$\forall A, B, C: \paren {A \cap B} \cap \paren {A \cap C} = A \cap B \cap C = \paren {A \cap C} \cap \paren {B \cap C}$ where $A, B, C$ are sets.
Inclusion Mapping is Restriction of Identity
Let $T$ be a set. Let $S \subseteq T$ be a subset of $T$. Let $i_S: S \to T$ be the inclusion mapping on $S$. Then $i_S$ is the restriction of the identity mapping $I_T: T \to T$ on $T$.
Preimage of Subset under Inclusion Mapping
Let $S$ be a set. Let $H \subseteq S$ be a subset of $S$. Let $i_H: H \to S$ be the inclusion mapping on $H$. Let $T \subseteq S$. Then: :$i_H^{-1} \left({T}\right) = T \cap H$ where $i_H^{-1} \left({T}\right)$ is the preimage of $T$ under $i_H$.
Inclusion Mapping is Continuous
Let $T = \struct {S, \tau}$ be a topological space. Let $T_H = \struct {H, \tau_H}$ be a topological subspace of $T$ where $H \subseteq S$. Let $i_H: H \to S$ be the inclusion mapping on $H$. Then $i_H$ is a $\struct {\tau_H, \tau}$-continuous mapping.
Continuity of Composite with Inclusion/Mapping on Inclusion
If $f$ is $\tuple {\tau, \tau'}$-continuous, then $f \circ i$ is $\tuple {\tau_H, \tau'}$-continuous
Continuity of Composite with Inclusion/Inclusion on Mapping
$g$ is $\left({\tau', \tau_H}\right)$-continuous iff $i \circ g$ is $\left({\tau', \tau}\right)$-continuous.
Continuity of Composite with Inclusion/Uniqueness of Induced Topology
The induced topology $\tau_H$ is the ''only'' topology on $H$ satisfying Continuity of Composite with Inclusion: Inclusion on Mapping for all possible $g$.
Products of Open Sets form Local Basis in Product Space
Let $T_1 = \struct{A_1, \tau_1}$ and $T_2 = \struct{A_2, \tau_2}$ be topological spaces. Let $\struct{T, \tau} = T_1 \times T_2$ be the product space of $T_1$ and $T_2$. Let $\tuple{x, y} \in A_1 \times A_2$. Let $W \in \tau$ be an open set of $T$ such that $\tuple{x, y} \in W$. Then: :$\exists U_1 \in \tau_1, U_2 \in \tau_2: \tuple{x, y} \in U_1 \times U_2 \subseteq W$ That is, products of open sets from $T_1$ and $T_2$ form a local basis at $\tuple{x, y}$.
Open Real Intervals are Homeomorphic
Consider the real numbers $\R$ as a metric space under the Euclidean metric. Let $I_1 := \left({a \,.\,.\, b}\right)$ and $I_2 := \left({c \,.\,.\, d}\right)$ be non-empty open real intervals. Then $I_1$ and $I_2$ are homeomorphic.
Half-Open Real Interval is not Closed in Real Number Line
Let $\R$ be the real number line considered as an Euclidean space. Let $\left[{a \,.\,.\, b}\right) \subset \R$ be a half-open interval of $\R$. Then $\left[{a \,.\,.\, b}\right)$ is not a closed set of $\R$. Similarly, the half-open interval $\left({a \,.\,.\, b}\right] \subset \R$ is not a closed set of $\R$.
Underlying Set of Topological Space is Clopen
Let $T = \left({S, \tau}\right)$ be a topological space. Then the underlying set $S$ of $T$ is both open and closed in $T$.
Empty Set is Closed in Topological Space
Let $T = \struct {S, \tau}$ be a topological space. Then $\O$ is closed in $T$.
Basis induces Local Basis
Let $\left({S, \tau}\right)$ be a topological space. Let $\mathcal B$ be an analytic basis for $\tau$. Let $x \in S$, and denote with $\mathcal B_x$ the set $\left\{{B \in \mathcal B: x \in B}\right\}$. Then $\mathcal B_x$ is a local basis at $x$.
