Definition:Composable Morphisms

Let $\mathbf C$ be a metacategory. Let $f, g \in \mathbf C_1$ be morphisms of $\mathbf C$. Then $f$ is said to be '''composable with $g$''' iff: :$\operatorname{cod} f = \operatorname{dom} g$ that is, iff the codomain of $f$ is the domain of $g$. When the order of composition is to be made more explicit, one says that '''$\left({g, f}\right)$ is a composable pair'''. The collection of all such '''composable pairs''' in $\mathbf C$ is denoted $\mathbf C_2$.


Definition:Category with Products/Binary

Let $\mathbf C$ be a metacategory. Then $\mathbf C$ is said to '''have binary products''' or to be a '''(meta)category with binary products''' iff: :For all objects $C, D \in \mathbf C_0$, there is a binary product $C \times D$ for $C$ and $D$.


Definition:Cofinal Relation on Ordinals

Let $x$ and $y$ be ordinals. Then $y$ is said to be '''cofinal''' with respect to $x$ {{iff}} there exists a mapping $f: y \to x$ such that: :$(1): \quad y \le x$ :$(2): \quad f$ is strictly increasing. :$(3): \quad$ For all $a \in x$, there is some $b \in y$ such that $f \left({b}\right) \ge a$.


Definition:Regular Cardinal

Let $\kappa$ be an infinite cardinal. Then $\kappa$ is a '''regular cardinal''' {{iff}}: :$\operatorname{cf} \left({\kappa}\right) = \kappa$ That is, {{iff}} the cofinality of $\kappa$ is equal to itself.


Definition:Singular Cardinal

Let $\kappa$ be an infinite cardinal. Then $\kappa$ is a '''singular cardinal''' {{iff}} $\operatorname{cf} \left({\kappa}\right) < \kappa$. That is, the cofinality of $\kappa$ is less than itself.


Definition:Weakly Inaccessible Cardinal

An infinite cardinal $\aleph_\kappa$ is '''weakly inaccessible''' {{iff}}: :$(1): \quad \kappa$ is a limit ordinal. :$(2): \quad \aleph_\kappa$ is a regular cardinal.


Definition:Strongly Inaccessible Cardinal

An infinite cardinal $\aleph_\kappa$ is called a '''strongly inaccessible cardinal''' {{iff}}: :$(1): \quad \aleph_\kappa$ is a weakly inaccessible cardinal :$(2): \quad \forall x \in \kappa: \left|{\mathcal P \left({x}\right)}\right| \in \kappa$


Definition:Greatest Common Divisor/Integers/General Definition

Let $S = \set {a_1, a_2, \ldots, a_n} \subseteq \Z$ such that $\exists x \in S: x \ne 0$ (that is, at least one element of $S$ is non-zero). Then the '''greatest common divisor''' of $S$: :$\gcd \paren S = \gcd \set {a_1, a_2, \ldots, a_n}$ is defined as the largest $d \in \Z_{>0}$ such that: :$\forall x \in S: d \divides x$


Definition:Inverse Morphism

A morphism $g: Y \to X$ is said to be an '''inverse (morphism)''' for $f$ {{iff}}: :$g \circ f = I_X$ :$f \circ g = I_Y$ where $I_X$ denotes the identity morphism on $X$.


Definition:Population

The '''population''' of a statistical study is everything in the universe of discourse that is relevant to the study. It includes all objects, measures, and observations under discussion. === Finite Population === {{:Definition:Population/Finite}} === Infinite Population === {{:Definition:Population/Infinite}}


Definition:Sample

A '''sample''' is a non-empty, finite, proper subset of a population.


Definition:Individual

An '''individual''' is an object in a population.


