hcju/proofwikips huggingface dataset:
Definition:Interior (Topology)/Definition 1
The '''interior''' of $H$ is the union of all subsets of $H$ which are open in $T$. That is, the '''interior''' of $H$ is defined as: :$\displaystyle H^\circ := \bigcup_{K \mathop \in \mathbb K} K$ where $\mathbb K = \set {K \in \tau: K \subseteq H}$.
Definition:Interior (Topology)/Definition 2
The '''interior''' of $H$ is defined as the largest open set of $T$ which is contained in $H$.
Definition:Interior Point (Topology)
Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$. === Definition 1 === {{:Definition:Interior Point (Topology)/Definition 1}} === Definition 2 === {{:Definition:Interior Point (Topology)/Definition 2}} === Definition 3 === {{:Definition:Interior Point (Topology)/Definition 3}}
Definition:Nowhere Dense/Definition 1
$H$ is '''nowhere dense''' in $T$ : :$\paren {H^-}^\circ = \O$ where $H^-$ denotes the closure of $H$ and $H^\circ$ denotes its interior.
Definition:Nowhere Dense/Definition 2
$H$ is '''nowhere dense''' in $T$ : :$H^-$ contains no open set of $T$ which is non-empty where $H^-$ denotes the closure of $H$.
Definition:Category of Cones
Let $\mathbf C$ be a metacategory. Let $D: \mathbf J \to \mathbf C$ be a $\mathbf J$-diagram in $\mathbf C$. The '''category of cones to $D$''', denoted $\mathbf{Cone} \left({D}\right)$, is defined as follows: {{DefineCategory | ob = The cones to $D$ | mor = The morphisms of cones | comp = Composition in $\mathbf C$ |id = $\operatorname{id}_{\left({C, c_j}\right)} := \operatorname{id}_C$, for a cone $\left({C, c_j}\right)$ }}
Definition:Limit (Category Theory)
Let $\mathbf C$ be a metacategory. Let $D: \mathbf J \to \mathbf C$ be a $\mathbf J$-diagram in $\mathbf C$. Let $\mathbf{Cone} \left({D}\right)$ be the category of cones to $D$. A '''limit for $D$''' is a terminal object in $\mathbf{Cone} \left({D}\right)$. It is denoted by $\varprojlim_j D_j$; the associated morphisms $p_i: \varprojlim_j D_j \to D_i$ are usually left implicit.
Definition:Identification Topology/Identification Mapping
The mapping $f: S_1 \to S_2$ in this context is called the '''identification mapping'''.
Definition:Product (Category Theory)/General Definition
Let $\mathbf C$ be a metacategory. Let $\mathcal C$ be any collection of objects of $\mathbf C$. Let $\mathbf{Dis} \left({\mathcal C}\right)$ be the discrete category on $\mathcal C$, considered as a subcategory of $\mathbf C$. A '''product''' for $\mathcal C$, denoted $\displaystyle \prod \mathcal C$, is a limit for the inclusion functor $D: \mathbf{Dis} \left({\mathcal C}\right) \to \mathbf C$, considered as a diagram. For an object $C$ in $\mathcal C$, the associated morphism $\displaystyle \prod \mathcal C \to C$ is denoted $\operatorname{pr}_C$ and called the '''projection on $C$'''. The whole construction is pictured in the following commutative diagram: ::$\begin{xy}\xymatrix@R-.5em@L+3px{ & &  A   \ar@{-->}[dd]   \ar[dddl]_*+{a_C}   \ar[dddr]^*+{a_C'} \\ \\ & &  \displaystyle \prod \mathcal C   \ar[dl]^*{\operatorname{pr}_C}   \ar[dr]_*{\operatorname{pr}_{C'}} \\  \mathbf{Dis} \left({\mathcal C}\right) &  C &  \dots \quad \dots &  C' }\end{xy}$ === Finite Product === {{:Definition:Product (Category Theory)/General Definition/Finite Product}}
Definition:Product (Category Theory)/Binary Product