Supremum of Bounded Above Set of Reals is in Closure
Let $\R$ be the real number line under the Euclidean metric. Let $H \subseteq \R$ be a bounded above subset of $\R$ such that $H \ne \varnothing$. Let $u = \map \sup H$ be the supremum of $H$. Then: :$u \in \map \cl H$ where $\map \cl H$ denotes the closure of $H$ in $\R$.
Topological Space is Open Neighborhood of Subset
Let $T = \left({S, \tau}\right)$ be a topological space. Let $H \subseteq S$ be a subset of $S$. Then $S$ is an open neighborhood of $H$.
Open Superset is Open Neighborhood
Let $T = \struct {S, \tau}$ be a topological space. Let $U \in \tau$ be an open set of $T$. Let $H \subseteq U$. Then $U$ is an open neighborhood of $H$.
Category has Products and Equalizers iff Pullbacks and Terminal Object
Let $\mathbf C$ be a metacategory. Then the following are equivalent: :$(1): \mathbf C$ has all finite products and equalizers. :$(2): \mathbf C$ has all pullbacks and a terminal object.
Infimum of Bounded Below Set of Reals is in Closure
Let $\R$ be the real number line with the  usual (Euclidean} metric. Let $H \subseteq \R$ be a bounded below subset of $\R$ such that $H \ne \O$. Let $l = \map \inf H$ be the infimum of $H$. Then: :$l \in \map \cl H$ where $\map \cl H$ denotes the closure of $H$ in $\R$.
Nowhere Dense iff Complement of Closure is Everywhere Dense/Corollary
Let $H$ be a closed set of $T$. Then $H$ is nowhere dense in $T$  $S \setminus H$ is everywhere dense in $T$.
Equivalence of Definitions of Topology Generated by Synthetic Basis
Let $S$ be a set. Let $\BB$ be a synthetic basis on $S$. {{TFAE|def = Topology Generated by Synthetic Basis}}
Union from Synthetic Basis is Topology
Let $\BB$ be a synthetic basis on a set $X$. Let $\displaystyle \tau = \set {\bigcup \AA: \AA \subseteq \BB}$. Then $\tau$ is a topology on $X$.
Product as Limit
Let $\mathbf C$ be a metacategory. Let $C_1, C_2$ be objects of $\mathbf C$. Let their binary product $C_1 \times C_2$ exist in $\mathbf C$. Then $C_1 \times C_2$ is the limit of the diagram $D: 2 \to \mathbf C$ defined by: :$D_0 := C_1, D_1 := C_2$ where $2$ is the discrete category with two objects $0, 1$.
Equalizer as Limit
Let $\mathbf C$ be a metacategory. Let $f_1, f_2: C_1 \to C_2$ be morphisms of $\mathbf C$. Let their equalizer $e: E \to C_1$ exist in $\mathbf C$. Then $\left({E, e}\right)$ is the limit of the diagram $D: \mathbf J \to \mathbf C$ defined by: ::$\begin{xy}\xymatrix@+1em@L+3px{  \mathbf{J}: &  \ast   \ar[r]<2pt>   \ar[r]<-2pt> &  \star }\end{xy}$ ::$\begin{xy}\xymatrix@+1em@L+3px{  D: &  C_1   \ar[r]<2pt>^*+{f_1}   \ar[r]<-2pt>_*+{f_2} &  C_2 }\end{xy}$
Terminal Object as Limit
Let $\mathbf C$ be a metacategory. Let $\mathbf C$ have a terminal object $1$. Then $1$ is the limit of the unique diagram $D: \mathbf 0 \to \mathbf C$, where $\mathbf 0$ is the zero category.
