hcju/proofwikips huggingface dataset:
Definition:Graham's Number
Let $3 \uparrow 3$ denote Knuth's notation for powers: :$3 \uparrow 3 := 3^3$ Further, let: :$3 \uparrow \uparrow 3 := 3 \uparrow \paren {3 \uparrow 3} = 3^{\paren {3^3} }$ and so define: :$3 \underbrace {\uparrow \uparrow \ldots \uparrow \uparrow}_n 3 := 3 \underbrace {\uparrow \uparrow \ldots \uparrow \uparrow}_{n - 1} \paren {3 \underbrace {\uparrow \uparrow \ldots \uparrow \uparrow}_{n - 1} 3}$ Thus, for example: :$3 \uparrow \uparrow \uparrow \uparrow 3 = 3 \uparrow \uparrow \uparrow \paren {3 \uparrow \uparrow \uparrow 3}$ Let: : $n_1 := 3 \underbrace {\uparrow \uparrow \ldots \uparrow \uparrow}_{3 \uparrow \uparrow \uparrow \uparrow 3} 3$ That is, a total of $3 \uparrow \uparrow \uparrow \uparrow 3$ instances of $\uparrow$. Similarly, let: :$n_2 := 3 \underbrace {\uparrow \uparrow \ldots \uparrow \uparrow}_{n_1} 3$ In general for $m \ge 2$: :$n_m := 3 \underbrace {\uparrow \uparrow \ldots \uparrow \uparrow}_{n_{m - 1} } 3$ and so specifically: :$n_{63} := 3 \underbrace {\uparrow \uparrow \ldots \uparrow \uparrow}_{n_{62} } 3$ It is this $n_{63}$ which is defined as '''Graham's number'''. {{NamedforDef|Ronald Lewis Graham|cat = Graham}}
Definition:Metrizable Topology/Linguistic Note
The UK English spelling of '''metrizable''' is '''metrisable''', but it is rarely found.
Definition:Noetherian Topological Space/Definition 1
A topological space $T = \left({S, \tau}\right)$ is '''Noetherian'''  its set of closed sets, ordered by inclusion, satisfies the descending chain condition.
Definition:Noetherian Topological Space/Definition 2
A topological space $T = \left({S, \tau}\right)$ is '''Noetherian'''  its set of open sets, ordered by inclusion, satisfies the ascending chain condition.
Definition:Everywhere Dense/Definition 1
The subset $H$ is '''(everywhere) dense in $T$''' : :$H^- = S$ where $H^-$ is the closure of $H$.
Definition:Everywhere Dense/Definition 2
The subset $H$ is '''(everywhere) dense in $T$'''  the intersection of $H$ with every open subset of $T$ is non-empty: :$\forall U \in \tau: H \cap U \ne \O$
Definition:Product Space (Topology)/Factor Space
Each of the topological spaces $\struct{S_i, \tau_i}$ are called the '''factors''' of $\struct{S, \tau}$, and can be referred to as '''factor spaces'''.
Definition:Finer Filter on Set
Let $\FF \subseteq \FF'$. Then  $\FF'$ is '''finer''' than $\FF$.
Definition:Coarser Filter on Set
Let $\FF \subseteq \FF'$. Then $\FF$ is '''coarser''' than $\FF$.
Definition:Finer Filter on Set/Strictly Finer
Let $\FF \subset \FF'$, that is, $\FF \subseteq \FF'$ but $\FF \ne \FF'$. Then $\FF'$ is '''strictly finer''' than $\FF$.
Definition:Comparable Filters on Set
Let $S$ be a set. Let $\powerset S$ be the power set of $S$. Let $\FF, \FF' \subset \powerset S$ be two filters on $S$. $\FF$ and $\FF'$ are '''comparable'''  one is finer (or coarser) than the other.
Definition:Ultrafilter on Set/Definition 2
Let $S$ be a set. Let $\FF \subseteq \powerset S$ be a filter on $S$. Then $\FF$ is an '''ultrafilter (on $S$)''' : :for every $A \subseteq S$ and $B \subseteq S$ such that $A \cap B = \O$ and $A \cup B \in \FF$, either $A \in \FF$ or $B \in \FF$.
