hcju/proofwikips huggingface dataset:
Definition:Integral Element of Ring Extension
Let $A$ be a commutative ring with unity. Let $R \subseteq A$ be a subring. Then $a \in A$ is said to be '''integral''' over $R$  is is a root of a monic nonzero polynomial over $R$.
Definition:Ring Extension
Let $R$ and $S$ be commutative rings with unity. Let $\phi : R \to S$ be a ring monomorphism. Then $\phi : R \to S$ is a '''ring extension''' of $R$.
Definition:Faithful Module
Let $R$ be a commutative ring with unity. Let $M$ be an $R$-module. Then $M$ is '''faithful'''  its annihilator is zero.
Definition:Integral Closure
Let $A$ be an extension of a commutative ring with unity $R$. Let $C$ be the set of all elements of $A$ that are integral over $R$. Then $C$ is called the '''integral closure''' of $R$ in $A$.
Definition:Integrally Closed
=== Ring Extension === {{Definition:Integrally Closed in Ring Extension}} === Integral Domain === {{Definition:Integrally Closed Integral Domain}}
Definition:Multiplicatively Closed Subset of Ring
Let $\left({A, +, \circ}\right)$ be a ring with unity $1_A$ and zero $0_A$. Let $S \subseteq A$ be a subset. Then $S$ is '''multiplicatively closed''' : :$(1): \quad 1_A \in S$ :$(2): \quad x, y \in S \implies x \circ y \in S$
Definition:Localization of Ring
Let $A$ be a commutative ring with unity. Let $S \subseteq A$ be a multiplicatively closed subset of $A$. A '''localization of $A$ at $S$''' is a pair $\struct {A_S, \iota}$ where: :$A_S$ is a commutative ring with unity, the actual '''localization''' :$\iota: A \to A_S$ is a ring homomorphism, the '''localization homomorphism''' such that: :$(1): \quad \map \iota S \subseteq A_S^\times$, where $A_S^\times$ is the group of units of $A_S$ :$(2): \quad$ For every pair $\tuple {B, g}$ where: ::::$B$ is any ring with unity ::::$g: A \to B$ is a ring homomorphism such that $\map g S \subseteq B^\times$ :::there exists a unique ring homomorphism $h: A_S \to B$ such that: ::::$g = h \circ \iota$ That is, the following diagram commutes:  :325px
Definition:Prime Spectrum of Ring
Let $A$ be a commutative ring with unity. The '''prime spectrum''' or '''spectrum''' of $A$ is the set of prime ideals $\mathfrak p$ of $A$: :$\Spec A = \set {\mathfrak p \lhd A: \mathfrak p \text{ is prime} }$
Definition:Maximal Spectrum of Ring
Let $A$ be a commutative ring with unity. The '''maximal spectrum''' of $A$ is the set of maximal ideals of $A$: :$\operatorname{Max} \: \Spec A = \set {\mathfrak m \lhd A : \mathfrak m \text { is maximal} }$ where $I \lhd A$ indicates that $I$ is an ideal of $A$. The notation $\operatorname {Max} \: \Spec A$ is also a shorthand for the locally ringed space :$\struct {\operatorname {Max} \: \Spec A, \tau, \mathcal O_{\map {\operatorname {Max Spec} } A} }$ where: :$\tau$ is the Zariski topology on $\map {\operatorname {Max Spec} } A$ :$\mathcal O_{\map {\operatorname {Max Spec} } A}$ is the structure sheaf of $\map {\operatorname {Max Spec} } A$
Definition:Affine Algebraic Set
Let $k$ be a field. Let $A = k[X_1,\ldots,X_n]$ be the ring of polynomial functions in $n$ variables over $k$. Then a subset $X \subseteq k^n$ is an '''affine algebraic set''' if it is the zero locus of some set $T \subseteq A$.
