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 $\alpha \in S$''' {{iff}}: :$\forall U \in \tau: \alpha \in U \implies \paren {\exists N \in \R_{>0}: \forall n \in \N: n > N \implies x_n \in U}$


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 $\alpha \in S$''' {{iff}}: :$\forall U \in \tau: \alpha \in U \implies \set {n \in \N: x_n \notin U}$ 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.


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 $T$ {{iff}}: :$\paren {H^-}^\circ = \O$ where $H^-$ denotes the closure of $H$ and $H^\circ$ denotes its interior.


Definition:Nowhere Dense/Definition 2

$H$ is '''nowhere dense''' in $T$ {{iff}}: :$H^-$ 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'''.


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''' {{iff}} $\mathbf J$ 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''' {{iff}}: :$\forall x, y \in S, x \ne y: \exists U, V \in \tau: x \in U, y \in V: U \cap V = \O$  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''' {{iff}} 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''' {{iff}} $\mathbf J$ is a finite category.


Definition:T3 Space/Definition 1

$T = \struct {S, \tau}$ is a '''$T_3$ space''' {{iff}}: :$\forall F \subseteq S: \relcomp S F \in \tau, y \in \relcomp S F: \exists U, V \in \tau: F \subseteq U, y \in V: U \cap V = \O$  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 '''$T_3$''' {{iff}} 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 '''$T_3$''' {{iff}} 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''' {{iff}} 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''' {{iff}} no two points can be limit points of each other.


=== 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''' {{iff}}: :$\forall A, B \in \map \complement \tau, A \cap B = \O: \exists U, V \in \tau: A \subseteq U, B \subseteq V, U \cap V = \O$  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 '''$T_4$''' {{iff}} 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''' {{iff}}: :$\forall A, B \subseteq S, A^- \cap B = A \cap B^- = \O: \exists U, V \in \tau: A \subseteq U, B \subseteq V, U \cap V = \O$    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''' {{iff}}: :$\forall Y,A \subseteq S: (A \subseteq Y^\circ \wedge A^- \subseteq Y) \implies \exists N \subseteq Y: \relcomp S N \in \tau: \exists U \in \tau: A \subseteq U \subseteq N$ That is: :$\struct {S, \tau}$ is a '''$T_5$ space''' {{iff}} 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''' {{iff}} both $f$ and $f^{-1}$ are continuous.


Definition:Homeomorphism/Topological Spaces/Definition 2

$f$ is a '''homeomorphism''' {{iff}}: :$\forall U \subseteq S_\alpha: U \in \tau_\alpha \iff f \sqbrk U \in \tau_\beta$


Definition:Homeomorphism/Topological Spaces/Definition 3

$f$ is a '''homeomorphism''' {{iff}} $f$ is both an open mapping and a continuous mapping.


Definition:Homeomorphism/Topological Spaces/Definition 4

$f$ is a '''homeomorphism''' {{iff}} $f$ 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''' {{iff}} $\mathcal C$ is a finite set.


Definition:Cover of Set/Countable

A cover $\mathcal C$ for $S$ is a '''countable cover''' {{iff}} $\mathcal C$ 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 $T$''' {{iff}} $T_H$ is itself a compact topological space.


Definition:Compact Space/Topology/Subspace/Definition 2

$H$ is '''compact in $T$''' {{iff}} 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: :$\operatorname{ev} \left({f, t}\right) := f \left({t}\right)$


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 $k \in \N$. Category:Definitions/Manifolds nq6z6g1sej00ysl0zvoltq30dtnffsn


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 $d$''' {{iff}} 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''' {{iff}} $\mathbf C$ 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 $T$) {{iff}} 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''' {{iff}} it admits no separation.


Definition:Connected (Topology)/Topological Space/Definition 2

$T$ is '''connected''' {{iff}} it has no two disjoint nonempty closed sets whose union is $S$.


Definition:Connected (Topology)/Topological Space/Definition 3

$T$ is '''connected''' {{iff}} its only subsets whose boundary is empty are $S$ and $\varnothing$.


Definition:Connected (Topology)/Topological Space/Definition 4

$T$ is '''connected''' {{iff}} its only clopen sets are $S$ and $\O$.


Definition:Connected (Topology)/Topological Space/Definition 5

$T$ is '''connected''' {{iff}} there are no two non-empty separated sets whose union is $S$.


Definition:Connected (Topology)/Topological Space/Definition 6

$T$ is '''connected''' {{iff}} 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 $g$.


Definition:Smooth Path/Complex

Let $\left[{a \,.\,.\, b}\right]$ be a closed real interval. Let $\gamma: \left[{a \,.\,.\, b}\right] \to \C$ be a path in $\C$. That is, $\gamma$ is a continuous complex-valued function from $\left[{a \,.\,.\, b}\right]$ to $\C$. Define the real function $x : \left[{a \,.\,.\, b}\right] \to \R$ by: :$\forall t \in \left[{a \,.\,.\, b}\right]: x \left({t}\right) = \operatorname{Re} \left({\gamma \left({t}\right)}\right)$ Define the real function $y: \left[{a \,.\,.\, b}\right] \to \R$ by: :$\forall t \in \left[{a \,.\,.\, b}\right]: y \left({t}\right) = \operatorname{Im} \left({\gamma \left({t}\right)}\right)$ where: : $\operatorname{Re} \left({\gamma \left({t}\right)}\right)$ denotes the real part of the complex number $\gamma \left({t}\right)$ : $\operatorname{Im} \left({\gamma \left({t}\right)}\right)$ denotes the imaginary part of $\gamma \left({t}\right)$. Then $\gamma$ is a '''smooth path (in $\C$)''' {{iff}}: :$(1): \quad$ Both $x$ and $y$ are continuously differentiable :$(2): \quad$ For all $t \in \left[{a \,.\,.\, b}\right]$, either $x' \left({t}\right) \ne 0$ or $y' \left({t}\right) \ne 0$.


Definition:Contour/Complex Plane

Let $C_1, \ldots, C_n$ be directed smooth curves in the complex plane $\C$. For each $i \in \left\{ {1, \ldots, n}\right\}$, let $C_i$ be parameterized by the smooth path $\gamma_i: \left[{a_i \,.\,.\, b_i}\right] \to \C$. For each $i \in \left\{ {1, \ldots, n - 1}\right\}$, let the endpoint of $\gamma_i$ equal the start point of $\gamma_{i + 1}$: :$\gamma_i \left({b_i}\right) = \gamma_{i + 1} \left({a_{i + 1} }\right)$ Then the finite sequence $\left\langle{C_1, \ldots, C_n}\right\rangle$ is a '''contour'''.


Definition:Connected (Topology)/Set/Definition 1

$H$ is a '''connected set of $T$''' {{iff}} it is not the union of any two non-empty separated sets of $T$.


Definition:Connected (Topology)/Set/Definition 2

$H$ is a '''connected set of $T$''' {{iff}} it is not '''disconnected in $T$'''.


Definition:Connected (Topology)/Set/Definition 3

$H$ is a '''connected set of $T$''' {{iff}}: :the topological subspace $\struct {H, \tau_H}$ of $T$ is a connected topological space.


Definition:Contour/Closed/Complex Plane

$C$ is a '''closed contour''' {{iff}} the start point of $C$ is equal to the end point of $C$: :$\gamma_1 \left({a_1}\right) = \gamma_n \left({b_n}\right)$


Definition:Complex Riemann Integral

Let $\mathbb I := \closedint a b$ be a closed real interval. Let $f: \mathbb I \to \C$ be a bounded complex function. Define the real function $x: \mathbb I \to \R$ by: :$\forall t \in \mathbb I: \map x t = \map \Re {\map f t}$ Define the real function $y: \mathbb I \to \R$ by: :$\forall t \in \mathbb I: \map y t = \map \Im {\map f t}$ Here: :$\map \Re {\map f t}$ denotes the real part of the complex number $\map f t$ :$\map \Im {\map f t}$ denotes the imaginary part of $\map f t$. Suppose that both $x$ and $y$ are Riemann integrable over $\mathbb I$. Then the '''complex (Riemann) integral of $f$ over $\mathbb I$''' is defined as: :$\displaystyle \int_a^b \map f t \rd t = \int_a^b \map \Re {\map f t} \rd t + i \int_a^b \map \Im {\map f t} \rd t$ $f$ is formally defined as '''(properly) complex integrable over $\mathbb I$ in the sense of Riemann''', or '''(properly) complex Riemann integrable over $\mathbb I$'''. More usually (and informally), we say: :'''$f$ is (Riemann) complex integrable over $\mathbb I$.'''


Definition:Path (Topology)/Initial Point

The '''initial point''' of $\gamma$ is $\map \gamma a$. That is, the path '''starts''' (or '''begins''') at $\map \gamma a$.


Definition:Compatible Atlases

Let $M$ be a topological space. Let $\mathscr F, \mathscr G$ be $d$-dimensional atlases of class $C^k$ on $M$. === Definition 1 === {{:Definition:Compatible Atlases/Definition 1}} === Definition 2 === {{:Definition:Compatible Atlases/Definition 2}}


Definition:Path (Topology)/Final Point

The '''final point''' of $\gamma$ is $\map \gamma b$. That is, the path '''ends''' (or '''finishes''') at $\map \gamma b$.


Definition:Coordinate System/Coordinate Function

Let $\left \langle {a_n} \right \rangle$ be a coordinate system of a unitary $R$-module $G$. For each $x \in G$ let $x_1, x_2, \ldots, x_n$ be the coordinates of $x$ relative to $\left \langle {a_n} \right \rangle$. Then for $i = 1, \ldots, n$ the mapping $f_i : G \to R$ defined by $f_i \left({x}\right) = x_i$ is called the '''$i$-th coordinate function on $G$ relative to $\left \langle {a_n} \right \rangle$'''.


Definition:Component (Topology)/Definition 1

From Connectedness of Points is Equivalence Relation, $\sim$ is an equivalence relation. From the Fundamental Theorem on Equivalence Relations, the points in $T$ can be partitioned into equivalence classes. These equivalence classes are called the '''(connected) components''' of $T$. If $x \in S$, then the '''component of $T$ containing $x$''' (that is, the set of points $y \in S$ with $x \sim y$) is denoted by $\map {\operatorname {Comp}_x} T$.


Definition:Component (Topology)/Definition 2

The '''component of $T$ containing $x$''' is defined as: : $\displaystyle \operatorname{Comp}_x \left({T}\right) = \bigcup \left\{{A \subseteq S: x \in A \land A}\right.$ is connected $\left.\right\}$


Definition:Component (Topology)/Definition 3

The '''component of $T$ containing $x$''' is defined as: : the maximal connected set of $T$ that contains $x$.


Definition:Closed Ball/Metric Space/Radius

The value $\epsilon$ is referred to as the '''radius''' of $\map { {B_\epsilon}^-} a$.