Definition:Ultrafilter on Set/Definition 3
Let $S$ be a set. Let $\FF \subseteq \powerset S$ be a filter on $S$. Then $\FF$ is an '''ultrafilter (on $S$)''' : :for every $A \subseteq S$, either $A \in \FF$ or $\relcomp S A \in \FF$ where $\relcomp S A$ is the relative complement of $A$ in $S$, that is, $S \setminus A$.
Definition:Ring under Ring
Let $A$ be a commutative ring with unity. A '''ring under $A$''' is a pair $(B, f)$ where: * $B$ is a commutative ring with unity * $f : A \to B$ is a unital ring homomorphism
Definition:Principal Ultrafilter/Nonprincipal
Let $\FF \subset \powerset S$ be an ultrafilter on $S$ which does not have a cluster point. Then $\FF$ is a '''nonprincipal ultrafilter'''  on $S$.
Definition:Limit Point/Filter Basis/Definition 1
A point $x \in S$ is called a '''limit point of $\BB$'''  $\FF$ converges on $x$. $\BB$ is likewise said to converge on $x$.
Definition:Limit Point/Filter Basis/Definition 2
A point $x \in S$ is called a '''limit point of $\BB$'''  every neighborhood of $x$ contains a set of $\BB$.
Definition:Determinant
=== Determinant of Matrix === {{:Definition:Determinant of Matrix}} === Determinant of Linear Operator === {{:Definition:Determinant of Linear Operator}}
Definition:Compact Space/Topology/Definition 1
A topological space $T = \struct {S, \tau}$ is '''compact'''  every open cover for $S$ has a finite subcover.
Definition:Compact Space/Topology/Definition 2
A topological space $T = \struct {S, \tau}$ is '''compact'''  it satisfies the Finite Intersection Axiom.
Definition:Compact Space/Topology/Definition 3
A topological space $T = \struct {S, \tau}$ is '''compact'''  $\tau$ has a sub-basis $\BB$ such that: :from every cover of $S$ by elements of $\BB$, a finite subcover of $S$ can be selected.
Definition:Compact Space/Topology/Definition 4
A topological space $T = \left({S, \tau}\right)$ is '''compact'''  every filter on $S$ has a limit point in $S$.
Definition:Countably Compact Space/Definition 1
A topological space $T = \struct {S, \tau}$ is '''countably compact''' : :every countable open cover of $T$ has a finite subcover.
Definition:Countably Compact Space/Definition 2
A topological space $T = \struct {S, \tau}$ is '''countably compact''' : :every countable set of closed sets of $T$ whose intersection is empty has a finite subset whose intersection is empty. That is, $T$ satisfies the countable finite intersection axiom.
Definition:Countably Compact Space/Definition 3
A topological space $T = \struct {S, \tau}$ is '''countably compact''' : :every infinite sequence in $S$ has an accumulation point in $S$.
Definition:Countably Compact Space/Definition 5
A topological space $T = \struct {S, \tau}$ is '''countably compact''' : :every infinite subset of $S$ has an $\omega$-accumulation point in $S$.
Definition:Normed Algebra
Let $R$ be a division ring with norm $\norm {\,\cdot\,}_R$. A '''normed algebra''' over $R$ is a pair $\struct {A, \norm{\,\cdot\,} }$ where: :$A$ is a algebra over $R$ :$ \norm{\,\cdot\,} $ is a norm on $A$.
Definition:Normed Ring
Let $\norm {\, \cdot \,}$ be a norm on $R$. Then $\struct {R, \norm {\, \cdot \,} }$ is a '''normed ring'''.
