hcju/proofwikips huggingface dataset:
Definition:Cauchy Sequence/Metric Space
Let $M = \struct {A, d}$ be a metric space. Let $\sequence {x_n}$ be a sequence in $M$. Then $\sequence {x_n}$ is a '''Cauchy sequence''' : :$\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall m, n \in \N: m, n \ge N: \map d {x_n, x_m} < \epsilon$
Definition:Net (Metric Space)/Finite Net
A '''finite $\epsilon$-net for $M$''' is an $\epsilon$-net for $M$ which is a finite set.
Definition:Cauchy Sequence/Real Numbers
Let $\sequence {x_n}$ be a sequence in $\R$. Then $\sequence {x_n}$ is a '''Cauchy sequence''' : : $\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall m, n \in \N: m, n \ge N: \size {x_n - x_m} < \epsilon$
Definition:Cauchy Sequence/Rational Numbers
Let $\left \langle {x_n} \right \rangle$ be a rational sequence. Then $\left \langle {x_n} \right \rangle$ is a '''Cauchy sequence''' : : $\forall \epsilon \in \Q_{>0}: \exists N \in \N: \forall m, n \in \N: m, n \ge N: \left|{x_n - x_m}\right| < \epsilon$ where $\Q_{>0}$ denotes the set of all strictly positive rational numbers.
Definition:Euclidean Space/Real
Let $\R^n$ be an $n$-dimensional real vector space. Let the Euclidean metric $d$ be applied to $\R^n$. Then $\struct {\R^n, d}$ is a '''Euclidean $n$-space'''.
Definition:Euclidean Space/Rational
Let $\Q^n$ be an $n$-dimensional vector space of rational numbers. Let the Euclidean Metric $d$ be applied to $\Q^n$. Then $\left({\Q^n, d}\right)$ is a Euclidean $n$-space.
Definition:Euclidean Space/Complex
Let $\C$ be the complex plane. Let $d$ be the Euclidean metric on $\C$. Then $\left({\C, d}\right)$ is a Euclidean space.
Definition:Euclidean Space/Euclidean Topology
Let $S$ be one of the standard number fields $\Q$, $\R$, $\C$. Let $S^n$ be a cartesian space for $n \in \N_{\ge 1}$. Let $M = \left({S^n, d}\right)$ be a Euclidean space. The topology $\tau_d$ induced by the Euclidean metric $d$ is called the '''Euclidean topology'''.
Definition:Discriminant of Polynomial/Quadratic Equation
The expression $b^2 - 4 a c$ is called the '''discriminant''' of the equation.
Definition:Linearly Ordered Space
Let $\struct {S, \preceq}$ be a linearly ordered set. Let $\tau$ be the order topology on $S$. Then $\struct {S, \preceq, \tau}$ is a '''linearly ordered space'''.
Definition:Bounded Mapping/Real-Valued/Attaining its Bounds
Let $f: S \to \R$ be a bounded real-valued function. Let $T$ be a subset of $S$. Suppose that: :$\exists a, b \in T: \forall x \in S: f \left({a}\right) \le f \left({x}\right) \le f \left({b}\right)$ Then $f$ '''attains its bounds on $T$'''.
Definition:Bounded Mapping/Real-Valued
{{:Definition:Bounded Mapping/Real-Valued/Definition 1}}
Definition:Bounded Mapping/Complex-Valued
Let $f: S \to \C$ be a complex-valued function. Then $f$ is '''bounded'''  the real-valued function $\cmod f: S \to \R$ is bounded, where $\cmod f$ is the modulus of $f$. That is, $f$ is '''bounded''' if there is a constant $K \ge 0$ such that $\cmod {f \paren z} \le K$ for all $z \in S$.
Definition:Continuous Mapping (Metric Space)/Space/Definition 2
$f$ is '''continuous from $\left({A_1, d_1}\right)$ to $\left({A_2, d_2}\right)$''' : :for every $U \subseteq A_2$ which is open in $M_2$, $f^{-1} \left[{U}\right]$ is open in $M_1$.