Definition:Object (Category Theory)

Let $\mathbf C$ be a metacategory. An '''object''' of $\mathbf C$ is an object which is considered to be atomic from a category theoretic perspective. It is a conceptual device introduced mainly to make the discussion of morphisms more convenient. '''Objects''' in a general metacategory are usually denoted with capital letters like $A,B,C,X,Y,Z$. The collection of '''objects''' of $\mathbf C$ is denoted $\mathbf C_0$. That '''objects''' don't play an important role in category theory is apparent from the fact that the notion of a metacategory can be described while avoiding to mention '''objects''' altogether. {{wtd|That is a big claim which needs a page to back it up; this is demonstrated eg. in 'Categories for the Working Mathematician' by MacLane}} Nonetheless the notion of '''object''' is one of the two basic concepts of metacategories and as such of category theory.


Definition:Maximal Subgroup

Let $G$ be a group. Let $M \le G$ be a proper subgroup of $G$. Then $M$ is a '''maximal subgroup of $G$''' {{iff}}: :For every subgroup $H$ of $G$, $M \subseteq H \subseteq G$ means $M = H$ or $H = G$. That is, if there is no subgroup of $G$, except $M$ and $G$ itself, which contains $M$.


Definition:Join of Subgroups/General Definition

Let $H_1, H_2, \ldots, H_n$ be subgroups of $G$. Then the '''join''' of $H_1, H_2, \ldots, H_n$ is defined as: :$\displaystyle \bigvee_{k \mathop = 1}^n H_k := \left \langle {\bigcup_{k \mathop = 1}^n H_k}\right \rangle$ or: :$\displaystyle \bigvee_{k \mathop = 1}^n H_k := \bigcap \left\{{T: T \text { is a subgroup of } G: \bigcup_{k \mathop = 1}^n H_k \subseteq T}\right\}$


Definition:Simple Random Sample

Let $P$ be a population. Let $S \subsetneq P$ be a sample. Then $S$ is a '''simple random sample''' (of size $n$) iff it fulfils the following criteria: * The process used to select the individuals of $S$ from $P$ was random; * Every individual in $P$ had an equal chance of being selected to be in $S$; * Every $n$-combination of $P$ had an equal chance of being constructed as a potential $S$.


Definition:Stratified Random Sample

Let $P$ be a population. Let $S \subsetneq P$ be a sample. Then $S$ is a '''stratified random sample''' {{iff}}: :there exists a finite expansion $P = P_1 \mid P_2 \mid \cdots \mid P_k$ of $P$ such that $S$ is the union of $k$ simple random samples, one taken from each $P_i$.


Definition:Convenience Sample

Let $P$ be a population in the physical universe. Let $S \subsetneq P$ be a sample. Suppose that data about some individuals in $P$ are more easily accessible than data about others. Then $S$ is a '''convenience sample''' {{iff}} the primary criterion for the construction of $S$ is the higher accessibility of data of certain individuals in $P$ over others. That is, the objects in $S$ are the ones conveniently at hand.


Definition:Unambiguous

A statement is '''unambiguous''' if it is not ambiguous.


Definition:Stratified Random Sample/Strata

The components of $P$ are called '''strata''', singular '''stratum'''.


Definition:Dual Statement (Category Theory)