Definition:Convergent Product/Normed Algebra
Let $\mathbb K$ be a division ring with norm $\norm{\,\cdot\,}_{\mathbb K}$. Let $\struct{ A, \norm{\,\cdot\,} }$ be an associative normed unital algebra over $\mathbb K$. Let $\sequence{a_n}$ be a sequence in $A$. === Definition 1 === The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ is '''convergent'''  there exists $n_0\in\N$ such that: # $a_n$ is invertible for $n \geq n_0$ # the sequence of partial products of $\displaystyle \prod_{n \mathop = n_0}^\infty a_n$ converges to some invertible $b\in A^\times$. === Definition 2: for complete algebras === Let $\struct{ A, \norm{\,\cdot\,} }$ be complete. The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ is '''convergent'''  there exists $n_0\in\N$ such that: : the sequence of partial products of $\displaystyle \prod_{n \mathop = n_0}^\infty a_n$ converges to some invertible $a\in A^\times$.
Definition:Vectorization of Matrix
Let $S$ be a set. Let $m, n \ge 1$ be natural numbers. Let $A = \sqbrk {a_{i j} }$ be a $m \times n$ matrix over $S$. === Definition 1 === The '''vectorization''' of $A$ is the $m n \times 1$ column matrix: :$\map {\operatorname {vec} } A = \sqbrk {a_{11}, \ldots, a_{m1}, a_{12}, \ldots, a_{m2}, \ldots, a_{1n}, \ldots, a_{mn} }^\intercal$ informally obtained by stacking the columns of $A$. That is: :$\map {\operatorname {vec} } A_k = a_{\floor {k/m}, k \bmod m}$ where: :$\floor {\, \cdot \,}$ is the floor function :$\bmod$ is the modulo operation. === Definition 2 === Let $R$ be a ring with unity. Let $A$ be an $m\times n$ matrix over $R$. The '''vectorization''' of $A$ is its coordinate vector with respect to the standard matrix basis.
Definition:Block Matrix
Let $S$ be a set. Let $m, n \ge 1$ be positive integers. Let $A = \sqbrk {A_{i j} }$ be an $m \times n$ matrix of matrices over $S$. Let for every $i \in \set {1, \ldots, m}$, the elements of the $i$th row of $A$ have equal height $m_i$. Let for every $j \in \set {1,\ldots, n}$ the elements of the $j$th column of $A$ have equal width $n_i$. Define $M = \displaystyle \sum_{i \mathop = 1}^m m_i$ and $N = \displaystyle \sum_{i \mathop = 1}^n n_i$ as indexed summations. Let more generally $M_k = \displaystyle \sum_{i \mathop = 1}^k m_i$ and $N_l = \displaystyle \sum_{i \mathop = 1}^l n_i$ for $k \in \set {0, \ldots, m}$ and $l \in \set {0, \ldots, n}$. Then the '''block matrix''' of $A$ is the $M \times N$ matrix $\sqbrk {b_{i j} }$ over $S$ defined as the union of the mappings: :$b_{i j} = \sqbrk {A_{kl} }_{i - M_{k - 1}, j - N_{l - 1} }$ on $\set {M_{k - 1}, \ldots, M_k} \times \set {N_{l - 1}, \ldots, N_l}$ for $k \in \set {1, \ldots, m}$ and $ l \in \set {1, \ldots, n}$.
Definition:Standard Matrix Basis
Let $R$ be a ring with unity. Let $m,n\geq1$ be positive integers. Let $i, j \in \left\{ {1, \ldots, m}\right\} \times \{ 1, \ldots, n\}$. The '''standard matrix basis''' of $m\times n$ matrices over $R$ is the ordered basis of standard basis matrices ordered by the colexicographic order on $\left\{ {1, \ldots, m}\right\} \times \{ 1, \ldots, n\}$.
Definition:Sigma-Locally Compact Space
$T$ is '''$\sigma$-locally compact''' : : $T$ is $\sigma$-compact : $T$ is locally compact That is, $T$ is '''$\sigma$-locally compact''' : : $T$ is the union of countably many compact subspaces : every point of $S$ has a local basis $\mathcal B$ such that all elements of $\mathcal B$ are  compact.
Definition:Normed Unital Algebra
Let $R$ be a division ring with norm $\norm {\,\cdot\,}_R$. A '''normed unital algebra''' over $R$ is a pair $\struct {A, \norm {\,\cdot\,} }$ where: :$A$ is a unital algebra over $R$ :$\norm {\,\cdot\,}$ is a norm on $A$.