Pullback as Limit
Let $\mathbf C$ be a metacategory. Let $f_1: A \to C$ and $f_2: B \to C$ be morphisms of $\mathbf C$. Let their pullback: ::$\begin{xy}\xymatrix@+1em@L+3px{  P   \ar[r]^*+{p_2}   \ar[d]_*+{p_1} &  A   \ar[d]^*+{f_1} \\  B   \ar[r]_*+{f_2} &  C }\end{xy}$ exist in $\mathbf C$. Then $\left({P, p_1, p_2}\right)$ is the limit of the diagram $D: \mathbf J \to \mathbf C$ defined by: ::$\begin{xy}\xymatrix@+1em@L+3px{  \save[]+<0em,-2em>*{\mathbf{J}:} \restore & &  \cdot   \ar[d] \\ &  \cdot   \ar[r] &  \cdot }\end{xy}$ ::$\begin{xy}\xymatrix@+1em@L+3px{  \save[]+<0em,-2em>*{D:} \restore & &  A   \ar[d]^*+{f_1} \\ &  B   \ar[r]_*+{f_2} &  C }\end{xy}$
Identification Topology is Topology
Let $T_1 = \struct {S_1, \tau_1}$ be a topological space. Let $S_2$ be a set. Let $f: S_1 \to S_2$ be a mapping. Let $\tau_2$ be the identification topology on $S_2$ with respect to $f$ and $\struct {S_1, \tau_1}$. Then $\tau_2$ is a topology on $S_2$.
Identification Mapping is Continuous
Let $T_1 = \struct {S_1, \tau_1}$ be a topological space. Let $S_2$ be a set. Let $f: S_1 \to S_2$ be a mapping. Let $\tau_2$ be the identification topology on $S_2$ with respect to $f$ and $\struct {S_1, \tau_1}$. Then the identification mapping $f$ is continuous.
Existence and Uniqueness of Identification Topology
Let $T_1 = \left({S_1, \tau_1}\right)$ be a topological space. Let $S_2$ be a set. Let $f: S_1 \to S_2$ be a mapping. Let $\tau_2$ be the identification topology on $S_2$ with respect to $f$ and $\left({S_1, \tau_1}\right)$. Then the identification topology on $S_2$ with respect to $f$ and $\left({S_1, \tau_1}\right)$ always exists and is unique.
Identification Topology is Finest Topology for Mapping to be Continuous
Let $T_1 := \struct {S_1, \tau_1}$ be a topological space. Let $S_2$ be a set. Let $f: S_1 \to S_2$ be a mapping. Let $\tau_2$ be the identification topology on $S_2$ with respect to $f$ and $\struct {S_1, \tau_1}$. Let $T_2 := \struct {S_2, \tau_2}$ be the resulting topological space. Then $\tau_2$ is the finest topology on $S_2$ such that $f: T_1 \to T_2$ is continuous.
Identification Topology equals Quotient Topology on Induced Equivalence
Let $T_1 = \struct {S_1, \tau_1}$ be a topological space. Let $S_2$ be a set. Let $f: S_1 \to S_2$ be a mapping. Let $\tau_2$ be the identification topology on $S_2$ with respect to $f$ and $\struct {S_1, \tau_1}$: :$\tau_2 = \set {V \in \powerset {S_2}: f^{-1} \sqbrk V \in \tau_1}$ Let $\RR_f \subseteq S_1 \times S_1$ be the equivalence on $S_1$ induced by $f$: :$\tuple {s_1, s_2} \in \RR_f \iff \map f {s_1} = \map f {s_2}$ Let $\tau_{\RR_f}$ be the quotient topology on $S / \RR_f$ by $q_{\RR_f}$: :$\tau_{\RR_f} := \set {U \subseteq S / \RR_f: q_{\RR_f}^{-1} \sqbrk U \in \tau_1}$ Then: :$\tau_2$ is homeomorphic to $\tau_{\RR_f}$.
Continuity of Composite with Identification Mapping
Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces. Let $f_1: T_1 \to T_2$ be a mapping. Let $S_3$ be a set. Let $p: S_1 \to S_3$ be a mapping. Let $\tau_3$ be the identification topology on $S_3$ with respect to $p$ and $T_1$. Let $T_3 = \struct {S_3, \tau_3}$ be the resulting topological space. Let $f_2: S_3 \to S_2$ be a mapping such that: :$f_1 = f_2 \circ p$ where $f_2 \circ p$ is the composition of $f_2$ with $p$. Then $f_1$ is a continuous mapping  $f_2$ is a continuous mapping.