Definition:Irreducible Space
=== Definition 1 === {{Definition:Irreducible Space/Definition 1}} === Definition 2 === {{Definition:Irreducible Space/Definition 2}} === Definition 3 === {{Definition:Irreducible Space/Definition 3}} === Definition 4 === {{Definition:Irreducible Space/Definition 4}} === Definition 5 === {{Definition:Irreducible Space/Definition 5}} === Definition 6 === {{Definition:Irreducible Space/Definition 6}} === Definition 7 === {{Definition:Irreducible Space/Definition 7}}
Definition:Zero Locus of Set of Polynomials
Let $I \subseteq A$ be a set. Then the '''zero locus''' of $I$ is the set: :$\map V I = \set {x \in k^n : \forall f \in I: \map f x = 0}$
Definition:Locally Euclidean Space
Let $M$ be a topological space. Let $d \in \N$ be a natural number. Then $M$ is a '''locally Euclidean space''' of dimension $d$ if each point in $M$ has an open neighbourhood homeomorphic to an open subset of Euclidean space $\R^d$. === Complex Locally Euclidean Space === {{:Definition:Locally Euclidean Space/Complex}}
Definition:Chart
Let $M$ be a locally Euclidean space, or complex locally Euclidean space of dimension $d$. A '''$d$-dimensional chart''' of $M$ is an ordered pair $\struct {U, \phi}$, where: :$U$ is an open subset of $M$ :$\phi: U \to D$ is a homeomorphism of $U$ onto an open subset $D$ of Euclidean space $\R^d$.
Definition:Differentiable Structure
Let $M$ be a topological space. Let $d$ be a natural number. Let $k \ge 1$ be a natural number. A '''$d$-dimensional differentiable structure of class $\mathcal C^k$''' on $M$ is a non-empty equivalence class of the set of $d$-dimensional $\mathcal C^k$-atlases on $M$ under the equivalence relation of compatibility.
Definition:Atlas
Let $M$ be a topological space. An '''atlas of class $C^k$ and dimension $d$''' on $M$ is a set of $d$-dimensional charts $\mathscr F = \family {\struct {U_\alpha, \phi_\alpha}: \alpha \in A}$ indexed by some set $A$ such that: :$(1): \quad \displaystyle \bigcup_{\alpha \mathop \in A} U_\alpha = M$ :$(2): \quad$ Every two charts $\struct {U, \phi}$ and $\struct {V, \psi}$ are $C^k$-compatible. === Smooth Atlas === {{:Definition:Atlas/Smooth Atlas}}
Definition:Connected Between Two Points
Let $T = \struct {S, \tau}$ be a topological space. Let $a, b \in S$. $T$ is '''connected between (the) two points $a$ and $b$'''  each separation of $T$ includes a single open set $U \in \tau$ which contains both $a$ and $b$.
Definition:Quasicomponent
Let $T = \struct {S, \tau}$ be a topological space. Let the relation $\sim$ be defined on $T$ as follows: :$x \sim y \iff T$ is connected between the two points $x$ and $y$ That is,  each separation of $T$ includes a single open set $U \in \tau$ which contains both $x$ and $y$. We have that $\sim$ is an equivalence relation, so from the Fundamental Theorem on Equivalence Relations, the points in $T$ can be partitioned into equivalence classes. These equivalence classes are called the '''quasicomponents''' of $T$.
Definition:Arc (Topology)
Let $T$ be a topological space. Let $\mathbb I \subset \R$ be the closed unit interval $\closedint 0 1$. Let $a, b \in T$. An '''arc from $a$ to $b$''' is a path $f: \mathbb I \to T$ such that $f$ is injective. That is, an '''arc from $a$ to $b$''' is a continuous injection $f: \mathbb I \to T$ such that $\map f 0 = a$ and $\map f 1 = b$. The mapping $f$ can be described as an '''arc (in $T$) joining $a$ and $b$'''.
Definition:Path Component
Let $T$ be a topological space.