Definition:Strongly Locally Compact Space/Definition 1
The space $T$ is '''strongly locally compact''' : :every point of $S$ is contained in an open set whose closure is compact.
Definition:Strongly Locally Compact Space/Definition 2
The space $T$ is '''strongly locally compact''' : :every point has a closed compact neighborhood. That is: :every point of $S$ is contained in an open set which is contained in a closed compact subspace.
Definition:Ringed Space/Structure Sheaf
Let $\left({X, \mathcal O_X}\right)$ be a ringed space. The '''structure sheaf''' of $\left({X, \mathcal O_X}\right)$ is the term $\mathcal O_X$.
Definition:Category of Rings
The '''category of rings''', denoted $\mathbf{Ring}$, is the category with: {{DefineCategory | ob = rings | mor = ring homomorphisms |id = identity mappings |comp = composition of mappings }} === Category of rings with unity === {{Definition:Category of Rings with Unity}}
Definition:Category of Rings with Unity
The '''category of rings with unity''' is the category with: {{DefineCategory | ob = rings with unity | mor = unital ring homomorphisms |id = identity mappings |comp = composition of mappings }}
Definition:Morphism of Ringed Spaces
Let $(X, \mathcal O_X)$ and $(Y, \mathcal O_Y)$ be ringed spaces. === Definition 1 === A '''morphism of ringed spaces''' from $(X, \mathcal O_X)$ to $(Y, \mathcal O_Y)$ is a pair $(f, f^\sharp)$ where: * $f : X \to Y$ is continuous * $f^\sharp : \mathcal O_Y \to f_* \mathcal O_X$ is a morphism of sheaves to the direct image sheaf of $\mathcal O_X$ via $f$ === Definition 2 === A '''morphism of ringed spaces''' from $(X, \mathcal O_X)$ to $(Y, \mathcal O_Y)$ is a pair $(f, f^\sharp)$ where: * $f : X \to Y$ is continuous * $f^\sharp : f^{-1}\mathcal O_Y \to \mathcal O_X$ is a morphism of sheaves from the inverse image sheaf of $\mathcal O_Y$ via $f$ === Definition 3 === {{definition wanted|a description using open subsets, between the two above}}
Definition:Restriction of Presheaf to Open Set
Let $C$ be a category. Let $X$ be a topological space. Let $\mathcal F$ be a $C$-valued presheaf on $X$. Let $U \subset X$ be open. The '''restriction''' of $\mathcal F$ to $U$ is the restriction of the contravariant functor $\mathcal F$ to the subcategory $U$.
Definition:Scheme
A '''scheme''' is a locally ringed space $\struct {X, \OO_X}$ such that every point $x \in X$ has an open neighborhood $U$ such that the restriction of $\struct {X, \OO_X}$ to $U$ is an affine scheme.
Definition:Affine Scheme
An '''affine scheme''' is a ringed space which is isomorphic to the spectrum of a commutative ring with unity.
Definition:Sum of Ideals of Ring
Let $R$ be a ring. === Two ideals === Let $I$ and $J$ be ideals of $R$. Their '''sum''' is the ideal equal to their subset sum: :$I + J  = \set {i + j : i \in I \land j \in J}$ === Multiple Ideals === {{definition wanted|1) explicit description, and 2) using the iterated induced operation on subsets}}
Definition:Unique up to Unique Morphism
Let $\mathbf C$ be a category. Let $S \subseteq \operatorname{Ob}(\mathbf C)$ be a subclass of its objects. === Definition 1 === The class $S$ is '''unique up to unique morphism'''  for all objects $s,t \in S$ there is a unique morphism from $s$ to $t$. === Definition 2 === The class $S$ is '''unique up to unique morphism'''  for all objects $s,t \in S$ there is a unique morphism from $s$ to $t$, and it is an isomorphism.