Definition:Composition Functor on Categories of Subobjects
Let $\mathbf C$ be a metacategory. Let $C$ and $D$ be objects of $\mathbf C$. Let $\mathbf{Sub}_{\mathbf C} \left({C}\right)$ and $\mathbf{Sub}_{\mathbf C} \left({D}\right)$ be the associated categories of subobjects. Let $g: C \to D$ be a monomorphism of $\mathbf C$. Then $g$ defines a '''composition functor''' $g_* : \mathbf{Sub}_{\mathbf C} \left({C}\right) \to \mathbf{Sub}_{\mathbf C} \left({D}\right)$: {{begin-axiom}} {{axiom|lc= Object functor:        |m = g_* f := g \circ f        |rc= The composition $\circ$ is taken in $\mathbf C$ }} {{axiom|lc= Morphism functor:        |m = g_* a := a }} {{end-axiom}} That it is in fact a functor is shown on Composition Functor on Categories of Subobjects is Functor. The effect of $g_*$ is captured in the following commutative diagram: ::$\begin{xy} <-3em,0em>*+{X} = "X", <3em,0em>*+{X'} = "X2", <0em,-4em>*+{C} = "C", <0em,-8em>*+{D} = "D", "X";"X2" **@{-} ?>*@{>} ?*!/_1em/{a}, "X";"C" **@{-} ?>*@{>} ?<>(.4)*!/^.6em/{f}, "X2";"C" **@{-} ?>*@{>} ?<>(.4)*!/_.6em/{f'}, "C";"D" **@{-} ?>*@{>} ?<>(.4)*!/^.6em/{g}, "X";"D" **\crv{<-5em,-4em>} ?>*@{>} ?*!/^1.6em/{g_* f = \\ g \circ f}, "X2";"D" **\crv{<5em,-4em>} ?>*@{>} ?*!/_1.6em/{g_* f' = \\ g \circ f'}, \end{xy}$
Definition:Local Membership Relation
Let $\mathbf C$ be a metacategory. Let $C$ be an object of $\mathbf C$. The '''local membership relation''' $\in_C$ between variable elements $z: Z \to C$ and subobjects $m: M \to C$ of $C$ is defined by: :$z \in_C m$ iff there exists an $f: Z \to M$: $z = m \circ f$. If $z \in_C m$, one says that $z$ is a '''local member''' of $m$.
Definition:Lipschitz Continuity
Let $M = \struct {A, d}$ and $M' = \struct {A', d'}$ be metric spaces. Let Let $f: A \to A'$ be a mapping. Then $f$ is a '''Lipschitz continuous mapping'''  there exists a positive real number $K \in \R_{\ge 0}$ such that: :$\forall x, y \in A: \map {d'} {\map f x, \map f y} \le K \map d {x, y}$ That is, the distance between the images of two points lies within a fixed multiple of the distance between the points. === At a Point === {{:Definition:Lipschitz Continuity/Point}} === Lipschitz Constant === {{:Definition:Lipschitz Continuity/Lipschitz Constant}} === Real Function === The concept can be directly applied to the real numbers considered as a metric space under the usual topology: {{:Definition:Lipschitz Continuity/Real Function}}
Definition:Lipschitz Continuity/Lipschitz Constant
Let $f: A \to A'$ be a (Lipschitz continuous) mapping such that: : $\forall x, y \in A: d\,' \left({f \left({x}\right), f \left({y}\right)}\right) \le K d \left({x, y}\right)$ where $K \in \R_{\ge 0}$ is a positive real number. Then $K$ is '''a Lipschitz constant for $f$'''.
Definition:Lipschitz Continuity/Real Function
Let $A \subseteq \R$. Let $f: A \to \R$ be a real function. Let $I \subseteq A$ be a real interval on which: :$\exists K \in \R_{\ge 0}: \forall x, y \in I: \size {\map f x - \map f y} \le K \size {x - y}$ Then $f$ is '''Lipschitz continuous on $I$'''.