Definition:Arc-Connected
=== Points in Topological Space === {{:Definition:Arc-Connected/Points}} === Subset of Topological Space === {{:Definition:Arc-Connected/Subset}} === Topological Space === {{:Definition:Arc-Connected/Topological Space}}
Definition:Arc Component
Let $T$ be a topological space. Let us define the relation $\sim$ on $T$ as follows: :$x \sim y \iff x$ and $y$ are arc-connected. We have that $\sim $ is an equivalence relation, so from the Fundamental Theorem on Equivalence Relations, the points in $T$ can be partitioned into equivalence classes. These equivalence classes are called the '''arc components of $T$'''.
Definition:Ring of Integers of Number Field
Let $K$ be a number field. The '''ring of integers''' of $K$, usually denoted $\mathcal O_K$ or $\mathfrak o_K$, is the integral closure of $\Z$ in $K$.
Definition:GCD Domain
A '''GCD domain''' is an integral domain in which any two non-zero elements have a greatest common divisor.
Definition:Ascending Chain Condition/Module
Let $R$ be a commutative ring with unity. Let $M$ be an $R$-module. Let $\left({D, \subseteq}\right)$ be a set of submodules of $M$ ordered by inclusion. Then $M$ is said to have the '''ascending chain condition on submodules''' : :''Every increasing sequence $N_1 \subseteq N_2 \subseteq N_3 \subseteq \cdots$ with $N_i \in D$ eventually stabilizes: $\exists k \in \N: \forall n \in \N, n \ge k: N_n = N_{n+1}$''
Definition:Descending Chain Condition
Let $A$ be a commutative ring with unity. Let $M$ be an $A$-module. Let $(D,\supseteq)$ be a set of submodules of $M$ ordered by inclusion. Then the hypothesis :''Every increasing sequence $N_1 \supseteq N_2 \supseteq N_3 \supseteq \cdots$ with $N_i \in D$ eventually terminates: there is $k \in \N$ such that $N_k = N_{k+1} = \cdots$'' is called the '''descending chain condition''' on the submodules in $D$.
Definition:Maximal Condition
Let $A$ be a commutative ring with unity. Let $M$ be an $A$-module. Let $(D,\subseteq)$ be the set of submodules of $M$ ordered by inclusion. Then the hypothesis :''Every non-empty subset of $D$ has a maximal element'' is called the '''maximal condition''' on submodules.
Definition:Minimal Condition
=== Ordered  set === {{Definition:Minimal Condition/Ordered Set}} === Minimal condition on subsets === Let $S$ be a set. Let $F$ be a set of subsets of $S$. Then $S$ '''satisfies the minimal condition on $F$'''  $F$, ordered by inclusion satisfies the minimal condition. ==== Minimal condition on submodules ==== Let $A$ be a commutative ring with unity. Let $M$ be an $A$-module. Let $(D,\supseteq)$ be the set of submodules of $M$ ordered by inclusion. Then the hypothesis :''Every non-empty subset of $D$ has a minimal element'' is called the '''minimal condition''' on submodules.
Definition:Noetherian Module
Let $A$ be a commutative ring with unity. Let $M$ be an $A$-module. === Definition 1 === {{:Definition:Noetherian Module/Definition 1}} === Definition 2 === {{:Definition:Noetherian Module/Definition 2}} === Definition 3 === {{:Definition:Noetherian Module/Definition 3}}
Definition:Artinian Module
Let $A$ be a commutative ring with unity. Let $M$ be an $A$-module Then $M$ is a '''Artinian module''' if either of the following conditions hold: :$(1): \quad$ $M$ satisfies the descending chain condition on submodules :$(2): \quad$ $M$ satisfies the minimal condition on submodules.
Definition:Artinian Ring
Let $A$ be a commutative ring with unity. Then $A$ is '''Artinian''' if it is Artinian as an $A$-module {{NamedforDef|Emil Artin|cat=Artin}}
Definition:Prime Element of Ring
Let $R$ be a commutative ring. Let $p \in R \setminus \set 0$ be any non-zero element of $R$. Then $p$ is a '''prime element of $R$''' : :$(1): \quad p$ is not a unit of $R$  :$(2): \quad$ whenever $a, b \in R$ such that $p$ divides $a b$, then either $p$ divides $a$ or $p$ divides $b$.