Definition:Continuous Mapping (Metric Space)/Point/Definition 1

'''$f$ is continuous at (the point) $a$ (with respect to the metrics $d_1$ and $d_2$)''' {{iff}}: :$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in A_1: \map {d_1} {x, a} < \delta \implies \map {d_2} {\map f x, \map f a} < \epsilon$ where $\R_{>0}$ denotes the set of all strictly positive real numbers.


Definition:Continuous Mapping (Metric Space)/Point/Definition 2

$f$ is '''continuous at (the point) $a$ (with respect to the metrics $d_1$ and $d_2$)''' {{iff}}: :$(1): \quad$ The limit of $f \left({x}\right)$ as $x \to a$ exists :$(2): \quad \displaystyle \lim_{x \to a} f \left({x}\right) = f \left({a}\right)$.


Definition:Continuous Mapping (Metric Space)/Point/Definition 3

$f$ is '''continuous at (the point) $a$ (with respect to the metrics $d_1$ and $d_2$)''' {{iff}}: :$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: f \left[{B_\delta \left({a; d_1}\right)}\right] \subseteq B_\epsilon \left({f \left({a}\right); d_2}\right)$ where $B_\epsilon \left({f \left({a}\right); d_2}\right)$ denotes the open $\epsilon$-ball of $f \left({a}\right)$ with respect to the metric $d_2$, and similarly for $B_\delta \left({a; d_1}\right)$.


Definition:Continuous Mapping (Metric Space)/Space/Definition 1

$f$ is '''continuous from $\left({A_1, d_1}\right)$ to $\left({A_2, d_2}\right)$''' {{iff}} it is continuous at every point $x \in A_1$.


Definition:Limit of Function (Metric Space)/Epsilon-Delta Condition

:$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: 0 < d_1 \left({x, c}\right) < \delta \implies d_2 \left({f \left({x}\right), L}\right) < \epsilon$ That is, for every real positive $\epsilon$ there exists a real positive $\delta$ such that ''every'' point in the domain of $f$ within $\delta$ of $c$ has an image within $\epsilon$ of some point $L$ in the codomain of $f$.


Definition:Limit of Function (Metric Space)/Epsilon-Ball Condition

:$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: f \left({B_\delta \left({c; d_1}\right) \setminus \left\{{c}\right\}}\right) \subseteq B_\epsilon \left({L; d_2}\right)$. where: : $B_\delta \left({c; d_1}\right) \setminus \left\{{c}\right\}$ is the deleted $\delta $-neighborhood of $c$ in $M_1$ : $B_\epsilon \left({L; d_2}\right)$ is the open $\epsilon$-ball of $L$ in $M_2$. That is, for every open $\epsilon$-ball of $L$ in $M_2$, there exists a deleted $\delta $-neighborhood of $c$ in $M_1$ whose image is a subset of that open $\epsilon$-ball.


Definition:Totally Bounded Metric Space/Definition 1

A metric space $M = \struct {A, d}$ is '''totally bounded''' {{iff}}: :for every $\epsilon \in \R_{>0}$ there exists a finite $\epsilon$-net for $M$. That is, $M$ is '''totally bounded''' {{iff}}: :for every $\epsilon \in \R_{>0}$ there exists a finite set of points $x_1, \ldots, x_n \in A$ such that: ::$\displaystyle A = \bigcup_{i \mathop = 1}^n \map {B_\epsilon} {x_i}$ :where $\map {B_\epsilon} {x_i}$ denotes the open $\epsilon$-ball of $x_i$. That is: $M$ is '''totally bounded''' {{iff}}, given any $\epsilon \in \R_{>0}$, one can find a finite number of open $\epsilon$-balls which cover $A$.


Definition:Totally Bounded Metric Space/Definition 2

A metric space $M = \struct {A, d}$ is '''totally bounded''' {{iff}}: :for every $\epsilon \in \R_{>0}$ there exist finitely many points $x_0, \dots, x_n \in A$ such that: ::$\displaystyle \inf_{0 \mathop \le i \mathop \le n} \map d {x_i, x} \le \epsilon$ :for all $x \in A$.


Definition:Uniform Convergence/Metric Space

Let $S$ be a set. Let $M = \left({A, d}\right)$ be a metric space. Let $\left \langle {f_n} \right \rangle$ be a sequence of mappings $f_n: S \to A$. Let: : $\forall \epsilon \in \R_{>0}: \exists N \in \R: \forall n \ge N, \forall x \in S: d \left({f_n \left({x}\right), f \left({x}\right)}\right) < \epsilon$ Then '''$\left \langle {f_n} \right \rangle$ converges to $f$ uniformly on $S$ as $n \to \infty$'''.


Definition:Uniform Convergence/Real Numbers

Let $\sequence {f_n}$ be a sequence of real functions defined on $D \subseteq \R$. Let: :$\forall \epsilon \in \R_{>0}: \exists N \in \R: \forall n \ge N, \forall x \in D: \size {\map {f_n} x - \map f x} < \epsilon$ That is:  :$\displaystyle \forall \epsilon \in \R_{>0}: \exists N \in \R: \forall n \ge N: \sup_{x \mathop \in D} \size {\map {f_n} x - \map f x} < \epsilon$ Then '''$\sequence {f_n}$ converges to $f$ uniformly on $D$ as $n \to \infty$'''.


Definition:Real Function/Domain

Let $f: S \to \R$ be a real function. The domain of $f$ is the set $S$. It is frequently the case that $S$ is not explicitly specified. If this is so, then it is understood that the domain is to consist of ''all'' the values in $\R$ for which the function is defined. This often needs to be determined as a separate exercise in itself, by investigating the nature of the function in question.


Definition:Modulo Operation/Modulo Zero

:$\forall x \in \R: x \bmod 0 = x$ This can be considered as a '''special case''' of the modulo operation, but it is interesting to note that most of the results concerning the modulo operation still hold.


Definition:Modulo Operation/Modulo One

:$x \bmod 1 = x - \left \lfloor {x}\right \rfloor$ from which it follows directly that: :$x = \left \lfloor {x}\right \rfloor + \left({x \bmod 1}\right)$


Definition:Symmetric Difference/Definition 1

:$S * T := \paren {S \setminus T} \cup \paren {T \setminus S}$


Definition:Symmetric Difference/Definition 2

:$S * T = \paren {S \cup T} \setminus \paren {S \cap T}$


Definition:Symmetric Difference/Definition 3

: $S * T = \left({S \cap \overline T}\right) \cup \left({\overline S \cap T}\right)$


Definition:Symmetric Difference/Definition 4

:$S * T = \paren {S \cup T}\cap \paren {\overline S \cup \overline T}$


Definition:Standard Deviation

Let $X$ be a random variable. Then the '''standard deviation of $X$''', written $\sigma_X$ or $\sigma$, is defined as the principal square root of the variance of $X$: :$\sigma_X := \sqrt {\var X}$


Definition:Transfinite Ordinal

Let $\alpha$ be an ordinal. Then $\alpha$ is said to be '''transfinite''' {{iff}} it is an infinite set.


Definition:Characteristic Function of Random Variable

Let $X$ be a random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$. The '''characteristic function of $X$''' is the mapping $\phi: \R \to \C$ defined by: :$\map \phi t = \expect {e^{i t X} }$ where: :$i$ is the imaginary unit :$\expect \cdot$ denotes expectation.


Definition:Transfinite Sequence

Let $\alpha$ be an infinite ordinal. Let $\left({x_\beta}\right)_{\beta \mathop \in \alpha}$ be an $\alpha$-indexed family. Then $\left({x_\beta}\right)_{\beta \mathop \in \alpha}$ is called a '''transfinite sequence'''.


Definition:Nest

Let $S$ be a set. Let $\powerset S$ be its power set. Let $N \subseteq \powerset S$ be a subset of $\powerset S$. Then $N$ is a '''nest''' {{iff}}: :$\forall X, Y \in N: X \subseteq Y$ or $Y \subseteq X$ === Class Theory === {{:Definition:Nest/Class Theory}}


Definition:Ideal (Order Theory)

Let $\left({S, \preceq}\right)$ be an ordered set. Let $I \subseteq S$ be a non-empty subset of $S$. Then $I$ is an '''ideal''' of $S$ {{iff}} it is both a lower set and a directed set. That is, $I$ is an '''ideal''' {{iff}}: :$\forall x \in I, y \in P: y \preceq x \implies y \in I$ :$\forall x, y \in I: \exists z \in I: x \preceq z$ and $y \preceq z$ Category:Definitions/Order Theory qc6oyt63p8ea0goz322i0fy4l2z3f9j


Definition:Characteristic Function (Set Theory)/Set/Support

Let $S$ be a set Let $E \subseteq S$ be a subset. Let $\chi_E: S \to \set {0, 1}$ be the characteristic function of $E$. The '''support''' of $\chi_E$, denoted $\operatorname{supp} \chi_E$, is the set $E$. That is: :$\operatorname{supp} \chi_E = \set {x \in S: \map {\chi_E} x = 1}$


Definition:Contour/Length/Complex Plane

Let $C$ be a '''contour''' in $C$ defined by the (finite) sequence $\left\langle{C_1, \ldots, C_n}\right\rangle$ of directed smooth curves in $\C$. Let $C_i$ be parameterized by the smooth path $\gamma_i: \left[{a_i\,.\,.\,b_i}\right] \to \C$ for all $i \in \left\{ {1, \ldots, n}\right\}$. The '''length''' of $C$ is defined as: :$\displaystyle L \left({C}\right) := \sum_{i \mathop = 1}^n \int_{a_i}^{b_i} \left\vert{\gamma_i' \left({t}\right) }\right\vert \rd t$


Definition:Two Ring

Let $\struct {\Z_2, +_2, \times_2}$ be the ring of integers modulo $2$. Then $\struct {\Z_2, +_2, \times_2}$ is referred to as the '''two ring'''.


Definition:Idempotent Ring

Let $\struct {R, +, \circ}$ be a ring. $R$ is an '''idempotent ring''' {{iff}} its ring product $\circ$ is an idempotent operation.