Definition:Lipschitz Equivalence/Metrics
=== Definition 1 === {{:Definition:Lipschitz Equivalence/Metrics/Definition 1}} === Definition 2 === {{:Definition:Lipschitz Equivalence/Metrics/Definition 2}}
Definition:Pullback (Category Theory)
Let $\mathbf C$ be a metacategory. Let $f: A \to C$ and $g: B \to C$ be morphisms with common codomain. A '''pullback''' of $f$ and $g$ is a commutative diagram: ::$\begin{xy}\xymatrix{  P   \ar[r]^*+{p_1}   \ar[d]_*+{p_2} &  A   \ar[d]^*+{f} \\  B   \ar[r]_*+{g} &  C }\end{xy}$ such that $f \circ p_1 = g \circ p_2$, subject to the following UMP: :For any commutative diagram: :::$\begin{xy}\xymatrix{  Q   \ar[r]^*+{q_1}   \ar[d]_*+{q_2} &  A   \ar[d]^*+{f} \\  B   \ar[r]_*+{g} &  C }\end{xy}$ :there is a unique morphism $u: Q \to P$ making the following diagram commute: :::$\begin{xy}\xymatrix@+1em{  Q   \ar@/^/[rrd]^*+{q_1}   \ar@/_/[ddr]_*+{q_2}   \ar@{-->}[rd]^*+{u} \\ &  P   \ar[r]_*+{p_1}   \ar[d]^*+{p_2} &  A   \ar[d]^*+{f} \\ &  B   \ar[r]_*+{g} &  C }\end{xy}$ In this situation, $p_1$ is called the '''pullback of $f$ along $g$''' and may be denoted as $g^* f$. Similarly, $p_2$ is called the '''pullback of $g$ along $f$''' and may be denoted $f^* g$.
Definition:Homeomorphism/Topological Spaces
Let $T_\alpha = \left({S_\alpha, \tau_\alpha}\right)$ and $T_\beta = \left({S_\beta, \tau_\beta}\right)$ be topological spaces. Let $f: T_\alpha \to T_\beta$ be a bijection. === Definition 1 === {{:Definition:Homeomorphism/Topological Spaces/Definition 1}} === Definition 2 === {{:Definition:Homeomorphism/Topological Spaces/Definition 2}} === Definition 3 === {{:Definition:Homeomorphism/Topological Spaces/Definition 3}} === Definition 4 === {{:Definition:Homeomorphism/Topological Spaces/Definition 4}}
Definition:Homeomorphism/Manifolds
Let $X$ and $Y$ be manifolds. A '''homeomorphism''' of $X$ to $Y$ is a continuous bijection such that the inverse is also continuous.
Definition:Lipschitz Equivalence/Metric Spaces
Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces. Let $f: M_1 \to M_2$ be a mapping such that $\exists h, k \in \R_{>0}$ such that: :$\forall x, y \in A: h d_2 \left({f \left({x}\right), f \left({y}\right)}\right) \le d_1 \left({x, y}\right) \le k d_2 \left({f \left({x}\right), f \left({y}\right)}\right)$ Then $f$ is a '''Lipschitz equivalence''', and $M_1$ and $M_2$ are  described as '''Lipschitz equivalent'''.
Definition:Open Set Axioms
{{begin-axiom}} {{axiom | n = \text O 1         | t = The union of an arbitrary subset of $\tau$ is an element of $\tau$. }} {{axiom | n = \text O 2         | t = The intersection of any two elements of $\tau$ is an element of $\tau$. }} {{axiom | n = \text O 3         | t = $S$ is an element of $\tau$. }} {{end-axiom}}
Definition:Continuous Mapping (Topology)/Point
Let $x \in S_1$. === Definition using Open Sets === {{Definition:Continuous Mapping (Topology)/Point/Open Sets}} === Definition using Filters === {{Definition:Continuous Mapping (Topology)/Point/Filters}} If necessary, we can say that '''$f$ is $\left({\tau_1, \tau_2}\right)$-continuous at $x$'''.
Definition:Continuous Mapping (Topology)/Set
Let $S$ be a subset of $S_1$. The mapping $f$ is '''continuous on $S$'''  $f$ is continuous at every point $x \in S$.
Definition:Continuous Mapping (Topology)/Everywhere
=== Definition by Pointwise Continuity === {{Definition:Continuous Mapping (Topology)/Everywhere/Pointwise}} === Definition by Open Sets === {{Definition:Continuous Mapping (Topology)/Everywhere/Open Sets}}
Definition:Continuous Mapping (Topology)/Everywhere/Open Sets
The mapping $f$ is '''continuous on $S_1$''' : :$U \in \tau_2 \implies f^{-1} \sqbrk U \in \tau_1$ where $f^{-1} \sqbrk U$ denotes the preimage of $U$ under $f$.