Definition:Factorization Domain
A '''factorization domain''' is an integral domain in which every element can be written as a finite product of irreducible elements.
Definition:Primal
Let $A$ be an integral domain. An element $p \in A$ is called '''primal''' if whenever $a, b \in A$ are such that $p$ divides $a b$, then we can write $p = q_1 q_2$, with $q_1$ divides $a$ and $q_2$ divides $B$.
Definition:Schreier Domain
A ring $A$ is a '''Schreier domain''' if it is an integrally closed integral domain in which every non-zero element is primal. {{NamedforDef|Otto Schreier|cat = Schreier}}
Definition:Nagata Criterion
Let $A$ be a ring. The '''Nagata criterion''' reads as follows: : ''Let $S \subseteq A$ be a multiplicatively closed subset of $A$ generated by prime elements. If the localization $A_S$ is a UFD, then so is $A$''. By Localization of UFD is UFD, this is equivalent to: : ''Let $S \subseteq A$ be a multiplicatively closed subset of $A$ generated by prime elements. Then the localization $A_S$ is a UFD  $A$ is a UFD''. {{NamedforDef|Masayoshi Nagata|cat = Nagata M}}
Definition:Product of Ideals of Ring
Let $\left({R, +, \circ}\right)$ be a commutative ring. Let $I,J$ be ideals of $R$. === Definition 1 === The '''product of $I$ and $J$''' is the set of all finite sums: :$IJ = \{a_1 b_1 + \cdots + a_r b_r : a_i \in I, b_i \in J, r \in \N \}$ === Definition 2 === The '''product of $I$ and $J$''' is the ideal generated by their product as subsets.
Definition:Bézout Domain
=== Definition 1 === {{:Definition:Bézout Domain/Definition 1}} === Definition 2 === {{:Definition:Bézout Domain/Definition 2}}
Definition:Direct Sum of Modules
Let $A$ be a commutative ring with unity. {{explain|For this concept to be more easily understood, it is suggested that the ring be defined using its full specification, i.e. complete with operators, $\left({A, +, \circ}\right)$ and so on, and similarly that a symbol be used to make the module scalar product equally explicit.}} Let $\left\{ {M_i}\right\}_{i \in I}$ be a family of $A$-modules indexed by $I$. {{explain|Worth specifically stating exactly what $I$ is.}} Let $M = \displaystyle \prod_{i \mathop \in I} M_i$ be their direct product. The '''direct sum''' $\displaystyle \bigoplus_{i \mathop \in I} M_i$ is the submodule of $M$ the consisting of the elements of finite support.
Definition:Projective Module
Let $A$ be a ring with unity. Let $M$ be an $A$-module. $M$ is '''projective'''  there exists an $A$-module $N$ such that the direct sum $M \oplus N$ is a free module.
Definition:Hereditary Ring
Let $A$ be a commutative ring. We say that $A$ is '''hereditary''' if for every module $M$ over $A$, if $M$ is projective, then so is each submodule of $M$.
Definition:Semihereditary Ring
Let $A$ be a commutative ring. We say that $A$ is '''semihereditary''' if for every module $M$ over $A$, if $M$ is projective, then so is each finitely generated submodule of $M$.