Definition:Boolean Ring Axioms

{{begin-axiom}} {{axiom | n = \text A 0         | lc= Closure under addition         | q = \forall a, b \in R         | m = a * b \in R }} {{axiom | n = \text A 1         | q = \forall a, b, c \in R         | m = \paren {a * b} * c = a * \paren {b * c}         | lc= Associativity of addition }} {{axiom | n = \text A 2         | q = \forall a, b \in R         | m = a * b = b * a         | lc= Commutativity of addition }} {{axiom | n = \text A 3         | q = \exists 0_R \in R: \forall a \in R         | m = a * 0_R = a = 0_R * a         | lc= Identity element for addition: the zero }} {{axiom | n = \text {AC} 2         | q = \forall a \in R         | m = a * a = 0_R         | lc= Characteristic 2 for addition: }} {{axiom | n = \text M 0         | q = \forall a, b \in R         | m = a \circ b \in R         | lc= Closure under product }} {{axiom | n = \text M 1         | q = \forall a, b, c \in R         | m = \paren {a \circ b} \circ c = a \circ \paren {b \circ c}         | lc= Associativity of product }} {{axiom | n = \text M 2         | q = \exists 1_R \in R: \forall a \in R         | m = 1_R \circ a = a = a \circ 1_R         | lc= Identity element for product: the unity }} {{axiom | n = \text {MI}         | q = \forall a \in R         | m = a \circ a = a         | lc= Idempotence of product }} {{axiom | n = \text D         | q = \forall a, b, c \in R         | m = a \circ \paren {b * c} = \paren {a \circ b} * \paren {a \circ c},         | lc= Product is distributive over addition }} {{axiom | m = \paren {a * b} \circ c = \paren {a \circ c} * \paren {b \circ c} }} {{end-axiom}} These criteria are called the '''Boolean ring axioms'''.


Definition:Two-Valued Function

Let $2$ be the two ring. Let $f: X \to 2$ be a mapping with codomain $2$. Then $f$ is called a '''$2$-valued function'''.


Definition:Ring of Idempotents

Let $\left({R, +, \circ}\right)$ be a commutative ring. Let $A$ be the set of all idempotent elements of $R$ under $\circ$: :$A = \left\{{x \in R: x \circ x = x}\right\}$ Define a binary operation $\circ$ on $A$ by: :$\forall x, y \in A: x \oplus y := x + y - 2 x \circ y$ Denote also with $\circ$ the restriction of $\circ$ to $A$. The algebraic structure $\left({A, \oplus, \circ}\right)$ is called the '''ring of idempotents''' of $R$. It is an idempotent ring, as shown on Ring of Idempotents is Idempotent Ring.


Definition:Algebraic Element of Ring Extension

Let $\struct {R, +, \circ}$ be a commutative ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $\struct {D, +, \circ}$ be an integral subdomain of $R$. Let $x \in R$. Then $x$ is '''algebraic over $D$''' {{iff}}: :$\exists \map f x$ over $D$ such that $\map f x = 0_R$ where $\map f x$ is a non-null polynomial in $x$ over $D$.


Let $E / F$ be a field extension. Let $\alpha \in E$. === Definition 1 === {{:Definition:Algebraic Element of Field Extension/Definition 1}} === Definition 2 === {{:Definition:Algebraic Element of Field Extension/Definition 2}}


Definition:Algebraic Field Extension

A field extension $E / F$ is said to be '''algebraic''' {{iff}}: : $\forall \alpha \in E: \alpha$ is algebraic over $F$


Definition:Prime Ideal (Order Theory)

Let $I$ be an ideal in an ordered set $S$. Then $I$ is a '''prime ideal''' in $S$ {{iff}} $S \setminus I$ is a filter.


Definition:Distributive Lattice

Let $\left({S, \vee, \wedge, \preceq}\right)$ be a lattice. Then $\left({S, \vee, \wedge, \preceq}\right)$ is '''distributive''' {{iff}} one (hence all) of the following equivalent statements holds: {{begin-axiom}} {{axiom | n = 1         | q = \forall x, y, z \in S         | m = x \wedge \left({y \vee z}\right) = \left({x \wedge y}\right) \vee \left({x \wedge z}\right) }} {{axiom | n = 1'         | q = \forall x, y, z \in S         | m = \left({x \vee y}\right) \wedge z = \left({x \wedge z}\right) \vee \left({y \wedge z}\right) }} {{axiom | n = 2         | q = \forall x, y, z \in S         | m = x \vee \left({y \wedge z}\right) = \left({x \vee y}\right) \wedge \left({x \vee z}\right) }} {{axiom | n = 2'         | q = \forall x, y, z \in S         | m = \left({x \wedge y}\right) \vee z = \left({x \vee z}\right) \wedge \left({y \vee z}\right) }} {{end-axiom}}


Definition:Stone Space

Let $B$ be a Boolean algebra. The '''Stone space of $B$''' is the topological space: :$\map S B = \struct {U, \tau}$ where: :$U$ is the set of ultrafilters in $B$ :$\tau$ is the topology generated by the basis consisting of all sets of the form: :$\set {x \in \map S B: b \in x}$ :for some $b \in B$.


Definition:Contour Integral/Complex

Let $C$ be a contour defined by a finite sequence $C_1, \ldots, C_n$ of directed smooth curves. Let $C_i$ be parameterized by the smooth path $\gamma_i: \closedint {a_i} {b_i} \to \C$ for all $i \in \set {1, \ldots, n}$. Let $f: \Img C \to \C$ be a continuous complex function, where $\Img C$ denotes the image of $C$. The '''contour integral of $f$ along $C$''' is defined by: :$\displaystyle \int_C \map f z \rd z = \sum_{i \mathop = 1}^n \int_{a_i}^{b_i} \map f {\map {\gamma_i} t} \map {\gamma_i'} t \rd t$ From Contour Integral is Well-Defined, it follows that the complex Riemann integral on the right side is defined and is independent of the parameterizations of $C_1, \ldots, C_n$. === Contour Integral along Closed Contour === {{:Definition:Contour Integral/Complex/Closed}}


Definition:Smooth Path/Closed/Complex Plane

Let $\gamma$ be a smooth path in $\C$. $\gamma$ is a '''closed smooth path''' {{iff}} $\gamma$ is a closed path. That is, {{iff}} $\gamma \left({a}\right) = \gamma \left({b}\right)$.


Definition:Ordered Sum/General Definition

Let $S_1, S_2, \ldots, S_n$ be tosets. Then we define $T_n$ as the ordered sum of $S_1, S_2, \ldots, S_n$ as: :$\forall n \in \N_{>0}: T_n = \begin{cases} S_1 & : n = 1 \\ T_{n-1} + S_n & : n > 1 \end{cases}$


Definition:Moore-Smith Sequence

Let $\left({S, \preceq}\right)$ be a directed set. Let $T$ be a set. Let $f: S \to T$ be a mapping. Then $f$ is a '''Moore-Smith sequence''' in $T$.


Definition:Ordered Square

Let: : $S = \left[{0 \,.\,.\, 1}\right] \times \left[{0 \,.\,.\, 1}\right]$ where $\left[{0 \,.\,.\, 1}\right]$ is the closed real interval from $0$ to $1$. Let $\preceq$ be the lexicographic ordering on $S$ induced by the usual ordering of $\left[{0 \,.\,.\, 1}\right]$. Let $\tau$ be the order topology on $S$ induced by $\preceq$. Then $\left({S, \tau}\right)$ is the '''ordered square'''.


Definition:Cartesian Product of Relations

Let $\left\langle{S_i}\right\rangle_{i \mathop \in I}$ and $\left\langle{T_i}\right\rangle_{i \mathop \in I}$ be families of sets indexed by $I$. For each $i \in I$, let $\mathcal R_i \subseteq S_i \times T_i$ be a relation from $S_i$ to $T_i$. Let $S$ and $T$ be the Cartesian product of $\left\langle{S_i}\right\rangle_{i \mathop \in I}$ and $\left\langle{T_i}\right\rangle_{i \mathop \in I}$ respectively: :$\displaystyle S = \prod_{i \mathop \in I} S_i$ :$\displaystyle T = \prod_{i \mathop \in I} T_i$ Then the '''product''' of the relations $\mathcal R_i$ is defined as the relation $\mathcal R \subseteq S \times T$ such that: :$x \mathrel{\mathcal R} y \iff \forall i \in I: x_i \mathrel{\mathcal R_i} y_i$


Definition:Time-Constructible Function

=== Definition 1 === {{:Definition:Time-Constructible Function/Definition 1}} === Definition 2 === {{:Definition:Time-Constructible Function/Definition 2}}


Definition:Boolean Algebra

=== Definition 1 === {{:Definition:Boolean Algebra/Definition 1}} === Definition 2 === {{:Definition:Boolean Algebra/Definition 2}} === Definition 3 === {{:Definition:Boolean Algebra/Definition 3}} The operations $\vee$ and $\wedge$ are called '''join''' and '''meet''', respectively. The identities $\bot$ and $\top$ are called '''bottom''' and '''top''', respectively. Also, $\neg a$ is called the '''complement''' of $a$. The operation $\neg$ is called '''complementation'''.


Definition:Concatenation of Contours

Let $\R^n$ be a real cartesian space of $n$ dimensions. Let $C$ and $D$ be contours in $\R^n$. Thus: :$C$ is a (finite) sequence of directed smooth curves $\sequence {C_1, \ldots, C_n}$ :$D$ is a (finite) sequence of directed smooth curves $\sequence {D_1, \ldots, D_m}$. Let $C_i$ be parameterized by the smooth path: :$\gamma_i: \closedint {a_i} {b_i} \to \R^n$ for all $i \in \set {1, \ldots, n}$ Let $D_i$ be parameterized by the smooth path: :$\sigma_i: \closedint {c_i} {d_i} \to \R^n$ for all $i \in \set {1, \ldots, m}$ Let $\map {\gamma_n} {b_n} = \map {\sigma_1} {c_1}$. Then the '''concatenation of the contours''' $C$ and $D$, denoted $C \cup D$, is the contour defined by the (finite) sequence: :$\sequence {C_1, \ldots, C_n, D_1, \ldots, D_m}$


Definition:Probability Density Function

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space. Let $X: \Omega \to \R$ be a continuous random variable on $\struct {\Omega, \Sigma, \Pr}$. Let $\Omega_X = \Img X$, the image of $X$. Then the '''probability density function''' of $X$ is the mapping $f_X: \R \to \closedint 0 1$ defined as: :$\forall x \in \R: \map {f_X} x = \begin {cases} \displaystyle \lim_{\epsilon \mathop \to 0^+} \frac {\map \Pr {x - \frac \epsilon 2 \le X \le x + \frac \epsilon 2} } \epsilon & : x \in \Omega_X \\ 0 & : x \notin \Omega_X \end {cases}$


Definition:Finite State Machine

A '''finite state machine''' is an ordered tuple: : $F = \left({ S, A, I, \Sigma, T }\right)$ where: : $S$ is the (finite) set of states : $A \subseteq S$ is the set of ''accepting'' states : $I \in S$ is the ''initial state'' : $\Sigma$ is the ''alphabet'' of symbols that can be fed into the machine : $T : \left({ S \times \Sigma }\right) \rightarrow S$ is the ''transition function''. A finite state machine operates as follows: :$(1): \quad$ At the beginning, the ''current state'' $s$ of the finite state machine is $I$. :$(2): \quad$ One by one, the ''input'' (a sequence of symbols from $\Sigma$) is fed into the machine. :$(3): \quad$ After each input symbol $\sigma$, the current state $s$ is set to the result of $T\left({s, \sigma}\right)$. If, at the end of processing an input word $w$, $s \in A$, the finite state machine is said to '''accept''' $w$, otherwise to '''reject''' it. The set of words $w$ ''accepted'' by the machine $F$ is called the '''accepted language''' $L\left({F}\right)$. Category:Definitions/Mathematical Logic 4y24bc1umm82j5akk79k7162mjwjpui