=== Morphisms-Only Category Theory === Let $\Sigma$ be a statement in the language of category theory. The '''dual statement''' $\Sigma^*$ of $\Sigma$ is the statement obtained from substituting: {{begin-eqn}} {{eqn|l = R_\circ \left({y, x, z}\right)      |o = \) for \(      |r = R_\circ \left({x, y, z}\right) }} {{eqn|l = \operatorname{dom}      |o = \) for \(      |r = \operatorname{cod} }} {{eqn|l = \operatorname{cod}      |o = \) for \(      |r = \operatorname{dom} }} {{end-eqn}} === Object Category Theory === In the more convenient description of metacategories by using objects, the '''dual statement''' $\Sigma^*$ of $\Sigma$ then becomes the statement obtained from substituting: {{begin-eqn}} {{eqn|l = f \circ g      |o = \) for \(      |r = g \circ f }} {{eqn|l = \operatorname{cod}      |o = \) for \(      |r = \operatorname{dom} }} {{eqn|l = \operatorname{dom}      |o = \) for \(      |r = \operatorname{cod} }} {{end-eqn}} {{finish|a page proving that these indeed are the resulting substitutions}} === Example === For example, if $\Sigma$ is the statement: :$\exists g: g \circ f = \operatorname{id}_{\operatorname{dom} f}$ describing that $f$ is a split mono, then $\Sigma^*$ becomes: :$\exists g: f \circ g = \operatorname{id}_{\operatorname{cod} f}$ which precisely expresses $f$ to be a split epi. For a set $\mathcal E$ of statements, write: :$\mathcal E^* := \left\{{\Sigma^*: \Sigma \in \mathcal E}\right\}$  for the set comprising of the dual statement of those in $\mathcal E$.


Definition:Cartesian Product/Factors

Let $S \times T$ be the cartesian product of $S$ and $T$. Then the sets $S$ and $T$ are called the '''factors''' of $S \times T$.


Definition:Metamodel

Let $\mathcal L$ be a formal system with deductive apparatus $\mathcal D$. A '''metamodel''' $\mathcal M$ for $\mathcal L$ is an interpretation for $\mathcal L$. Thus, it assigns meanings to all of the undefined terms of $\mathcal L$. Further, it specifies a means for deciding if a sentence of $\mathcal L$ is to be considered true in $\mathcal M$. It is then required that all the sentences derivable by the deductive apparatus $\mathcal D$ are true in $\mathcal M$. From the above specification, it follows that a '''metamodel''' $\mathcal M$ is a conceptual device, in that it does not require to be founded on rigorous mathematical foundations. Thus, $\mathcal M$ need not be a set or some other rigid notion, just something that one can contemplate, denote and convey to others. For example, even without rigid mathematical foundations it is clear to all sane people that there is a concept "set" that one can reason about. One could then try to determine if the collection of these "sets" is a '''metamodel''' for the language of set theory. As another example there is the notion of metacategory, which can be described as being any '''metamodel''' for the language of category theory.


Definition:QED

The initials of '''Quod Erat Demonstrandum''', which is {{WP|Latin|Latin}} for '''which was to be demonstrated'''. These initials were traditionally added to the end of a proof, after the last line which is supposed to contain the statement that was to be proved.


Definition:Real Interval/Unit Interval

A '''unit interval''' is a real interval whose endpoints are $0$ and $1$: {{begin-eqn}} {{eqn | l = \openint 0 1       | o = :=       | r = \set {x \in \R: 0 < x < 1} }} {{eqn | l = \hointr 0 1       | o = :=       | r = \set {x \in \R: 0 \le x < 1} }} {{eqn | l = \hointl 0 1       | o = :=       | r = \set {x \in \R: 0 < x \le 1} }} {{eqn | l = \closedint 0 1       | o = :=       | r = \set {x \in \R: 0 \le x \le 1} }} {{end-eqn}}


Definition:Inductive Set/Subset of Real Numbers

Let $I$ be a subset of the real numbers $\R$. Then $I$ is an '''inductive set''' {{iff}}: :$1 \in I$  and :$x \in I \implies \paren {x + 1} \in I$


Definition:Language of Category Theory

The language of '''(morphisms-only) category theory''' is a specific instance of the language of predicate logic. As such, it is considered to have: * unary function symbols $\operatorname{dom}$ and $\operatorname{cod}$, called the '''domain''' and '''codomain''' symbols * A ternary relation symbol $R_\circ$, called the '''composition relation''' symbol All further descriptions required to determine a formal language are inherited from the definition of the language of predicate logic.