Definition:Discrete Topology/Finite
Let $S$ be a finite set. Then $\tau = \powerset S$ is a '''finite discrete topology''', and $\struct {S, \tau} = \struct {S, \powerset S}$ is a '''finite discrete space'''.
Definition:Discrete Topology/Countable
Let $S$ be a countably infinite set. Then $\tau = \powerset S$ is a '''countable discrete topology''', and $\struct {S, \tau} = \struct {S, \powerset S}$ is a '''countable discrete space'''.
Definition:Discrete Topology/Uncountable
Let $S$ be an uncountably infinite set. Then $\tau = \powerset S$ is an '''uncountable discrete topology''', and $\struct {S, \tau} = \struct {S, \powerset S}$ is an '''uncountable discrete space'''.
Definition:Discrete Topology/Infinite
Let $S$ be an infinite set. Then $\tau = \powerset S$ is an '''infinite discrete topology''', and $\struct {S, \tau} = \struct {S, \powerset S}$ is an '''infinite discrete space'''.
Definition:Finer Topology/Strictly Finer
Let $\tau_1 \supsetneq \tau_2$. $\tau_1$ is said to be '''strictly finer''' than $\tau_2$. This can be expressed as: :$\tau_1 > \tau_2 := \tau_1 \supsetneq \tau_2$
Definition:Coarser Topology/Strictly Coarser
Let $\tau_1 \subsetneq \tau_2$. Then $\tau_1$ is said to be '''strictly coarser''' than $\tau_2$. This can be expressed as: :$\tau_1 < \tau_2 := \tau_1 \subsetneq \tau_2$
Definition:Uniform Continuity/Metric Space
Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces. Then a mapping $f: A_1 \to A_2$ is '''uniformly continuous on $A_1$''' : :$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x, y \in A_1: \map {d_1} {x, y} < \delta \implies \map {d_2} {\map f x, \map f y} < \epsilon$ where $\R_{>0}$ denotes the set of all strictly positive real numbers.
Definition:Uniform Continuity/Real Numbers
Let $I \subseteq \R$ be a real interval. A real function $f: I \to \R$ is said to be '''uniformly continuous''' on $I$ : :for every $\epsilon > 0$ there exists $\delta > 0$ such that the following property holds: ::for every $x, y \in I$ such that $\size {x - y} < \delta$ it happens that $\size {\map f x - \map f y} < \epsilon$. Formally: $f: I \to \R$ is '''uniformly continuous'''  the following property holds: :$\forall \epsilon > 0: \exists \delta > 0: \paren {x, y \in I, \size {x - y} < \delta \implies \size {\map f x - \map f y} < \epsilon}$
Definition:Compact Space/Metric Space
Let $M = \left({A, d}\right)$ be a metric space. Let $\tau$ denote the topology on $A$ induced by $d$. Then $M$ is '''compact'''  $\left({A, \tau}\right)$ is a compact topological space.
Definition:Upper Integral
Let $\closedint a b$ be a closed real interval. Let $f: \closedint a b \to \R$ be a bounded real function. The '''upper integral of $f$ over $\closedint a b$''' is defined as: :$\displaystyle \overline {\int_a^b} \map f x \rd x = \inf_P \map U P$ where: :the infimum is taken over all subdivisions $P$ of $\closedint a b$ :$\map U P$ denotes the upper sum of $f$ on $\closedint a b$ belonging to $P$.
Definition:Lower Integral
Let $\closedint a b$ be a closed real interval. Let $f: \closedint a b \to \R$ be a bounded real function. The '''lower integral of $f$ over $\closedint a b$''' is defined as: :$\displaystyle \underline {\int_a^b} \map f x \rd x = \sup_P \map L P$ where: :the supremum is taken over all subdivisions $P$ of $\closedint a b$ :$\map L P$ denotes the lower sum of $f$ on $\closedint a b$ belonging to $P$.