Let $\mathbf C$ be a metacategory. Let $A$ and $B$ be objects of $\mathbf C$. A '''(binary) product diagram''' for $A$ and $B$ comprises an object $P$ and morphisms $p_1: P \to A$, $p_2: P \to B$: ::$\begin{xy}\xymatrix@+1em@L+3px{  A &  P   \ar[l]_*+{p_1}   \ar[r]^*+{p_2} &  B }\end{xy}$ subjected to the following universal mapping property: :For any object $X$ and morphisms $x_1, x_2$ like so: ::$\begin{xy}\xymatrix@+1em@L+3px{  A &  X   \ar[l]_*+{x_1}   \ar[r]^*+{x_2} &  B }\end{xy}$ :there is a unique morphism $u: X \to P$ such that: ::$\begin{xy}\xymatrix@+1em@L+3px{ &  X   \ar[ld]_*+{x_1}   \ar@{-->}[d]^*+{u}   \ar[rd]^*+{x_2} \\  A &  P   \ar[l]^*+{p_1}   \ar[r]_*+{p_2} &  B }\end{xy}$ :is a commutative diagram, i.e., $x_1 = p_1 \circ u$ and $x_2 = p_2 \circ u$. In this situation, $P$ is called a '''(binary) product of $A$ and $B$''' and may be denoted $A \times B$. Generally, one writes $\left\langle{x_1, x_2}\right\rangle$ for the unique morphism $u$ determined by above diagram. The morphisms $p_1$ and $p_2$ are often taken to be implicit. They are called '''projections'''; if necessary, $p_1$ can be called the '''first projection''' and $p_2$ the '''second projection'''. {{expand|the projection definition may merit its own, separate page}}
Definition:Product (Category Theory)/General Definition/Finite Product
Let $\displaystyle \prod \mathcal C$ be a product for a finite set $\mathcal C$ of objects of $\mathbf C$. Then $\displaystyle \prod \mathcal C$ is called a '''finite product'''.
Definition:Finite Category
Let $\mathbf C$ be a metacategory. Then $\mathbf C$ is said to be a '''finite category''' iff both the collection of its objects $\mathbf C_0$ and the collection of its morphisms $\mathbf C_1$ are finite sets.
Definition:Limit (Category Theory)/Finite Limit
Let $\varprojlim_j D_j$ be a limit for $D$. Then $\varprojlim_j D_j$ is called a '''finite limit'''  $\mathbf J$ is a finite category.
Definition:Functor Preserving Limits
Let $\mathbf C$, $\mathbf D$ and $\mathbf J$ be metacategories. Let $F: \mathbf C \to \mathbf D$ be a functor. Then $F$ '''preserves limits of type $\mathbf J$''' iff for all diagrams $D: \mathbf J \to \mathbf C$ with limit ${\varprojlim \,}_j \, D_j$: :$F \left({{\varprojlim \,}_j \, D_j}\right) \cong {\varprojlim \,}_j \, F D_j$ where $F D: \mathbf J \to \mathbf D$ is the composition of $F$ with $D$.
Definition:Continuous Functor
Let $\mathbf C$, $\mathbf D$ be metacategories. Let $F: \mathbf C \to \mathbf D$ be a functor. Then $F$ is '''continuous''' iff for all diagrams $D: \mathbf J \to \mathbf C$ with limit ${\varprojlim \,}_j \, D_j$: :$F \left({{\varprojlim \,}_j \, D_j}\right) \cong {\varprojlim \,}_j \, F D_j$ where $F D: \mathbf J \to \mathbf D$ is the diagram obtained by composition of $F$ with $D$, and $\mathbf J$ is an arbitrary metacategory.
Definition:Hausdorff Space/Definition 1
$\struct {S, \tau}$ is a '''Hausdorff space''' or '''$T_2$ space''' : :$\forall x, y \in S, x \ne y: \exists U, V \in \tau: x \in U, y \in V: U \cap V = \O$  That is: :for any two distinct elements $x, y \in S$ there exist disjoint open sets $U, V \in \tau$ containing $x$ and $y$ respectively.
Definition:Hausdorff Space/Definition 2
$\struct {S, \tau}$ is a '''Hausdorff space''' or '''$T_2$ space'''  each point is the intersection of all its closed neighborhoods.