Definition:Prüfer Domain
Let $A$ be an integral domain. Then $A$ is a '''Prüfer domain''' if every finitely generated ideal of $A$ is projective. {{NamedforDef|Heinz Prüfer|cat = Prüfer}}
Definition:Height (Ring Theory)
Let $A$ be a commutative ring. Let $\mathfrak p$ be a prime ideal in $A$. The '''height''' of $\mathfrak p$ is the supremum over all $n$ such that there exists a chain of prime ideals: :$\mathfrak p_0 \subsetneqq \mathfrak p_1 \subsetneqq \cdots \subsetneqq \mathfrak p$ {{explain|make this precise, as in Definition:Krull Dimension of Ring}}
Definition:Ultraconnected Space
=== Definition 1 === {{:Definition:Ultraconnected Space/Definition 1}} === Definition 2 === {{:Definition:Ultraconnected Space/Definition 2}} === Definition 3 === {{:Definition:Ultraconnected Space/Definition 3}}
Definition:Addition of Polynomials/Polynomial Forms
Let: : $\displaystyle f = \sum_{k \mathop \in Z} a_k \mathbf X^k$ : $\displaystyle g = \sum_{k \mathop \in Z} b_k \mathbf X^k$ be polynomials in the indeterminates $\left\{{X_j: j \in J}\right\}$ over $R$. {{explain|What is $Z$ in the above? Presumably the integers, in which case they need to be denoted $\Z$ and limited in domain to non-negative? However, because $Z$ is used elsewhere in the exposition of polynomials to mean something else (I will need to hunt around to find out exactly what), I can not take this assumption for granted.}} The operation '''polynomial addition''' is defined as: :$\displaystyle f + g := \sum_{k \mathop \in Z} \left({a_k + b_k}\right) \mathbf X^k$ The expression $f + g$ is known as the '''sum''' of $f$ and $g$.
Definition:Multiplication of Polynomials/Polynomial Forms
Let $\displaystyle f = \sum_{k \mathop \in Z} a_k \mathbf X^k$, $\displaystyle g = \sum_{k \mathop \in Z} b_k \mathbf X^k$ be polynomial forms in the indeterminates $\left\{{X_j: j \in J}\right\}$ over $R$. The '''product of $f$ and $g$''' is defined as: :$\displaystyle f \circ g := \sum_{k \mathop \in Z} c_k \mathbf X^k$  where: :$\displaystyle c_k = \sum_{\substack{p + q \mathop = k \\ p, q \mathop \in Z}} a_p b_q$
Definition:Trivial Topological Space
A '''trivial topological space''' is a topological space with only one element. The open sets of a '''trivial topological space''' $T = \left({\left\{{s}\right\}, \tau}\right)$ are $\varnothing$ and $\left\{{s}\right\}$.
Definition:Locally Connected Space
=== Definition 1: Using Local Bases === {{:Definition:Locally Connected Space/Definition 1}} === Definition 2: Using Neighborhood Bases === {{:Definition:Locally Connected Space/Definition 2}} === Definition 3: Using (Global) Basis === {{:Definition:Locally Connected Space/Definition 3}} === Definition 4: Using Open Components === {{:Definition:Locally Connected Space/Definition 4}}
Definition:Locally Path-Connected Space
=== Definition 1: using Local Bases === {{:Definition:Locally Path-Connected Space/Definition 1}} === Definition 2: using Neighborhood Bases === {{:Definition:Locally Path-Connected Space/Definition 2}} === Definition 3: using (Global) Basis === {{:Definition:Locally Path-Connected Space/Definition 3}} === Definition 4: using Open Path Components === {{:Definition:Locally Path-Connected Space/Definition 4}}
Definition:Locally Arc-Connected Space
A topological space $T = \struct {S, \tau}$ is '''locally arc-connected'''  it has a basis consisting entirely of arc-connected sets.
Definition:Partition of Unity (Topology)
Let $X$ be a topological space. Let $\AA = \set {\phi_\alpha : \alpha \in A}$ be a collection of continuous mappings $X \to \R$ such that: :$(1): \quad$ The set $\set {\map {\operatorname {supp} } {\phi_\alpha}^\circ: \alpha \in A}$ of interiors of the supports is a locally finite cover of $X$ :$(2): \quad \forall x \in X: \forall \alpha \in A: \map {\phi_\alpha} x \ge 0$ :$(3): \quad \displaystyle \forall x \in X: \sum_{\alpha \mathop \in A} \map {\phi_\alpha} x = 1$ Then $\set {\phi_\alpha: \alpha \in A}$ is a '''partition of unity''' on $X$. === Subordinate to Cover === {{:Definition:Partition of Unity (Topology)/Subordinate}}
Definition:Vector Space on Field Extension
Let $\struct {K, +, \times}$ be a field. Let $L / K$ be a field extension over $K$. Then $\struct {L, +, \times}_K$ is the '''vector space on $L$ over $K$'''.