Young's Inequality for Increasing Functions
Let $a_0$ and $b_0$ be strictly positive real numbers. Let $f: \closedint 0 {a_0} \to \closedint 0 {b_0}$ be a strictly increasing bijection. Let $a$ and $b$ be real numbers such that $0 \le a \le a_0$ and $0 \le b \le b_0$. Then: :$\displaystyle ab \le \int_0^a \map f u \rd u + \int_0^b \map {f^{-1}} v \rd v$ where $\displaystyle \int$ denotes the definite integral.
Set is Closed iff Equals Topological Closure
Let $T$ be a topological space. Let $H \subseteq T$. Then $H$ is closed in $T$ : :$H = \map \cl H$
Equivalence of Definitions of Compact Topological Subspace
{{TFAE|def = Compact Subspace}} Let $T = \left({S, \tau}\right)$ be a topological space. Let $T_H = \left({H, \tau_H}\right)$ be a topological subspace of $T$, where $H \subseteq S$.
Covariant Hom Functor is Continuous
Let $\mathbf{Set}$ be the category of sets. Let $\mathbf C$ be a locally small category. Let $C$ be an object of $\mathbf C$, and let $\hom \paren {C, \cdot}: \mathbf C \to \mathbf{Set}$ be the covariant hom functor based at $C$. Then $\hom \paren {C, \cdot}$ is a continuous functor.
Contravariant Hom Functor maps Colimits to Limits
Let $\mathbf{Set}$ be the category of sets. Let $\mathbf C$ be a locally small category. Let $C$ be an object of $\mathbf C$. Let $\map \hom {\cdot, C}: \mathbf C \to \mathbf{Set}$ be the contravariant hom functor based at $C$. Then $\map \hom {\cdot, C}$ maps every colimit to a limit, in that: :$\map \hom { {\varinjlim \,}_j \,D_j, C} \cong {\varprojlim \,}_j \, \map \hom {D_j, C}$ for every diagram $D: \mathbf J \to \mathbf C$.
Composite Mapping is Mapping
Let $S_1, S_2, S_3$ be sets. Let $f: S_1 \to S_2$ and $g: S_2 \to S_3$ be mappings. Then the composite mapping $g \circ f$ is also a mapping.
Normal Space is Regular Space
Let $\struct {S, \tau}$ be a normal space. Then $\struct {S, \tau}$ is also a regular space.
Product of Hausdorff Factor Spaces is Hausdorff/General Result
Let $\SS = \family {\struct {S_\alpha, \tau_\alpha} }$ be an indexed family of topological spaces for $\alpha$ in some indexing set $I$. Let $\displaystyle T = \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\SS$. Let each of $\struct {S_\alpha, \tau_\alpha}$ for $\alpha \in I$ be $T_2$ (Hausdorff) spaces. Then $T$ is a $T_2$ (Hausdorff) space.
Product of Hausdorff Factor Spaces is Hausdorff
Let $T_\alpha = \struct {S_\alpha, \tau_\alpha}$ and $T_\beta = \struct {S_\beta, \tau_\beta}$ be topological spaces. Let $T = T_\alpha \times T_\beta$ be the product space of $T_\alpha$ and $T_\beta$ Let $T_\alpha$ and $T_\beta$ both be $T_2$ (Hausdorff) spaces. Then $T$ is also a $T_2$ (Hausdorff) space.
Domain of Continuous Injection to Hausdorff Space is Hausdorff
Let $T_\alpha = \struct {S_\alpha, \tau_\alpha}$ and $T_\beta = \struct {S_\beta, \tau_\beta}$ be topological spaces. Let $f: S_\alpha \to S_\beta$ be a continuous mapping which is an injection. If $T_\beta$ is a $T_2$ (Hausdorff) space, then $T_\alpha$ is also a $T_2$ (Hausdorff) space.