Definition:Abstract Geometry
Let $P$ be a set and $L$ be a set of subsets of $P$. Then $\left({P, L}\right)$ is an '''abstract geometry''' : {{begin-axiom}} {{axiom | n = 1         | q = \forall A, B \in P         | m = \exists l \in L: A, B \in l }} {{axiom | n = 2         | q = \forall l \in L         | m = \exists A, B \in P: A, B \in l \land A \ne B }} {{end-axiom}}
Definition:Euclidean Plane
For any real number $a$ let: :$L_a = \set {\tuple {x, y} \in \R^2: x = a}$ Furthermore, define: :$L_A = \set {L_a: a \in \R}$ For any two real numbers $m$ and $b$ let: :$L_{m, b} = \set {\tuple {x, y} \in \R^2: y = m x + b}$ Furthermore, define: :$L_{M, B} = \set {L_{m, b}: m, b \in \R}$ Finally let: :$L_E = L_A \cup L_{M, B}$ The abstract geometry $\struct {\R^2, L_E}$ is called the '''Euclidean plane'''.
Definition:Multiple Edge/Multiplicity
The '''multiplicity''' of an edge is the number of edges having the same pair of endvertices.
Definition:Poincaré Plane
Let: :$\H = \set {\tuple {x, y} \in \R^2: y > 0}$ Let $a \in \R$ be a real number. Let: :${}_a L := \set {\tuple {x, y} \in \H: x = a}$ Define: :${}_A L := \set{ {}_a L: a \in \R}$ Let $c \in \R$ be a real number and $r \in \R_{>0}$ be a strictly positive real number. Let: :${}_c L_r := \set {\tuple {x, y} \in \H: \paren {x - c}^2 + y^2 = r^2}$ Define: :${}_C L_R := \set { {}_c L_r: c \in \R \land r \in \R_{>0} }$ Finally let: :$L_H = {}_A L \cup {}_C L_R$ The abstract geometry $\struct {\H, L_H}$ is called the '''Poincaré plane'''. This is shown to be an abstract geometry in Poincaré Plane is Abstract Geometry.
Definition:Topology Generated by Synthetic Sub-Basis/Definition 1
Define: :$\displaystyle \BB = \set {\bigcap \FF: \FF \subseteq \SS, \FF \text{ is finite} }$ That is, $\BB$ is the set of all finite intersections of sets in $\SS$. Note that $\FF$ is allowed to be empty in the above definition. The '''topology generated by $\SS$''', denoted $\map \tau \SS$, is defined as: :$\displaystyle \map \tau \SS = \set {\bigcup \AA: \AA \subseteq \BB}$
Definition:Topology Generated by Synthetic Sub-Basis/Definition 2
The '''topology generated by $\mathcal S$''', denoted $\tau \left({\mathcal S}\right)$, is defined as the unique topology on $X$ that satisfies the following axioms: :$\left({1}\right): \quad \mathcal S \subseteq \tau \left({\mathcal S}\right)$ :$\left({2}\right): \quad$ For any topology $\mathcal T$ on $X$, the implication $\mathcal S \subseteq \mathcal T \implies \tau \left({\mathcal S}\right) \subseteq \mathcal T$ holds. That is, $\tau \left({\mathcal S}\right)$ is the coarsest topology on $X$ for which every element of $\mathcal S$ is open.
Definition:Balanced String
Let $S$ be a string in an alphabet containing the left and right brackets $($ and $)$. Then $S$ is said to be '''balanced''' iff it contains equally many left and right brackets.
Definition:Sequential Continuity
Let $X$ and $Y$ be topological spaces. Let $f: X \to Y$ be a mapping. === At a Point === {{:Definition:Sequential Continuity/Point}} === On a Domain === {{:Definition:Sequential Continuity/Domain}}
Definition:Initial Topology/Definition 1
Let: :$\mathcal S = \left\{{f_i^{-1} \left[{U}\right]: i \in I, U \in \tau_i}\right\} \subseteq \mathcal P \left({X}\right)$ where $f_i^{-1} \left[{U}\right]$ denotes the preimage of $U$ under $f_i$. The topology $\tau$ on $X$ generated by $\mathcal S$ is called the '''initial topology on $X$ with respect to $\left \langle {f_i}\right \rangle_{i \mathop \in I}$'''.
Definition:Initial Topology/Definition 2
Let $\tau$ be the coarsest topology on $X$ such that each $f_i: X \to Y_i$ is $\left({\tau, \tau_i}\right)$-continuous. Then $\tau$ is known as the '''initial topology on $X$ with respect to $\left \langle {f_i} \right \rangle_{i \mathop \in I}$'''.