Definition:Totally Pathwise Disconnected Space
=== Definition 1 === {{:Definition:Totally Pathwise Disconnected Space/Definition 1}} === Definition 2 === {{:Definition:Totally Pathwise Disconnected Space/Definition 2}}
Definition:Characteristic Polynomial
Let $K$ be a field. Let $L / K$ be a finite field extension of $K$. Then by Vector Space on Field Extension is Vector Space, $L$ is naturally a vector space over $K$. Let $\alpha \in L$, and $\theta_\alpha$ be the linear operator: :$\theta_\alpha: L \to L : \beta \mapsto \alpha \beta$ The '''characteristic polynomial''' of $\alpha$ with respect to the extension $L / K$ is: :$\det \sqbrk {X I_L - \theta_\alpha}$ where: :$\det$ denotes the determinant of a linear operator :$X$ is an indeterminate  :$I_L$ is the identity mapping on $L$.
Definition:Determinant/Linear Operator
Let $V$ be a finite-dimensional vector space over a field $K$. Let $A: V \to V$ be a linear operator of $V$. The '''determinant''' $\det \left({A}\right)$ of $A$ is defined to be the determinant of any matrix of $A$ relative to some basis.
Definition:Trace (Linear Algebra)
=== Matrix === {{:Definition:Trace (Linear Algebra)/Matrix}} === Linear Operator === {{:Definition:Trace (Linear Algebra)/Linear Operator}}
Definition:Totally Separated Space
=== Definition 1 === {{:Definition:Totally Separated Space/Definition 1}} === Definition 2 === {{:Definition:Totally Separated Space/Definition 2}}
Definition:Grothendieck Universe
A '''Grothendieck universe''' is a set (not a class) which has the properties expected of the universe $\mathbb U$ of sets in the sense of the Zermelo-Fraenkel axioms with the following properties: :$(1): \quad \mathbb U$ is a transitive set: if $u \in \mathbb U$ and $x \in u$ then $x \in \mathbb U$ :$(2): \quad$ If $ u, v \in \mathbb U$ then $\set {u, v} \in \mathbb U$ :$(3): \quad$ If $u \in \mathbb U$ then the power set $\powerset u \in \mathbb U$ :$(4): \quad$ If $A \in \mathbb U$, and $\set {u_\alpha: \alpha \in A}$ is a collection of elements of $\mathbb U$, then $\displaystyle \bigcup_{\alpha \mathop \in A} u_\alpha \in \mathbb U$ {{explain|Tidy up language of that last axiom -- when "collection" is used, does this mean set? If so, say so -- if not, then link to exactly what concept is meant. This area of mathematics is such that differences between sets and e.g. classes is important to distinguish, and undefined terms such as "collection" are not to be tolerated.}}
Definition:Morphism
Let $\mathbf C$ be a metacategory. A '''morphism''' of $\mathbf C$ is an object $f$, together with: * A domain $\operatorname{dom} f$, which is an object of $\mathbf C$ * A codomain $\operatorname{cod} f$, also an object of $\mathbf C$
Definition:Hom Class
Let $\mathbf C$ be a metacategory. Let $C$ and $D$ be objects of $\mathbf C$. The collection of morphisms $f: C \to D$ is called a '''hom class''' and is denoted $\operatorname{Hom}_{\mathbf C} \left({C, D}\right)$.
Definition:Small Category
Let $\mathbf C$ be a metacategory. Then $\mathbf C$ is said to be '''small'''  both of the following hold: :The collection of objects $\mathbf C_0$ is a set; :The collection of morphisms $\mathbf C_1$ is a set.