Quotient Space of Hausdorff Space is not necessarily Hausdorff
Let $T = \struct {S, \tau}$ be a Hausdorff space. Let $\RR \subseteq S \times S$ be an equivalence relation on $S$. Let $T_\RR := \struct {S / \RR, \tau_\RR}$ be the quotient space of $S$ by $\RR$. Then $T_\RR$ is not necessarily also a Hausdorff space.
Topological Product of Compact Spaces/Finite Product
Let $T_1, T_2, \ldots, T_n$ be topological spaces. Let $\displaystyle \prod_{i \mathop = 1}^n T_i$ be the product space of $T_1, T_2, \ldots, T_n$. Then $\displaystyle \prod_{i \mathop = 1}^n T_i$ is compact  all of $T_1, T_2, \ldots, T_n$ are compact.
Uniform Continuity on Metric Space does not imply Compactness
Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces. Let $f: A_1 \to A_2$ be a uniformly continuous mapping on $A_1$. Then $M_1$ does not necessarily have to be a compact metric space.
Uniformly Continuous Function is Continuous/Metric Space
Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_1, d_1}\right)$ be metric spaces. Let the mapping $f: M_1 \to M_2$ be uniformly continuous on $M_1$. Then $f$ is continuous on $M_1$.
Uniformly Continuous Function is Continuous/Real Function
Let $I$ be an interval of $\R$. Let $f: I \to \R$ be a uniformly continuous real function on $I$. Then $f$ is continuous on $I$.
Category of Finite Sets is Cartesian Closed
Let $\mathbf{Finset}$ be the category of finite sets. Then $\mathbf{Finset}$ is Cartesian closed.
Components of Separation are Clopen
Let $T = \left({S, \tau}\right)$ be a topological space. Let $A \mid B$ be a separation of $T$. Then both $A$ and $B$ are clopen in $T$.
Equivalence of Definitions of Connected Topological Space/No Separation iff No Union of Closed Sets
{{TFAE|def = Connected Topological Space}} Let $T = \left({S, \tau}\right)$ be a topological space.
Complements of Components of Two-Component Partition form Partition
Let $S$ be a set with at least two elements. Let $A, B \subseteq S$. Let $\complement_S$ denote the complement relative to $S$. $A \mid B$ is a partition of $S$  $\relcomp S A \mid \relcomp S B$ is a partition of $S$.
Sigma-Algebra Closed under Finite Intersection
Let $A_1, \ldots, A_n \in \Sigma$. Then $\displaystyle \bigcap_{k \mathop = 1}^n A_k \in \Sigma$.
Equivalence of Definitions of Connected Topological Space/No Union of Closed Sets implies No Subsets with Empty Boundary
Let $T = \left({S, \tau}\right)$ be a topological space. Let $T$ have no two disjoint non-empty closed sets whose union is $S$. Then the only subsets of $S$ whose boundary is empty are $S$ and $\varnothing$.
Equivalence of Definitions of Connected Topological Space/No Subsets with Empty Boundary implies No Clopen Sets
Let $T = \left({S, \tau}\right)$ be a topological space. Let $T$ be such that the only subsets of $S$ whose boundary is empty are $S$ and $\varnothing$. Then the only clopen sets of $T$ are $S$ and $\varnothing$.
Equivalence of Definitions of Connected Topological Space/No Clopen Sets implies No Union of Separated Sets
Let $T = \struct {S, \tau}$ be a topological space. Let the only clopen sets of $T$ be $S$ and $\O$. Then there are no two non-empty separated sets of $T$ whose union is $S$.
Equivalence of Definitions of Connected Topological Space/No Union of Separated Sets implies No Continuous Surjection to Discrete Two-Point Space
Let $T = \struct {S, \tau}$ be a topological space. Let $T$ be such that there are no two non-empty separated sets whose union is $S$. Then there exists no continuous surjection from $T$ onto a discrete two-point space.