Definition:Regular Expression

A '''regular expression''' is an algebraic structure on an ''alphabet'' $\Sigma$ defined as follows: * The empty-set regular expression, $\varnothing$, is a regular expression. * The empty-word regular expression, $\epsilon$, is a regular expression. * Every $\sigma \in \Sigma$ is a regular expression. (These are called ''literals''.) * If $R_1$ and $R_2$ are regular expressions, $R_1 R_2$ is a regular expression (''concatenation''). * If $R_1$ and $R_2$ are regular expressions, $R_1 \mid R_2$ is a regular expression (''alternation''). * If $R$ is a regular expression, $R^*$ is a regular expression (''Kleene star''). Every regular expression $R$ on an alphabet $\Sigma$ defines a language $L \left({R}\right) \subseteq \Sigma^*$, where $\Sigma^*$ is the set of all (finite-length) ''words'' (sequences) of symbols in $\Sigma$: * $ L \left({\varnothing}\right) = \varnothing $ (the empty set). * $ L \left({\epsilon}\right) = \left\{ { \left[{}\right] }\right\} $ (the set containing only the empty word). * If $R$ is a ''literal'' $\sigma$, $ L \left({R}\right) = \left\{ { \left[{ \sigma }\right] }\right\} $ (i.e., the set containing only the single-symbol word “$\sigma$”). * If $R$ is a ''concatenation'' $R_1 R_2$, $ L \left({R}\right) = \left\{ { w_1 w_2 : w_1 \in L \left({R_1}\right), w_2 \in L \left({R_2}\right) }\right\} $, where $w_1 w_2$ is the concatenation of the words $w_1$ and $w_2$. * If $R$ is an ''alternation'' $R_1 \mid R_2$, $ L \left({R}\right) = L \left({R_1}\right) \cup L \left({R_2}\right)$. * If $R$ is a ''Kleene star'' $R_1^*$, $ L\left({R}\right) $ is the smallest set satisfying the following: ** $\left[{}\right] \in L \left({R}\right) $ (the empty word is in the set); ** if $w_1 \in L \left({R}\right)$ and $w_2 \in L \left({R_1}\right)$, then $w_1 w_2 \in L \left({R}\right)$. Category:Definitions/Formal Systems 5tdtcn3ny6g9fhsewv68p3ahwc0e47x


Definition:Directed Smooth Curve

Let $\R^n$ be a real cartesian space of $n$ dimensions. Let $\rho: \left[{a \,.\,.\, b}\right] \to \R^n$ be a smooth path in $\R^n$. The '''directed smooth curve''' with '''parameterization''' $\rho$ is defined as an equivalence class of smooth paths as follows: A smooth path $\sigma: \left[{a \,.\,.\, b}\right] \to \R^n$ belongs to the equivalence class of $\rho$ {{iff}}: : there exists a bijective differentiable strictly increasing real function: :: $\phi: \left[{c \,.\,.\, d}\right] \to \left[{a \,.\,.\, b}\right]$ : such that $\sigma = \rho \circ \phi$. </onlyinclude> It follows from Directed Smooth Curve Relation is Equivalence and Fundamental Theorem on Equivalence Relations that this does in fact define an equivalence class. If a '''directed smooth curve''' is only defined by a smooth path $\rho$, then it is often denoted with the same symbol $\rho$. === Parameterization === {{:Definition:Directed Smooth Curve/Parameterization}} === Endpoints === {{:Definition:Directed Smooth Curve/Endpoints}}


Definition:Smooth Path/Simple/Complex Plane

Let $\gamma : \left[{a \,.\,.\, b}\right] \to \C$ be a '''smooth path in $\C$'''. $\gamma$ is a '''simple smooth path''' {{iff}}: :$(1): \quad \gamma$ is injective on the half-open interval $\left[{a \,.\,.\, b}\right)$ :$(2): \quad \forall t \in \left({a \,.\,.\, b}\right): \gamma \left({t}\right) \ne \gamma \left({b}\right)$ That is, if $t_1, t_2 \in \left({a \,.\,.\, b}\right)$ with $t_1 \ne t_2$, then $\gamma \left({a}\right) \ne \gamma \left({t_1}\right) \ne \gamma \left({t_2}\right) \ne \gamma \left({b}\right)$.


Definition:Arc-Connected/Points

Let $T = \struct {S, \tau}$ be a topological space. Let $a, b \in S$ be such that there exists an arc from $a$ to $b$. That is, there exists a continuous injection $f: \closedint 0 1 \to S$ such that $\map f 0 = a$ and $\map f 1 = b$. Then $a$ and $b$ are '''arc-connected'''. It is also declared that any point $a$ is '''arc-connected''' to itself.


Definition:Arc-Connected/Topological Space

Let $T = \struct {S, \tau}$ be a topological space. Then $T$ is '''arc-connected''' {{iff}} every two points in $T$ are arc-connected in $T$. That is, $T$ is '''arc-connected''' {{iff}}: :for every $x, y \in X, \exists$ a continuous injection $f: \closedint 0 1 \to X$ such that $\map f 0 = x$ and $\map f 1 = y$.


Definition:Arc-Connected/Subset

Let $T = \left({S, \tau}\right)$ be a topological space. Let $U \subseteq S$ be a subset of $T$. Let $T\,' = \left({U, \tau_U}\right)$ be the subspace of $T$ induced by $U$. Then $U$ is '''arc-connected in $T$''' {{iff}} every two points in $U$ are arc-connected in $T\,'$. That is, $U$ is '''arc-connected''' {{iff}}: :for every $x, y \in U$, there exists a continuous injection $f: \left[{0 \,.\,.\, 1}\right] \to S$ such that $f \left({0}\right) = x$ and $f \left({1}\right) = y$.


Definition:Mistake/Spello

A '''spello''' (abbreviation for '''spelling mistake''') is a mistake caused by inadequate spelling skills of the writer. These can sometimes be corrected by the writer having the self-doubt necessary to occasion the act of referring to a reliable manual, for example, a dictionary. A competent proof-reader (that is, one whose spelling skills are more comprehensive than the writer) is usually an adequate resource for reducing spelloes.


Definition:Mistake/Typo

A '''typo''' (abbreviation for '''typographical error''') is a mistake arising from pressing the wrong button on the machine via which you are transcribing your thoughts onto a storage medium. These can be corrected by assiduous checking by the writer.


Definition:Mistake/Homophone Horror

A '''homophone horror''' is a mistake caused by the use of a word which sounds the same as, but is spelled differently to, the word intended. A proficient proof-reader is needed to reduce instances of this sort of mistake.


Definition:Directed Smooth Curve/Parameterization/Complex Plane

Let $C$ be a directed smooth curve in the complex plane $\C$. Let $\gamma: \left[{a \,.\,.\, b}\right] \to \C$ be a smooth path in $\C$. Then $\gamma$ is a '''parameterization''' of $C$ {{iff}} $\gamma$ is an element of the equivalence class that constitutes $C$.


Definition:Directed Smooth Curve/Endpoints/Complex Plane

Let $C$ be a directed smooth curve in the complex plane $\C$. Let $C$ be parameterized by a smooth path $\gamma: \left[{a \,.\,.\, b}\right] \to \C$. Then: : $\gamma \left({a}\right)$ is the '''start point''' of $C$ : $\gamma \left({b}\right)$ is the '''end point''' of $C$. Collectively, $\gamma \left({a}\right)$ and $\gamma \left({b}\right)$ are known as the '''endpoints''' of $\rho$.


Definition:Integral Domain/Definition 1

An '''integral domain''' $\struct {D, +, \circ}$ is: :a commutative ring which is non-null :with a unity :in which there are no (proper) zero divisors, that is: ::: $\forall x, y \in D: x \circ y = 0_D \implies x = 0_D \text{ or } y = 0_D$ that is (from the Cancellation Law of Ring Product of Integral Domain) in which all non-zero elements are cancellable.


Definition:Integral Domain/Definition 2

An '''integral domain''' $\left({D, +, \circ}\right)$ is a commutative ring such that $\left({D^*, \circ}\right)$ is a monoid, all of whose elements are cancellable. In this context, $D^*$ denotes the ring $D$ without zero: $D \setminus \left\{{0_D}\right\}$.


Definition:Integral Domain Axioms

An integral domain is an algebraic structure $\struct {D, *, \circ}$, on which are defined two binary operations $\circ$ and $*$, which satisfy the following conditions: {{begin-axiom}} {{axiom | n = A0         | q = \forall a, b \in D         | m = a * b \in D         | lc= Closure under addition }} {{axiom | n = A1         | q = \forall a, b, c \in D         | m = \paren {a * b} * c = a * \paren {b * c}         | lc= Associativity of addition }} {{axiom | n = A2         | q = \forall a, b \in D         | m = a * b = b * a         | lc= Commutativity of addition }} {{axiom | n = A3         | q = \exists 0_D \in D: \forall a \in D         | m = a * 0_D = a = 0_D * a         | lc= Identity element for addition: the zero }} {{axiom | n = A4         | q = \forall a \in D: \exists a' \in D         | m = a * a' = 0_D = a' * a         | lc= Inverse elements for addition: negative elements }} {{axiom | n = M0         | q = \forall a, b \in D         | m = a \circ b \in D         | lc= Closure under product }} {{axiom | n = M1         | q = \forall a, b, c \in D         | m = \paren {a \circ b} \circ c = a \circ \paren {b \circ c}         | lc= Associativity of product }} {{axiom | n = D         | q = \forall a, b, c \in D         | m = a \circ \paren {b * c} = \paren {a \circ b} * \paren {a \circ c}         | lc= Product is distributive over addition }} {{axiom | m = \paren {a * b} \circ c = \paren {a \circ c} * \paren {b \circ c} }} {{axiom | n = C         | q = \forall a, b \in D         | m = a \circ b = b \circ a         | lc= Product is commutative }} {{axiom | n = U         | q = \exists 1_D \in D: \forall a \in D         | m = a \circ 1_D = a = 1_D \circ a         | lc= Identity element for product: the unity }} {{axiom | n = ZD         | q = \forall a, b \in D         | m = a \circ b = 0_D \iff a = 0 \lor b = 0         | lc= No proper zero divisors }} {{end-axiom}} These criteria are called the '''integral domain axioms'''.