Definition:Cocone
Let $\mathbf C$ be a metacategory. Let $D: \mathbf J \to \mathbf C$ be a $\mathbf J$-diagram in $\mathbf C$. A '''cocone from $D$''' comprises an object $C$ of $\mathbf C$, and a morphism: :$c_j: D_j \to C$ for each object of $\mathbf J$, such that for each morphism $\alpha: i \to j$ of $\mathbf J$: ::$\begin{xy}\xymatrix@+0.5em@L+2px{  D_i   \ar[r]^*+{D_\alpha}   \ar[dr]_*+{c_i} &  D_j   \ar[d]^*+{c_j} \\ &  C }\end{xy}$ is a commutative diagram. {{expand|when time comes, add def "cocone is natural trafo $D \to \kappa_C$ with $\kappa_C$ constant functor"}}
Definition:Morphism of Cocones
Let $\mathbf C$ be a metacategory. Let $D: \mathbf J \to \mathbf C$ be a $\mathbf J$-diagram in $\mathbf C$. Let $\left({C, c_j}\right)$ and $\left({C', c'_j}\right)$ be cocones from $D$. Let $f: C \to C'$ be a morphism of $\mathbf C$. Then $f$ is a '''morphism of cones''' iff, for all objects $j$ of $\mathbf J$: ::$\begin{xy}\xymatrix@+0.5em@L+3px{  D_j   \ar[d]_*+{c_j}   \ar[dr]^*+{c'_j} \\  C   \ar[r]_*+{f} &  C' }\end{xy}$ is a commutative diagram.
Definition:Category of Cocones
Let $\mathbf C$ be a metacategory. Let $D: \mathbf J \to \mathbf C$ be a $\mathbf J$-diagram in $\mathbf C$. The '''category of cocones from $D$''', denoted $\mathbf{Cocone} \left({D}\right)$, is the category with: {{DefineCategory |ob = cocones to $D$ |mor = morphisms of cocones |comp = Composition in $\mathbf C$ |id = $\operatorname{id}_{\left({C, c_j}\right)} := \operatorname{id}_C$, for a cocone $\left({C, c_j}\right)$ }}
Definition:Colimit
Let $\mathbf C$ be a metacategory. Let $D: \mathbf J \to \mathbf C$ be a $\mathbf J$-diagram in $\mathbf C$. Let $\mathbf{Cocone} \left({D}\right)$ be the category of cocones from $D$. A '''colimit for $D$''' is an initial object in $\mathbf{Cocone} \left({D}\right)$. It is denoted by $\varinjlim_j D_j$; the associated morphisms $\iota_i: D_i \to \varinjlim_j D_j$ are usually left implicit. === Finite Colimit === {{:Definition:Colimit/Finite Colimit}}
Definition:Colimit/Finite Colimit
Let $\varinjlim_j D_j$ be a colimit for $D$. Then $\varinjlim_j D_j$ is called a '''finite colimit'''  $\mathbf J$ is a finite category.
Definition:T3 Space/Definition 1
$T = \struct {S, \tau}$ is a '''$T_3$ space''' : :$\forall F \subseteq S: \relcomp S F \in \tau, y \in \relcomp S F: \exists U, V \in \tau: F \subseteq U, y \in V: U \cap V = \O$  That is, for any closed set $F \subseteq S$ and any point $y \in S$ such that $y \notin F$ there exist disjoint open sets $U, V \in \tau$ such that $F \subseteq U$, $y \in V$.
Definition:T3 Space/Definition 2
$T = \struct {S, \tau}$ is '''$T_3$'''  each open set contains a closed neighborhood around each of its points: :$\forall U \in \tau: \forall x \in U: \exists N_x: \relcomp S {N_x} \in \tau: \exists V \in \tau: x \in V \subseteq N_x \subseteq U$ where $N_x$ denotes a neighborhood of $x$.