Definition:Morphisms-Only Metacategory

A '''morphisms-only metacategory''' is a metamodel for the language of category theory subject to the following axioms: {{:Axiom:Axioms for Morphisms-Only Category Theory}}


Definition:Mapping/Defined

A mapping $f \subseteq S \times T$ is '''defined''' at $x \in S$ {{iff}}: :$\exists y \in T: \tuple {x, y} \in f$ If for some $x \in S$, one has: :$\forall y \in T: \tuple {x, y} \notin f$ then $f$ is '''not defined''' or ('''undefined''') at $x$, and indeed, $f$ is not technically a mapping at all.


Definition:Relation/Truth Set

Let $\RR$ be a relation on $S \times T$. The '''truth set''' of $\RR$ is the set of all ordered pairs $\tuple {s, t}$ of $S \times T$ such that $s \mathrel \RR t$: : $\map \TT \RR = \set {\tuple {s, t}: s \mathrel \RR t}$


Definition:Relation/General Definition

Let $\displaystyle \Bbb S = \prod_{i \mathop = 1}^n S_i = S_1 \times S_2 \times \ldots \times S_n$ be the cartesian product of $n$ sets $S_1, S_2, \ldots, S_n$. An '''$n$-ary relation on $\Bbb S$''' is an ordered $n + 1$-tuple $\RR$ defined as: :$\RR := \struct {S_1, S_2, \ldots, S_n, R}$ where $R$ is an arbitrary subset $R \subseteq \Bbb S$. To indicate that $\tuple {s_1, s_2, \ldots, s_n} \in R$, we write: :$\map \RR {s_1, s_2, \ldots, s_n}$


Definition:Relation/Unary Relation

As a special case of an $n$-ary relation on $S$, note that when $n = 1$ we define a '''unary relation''' on $S$ as: :$\RR \subseteq S$ That is, a '''unary relation''' is a subset of $S$.


Definition:Random Selection

A manner of selecting objects from some larger collection of objects is said to be '''random''' if the selection is made according to chance. That is, there is no strict rule or procedure that predictably determines the outcome of the selection.  See experiment and random variable for a  precise mathematical treatment of randomness.


Definition:Agreement/Relations

Let: :$(1): \quad \mathcal R_1 \subseteq S_1 \times T_1$ be a relation on $S_1 \times T_1$ :$(2): \quad \mathcal R_2 \subseteq S_2 \times T_2$ be a relation on $S_2 \times T_2$ :$(3): \quad X \subseteq S_1 \cap S_2$ Let: :$\forall s \in X: \mathcal R_1 \left ({s}\right) = \mathcal R_2 \left ({s}\right)$ where $\mathcal R_1 \left ({s}\right)$ denotes the image of $s$ under $\mathcal R$: :$\mathcal R_1 \left ({s}\right) := \left\{ {t \in T: s \mathrel{\mathcal R_1} t}\right\}$ Then the relations $\mathcal R_1$ and $\mathcal R_2$ are said to '''agree on''' or '''be in agreement on''' $X$.


Definition:Agreement/Mappings

Let: :$(1): \quad f_1: S_1 \to T_1$ be a mapping from $S_1$ to $T_1$ :$(2): \quad f_2: S_2 \to T_2$ be a mapping from $S_2$ to $T_2$ :$(3): \quad X \subseteq S_1 \cap S_2$ Let: :$\forall s \in X: \map {f_1} s = \map {f_2} s$ Then the mappings $f_1$ and $f_2$ are said to '''agree on''' or '''be in agreement on''' $X$.


Definition:Combinable/Mappings

Let: :$(1): \quad f_1: S_1 \to T_1$ be a mapping from $S_1$ to $T_1$ :$(2): \quad f_2: S_2 \to T_2$ be a mapping from $S_2$ to $T_2$ If $f_1$ and $f_2$ agree on $S_1 \cap S_2$, they are said to be '''combinable'''.


Definition:Dedekind Complete Set

Let $\struct {S, \preceq}$ be an ordered set. Then $\struct {S, \preceq}$ is '''Dedekind complete''' {{iff}} every non-empty subset of $S$ that is bounded above admits a supremum (in $S$).