Definition:Triangular Matrix/Upper Triangular Matrix

An '''upper triangular matrix''' is a matrix in which all the lower triangular elements are zero. That is, all the non-zero elements are on the main diagonal or in the upper triangle. That is, $\mathbf U$ is '''upper triangular''' {{iff}}: :$\forall a_{ij} \in \mathbf U: i > j \implies a_{ij} = 0$


Definition:Triangular Matrix/Lower Triangular Matrix

A '''lower triangular matrix''' is a matrix in which all the upper triangular elements are zero. That is, all the non-zero elements are on the main diagonal or in the lower triangle. That is, $\mathbf L$ is '''lower triangular''' {{iff}}: :$\forall a_{i j} \in \mathbf U: i < j \implies a_{i j} = 0$


Definition:Contour/Image/Complex Plane

Let $C$ be a '''contour''' in $\C$ defined by the (finite) sequence $\left\langle{C_1, \ldots, C_n}\right\rangle$ of directed smooth curves in $\C$. Let $C_i$ be parameterized by the smooth path $\gamma_i: \left[{a_i\,.\,.\,b_i}\right] \to \C$ for all $i \in \left\{ {1, \ldots, n}\right\}$. The '''image of $C$''' is defined as: :$\displaystyle \operatorname{Im} \left({C}\right) := \bigcup_{i \mathop = 1}^n \operatorname{Im} \left({\gamma_i}\right)$ where $\operatorname{Im} \left({\gamma_i}\right)$ denotes the image of $\gamma_i$. If $\operatorname{Im} \left({C}\right) \subseteq D$, where $D$ is a subset of $\C$, we say that $C$ is a '''contour in $D$'''.


Definition:Contour/Parameterization/Complex Plane

Let $C_1, \ldots, C_n$ be directed smooth curves in the complex plane $\C$. Let $C_i$ be parameterized by the smooth path $\gamma_i: \left[{a_i \,.\,.\, b_i}\right] \to \C$ for all $i \in \left\{ {1, \ldots, n}\right\}$. Let $C$ be the contour defined by the finite sequence $C_1, \ldots, C_n$. The '''parameterization of $C$''' is defined as the function $\gamma: \left[{a_1 \,.\,.\, c_n}\right] \to \C$ with: :$\gamma \restriction_{\left[{c_i \,.\,.\, c_{i + 1} }\right] } \left({t}\right) = \gamma_i \left({t}\right)$ where $\displaystyle c_i = a_1 + \sum_{j \mathop = 1}^i b_j - \sum_{j \mathop = 1}^i a_j$ for $i \in \left\{ {0, \ldots, n}\right\}$. Here, $\gamma \restriction_{\left[{c_i \,.\,.\, c_{i + 1} }\right] }$ denotes the restriction of $\gamma$ to $\left[{c_i \,.\,.\, c_{i + 1} }\right]$.


Definition:Contour/Endpoints/Complex Plane

Let $C_1, \ldots, C_n$ be directed smooth curves in $\C$. Let $C_i$ be parameterized by the smooth path $\gamma_i: \left[{a_i \,.\,.\, b_i}\right] \to \C$ for all $i \in \left\{ {1, \ldots, n}\right\}$. Let $C$ be the contour defined by the finite sequence $C_1, \ldots, C_n$. The '''start point''' of $C$ is $\gamma_1 \left({a_1}\right)$. The '''end point''' of $C$ is $\gamma_n \left({b_n}\right)$. Collectively, $\gamma_1 \left({a_1}\right)$ and $\gamma_n \left({b_n}\right)$ are referred to as the '''endpoints''' of $C$.


Definition:Contour/Simple/Complex Plane

Let $C_1, \ldots, C_n$ be directed smooth curves in the complex plane $\C$. Let $C_i$ be parameterized by the smooth path $\gamma_i: \left[{a_i \,.\,.\, b_i}\right] \to \C$ for all $i \in \left\{ {1, \ldots, n}\right\}$. Let $C$ be the contour defined by the finite sequence $C_1, \ldots, C_n$. $C$ is a '''simple contour''' {{iff}}: :$(1): \quad$ For all $i,j \in \left\{ {1, \ldots, n}\right\}, t_1 \in \left[{a_i \,.\,.\, b_i}\right), t_2 \in \left[{a_j \,.\,.\, b_j}\right)$ with $t_1 \ne t_2$, we have $\gamma_i \left({t_1}\right) \ne \gamma_j \left({t_2}\right)$. :$(2): \quad$ For all $k \in \left\{ {1, \ldots, n}\right\}, t \in \left[{a_k \,.\,.\, b_k}\right)$ where either $k \ne 1$ or $t \ne a_1$, we have $\gamma_k \left({t}\right) \ne \gamma_n \left({b_n}\right)$.


Definition:Quotient Epimorphism/Group

Let $G$ be a group. Let $N$ be a normal subgroup of $G$. Let $G / N$ be the quotient group of $G$ by $N$. The mapping $q_N: G \to G / N$ defined as: :$\forall x \in G: \map {q_N} x = x N$ is known as the '''quotient (group) epimorphism''' from $G$ to $G / N$.


Definition:Quotient Epimorphism/Ring

Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$ and whose unity is $1_R$. Let $J$ be an ideal of $R$. Let $\struct {R / J, +, \circ}$ be the quotient ring defined by $J$. The mapping $\phi: R \to R / J$ given by: : $\forall x \in R: \map \phi x = x + J$ is known as the '''quotient (ring) epimorphism from $\struct {R, +, \circ}$ (on)to $\struct {R / J, +, \circ}$'''.


Definition:Integrable Function/p-Integrable

Let $\struct {X, \Sigma, \mu}$ be a measure space. Let $f \in \MM_{\overline \R}, f: X \to \overline \R$ be a measurable function. Let $p \ge 1$ be a real number. Then $f$ is said to be '''$p$-integrable in respect to $\mu$''' {{iff}}: :$\displaystyle \int \size f^p \rd \mu < +\infty$ is $\mu$-integrable.


Definition:Derivative of Smooth Path/Complex Plane

Let $\gamma: \left[{a \,.\,.\, b}\right] \to \C$ be a '''smooth path in $\C$'''. Define the real function $x : \left[{a \,.\,.\, b}\right] \to \R$ by: :$\forall t \in \left[{a \,.\,.\, b}\right]: x \left({t}\right) = \operatorname{Re} \left({\gamma \left({t}\right)}\right)$ Define the real function $y: \left[{a \,.\,.\, b}\right] \to \R$ by: :$\forall t \in \left[{a \,.\,.\, b}\right]: y \left({t}\right) = \operatorname{Im} \left({\gamma \left({t}\right)}\right)$ where: : $\operatorname{Re} \left({\gamma \left({t}\right)}\right)$ denotes the real part of the complex number $\gamma \left({t}\right)$ : $\operatorname{Im} \left({\gamma \left({t}\right)}\right)$ denotes the imaginary part of $\gamma \left({t}\right)$. It follows from the definition of a  smooth path that both $x$ and $y$ are continuously differentiable. Let $x' \left({t}\right)$ and $y' \left({t}\right)$ denote the derivative of $x$ and $y$ {{WRT|Differentiation}} $t$. The '''derivative of $\gamma$''' is the continuous complex function $\gamma': \left[{a \,.\,.\, b}\right] \to \C$ defined by: :$\forall t \in \left[{a \,.\,.\, b}\right]: \gamma' \left({t}\right) = x' \left({t}\right) + i y' \left({t}\right)$


Definition:Ring Direct Product

{{:Definition:Ring Direct Product/Finite Case}}


Definition:Reversed Contour

Let $\R^n$ be a real cartesian space of $n$ dimensions. Let $C$ be a contour in $\R^n$. Then $C$ is defined as a concatenation of a finite sequence $C_1, \ldots, C_n$ of directed smooth curves in $\R^n$. The '''reversed contour of $C$''' is denoted $-C$ and is defined as the concatenation of the finite sequence: :$-C_n, -C_{n - 1}, \ldots, -C_1$ where $-C_i$ is the reversed directed smooth curve of $C_i$ for all $i \in \left\{ {1, \ldots, n}\right\}$.


Definition:Reversed Directed Smooth Curve

Let $\R^n$ be a real cartesian space of $n$ dimensions. Let $C$ be a directed smooth curves in $\R^n$. Let $C$ be parameterized by the smooth path $\gamma: \left[{a \,.\,.\, b}\right] \to \R^n$. The '''reversed directed smooth curve of $C$''' is denoted $-C$ and is defined as the directed smooth curve that is parameterized by: :$\rho: \left[{a \,.\,.\, b}\right] \to \R^n: \rho \left({t}\right) = \gamma \left({a + b - t}\right)$


Definition:Universal Net

Let $S$ be a non-empty set. Let $H$ be a set directed by $\preceq$. Let $f: H \to S$ be a net in $S$. Then $f$ is a '''universal net''' iff for each subset $A \subseteq S$, $f$ is eventually in $A$ or $f$ is eventually in $S \setminus A$. {{explain|"eventually"}} Category:Definitions/Set Theory Category:Definitions/Topology g6pdy53dz1lzccyidk0ptoihymj12jg


Definition:Subnet

Let $\left({E, \le}\right)$ and $\left({D,\preceq}\right)$ be directed sets. Let $A$ be a non-empty set. Let $T: E \to A$ and $S: D \to A$ be nets in $A$. Then $T$ is a '''subnet''' of $S$ iff there exists a cofinal mapping $N: E \to D$ such that $T = S \circ N$. That is, for each $m$ in $D$, there is an $n$ in $E$ such that for each $p \in E$, $p \ge n \implies N \left({p}\right) \succeq m$.


Definition:Ordered Tuple as Ordered Set/Ordered Triple

The '''ordered triple''' $\tuple {a, b, c}$ of elements $a$, $b$ and $c$ can be defined either as the ordered pair: :$\tuple {a, \tuple {b, c} }$ or as the ordered pair: :$\tuple {\tuple {a, b}, c}$ where $\tuple {a, b}$ and $\tuple {b, c}$ are themselves ordered pairs.


Definition:Ordered Tuple as Ordered Set/Ordered Quadruple

The '''ordered quadruple''' $\tuple {a, b, c, d}$ of elements $a$, $b$, $c$ and $d$ is defined either as the ordered pair: :$\tuple {a, \tuple {b, c, d} }$ or: :$\tuple {\tuple {a, b, c}, d}$ where $\tuple {a, b, c}$ and $\tuple {b, c, d}$ are themselves ordered triples.