Definition:T3 Space/Definition 3
$T = \struct {S, \tau}$ is '''$T_3$'''  each of its closed sets is the intersection of its closed neighborhoods: :$\forall H \subseteq S: \relcomp S H \in \tau: H = \bigcap \set {N_H: \relcomp S H \in \tau, \exists V \in \tau: H \subseteq V \subseteq N_H}$
Definition:Fréchet Space (Topology)/Definition 2
$\struct {S, \tau}$ is a '''Fréchet space''' or '''$T_1$ space'''  all points of $S$ are closed in $T$.
Definition:Separated Sets
Let $T = \left({S, \tau}\right)$ be a topological space. Let $A, B \subseteq S$. === Definition 1 === {{:Definition:Separated Sets/Definition 1}} === Definition 2 === {{:Definition:Separated Sets/Definition 2}} $A$ and $B$ are said to be '''separated sets (of $T$)'''.
Definition:Separated Points
Let $\struct {S, \tau}$ be a topological space. Let $x, y \in S$ such that both of the following hold: :$\exists U \in \tau: x \in U, y \notin U$ :$\exists V \in \tau: y \in V, x \notin V$ Then $x$ and $y$ are '''separated points'''.
Definition:Kolmogorov Space/Definition 2
$\struct {S, \tau}$ is a '''Kolmogorov space''' or '''$T_0$ space'''  no two points can be limit points of each other.
Definition:Direct Limit of Sequence of Groups
=== Definition 1 === {{:Definition:Direct Limit of Sequence of Groups/Definition 1}} === Definition 2 === {{:Definition:Direct Limit of Sequence of Groups/Definition 2}}
Definition:T4 Space/Definition 1
$T = \struct {S, \tau}$ is a '''$T_4$ space''' : :$\forall A, B \in \map \complement \tau, A \cap B = \O: \exists U, V \in \tau: A \subseteq U, B \subseteq V, U \cap V = \O$  That is, for any two disjoint closed sets $A, B \subseteq S$ there exist disjoint open sets $U, V \in \tau$ containing $A$ and $B$ respectively.
Definition:T4 Space/Definition 2
$T = \struct {S, \tau}$ is '''$T_4$'''  each open set $U$ contains a closed neighborhood of each closed set contained in $U$.
Definition:T5 Space/Definition 1
$\struct {S, \tau}$ is a '''$T_5$ space''' : :$\forall A, B \subseteq S, A^- \cap B = A \cap B^- = \O: \exists U, V \in \tau: A \subseteq U, B \subseteq V, U \cap V = \O$    That is: :$\struct {S, \tau}$ is a '''$T_5$ space''' when for any two separated sets $A, B \subseteq S$ there exist disjoint open sets $U, V \in \tau$ containing $A$ and $B$ respectively.
Definition:T5 Space/Definition 2
$\struct {S, \tau}$ is a '''$T_5$ space''' : :$\forall Y,A \subseteq S: (A \subseteq Y^\circ \wedge A^- \subseteq Y) \implies \exists N \subseteq Y: \relcomp S N \in \tau: \exists U \in \tau: A \subseteq U \subseteq N$ That is: :$\struct {S, \tau}$ is a '''$T_5$ space'''  every subset $Y \subseteq S$ contains a closed neighborhood of each $A \subseteq Y^\circ$ for which $A^- \subseteq Y$. In the above, $Y^\circ$ denotes the interior of $Y$ and $A^-$ denotes the closure of $A$.
Definition:Homeomorphism/Topological Spaces/Definition 1
$f$ is a '''homeomorphism'''  both $f$ and $f^{-1}$ are continuous.
Definition:Homeomorphism/Topological Spaces/Definition 2
$f$ is a '''homeomorphism''' : :$\forall U \subseteq S_\alpha: U \in \tau_\alpha \iff f \sqbrk U \in \tau_\beta$
Definition:Homeomorphism/Topological Spaces/Definition 3
$f$ is a '''homeomorphism'''  $f$ is both an open mapping and a continuous mapping.
Definition:Homeomorphism/Topological Spaces/Definition 4
$f$ is a '''homeomorphism'''  $f$ is both a closed mapping and a continuous mapping.