Definition:Set of Literals
Let $S$ be a set. A '''set of literals''' on $S$ is a triple $\struct {S^\pm, \iota, \theta}$ where: :$S^\pm$ is a set :$\iota : S \to S^\pm$ is a mapping, the '''canonical injection''' :$\theta : S^\pm \to S^\pm$ is an involution without fixed points, the '''inversion mapping''', and we also denote $\map \theta s = s^{-1}$ such that $S^\pm = \iota \sqbrk S \sqcup \theta \sqbrk {\iota \sqbrk S}$ is the disjoint union of the image of $S$ under $\iota$ and its image under $\theta$. Explicitly, $S^\pm$ can be defined as follows. Let $S^\pm = S \sqcup S = S \times \set 0 \cup S \times \set 1$ be the disjoint union of $S$ with $S$. Let $\iota: S \to S^\pm$ be the canonical mapping: :$s \mapsto \tuple {s, 0}$ Let $\theta : S^\pm \to S^\pm$ be the mapping: :$\tuple {s, i} \mapsto \tuple {s, 1 - i}$ {{tidy|The above phrasing is not up to par}}
Definition:Reduced Group Word on Set
Let $S$ be a set. Let $n \ge 0$ be a natural number. Let $w = w_1 \cdots w_i \cdots w_n$ be a group word on $S$ of length $n$. Then $w$ is '''reduced'''  $w_i \ne w_{i + 1}^{-1}$ for all $i \in \set {1, \ldots, n - 1}$
Definition:Elementary Reduction of Group Word on Set
Let $X$ be a set. Let $v$ and $w$ be group words on $X$. Let $n$ be the length of $v$. Then $w$ is an '''elementary reduction''' of $v$ : :$w$ has length $n - 2$ :There exists $k \in \set {1, \ldots, n - 1}$ such that: ::$v_k = v_{k + 1}^{-1}$ ::$w_i = \begin {cases} v_i & : i < k \\ v_{i + 2} & : i > k + 1 \end {cases}$ This is denoted $v \overset 1 \longrightarrow w$.
Definition:Reduction of Group Word on Set
Let $X$ be a set. Let $w$ be a group word on $X$. A '''reduction''' of $w$ is a finite sequence of group words $\tuple {w^{\paren 0}, \ldots, w^{\paren n} }$ such that: :$w^{\paren 0} = w$ :$w_{i + 1}$ is an elementary reduction of $w_i$, for all $i \in \set {0, \ldots, n - 1}$ :$w^{\paren n}$ is reduced
Definition:Reduced Form of Group Word
Let $X$ be a set. Let $w$ be a group word on $X$. The '''reduced form''' $\map {\operatorname {red} } w$ of $w$ is the unique reduced word for which there exists a reduction: :$w = w^{\paren 0} \to w^{\paren 1} \to \cdots \to w^{\paren n} = \map {\operatorname {red} } w$
Definition:Composition of Reduced Group Words
Let $X$ be a set. Let $v$ and $w$ be reduced group words on $X$. The '''composition''' of $v$ and $w$ is the reduced form of their concatenation: :$v \cdot w = \operatorname{red}(vw)$
Definition:Group of Reduced Group Words
Let $S$ be a set. The '''group of reduced group words''' on $S$ is the group $\left({F_S, \circ}\right)$ where: :$F_S$ is the set of reduced group words on $S$ :$\circ$ denotes composition of reduced group words.
Definition:Free Group on Set
Let $X$ be a set. A '''free group''' on $X$ is a certain $X$-pointed group, that is, a pair $(F, \iota)$ where: :$F$ is a group :$\iota : X \to F$ is a mapping that can be defined as follows: === Definition 1: by universal property === A '''free group''' on $X$ is an $X$-pointed group $(F, \iota)$ that satisfies the following universal property: :For every $X$-pointed group $(G, \kappa)$ there exists a unique group homomorphism $\phi : F \to G$ such that $\phi \circ \iota = \kappa$, that is, a morphism of pointed groups $F \to G$. === Definition 2: As the group of reduced group words === The '''free group''' on $X$ is the pair $(F, \iota)$ where: :$F$ is the group of reduced group words on $X$ :$\iota : X \to F$ is the canonical injection
Definition:Group Defined by Presentation
Let $S$ be a set. Let $R$ be a set of group words on $S$. Let $F_S$ be the group of reduced group words on $S$. Let $\operatorname{red} \left({R}\right)$ be the set of reduced forms of elements of $R$. Let $N$ be the normal subgroup generated by $\operatorname{red} \left({R}\right)$ in $F$. The '''group defined by the presentation''' $\langle S \mid R \rangle$ is the quotient group $F_S / N$.