Definition:Ordered Tuple as Ordered Set/Ordered Tuple

The '''ordered tuple''' $\tuple {a_1, a_2, \ldots, a_n}$ of elements $a_1, a_2, \ldots, a_n$ is defined as either the ordered pair: :$\tuple {a_1, \tuple {a_2, a_3, \ldots, a_n} }$ or as the ordered pair: :$\tuple {\tuple {a_1, a_2, \ldots, a_{n - 1} }, a_n}$ where $\tuple {a_2, a_3, \ldots, a_n}$ and $\tuple {a_1, a_2, \ldots, a_{n - 1} }$ are themselves ordered tuples.


Definition:Relation/Relation as Ordered Pair

Some sources define a relation between $S$ and $T$ as an ordered pair: :$\struct {S \times T, \map P {s, t} }$ where: :$S \times T$ is the Cartesian product of $S$ and $T$ :$\map P {s, t}$ is a propositional function on ordered pairs $\tuple {s, t}$ of $S \times T$. Note that this approach leaves the domain and codomain inadequately defined. This situation arises in the case that $S$ or $T$ are empty, whence it follows that $S \times T$ is empty, but $T$ or $S$ are not themselves uniquely determined.


Definition:Relation/Relation as Mapping

It is possible to define a relation as a mapping from the cartesian product $S \times T$ to the set of truth values $\set {\text {true}, \text {false} }$: :$\RR: S \times T \to \set {\text {true}, \text {false} }: \forall \tuple {s, t} \in S \times T: \map \RR {s, t} = \begin{cases} \text {true} & : \tuple {s, t} \in \RR \\ \text {false} & : \tuple {s, t} \notin \RR \end{cases}$ This is called the characteristic function of the relation. However, care needs to be taken that a mapping then cannot be defined as a special relation, as this would be circular.


Definition:Meet (Order Theory)

Let $\struct {S, \preceq}$ be an ordered set. Let $a, b \in S$, and suppose that their infimum $\inf \set {a, b}$ exists in $S$. Then $a \wedge b$, the '''meet of $a$ and $b$''', is defined as: :$a \wedge b = \inf \set {a, b}$ Expanding the definition of infimum, one sees that $c = a \wedge b$ {{iff}}: :$c \preceq a$ and $c \preceq b$ and $\forall s \in S: s \preceq a \land s \preceq b \implies s \preceq c$


Definition:Join (Order Theory)

Let $\left({S, \preceq}\right)$ be an ordered set. Let $a, b \in S$. Let their supremum $\sup \left\{{a, b}\right\}$ exist in $S$. Then the '''join of $a$ and $b$''' is defined as: :$a \vee b = \sup \left\{{a, b}\right\}$ Expanding the definition of supremum, one sees that $c = a \vee b$ {{iff}}: :$a \preceq c$ and $b \preceq c$ and $\forall s \in S: a \preceq s \land b \preceq s \implies c \preceq s$


Definition:Join Semilattice

{{:Definition:Join Semilattice/Definition 1}}


Definition:Meet Semilattice

Let $\struct {S, \preceq}$ be an ordered set. Suppose that for all $a, b \in S$: :$a \wedge b \in S$, where $a \wedge b$ is the meet of $a$ and $b$. Then the ordered structure $\struct {S, \wedge, \preceq}$ is called a '''meet semilattice'''.


Definition:Semilattice

Let $\struct {S, \circ}$ be a semigroup. Then $\struct {S, \circ}$ is called a '''semilattice''' {{iff}} $\circ$ is a commutative and idempotent operation.


Definition:Lattice

=== Definition 1 === {{:Definition:Lattice/Definition 1}} === Definition 2 === {{:Definition:Lattice/Definition 2}} === Definition 3 === {{:Definition:Lattice/Definition 3}}


Definition:Lattice/Definition 1

Let $\struct {S, \preceq}$ be an ordered set. Suppose that $S$ admits all finite non-empty suprema and finite non-empty infima. Denote with $\vee$ and $\wedge$ the join and meet operations on $S$, respectively. Then the ordered structure $\struct {S, \vee, \wedge, \preceq}$ is called a '''lattice'''.


Definition:Lattice/Definition 2

Let $\struct {S, \vee, \wedge, \preceq}$ be an ordered structure. Then $\struct {S, \vee, \wedge, \preceq}$ is called a '''lattice''' {{iff}}: :$\struct {S, \vee, \preceq}$ is a join semilattice :$\struct {S, \wedge, \preceq}$ is a meet semilattice


Definition:Lattice/Definition 3

Let $\struct {S, \vee}$ and $\struct {S, \wedge}$ be semilattices on a set $S$. Suppose that $\vee$ and $\wedge$ satisfy the absorption laws, that is, for all $a, b \in S$: :$a \vee \paren {a \wedge b} = a$ :$a \wedge \paren {a \vee b} = a$ Let $\preceq$ be the ordering on $S$ defined by: :$\forall a, b \in S: a \preceq b$ {{iff}} $a \vee b = b$ as on Semilattice Induces Ordering. Then the ordered structure $\struct {S, \vee, \wedge, \preceq}$ is called a '''lattice'''.


Definition:Bounded Lattice

=== Definition 1 === {{:Definition:Bounded Lattice/Definition 1}} === Definition 2 === {{:Definition:Bounded Lattice/Definition 2}} === Definition 3 === {{:Definition:Bounded Lattice/Definition 3}}


Definition:Bounded Lattice/Definition 1

Let $\left({S, \preceq}\right)$ be an ordered set. Let $S$ admit all finite suprema and finite infima. Let $\vee$ and $\wedge$ be the join and meet operations on $S$, respectively. Then the ordered structure $\left({S, \vee, \wedge, \preceq}\right)$ is a '''bounded lattice'''.


Definition:Bounded Lattice/Definition 2

Let $\left({S, \vee, \wedge, \preceq}\right)$ be a lattice. Let $\vee$ and $\wedge$ have identity elements $\bot$ and $\top$ respectively. Then $\left({S, \vee, \wedge, \preceq}\right)$ is a '''bounded lattice'''.


Definition:Lowest Common Multiple/Integers

For all $a, b \in \Z: a b \ne 0$, there exists a smallest $m \in \Z: m > 0$ such that $a \divides m$ and $b \divides m$. This $m$ is called the '''lowest common multiple of $a$ and $b$''', and denoted $\lcm \set {a, b}$.


Definition:Lowest Common Multiple/Integral Domain

Let $D$ be an integral domain and let $a, b \in A$ be nonzero. $l$ is the '''lowest common multiple''' of $a$ and $b$ {{iff}}: :$(1): \quad$ both $a$ and $b$ divide $l$ :$(2): \quad$ if $m$ is another element such that $a$ and $b$ divide $m$, then $l$ divides $m$.


Definition:Operation/Binary Operation/Product

Let $z = x \circ y$. Then $z$ is called the '''product''' of $x$ and $y$. This is an extension of the normal definition of product that is encountered in conventional arithmetic.


Definition:Diophantine Equation/Linear Diophantine Equation

A '''linear Diophantine equation''' is a Diophantine equation in which all the arguments appear to no higher than the first degree. For example: :$ax + by + c = 0$ :$a_1 x_1 + a_2 x_2 + \cdots + a_n x_n = b$


Definition:Absorption Law

Let $\struct {S, \circ, *}$ be an algebraic structure. Let both $\circ$ and $*$ be commutative. Then '''$\circ$ absorbs $*$''' {{iff}}: :$\forall a, b \in S: a \circ \paren {a * b} = a$ This equality is called the '''absorption law of $\circ$ for $*$'''. Category:Definitions/Abstract Algebra 5obgfea09xjreoixndqhmz2wsjj44jp


Definition:Supremum of Set/Finite Supremum

If $T$ is finite, $\sup T$ is called a '''finite supremum'''.


Definition:Infimum of Set/Finite Infimum

If $T$ is finite, $\inf T$ is called a '''finite infimum'''.


Definition:Empty Supremum

Let $\struct {S, \preceq}$ be an ordered set. Then the '''empty supremum''' is the supremum $\sup \O$.


Definition:Empty Infimum

Let $\struct {S, \preceq}$ be an ordered set. Then the '''empty infimum''' is the infimum $\inf \O$. By Infimum of Empty Set is Greatest Element, it exists {{iff}} $\struct {S, \preceq}$ has a greatest element.


Definition:Dual Ordering

Let $\struct {S, \preceq}$ be an ordered set. Let $\succeq$ be the inverse relation to $\preceq$. That is, for all $a, b \in S$: :$a \succeq b$ {{iff}} $b \preceq a$ Then $\succeq$ is called the '''dual ordering of $\preceq$'''. By Dual Ordering is Ordering, it is indeed an ordering. === Dual Ordered Set === {{:Definition:Dual Ordering/Dual Ordered Set}} === Notation for Inverse Ordering === {{:Definition:Inverse Relation/Ordering Notation}} === Notation for Inverse Strict Ordering === {{:Definition:Inverse Relation/Strict Ordering Notation}}


Definition:Dual Ordering/Dual Ordered Set

The ordered set $\left({S, \succeq}\right)$ is called the '''dual ordered set''' (or just '''dual''') '''of $\left({S, \preceq}\right)$'''.


Definition:Conditional/Necessary Condition

Let $p \implies q$ be a conditional statement. Then $q$ is a '''necessary condition''' for $p$. That is, if $p \implies q$, then it is ''necessary'' that $q$ be true for $p$ to be true. This is because unless $q$ is true, $p$ can ''not'' be true.


Definition:Conditional/Sufficient Condition

Let $p \implies q$ be a conditional statement. Then $p$ is a '''sufficient condition''' for $q$. That is, if $p \implies q$, then for $q$ to be true, it is ''sufficient'' to know that $p$ is true. This is because of the fact that if you know that $p$ is true, you know enough to know also that $q$ is true.


Definition:Dual Statement (Order Theory)

Let $\left({S, \preceq}\right)$ be an ordered set. Let $\succeq$ be the dual ordering to $\preceq$. Let $\Sigma$ be any statement pertaining to $\left({S, \preceq}\right)$ (be it in natural language or a formal language). The '''dual statement of $\Sigma$''', denoted $\Sigma^*$, is the statement obtained from replacing every reference to $\preceq$ in $\Sigma$ with a reference to its dual $\succeq$. This '''dual statement''' may then be turned into a statement about $\preceq$ again by applying the equivalences on Dual Pairs (Order Theory).