Definition:Category of Groups
The '''category of groups''', denoted $\mathbf{Grp}$, is the metacategory with: {{DefineCategory |ob   = groups |mor  = group homomorphisms |id   = identity mappings |comp = composition of mappings }}
Definition:Cover of Set/Finite
A cover $\mathcal C$ for $S$ is a '''finite cover'''  $\mathcal C$ is a finite set.
Definition:Cover of Set/Countable
A cover $\mathcal C$ for $S$ is a '''countable cover'''  $\mathcal C$ is a countable set.
Definition:Subcover/Finite
A '''finite subcover of $\mathcal U$ for $S$''' is a subcover $\mathcal V \subseteq \mathcal U$ which is finite.
Definition:Subcover/Countable
A '''countable subcover of $\mathcal U$ for $S$''' is a subcover $\mathcal V \subseteq \mathcal U$ which is countable.
Definition:Compact Space/Topology/Subspace/Definition 1
The topological subspace $T_H = \left({H, \tau_H}\right)$ is '''compact in $T$'''  $T_H$ is itself a compact topological space.
Definition:Compact Space/Topology/Subspace/Definition 2
$H$ is '''compact in $T$'''  every open cover $\mathcal C \subseteq \tau$ for $H$ has a finite subcover.
Definition:Direct Limit of Sequence of Groups/Definition 1
Let $\sequence {G_n}_{n \mathop \in \N}$ be a sequence of groups. For each $n \in \N$, let $g_n: G_n \to G_{n + 1}$ be a group homomorphism. A '''direct limit''' for the sequences $\sequence {G_n}_{n \mathop \in \N}$ and $\sequence {g_n}_{n \mathop \in \N}$ comprises: :$(1): \quad$ a group $G_\infty$ :$(2): \quad$ for each $n \in \N$, a group homomorphism $u_n: G_n \to G_\infty$ such that, for all $n \in \N$: :$u_{n + 1} \circ g_n = u_n$ and, for all groups $H$ together with group homomorphisms $h_n: G_n \to H$ satisfying $h_{n + 1} \circ g_n = h_n$, there exists a unique group homomorphism: :$h_\infty: G_\infty \to H$ such that for all $n \in \N$: :$h_n = h_\infty \circ u_n$
Definition:Direct Limit of Sequence of Groups/Definition 2
Let $\N$ be the order category on the natural numbers. Let $\mathbf{Grp}$ be the category of groups. Let $G: \N \to \mathbf{Grp}$ be an $\N$-diagram in $\mathbf{Grp}$. A '''direct limit''' for $G$ is a colimit ${\varinjlim \,}_n \, G_n$, and is denoted $G_\infty$.
Definition:Functor Creating Limits
Let $\mathbf C, \mathbf D$ and $\mathbf J$ be metacategories. Let $F: \mathbf C \to \mathbf D$ be a functor. Then $F$ is said to '''create limits of type $\mathbf J$''' iff: :For all $\mathbf J$-diagrams $C: \mathbf J \to \mathbf C$ in $\mathbf C$, given a limit $\left({{\varprojlim \,}_j \, FC_j, q_j}\right)$ for $FC: \mathbf J \to \mathbf D$ in $\mathbf D$, the limit: ::$\left({{\varprojlim \,}_j \, C_j, p_j}\right)$ :exists, and furthermore: ::$F \left({{\varprojlim \,}_j \, C_j}\right) = {\varprojlim \,}_j \, FC_j$ ::$F p_j = q_j$ :for all objects $j$ of $\mathbf J$.
Definition:Functor Creating Colimits
Let $\mathbf C, \mathbf D$ and $\mathbf J$ be metacategories. Let $F: \mathbf C \to \mathbf D$ be a functor. Then $F$ is said to '''create colimits of type $\mathbf J$''' iff: :For all $\mathbf J$-diagrams $C: \mathbf J \to \mathbf C$ in $\mathbf C$, given a colimit $\left({{\varinjlim \,}_j \, FC_j, q_j}\right)$ for $FC: \mathbf J \to \mathbf D$ in $\mathbf D$, the colimit: ::$\left({{\varinjlim \,}_j \, C_j, p_j}\right)$ :exists, and furthermore: ::$F \left({{\varinjlim \,}_j \, C_j}\right) = {\varinjlim \,}_j \, FC_j$ ::$F p_j = q_j$ :for all objects $j$ of $\mathbf J$.