Definition:Group Word on Set
Let $S$ be a set. A '''group word''' on $S$ is an ordered tuple on the set of literals $S^\pm$ of $S$.
Definition:Rank of Free Group
Let $F$ be a free group. The '''rank''' of $F$ is the dimension of its abelianization as a module over $\Z$.
Definition:Commutator Subgroup
Let $G$ be a group. Its '''commutator subgroup''' $\sqbrk {G, G}$ is the subgroup generated by all commutators. === Higher derived subgroups === Let $n \ge 0$ be a natural number. The $n$th '''derived subgroup''' $G^{\paren n}$ is recursively defined as: :$G^{\paren n} = \begin{cases} G & : n = 0 \\ \sqbrk {G^{\paren {n - 1} }, G^{\paren {n - 1} } } & : n \ge 1 \end{cases}$
Definition:Abelianization of Group
Let $G$ be a group. Its '''abelianization''' is the quotient by its commutator subgroup: :$G^{\mathrm {ab} } = G / \sqbrk {G, G}$
Definition:Cartesian Closed Set
Let $S$ be a set. Then $S$ is '''cartesian closed'''  the cartesian product $S \times S$ is a subset of $S$.
Definition:Group Relation on Set
Let $X$ be a set. A '''group relation''' on $X$ is a pair $(u, v)$ where $u$ and $v$ are group words on $X$. A '''group relation''' $(u, v)$ is also denoted $u=v$.
Definition:Generator of Algebraic Structure
Let $\left({A, \circ}\right)$ be an algebraic structure. Let $G \subset A$ be a subset. === Definition 1 === The subset $G$ is a '''generator''' of $A$  $A$ is the algebraic substructure generated by $G$. === Definition 2 === The subset $G$ is a '''generator''' of $A$ : * $\forall x, y \in G: x \circ y \in A$; * $\forall z \in A: \exists x, y \in W \left({G}\right): z = x \circ y$ where $W \left({G}\right)$ is the set of words of $G$. That is, every element in $A$ can be formed as the product of a finite number of elements of $G$. If $G$ is such a set, then we can write $A = \left \langle {G}\right \rangle$.
Definition:Cartesian Product/Uncountable
Let $I$ be an indexing set with uncountable cardinality. Let $\family {S_\alpha}_{\alpha \mathop \in I}$ be a family of sets indexed by $I$. The '''cartesian product''' of $\family {S_\alpha}$ is denoted: :$\displaystyle \prod_{\alpha \mathop \in I} S_\alpha$
Definition:Cauchy Sequence in Topological Group
Let $G$ be a topological group with identity element $e$. Let $\sequence {a_n}$ be a sequence in $G$. Then $\sequence {a_n}$ is a '''Cauchy sequence'''  for every neighborhood $U$ of $e$ there exists $N \in \N$ such that for all $m, n \ge N : a_n a_m^{-1} \in U$.