Definition:Relation Conversely Compatible with Operation

Let $\left({S, \circ}\right)$ be a closed algebraic structure. Let $\mathcal R$ be a relation in $S$. Then $\mathcal R$ is '''conversely compatible with $\circ$''' {{iff}}: :$\forall x, y, z \in S: \left({x \circ z}\right) \mathrel {\mathcal R} \left({y \circ z}\right) \implies x \mathrel {\mathcal R} y$ :$\forall x, y, z \in S: \left({z \circ x}\right) \mathrel {\mathcal R} \left({z \circ y}\right) \implies x \mathrel {\mathcal R} y$ Category:Definitions/Compatible Relations kl5soecrnof9n332oxdlch8n9yz685r


Definition:Relation Strongly Compatible with Operation

Let $\left({S, \circ}\right)$ be a closed algebraic structure. Let $\mathcal R$ be a relation in $S$. Then $\mathcal R$ is '''strongly compatible with $\circ$''' {{iff}}: :$\forall x, y, z \in S: x \mathrel {\mathcal R} y \iff \left({x \circ z}\right) \mathrel {\mathcal R} \left({y \circ z}\right)$ :$\forall x, y, z \in S: x \mathrel {\mathcal R} y \iff \left({z \circ x}\right) \mathrel {\mathcal R} \left({z \circ y}\right)$. That is, {{iff}} $\mathcal R$ is compatible with $\circ$ and conversely compatible with $\circ$.


Definition:Union of Relations

Let $S$ and $T$ be sets. Let $\mathcal R_1$ and $\mathcal R_2$ be relations on $S \times T$. The '''union of $\mathcal R_1$ and $\mathcal R_2$''' is the relation $\mathcal Q$ defined by: :$\mathcal Q := \mathcal R_1 \cup \mathcal R_2$ where $\cup$ denotes set union. Explicitly, for $s \in S$ and $t \in T$, we have: :$s \mathrel{\mathcal Q} t$ {{iff}} $s \mathrel{\mathcal R_1} t$ or $s \mathrel{\mathcal R_2} t$ === General Definition === {{:Definition:Union of Relations/General Definition}}


Definition:Union of Relations/General Definition

Let $\mathscr R$ be a collection of relations on $S \times T$. The '''union of $\mathscr R$''' is the relation $\mathcal R$ defined by: :$\mathcal R = \displaystyle \bigcup \mathscr R$ where $\bigcup$ denotes set union. Explicitly, for $s \in S$ and $t \in T$: :$s \mathrel{\mathcal R} t$ {{iff}} for some $\mathcal Q \in \mathscr R$, $s \mathrel{\mathcal Q} t$


Definition:Intersection of Relations

Let $S$ and $T$ be sets. Let $\mathcal R_1$ and $\mathcal R_2$ be relations on $S \times T$. The '''intersection of $\mathcal R_1$ and $\mathcal R_2$''' is the relation $\mathcal Q$ defined by: :$\mathcal Q := \mathcal R_1 \cap \mathcal R_2$ where $\cap$ denotes set intersection. Explicitly, for $s \in S$ and $t \in T$, we have: :$s \mathrel{\mathcal Q} t$ iff both $s \mathrel{\mathcal R_1} t$ and $s \mathrel{\mathcal R_2} t$ === General Definition === {{:Definition:Intersection of Relations/General Definition}}


Definition:Intersection of Relations/General Definition

Let $\mathscr R$ be a collection of relations on $S \times T$. The '''intersection of $\mathscr R$''' is the relation $\mathcal R$ defined by: :$\mathcal R = \displaystyle \bigcap \mathscr R$ where $\bigcap$ denotes set intersection. Explicitly, for $s \in S$ and $t \in T$: :$s \mathrel{\mathcal R} t$ {{iff}} for all $\mathcal Q \in \mathscr R$, $s \mathrel{\mathcal Q} t$


Definition:Complement (Lattice Theory)

Let $\left({S, \vee, \wedge, \preceq}\right)$ be a bounded lattice. Denote by $\bot$ and $\top$ the bottom and top of $S$, respectively. Let $a \in S$. Then $b \in S$ is called a '''complement of $a$''' {{iff}}: :$b \vee a = \top$ :$b \wedge a = \bot$ If $a$ has a unique '''complement''', it is denoted by $\neg a$. === Complemented Lattice === {{:Definition:Complemented Lattice}}


Definition:Finitely Generated Algebraic Structure

Let $\struct {A, \circ}$ be an algebraic structure. Let $\struct {A, \circ}$ have a generator which is finite. Then $\struct {A, \circ}$ is '''finitely generated'''.


Definition:Topological Group/Definition 2

Let the mapping $\psi: \left({G, \tau}\right) \times \left({G, \tau}\right) \to \left({G, \tau}\right)$ be defined as: :$\psi \left({x, y}\right) = x \odot y^{-1}$ $\left({G, \odot, \tau}\right)$ is a '''topological group''' {{iff}}: : $\psi$ is a continuous mapping where $\left({G, \tau}\right) \times \left({G, \tau}\right)$ is considered as $G \times G$ with the product topology.


Definition:Topological Group/Definition 1

$\left({G, \odot, \tau}\right)$ is a '''topological group''' {{iff}}: :$(1): \quad \odot: \left({G, \tau}\right) \times \left({G, \tau}\right) \to \left({G, \tau}\right)$ is a continuous mapping :$(2): \quad \phi: \left({G, \tau}\right) \to \left({G, \tau}\right)$ such that $\forall x \in G: \phi \left({x}\right) = x^{-1}$ is also a continuous mapping where $\left({G, \tau}\right) \times \left({G, \tau}\right)$ is considered as $G \times G$ with the product topology.


Definition:Topological Ring

Let $\struct{R, +, \circ}$ be a ring. Let $\tau$ be a topology over $R$. Then $\struct {R, +, \circ, \tau}$ is a '''topological ring''' {{iff}}: :$(1): \quad \struct {R, +, \tau}$ is a topological group :$(2): \quad \struct {R, \circ,\tau}$ is a topological semigroup.


Definition:Topological Field

Let $\struct {F, +, \circ}$ be a field with zero $0_F$. Let $\tau$ be a topology on $F$. Let $\struct {F, +, \circ, \tau}$ be a topological division ring. Then $\struct {F, +, \circ, \tau}$ is a '''topological field'''. That is, a topological field is a commutative topological division ring.


Definition:Field Zero

Let $\struct {F, +, \times}$ be a field. The identity for field addition is called the '''field zero''' (of $\struct {F, +, \times}$). It is denoted $0_F$ (or just $0$ if there is no danger of ambiguity).


Definition:Ray (Order Theory)

Let $\struct {S, \preccurlyeq}$ be a totally ordered set. Let $\prec$ be the reflexive reduction of $\preccurlyeq$. Let $a \in S$ be any point in $S$. === Open Ray === {{:Definition:Ray (Order Theory)/Open}} === Closed Ray === {{:Definition:Ray (Order Theory)/Closed}} === Upward-Pointing Ray === {{:Definition:Ray (Order Theory)/Upward-Pointing}} === Downward-Pointing Ray === {{:Definition:Ray (Order Theory)/Downward-Pointing}}


Definition:Complemented Lattice

Suppose that every $a \in S$ admits a complement. Then $\left({S, \vee, \wedge, \preceq}\right)$ is called a '''complemented lattice'''.


Definition:Top (Lattice Theory)

Let $\left({S, \vee, \wedge, \preceq}\right)$ be a lattice. === Definition 1 === {{:Definition:Top (Lattice Theory)/Definition 1}} === Definition 2 === {{:Definition:Top (Lattice Theory)/Definition 2}}


Definition:Top (Lattice Theory)/Definition 1

Let $S$ admit a greatest element $\top$. Then $\top$ is called the '''top''' of $S$.


Definition:Top (Lattice Theory)/Definition 2

Let $\wedge$ have an identity element $\top$. Then $\top$ is called the '''top''' of $S$.


Definition:Bottom (Lattice Theory)

Let $\left({S, \vee, \wedge, \preceq}\right)$ be a lattice. === Definition 1 === {{:Definition:Bottom (Lattice Theory)/Definition 1}} === Definition 2 === {{:Definition:Bottom (Lattice Theory)/Definition 2}}


Definition:Bottom (Lattice Theory)/Definition 1

Let $S$ admit a smallest element $\bot$. Then $\bot$ is called the '''bottom''' of $S$.


Definition:Bottom (Lattice Theory)/Definition 2

Let $\vee$ have an identity element $\bot$. Then $\bot$ is called the '''bottom''' of $S$.


Definition:Distance-Preserving Mapping

Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces, pseudometric spaces, or quasimetric spaces. Let $\phi: M_1 \to M_2$ be a mapping such that: : $\forall a, b \in M_1: d_1 \left({a, b}\right) = d_2 \left({\phi \left({a}\right), \phi \left({b}\right)}\right)$ Then $\phi$ is called a '''distance-preserving mapping'''.


Definition:Connected Domain (Complex Analysis)

Let $D \subseteq \C$ be a subset of the set of complex numbers. Then $D$ is a '''connected domain''' {{iff}} $D$ is open and connected.


Let $D \subseteq \C$ be a connected domain. Then $D$ is called a '''simply connected domain''' {{iff}} $D$ is simply connected. === Simply Connectedness Requirement ===  {{:Definition:Connected Domain (Complex Analysis)/Simply Connected Domain/Requirement}}


Definition:Topologist's Sine Curve

Let $G = \set {\tuple {x, \sin \dfrac 1 x} : 0 < x \le 1}$, considered as a subset of the real number plane with the usual (Euclidean) topology $\struct {\R^2, \tau_d}$. Then the set $G \cup \set {\tuple {0, 0} }$ is called the '''Topologist's Sine Curve'''.


Definition:Staircase Contour

Let $C$ be a contour that is a concatenation of the directed smooth curves $C_1, \ldots, C_n$. For all $i \in \set {1, \ldots, n}$, let it be possible for $C_i$ be parameterized by a smooth path $\gamma_i: \closedint 0 1 \to \C$ such that either: :$\map {\gamma_i} t = z_i + t r_i$ or :$\map {\gamma_i} t = z_i + i t r_i$ for some $z_i \in \C, r_i \in \R$ for all $t \in \closedint 0 1$. Then $C$ is called a '''staircase contour'''.


Definition:Differentiable Mapping/Complex Function/Point

Let $D\subset \C$ be an open set. Let $f : D \to \C$ be a complex function. Let $z_0 \in D$ be a point in $D$. Then $f$ is '''complex-differentiable''' at $z_0$ {{iff}} the limit: : $\displaystyle \lim_{h \to 0} \frac {f \left({z_0+h}\right) - f \left({z_0}\right)} h$ exists as a finite number.


Definition:Differentiable Mapping/Complex Function/Region

Let $D \subseteq \C$ be an open set. Let $f: D \to \C$ be a complex function. Then $f$ is '''complex-differentiable in $D$''' {{iff}} $f$ is complex-differentiable at every point in $D$.


Definition:Inequality

An '''inequality''' is a mathematical statement that two expressions relate in one of several conventional ways: : $a < b$ : $a \le b$ : $a > b$ : $a \ge b$ A statement of the form:  :$a \ne b$ may or may not be considered an '''inequality'''.


Definition:Convex Set (Vector Space)/Line Segment

The set: :$\set {t x + \paren {1 - t} y: t \in \closedint 0 1}$  is called the '''(straight) line segment joining $x$ and $y$'''. A convex set can thus be described as a set containing all '''straight line segments''' between its elements.