Definition:Product Space (Topology)/General Definition
Let $\family {\struct {S_i, \tau_i} }_{i \mathop \in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set. Let $S$ be the cartesian product of $\family {S_i}_{i \mathop \in I}$: :$\displaystyle S := \prod_{i \mathop \in I} S_i$ Let $\tau$ be the Tychonoff topology on $S$. From Natural Basis of Tychonoff Topology, $\tau$ is generated from: :the basis $\BB$ of cartesian products of the form $\displaystyle \prod_{i \mathop \in I} U_i$ where: ::for all $i \in I : U_i \in \tau_i$ ::for all but finitely many indices $i : U_i = S_i$ The topological space $\struct{X, \tau}$ is called the '''product space''' of $\family {\struct {S_i, \tau_i} }_{i \mathop \in I}$.
Definition:Neighborhood (Topology)/Set
Let $A \subseteq S$ be a subset of $S$. A '''neighborhood''' of $A$, which can be denoted $N_A$, is any subset of $S$ containing an open set of $T$ which itself contains $A$. That is: :$\exists U \in \tau: A \subseteq U \subseteq N_A \subseteq S$
Definition:Neighborhood (Topology)/Point
Let $z \in S$ be a point in a $S$. Let $N_z$ be a subset of $S$ which contains (as a subset) an open set of $T$ which itself contains (as an element) $z$. Then $N_z$ is a '''neighborhood''' of $z$. That is: :$\exists U \in \tau: z \in U \subseteq N_z \subseteq S$
Definition:Closed Neighborhood
If $\relcomp S {N_A} \in \tau$, that is if $N_A$ is closed in $S$, then $N_A$ is called a '''closed neighborhood'''.
Definition:Limit Point/Topology/Set/Definition 1
A point $x \in S$ is a '''limit point of $A$'''  every open neighborhood $U$ of $x$ satisfies: :$A \cap \paren {U \setminus \set x} \ne \O$ That is,  every open set $U \in \tau$ such that $x \in U$ contains some point of $A$ distinct from $x$.
Definition:Limit Point/Topology/Set/Definition 2
A point $x \in S$ is a '''limit point of $A$'''  : $x$ belongs to the closure of $A$ but is not an isolated point of $A$.
Definition:Limit Point/Topology/Set/Definition 3
A point $x \in S$ is a '''limit point of $A$'''  $x$ is an adherent point of $A$ but is not an isolated point of $A$.
Definition:Limit Point/Topology/Set/Definition 4
A point $x \in S$ is a '''limit point of $A$'''  $\left({S \setminus A}\right) \cup \left\{{x}\right\}$ is ''not'' a neighborhood of $x$.
Definition:Adherent Point/Definition 1
A point $x \in S$ is an '''adherent point of $A$'''  every open neighborhood $U$ of $x$ satisfies: :$A \cap U \ne \O$
Definition:Adherent Point/Definition 2
A point $x \in S$ is an '''adherent point of $A$'''  $x$ is an element of the closure of $A$.
Definition:Limit Point/Topology/Set/Definition 5
{{questionable|this definition is not equivalent to the others}} A point $x \in S$ is a '''limit point of $A$''' if there is a sequence $\left\langle{x_n}\right\rangle$ in $A$ such that $x$ is a limit point of $\left\langle{x_n}\right\rangle$, considered as sequence in $S$.
Definition:Convergent Sequence/Topology/Definition 1
Let $T = \struct {S, \tau}$ be a topological space. Let $\sequence {x_n}_{n \mathop \in \N}$ be an infinite sequence in $S$. Then $\sequence {x_n}$ '''converges to the limit $\alpha \in S$''' : :$\forall U \in \tau: \alpha \in U \implies \paren {\exists N \in \R_{>0}: \forall n \in \N: n > N \implies x_n \in U}$
Definition:Convergent Sequence/Topology/Definition 2
Let $T = \struct {S, \tau}$ be a topological space. Let $\sequence {x_n}_{n \mathop \in \N}$ be an infinite sequence in $S$. Then $\sequence {x_n}$ '''converges to the limit $\alpha \in S$''' : :$\forall U \in \tau: \alpha \in U \implies \set {n \in \N: x_n \notin U}$ is finite.