Definition:Semiring of Natural Numbers
The '''semiring of natural numbers''' is the semiring $\struct {\N, +, \times}$ where: :$\N$ is the set of natural numbers :$+$ is the addition :$\times$ is the multiplication
Definition:Localization of Ring at Prime Ideal
Let $A$ be a commutative ring with unity. Let $\mathfrak p$ be a prime ideal of $A$. The '''localization of $A$ at $\mathfrak p$''' is the localization of $A$ at the complement $A\setminus\mathfrak p$: :$A_{\mathfrak p} = (A\setminus\mathfrak p)^{-1}A$
Definition:Localization of Ring at Element
Let $A$ be a commutative ring with unity. Let $f\in A$ be an element. The '''localization of $A$ at $f$''' is the localization of $A$ at the set of powers $\{1, f, f^2, \ldots\}$: :$A_f = (\{1, f, f^2, \ldots\})^{-1}A$
Definition:Saturation of Multiplicatively Closed Subset of Ring
Let $A$ be a commutative ring with unity. Let $S \subset A$ be a multiplicatively closed subset. === Definition 1 === The '''saturation''' of $S$ is the smallest saturated multiplicatively closed subset of $A$ containing $S$. That is, it is the intersection of all saturated multiplicatively closed subsets containing $S$. {{circularStructure|this is a general issue with definitions of examples of closure operators}} === Definition 2 === The '''saturation''' of $S$ is the set of divisors of elements of $S$. === Definition 3 === The '''saturation''' of $S$ is the set of elements whose image in the localization $A_S$ is a unit of $A$. === Definition 4 === The '''saturation''' of $S$ is the complement relative to $A$ of the union of prime ideals that are disjoint from $S$: :$\map {\operatorname {Sat} } S = A \setminus \displaystyle \bigcup \set {\mathfrak p \in \operatorname{Spec} A: \mathfrak p \cap S = \O}$
Definition:Closure of Set under Closure Operator
Let $S$ be a set. Let $\operatorname{cl}$ be a closure operator on $S$. Let $T \subseteq S$ be a subset of $S$. The '''closure''' of $T$ is its image $\operatorname{cl}(T)$.
Definition:Weakly Locally Connected at Point
Let $T = \struct {S, \tau}$ be a topological space. Let $x \in S$. === Definition 1 === {{Definition:Weakly Locally Connected at Point/Definition 1}} === Definition 2 === {{Definition:Weakly Locally Connected at Point/Definition 2}}
Definition:Zero Morphism via Zero Object
Let $C$ be a category with has a zero object $0$. Let $a,b\in C$ be objects. The '''zero morphism $0$''' from $a$ to $b$ is the composition of the unique morphism $a \to 0$ and the unique morphism $0 \to b$: :$0 : a \to 0 \to b$
Definition:Pointed Category
Let $C$ be a category. Then $C$ is '''pointed'''  $C$ has a zero object.
Definition:Universal Object
Let $C$ be a category. A '''universal object''' of $C$ is an object that is initial or terminal.
Definition:Disconnected (Topology)/Topological Space
A topological space $T$ is '''disconnected'''  it is not connected.
Definition:Kernel (Category Theory)
Let $C$ be a category. Let $f : A \to B $ be a morphism. === Definition 1: for categories with initial objects === Let $C$ have an initial object $0$. A '''kernel''' of $f$ is a morphism $\map \ker f \to A$ which is a pullback of the unique morphism $0 \to B$ via $f$ to $A$. === Definition 2: for categories with zero objects === Let $C$ have an zero object $0$. A '''kernel''' of $f$ is the equalizer of $f$ and the zero morphism $0 : A \to B$.
Definition:Generated Field Extension
Let $E / F$ be a field extension. Let $S \subset E$ be a subset of $E$. === Definition 1 === {{:Definition:Generated Field Extension/Definition 1}} === Definition 2 === {{:Definition:Generated Field Extension/Definition 2}}
Definition:Non-Associative Ring
A '''non-associative ring''' is a ringoid $\struct {R, +, \times}$ such that $\struct {R, +}$ is an abelian group.
Definition:Nonassociative Ring with Unity
A '''non-associative ring with unity''' is a non-associative ring $(R, +, \times)$ such that $(R, \times)$ is a unitary magma.
Definition:Group Action of GL(2,Q) on Irrational Numbers
Let $\GL {2, \Q}$ be the general linear group on the field of rational numbers. Let $\R \setminus \Q$ be the set of irrational numbers. The '''standard group action''' of $\GL {2, \Q}$ on $\R \setminus \Q$ is the group action: :$\GL {2, \Q} \times \paren {\R \setminus \Q} \to \R \setminus \Q$: :$\paren {\begin {pmatrix} a & b \\ c & d \end {pmatrix}, x} \mapsto \dfrac {a x + b} {c x + d}$
Definition:Disconnected (Topology)/Set
{{:Definition:Disconnected (Topology)/Set/Definition 1}}