Definition:Uniform Equivalence/Metrics
Let $A$ be a set on which there are two metrics imposed: $d_1$ and $d_2$. Then $d_1$ and $d_2$ are '''uniformly equivalent''' iff the identity mapping of $A$ is uniformly $\left({d_1, d_2}\right)$-continuous and also uniformly $\left({d_2, d_1}\right)$-continuous.
Definition:Uniform Equivalence/Metric Spaces
Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces. Then the mapping $f: A_1 \to A_2$ is a '''uniform equivalence''' of $M_1$ with $M_2$ iff $f$ is a bijection such that $f$ and $f^{-1}$ are both uniformly continuous.
Definition:Evaluation Mapping
Let $S, T$ be sets, and let $S^T$ be the set of all mappings from $T$ to $S$. The '''evaluation mapping for $S^T$''' is the mapping $\operatorname{ev}: S^T \times T \to S$ defined by: :$\operatorname{ev} \left({f, t}\right) := f \left({t}\right)$
Definition:Exponential (Category Theory)
Let $\mathbf C$ be a metacategory with binary products. Let $B$ and $C$ be objects of $\mathbf C$. An '''exponential of $C$ by $B$''' consists of an object $C^B$ of $\mathbf C$ and a morphism: :$\epsilon: C^B \times B \to C$ subject to the following UMP: :For all objects $A$ and morphisms $f: A \times B \to C$, there exists a unique morphism: ::$\tilde f: A \to C^B$ :such that: :::$\begin{xy}\xymatrix@+1em@L+3px{  C^B \times B   \ar[r]^*+{\epsilon} &  C \\  A \times B   \ar[u]_*+{\tilde f \times \operatorname{id}_B \hskip{2cm}}   \ar[ur]_*+{f} }\end{xy}$ :is a commutative diagram, i.e. $\epsilon \circ \left({\tilde f \times \operatorname{id}_B}\right) = f$. === Evaluation Morphism === {{:Definition:Exponential (Category Theory)/Evaluation}} === Exponential Transpose === {{:Definition:Exponential (Category Theory)/Transpose}} === Category with Exponentials === {{:Definition:Category with Exponentials}}
Definition:Topological Manifold/Complex Manifold
Let $M$ be a second-countable, complex locally Euclidean space of dimension $d$.  Let $\mathscr F$ be a complex analytic differentiable structure on $M$. Then $\left({M, \mathscr F}\right)$ is called a '''complex manifold of dimension $d$'''.
Definition:Smooth Differentiable Structure
Let $M$ be a topological space. Let $d$ be a natural number. A '''$d$-dimensional smooth differentiable structure''' $\mathscr F$ on $M$ is a $d$-dimensional differentiable structure on $M$ which is of class $\mathcal C^k$ for every $k \in \N$.
Definition:Complex Analytic Differentiable Structure
Let $M$ be a locally Euclidean space of dimension $d$. Then a '''complex analytic structure''' $\mathscr F$ on $M$ is a collection of charts $\{(U_\alpha,\phi_\alpha) : \alpha \in A\}$ such that: :$(1): \quad \displaystyle \bigcup_{\alpha \in A} U_\alpha = M$ :$(2): \quad \phi_\alpha \circ \phi_\beta^{-1}$ is of biholomorphic as a map $\phi_\beta\left(U_\alpha \cap U_\beta\right) \to \phi_\alpha\left(U_\alpha \cap U_\beta\right)$ for all $\alpha,\beta \in A$ :$(3): \quad$ If $(U,\phi)$ is a chart such that $\phi \circ \phi_\alpha^{-1}$ and $\phi_\alpha \circ \phi^{-1}$ are biholomorphic for all $\alpha \in A$, then $(U,\phi) \in \mathscr F$.
Definition:Locally Euclidean Space/Complex
$M$ is a '''complex locally Euclidean space of dimension $d$'''  each point in $M$ has an open neighbourhood homeomorphic to an open subset of complex Euclidean space $\C^d$.