Definition:Boolean Lattice

=== Definition 1 === {{:Definition:Boolean Lattice/Definition 1}} === Definition 2 === {{:Definition:Boolean Lattice/Definition 2}} === Definition 3 === {{:Definition:Boolean Lattice/Definition 3}}


Definition:Boolean Lattice/Definition 1

A '''Boolean lattice''' is a complemented distributive lattice.


Definition:Boolean Lattice/Definition 2

An ordered structure $\left({S, \vee, \wedge, \preceq}\right)$ is a '''Boolean lattice''' {{iff}}: $(1): \quad \left({S, \vee, \wedge}\right)$ is a Boolean algebra $(2): \quad$ For all $a, b \in S$: $a \wedge b \preceq a \vee b$


Definition:Metalanguage/Formal Systems

In the context of formal systems, a '''metalanguage''' is a formal language used to specify another formal language.


Definition:Metalanguage/Object Language

The '''object language''' of a metalanguage is the language described by that metalanguage.


Definition:Metalanguage/Metasyntax

The syntax of a metalanguage is called a '''metasyntax''' of the object language of that metalanguage.


Definition:Metalanguage/Metasymbol

A '''metasymbol''' is a symbol used in a metalanguage to represent an arbitrary collation in the object language.


Definition:Theorem/Logic

A '''theorem''' in logic is a statement which can be shown to be the conclusion of a logical argument which depends on ''no'' premises except axioms. A sequent which denotes a theorem $\phi$ is written $\vdash \phi$, indicating that there are no premises. In this context, $\vdash$ is read as: :'''It is a theorem that ...'''


Definition:Theorem/Formal System

Let $\mathcal L$ be a formal language. Let $\mathscr P$ be a proof system for $\mathcal L$. A '''theorem of $\mathscr P$''' is a well-formed formula of $\mathcal L$ which can be deduced from the axioms and axiom schemata of $\mathscr P$ by means of its rules of inference. That a WFF $\phi$ is a '''theorem''' of $\mathscr P$ may be denoted as: :$\vdash_{\mathscr P} \phi$


Definition:Theorem/Mathematics

The term '''theorem''' is used throughout the whole of mathematics to mean a statement which has been proved to be true from whichever axioms relevant to that particular branch. Note that statements which are taken as axioms in one branch of mathematics may be theorems in others.


Definition:Distance/Sets/Real Numbers

Let $S, T$ be a subsets of the set of real numbers $\R$. Let $x \in \R$ be a real number. The '''distance between $x$ and $S$''' is defined and annotated $\displaystyle \map d {x, S} = \map {\inf_{y \mathop \in S} } {\map d {x, y} }$, where $\map d {x, y}$ is the distance between $x$ and $y$. The '''distance between $S$ and $T$''' is defined and annotated $\displaystyle \map d {S, T} = \map {\inf_{\substack {x \mathop \in S \\ y \mathop \in T} } } {\map d {x, y} }$.


Definition:Distance/Sets/Metric Spaces

Let $M = \left({A, d}\right)$ be a metric space. Let $x \in A$. Let $S, T$ be subsets of $A$. The '''distance between $x$ and $S$''' is defined and annotated $\displaystyle d \left({x, S}\right) = \inf_{y \mathop \in S} \left({d \left({x, y}\right)}\right)$. The '''distance between $S$ and $T$''' is defined and annotated $\displaystyle d \left({S, T}\right) = \inf_{\substack{x \mathop \in S \\ y \mathop \in T}} \left({d \left({x, y}\right)}\right)$.


Definition:Bounded Lattice/Definition 3

Let $\left({S, \wedge, \vee, \preceq}\right)$ be a lattice. Let $S$ be bounded in $\left({S,\preceq}\right)$. Then $\left({S, \wedge, \vee, \preceq}\right)$ is a '''bounded lattice'''.


Definition:Uniquely Complemented Lattice

Let $\left({S,\wedge,\vee,\preceq}\right)$ be a complemented lattice. If each element of $S$ has only one complement, then $\left({S,\wedge,\vee,\preceq}\right)$ is a '''uniquely complemented lattice'''. That is, a uniquely complemented lattice is a lattice in which each element has exactly one complement.


Definition:Boolean Algebra/Definition 3

A '''Boolean algebra''' is an algebraic structure $\struct {S, \vee, \wedge}$ such that: {{begin-axiom}} {{axiom | n = \text {BA} 0         | lc=         | t = $S$ is closed under both $\vee$ and $\wedge$ }} {{axiom | n = \text {BA} 1         | lc=         | t = Both $\vee$ and $\wedge$ are commutative }} {{axiom | n = \text {BA} 2         | lc=         | t = Both $\vee$ and $\wedge$ distribute over the other }} {{axiom | n = \text {BA} 3         | lc=         | t = Both $\vee$ and $\wedge$ have identities $\bot$ and $\top$ respectively }} {{axiom | n = \text {BA} 4         | lc=         | t = $\forall a \in S: \exists \neg a \in S: a \vee \neg a = \top, a \wedge \neg a = \bot$ }} {{end-axiom}}


Definition:Boolean Algebra/Definition 1

{{:Definition:Boolean Algebra/Axioms/Definition 1}}


Definition:Degenerate Boolean Algebra

Let $\struct {S, \vee, \wedge, \neg}$ be a Boolean algebra. Then $\struct {S, \vee, \wedge, \neg}$ is said to be '''degenerate''' {{iff}} $S$ is a singleton.


Definition:Exponential Function/Real/Inverse of Natural Logarithm

Consider the natural logarithm $\ln x$, which is defined on the open interval $\openint 0 {+\infty}$. From Logarithm is Strictly Increasing: :$\ln x$ is strictly increasing. From Inverse of Strictly Monotone Function: :the inverse of $\ln x$ always exists. The inverse of the natural logarithm function is called the '''exponential function''', which is denoted as $\exp$. Thus for $x \in \R$, we have: :$y = \exp x \iff x = \ln y$


Definition:Exponential Function/Real/Extension of Rational Exponential

Let $e$ denote Euler's number. Let $f: \Q \to \R$ denote the real-valued function defined as: :$f \left({ x }\right) = e^x$ That is, let $f \left({ x }\right)$ denote $e$ to the power of $x$, for rational $x$. Then $\exp : \R \to \R$ is defined to be the unique continuous extension of $f$ to $\R$. $\exp \left({ x }\right)$ is called the '''exponential of $x$'''.


Definition:Exponential Function/Real/Limit of Sequence

The '''exponential function''' can be defined as the following limit of a sequence: :$\exp x := \displaystyle \lim_{n \mathop \to +\infty} \paren {1 + \frac x n}^n$


Definition:Exponential Function/Real/Sum of Series

The '''exponential function''' can be defined as a power series: :$\exp x := \displaystyle \sum_{n \mathop = 0}^\infty \frac {x^n} {n!}$


Definition:Exponential Function/Real/Differential Equation

The '''exponential function''' can be defined as the unique solution $y = \map f x$ to the first order ODE: :$\dfrac {\d y} {\d x} = y$ satisfying the initial condition $\map f 0 = 1$.


Definition:Logical Not/Truth Table

The characteristic truth table of the negation operator $\neg p$ is as follows: :$\begin{array}{|c||c|} \hline p & \neg p \\ \hline \F & \T \\ \T & \F \\ \hline \end{array}$


Definition:Truth Value/Aristotelian Logic

In Aristotelian logic, a statement can be either true or false, and there is no undefined, in-between value. Whether it is true or false is called its '''truth value'''. Note that a statement's '''truth value''' may change depending on circumstances. Thus, the statement: :''It is currently raining on the grass outside my window'' has the truth value false, whereas it had the truth value true last week. The statement: :''I am listening to Shostakovich's 4th symphony'' is currently true, but that will last only for the next twenty minutes or so as I type. The '''truth values''' true and false are usually represented in one of two ways: :$\T$ for true and $\F$ for false; :$1$ for true and $0$ for false. There are advantages for both notations. In particular, the second lends itself to extending the discipline of logic into that of probability theory.


Definition:Conjunction/Truth Table

The characteristic truth table of the logical conjunction operator $p \land q$ is as follows: :$\begin{array}{|cc||c|} \hline p & q & p \land q \\ \hline \F & \F & \F \\ \F & \T & \F \\ \T & \F & \F \\ \T & \T & \T \\ \hline \end{array}$


Definition:Compactification

Let $\left({X, \tau_1}\right)$ be a topological space. Let $\left({Y, \tau_2}\right)$ be a compact space. Let $f: X \to Y$ be a topological embedding. Let $f \left({X}\right)$ be everywhere dense in $Y$. Then either $f$ or $\left({Y, \tau_2}\right)$ may be called a '''compactification''' of $\left({X, \tau_1}\right)$.


Definition:Disjunction/Truth Table

The characteristic truth table of the logical disjunction operator $p \lor q$ is as follows: :$\begin{array}{|cc||c|} \hline p & q & p \lor q \\ \hline \F & \F & \F \\ \F & \T & \T \\ \T & \F & \T \\ \T & \T & \T \\ \hline \end{array}$


Definition:Biconditional/Truth Table

The characteristic truth table of the biconditional operator $p \iff q$ is as follows: :$\begin{array}{|cc||c|} \hline p & q & p \iff q \\ \hline \F & \F & \T \\ \F & \T & \F \\ \T & \F & \F \\ \T & \T & \T \\ \hline \end{array}$


Definition:Exclusive Or/Truth Table

The characteristic truth table of the exclusive or operator $p \oplus q$ is as follows: :$\begin{array}{|cc||c|} \hline p & q & p \oplus q \\ \hline \F & \F & \F \\ \F & \T & \T \\ \T & \F & \T \\ \T & \T & \F \\ \hline \end{array}$


Definition:Conditional/Truth Table

The characteristic truth table of the conditional (implication) operator $p \implies q$ is as follows: :$\begin{array}{|cc||c|} \hline p & q & p \implies q \\ \hline \F & \F & \T \\ \F & \T & \T \\ \T & \F & \F \\ \T & \T & \T \\ \hline \end{array}$


Definition:Logical NAND/Truth Table

The characteristic truth table of the logical NAND operator $p \uparrow q$ is as follows: :$\begin{array}{|cc||c|} \hline p & q & p \uparrow q \\ \hline F & F & T \\ F & T & T \\ T & F & T \\ T & T & F \\ \hline \end{array}$


Definition:Logical NOR/Truth Table

The characteristic truth table of the logical NOR operator $p \downarrow q$ is as follows: :$\begin{array}{|cc||c|} \hline p & q & p \downarrow q \\ \hline F & F & T \\ F & T & F \\ T & F & F \\ T & T & F \\ \hline \end{array}$