Definition:Functor
Informally, a '''functor''' is a morphism of categories. It may be described as what one must define in order to define a '''natural transformation'''. This is formalized by defining the category of categories. === Covariant Functor === {{:Definition:Functor/Covariant}} === Contravariant Functor === {{:Definition:Functor/Contravariant}}
Definition:Composition of Morphisms
Let $\mathbf C$ be a metacategory. Let $\left({g, f}\right)$ be a pair of composable morphisms. Then the '''composition of $f$ and $g$''' is a morphism $g \circ f$ of $\mathbf C$ subject to: :$\operatorname{dom} \left({g \circ f}\right) = \operatorname{dom} f$ :$\operatorname{cod} \left({g \circ f}\right) = \operatorname{cod} g$ This '''composition of morphisms''' can be thought of as an abstraction of both composition of mappings and transitive relations.
Definition:Identity Morphism
Let $\mathbf C$ be a metacategory. Let $X$ be an object of $\mathbf C$. The '''identity morphism''' of $X$, denoted $\operatorname{id}_X$, is a morphism of $\mathbf C$ subject to: :$\operatorname{dom} \operatorname{id}_X = \operatorname{cod} \operatorname{id}_X = X$ :$f \circ \operatorname{id}_X = f$ :$\operatorname{id}_X \circ g = g$ whenever $X$ is the domain of $f$ or the codomain of $g$, respectively. In most metacategories, the '''identity morphisms''' can be viewed as a representation of "doing nothing", in a sense suitable to the metacategory under consideration.
Definition:Product Category
Let $\mathbf C$ and $\mathbf D$ be metacategories. The '''product category''' $\mathbf C \times \mathbf D$ is the category with: {{DefineCategory | ob = $\tuple {X, Y}$, for all $X \in \operatorname {ob} \mathbf C$, $Y \in \operatorname {ob} \mathbf D$ | mor = $\tuple {f, g}: \tuple {X, Y} \to \tuple {X', Y'}$ for all $f: X \to X'$ in $\mathbf C_1$ and $g: Y \to Y'$ in  $\mathbf D_1$ | comp = $\tuple {f, g} \circ \tuple {h, k} := \tuple {f \circ h, g \circ k}$, whenever this is defined | id = $\operatorname {id}_{\tuple {X, Y} } := \tuple {\operatorname {id}_X, \operatorname {id}_Y}$ }}
Definition:Extremally Disconnected Space
=== Definition 1 === {{:Definition:Extremally Disconnected Space/Definition 1}} === Definition 2 === {{:Definition:Extremally Disconnected Space/Definition 2}} === Definition 3 === {{:Definition:Extremally Disconnected Space/Definition 3}}
Definition:Zero Dimensional Space
Let $T = \struct {S, \tau}$ be a topological space. Then $T$ is zero dimensional  it has a basis whose sets are all both closed and open.
Definition:Scattered Space
Let $T = \struct {S, \tau}$ be a topological space. === Definition 1 === {{:Definition:Scattered Space/Definition 1}} === Definition 2 === {{:Definition:Scattered Space/Definition 2}}
Definition:Operator Domain
An operator domain is a notational tool for indexing a collection of operations on a set. Formally, an '''operator domain''' is a set $\Omega$  together with a mapping $a : \Omega \to \N_0$. We also use the notation, for $n \in \N_0$, $\Omega(n) = \{\omega \in \Omega : a(\omega) = n \}$. The elements of $\Omega$ are '''operators'''. If $\omega \in \Omega$, then $a(\omega)$ is the '''arity''' of the operator $\omega$. The elements of $\Omega(n)$ are the '''$n$-ary''' operators of $\Omega$. Note that an operator domain is an abstract labelling set, so the definitions here are purely formal; we have not identified any concrete operations. The latter is done using an Omega algebra.  16yueicrvs4xcp7n0b9p6dmg2elbm7e