Definition:Category with Products
Let $\mathbf C$ be a metacategory. Then $\mathbf C$ is said to '''have products''' or to be a '''(meta)category with products''' iff: :For all sets of objects $\mathcal C \subseteq \mathbf C_0$, there is a product $\displaystyle \prod \mathcal C$ for $\mathcal C$. === Category with Binary Products === {{:Definition:Category with Products/Binary}} === Category with Finite Products === {{:Definition:Category with Products/Finite}}
Definition:Category with Products/Finite
Let $\mathbf C$ be a metacategory. Then $\mathbf C$ is said to '''have products''' or to be a '''(meta)category with products''' iff: :For all finite sets of objects $\mathcal C \subseteq \mathbf C_0$, there is a product $\displaystyle \prod \mathcal C$ for $\mathcal C$.
Definition:Category with Exponentials
Suppose $\mathbf C$ has an exponential $C^B$ for all objects $B$ and $C$ of $\mathbf C$. Then $\mathbf C$ is called a '''category with exponentials'''.
Definition:Cartesian Closed Category
Let $\mathbf C$ be a metacategory. Then $\mathbf C$ is called '''Cartesian closed'''  $\mathbf C$ is a metacategory with finite products and with exponentials.
Definition:Topological Manifold
=== Differentiable Manifold === {{:Definition:Topological Manifold/Differentiable Manifold}} === Smooth Manifold === {{:Definition:Topological Manifold/Smooth Manifold}} === Complex Manifold === {{:Definition:Topological Manifold/Complex Manifold}}
Definition:Connected (Topology)/Topological Space
Let $T = \left({S, \tau}\right)$ be a non-empty topological space.
Definition:Connected (Topology)/Set
{{:Definition:Connected (Topology)/Set/Definition 1}}
Definition:Connected (Topology)/Points
Let $T = \struct {S, \tau}$ be a topological space. Let $a, b \in S$. Then $a$ and $b$ are '''connected''' (in $T$)  there exists a connected set in $T$ containing both $a$ and $b$.
Definition:Disconnected (Topology)
=== Topological Space === {{Definition:Disconnected (Topology)/Topological Space}} === Subset of Topological Space === {{Definition:Disconnected (Topology)/Set}} === Points in Topological Space === {{:Definition:Disconnected (Topology)/Points}}
Definition:Degenerate Connected Set/Non-Degenerate
A '''non-degenerate connected set''' of $T$ is a connected set of $T$ containing more than one element.
Definition:Connected (Topology)/Topological Space/Definition 1
$T$ is '''connected'''  it admits no separation.
Definition:Connected (Topology)/Topological Space/Definition 2
$T$ is '''connected'''  it has no two disjoint nonempty closed sets whose union is $S$.
Definition:Connected (Topology)/Topological Space/Definition 3
$T$ is '''connected'''  its only subsets whose boundary is empty are $S$ and $\varnothing$.
Definition:Connected (Topology)/Topological Space/Definition 4
$T$ is '''connected'''  its only clopen sets are $S$ and $\O$.
Definition:Connected (Topology)/Topological Space/Definition 5
$T$ is '''connected'''  there are no two non-empty separated sets whose union is $S$.
Definition:Connected (Topology)/Topological Space/Definition 6
$T$ is '''connected'''  there exists no continuous surjection from $T$ onto a discrete two-point space.
Definition:Exponential (Category Theory)/Evaluation
The morphism: :$\epsilon: C^B \times B \to C$ associated to $C^B$ is called the '''evaluation morphism'''.
Definition:Exponential (Category Theory)/Transpose
For a morphism $f: A \times B \to C$, the unique: :$\tilde f: A \to C^B$ provided by the UMP for $C^B$ is called the '''exponential transpose''' of $f$. For a morphism $g: A \to C^B$, the morphism $\bar g: A \times B \to C$ defined by: :$\bar g = \epsilon \circ \left({g \times \operatorname{id}_B}\right)$ is also called the '''exponential transpose''' of $g$.