Definition:Closure (Topology)/Definition 2
The '''closure of $H$ (in $T$)''' is defined as: :$\displaystyle H^- := \bigcap \left\{{K \supseteq H: K}\right.$ is closed in $\left.{T}\right\}$
Definition:Closure (Topology)/Definition 1
The '''closure of $H$ (in $T$)''' is defined as: :$H^- := H \cup H'$ where $H'$ is the derived set of $H$.
Definition:Pullback Functor
Let $\mathbf C$ be a metacategory having all pullbacks. Let $f: C \to D$ be a morphism of $\mathbf C$. Let $\mathbf C \mathop / C$ and $\mathbf C \mathop / D$ be the slice categories over $C$ and $D$, respectively. The '''pullback functor''' $f^* : \mathbf C \mathop / D \to \mathbf C \mathop / C$ associated to $f$ is defined by: {{begin-axiom}} {{axiom|lc= Object functor:        |m = f^* \alpha := f^* \alpha        |rc= $f^* \alpha$ is the pullback of $\alpha$ along $f$ }} {{axiom|lc= Morphism functor:        |m = f^* \gamma := \gamma'        |rc= $\gamma': f^* \alpha \to f^* \beta$ is as on Pullback of Commutative Triangle }} {{end-axiom}} Explicitly, $f^* \gamma$ is defined as the unique morphism fitting: ::$\begin{xy}\xymatrix@+1em@L+4px{  A'   \ar[rr]^*{f_\alpha}   \ar[dd]_*{f^* \alpha}   \ar@{-->}[rd]_*{f^* \gamma} & &  A   \ar[rd]^*{\gamma}   \ar[dd]^(.4)*{\alpha} \\ &  B'   \ar[ld]^*{f^* \beta}   \ar[rr] |{\hole} ^(.3)*{f_\beta} & &  B   \ar[ld]^*{\beta} \\  C   \ar[rr]_*{f} & &  D }\end{xy}$
Definition:Closure (Topology)/Definition 3
The '''closure of $H$ (in $T$)''', denoted $H^-$, is defined as the smallest closed set of $T$ that contains $H$.
Definition:Closure (Topology)/Definition 4
The '''closure of $H$ (in $T$)''' is defined as the union of $H$ and its boundary in $T$: :$H^- := H \cup \partial H$
Definition:Closure (Topology)/Definition 5
The '''closure of $H$ (in $T$)''' is the union of the set of all isolated points of $H$ and the set of all limit points of $H$: :$H^- := H^i \cup H'$
Definition:Diagram (Category Theory)
Let $\mathbf J$ and $\mathbf C$ be metacategories. A '''diagram of type $\mathbf J$''' in $\mathbf C$ is a functor $D: \mathbf J \to \mathbf C$. === Index Category === In this context, $\mathbf J$ is referred to as the '''index category'''. Its objects are typically denoted by lowercase letters, $i, j$ etc. Furthermore, one writes $D_i$ in place of the formally more correct $D \left({i}\right)$. Similarly, for $\alpha: i \to j$ a morphism one writes $D_\alpha$ in place of $D \left({\alpha}\right)$.
Definition:Cone (Category Theory)
Let $\mathbf C$ be a metacategory. Let $D: \mathbf J \to \mathbf C$ be a $\mathbf J$-diagram in $\mathbf C$. A '''cone to $D$''' comprises an object $C$ of $\mathbf C$, and a morphism: :$c_j: C \to D_j$ for each object of $\mathbf J$, such that for each morphism $\alpha: i \to j$ of $\mathbf J$: ::$\begin{xy}\xymatrix@+0.5em@L+2px{  C   \ar[d]_*+{c_i}   \ar[dr]^*+{c_j} \\  D_i   \ar[r]_*+{D_\alpha} &  D_j }\end{xy}$ is a commutative diagram. {{expand|when time comes, add def "cone is natural trafo $\kappa_C \to D$ with $\kappa_C$ constant functor"}}
Definition:Morphism of Cones
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 cones to $D$. Let $f: C \to C'$ be a morphism of $\mathbf C$. Then $f$ is a '''morphism of cones''' iff, for all objects $j$ of $\mathbf J$: ::$\begin{xy}\xymatrix@+0.5em@L+2px{  C   \ar[r]^*+{f}   \ar[dr]_*+{c_j} &  C'   \ar[d]^*+{c'_j} \\ &  D_j }\end{xy}$ is a commutative diagram.