Definition:Dedekind-MacNeille Completion

Let $\struct {S, \preceq}$ be an ordered set. For a subset $A \subseteq S$, let $A_+$ and $A_-$ be the sets of all upper and lower bounds for $A$ in $S$, respectively. {{explain|Review notation for upper and lower bounds}} The '''Dedekind-MacNeille completion''' of $\struct {S, \preceq}$ is defined as the set: :$\widehat S := \set {A \subseteq S: A = \paren {A_+}_-}$ ordered by inclusion ($\subseteq$). {{explain|ordered by inclusion in this context - exactly how does this work?}}


Definition:Dedekind Completion

Let $S$ be an ordered set. A '''Dedekind completion of $S$''' is a Dedekind complete ordered set $\tilde S$ together with an order embedding $\phi: S \to \tilde S$, subject to: :For all Dedekind complete ordered sets $X$, and for all order embeddings $f: S \to X$, there exists a unique order embedding $\tilde f: \tilde S \to X$ such that: ::$\tilde f \circ \phi = f$


Definition:Abnormal Subgroup

Let $G$ be a group. Let $H$ be a subgroup of $G$. Then $H$ is an '''abnormal subgroup''' {{iff}}: :$\forall g \in G: g \in \gen {H, H^g}$ where $\gen {H, H^g}$ is the subgroup of $G$ generated by $H$ and the conjugate of $H$ by $g$.


Definition:Self-Normalizing Subgroup

Let $G$ be a group. Let $H$ be a subgroup of $G$. Then $H$ is a '''self-normalizing subgroup''' {{iff}}: :$N_G \left({H}\right) = H$ where $N_G \left({H}\right)$ is the normalizer of $H$ in $G$.


Definition:Contranormal Subgroup

Let $G$ be a group. Let $H$ be a subgroup of $G$. Then $H$ is a '''contranormal subgroup''' {{iff}} the normal closure of $H$ is the whole of $G$.


Definition:Subnormal Subgroup

Let $G$ be a group. Let $H$ be a subgroup of $G$. Then $H$ is a '''$k$-subnormal subgroup of $G$''' {{iff}} there is a finite sequence of subgroups of $G$: :$H = H_0, H_1, H_2, \ldots, H_k = G$ such that $H_i$ is a normal subgroup of $H_{i+1}$ for all $i \in \left\{{0, 1, \ldots, k-1}\right\}$. That is, {{iff}} there exists a normal series: :$H = H_0 \lhd H_1 \lhd H_2 \lhd \cdots \lhd H_k = G$ where $\lhd$ denotes the relation of normality. Thus $H$ is a '''subnormal subgroup of $G$''' {{iff}} there exists $k \in \Z_{>0}$ such that $H$ is a '''$k$-subnormal subgroup of $G$'''.


Definition:Pronormal Subgroup

Let $G$ be a group. Let $H$ be a subgroup of $G$. Then $H$ is a '''pronormal subgroup in $G$''' iff each of its conjugates in $G$ is conjugate to it already in the subgroup generated by $H$ and its conjugate. That is, $H$ is '''pronormal in $G$''' iff: : $\forall g \in G: \exists k \in \left\langle{H, H^g}\right\rangle: H^k = H^g$ where: : $\left\langle{H, H^g}\right\rangle$ is the subgroup generated by $H$ and $H^g$ : $H^g$ is the conjugate of $H$ by $g$.


Definition:Paranormal Subgroup

Let $G$ be a group. Let $H$ be a subgroup of $G$. Then $H$ is a '''paranormal subgroup in $G$''' {{iff}} the subgroup generated by $H$ and any conjugate of $H$ is also generated by $H$ and a conjugate of $H$ within that generated subgroup. That is, $H$ is '''paranormal in $G$''' {{iff}}: : $\forall g \in G: \exists k \in \left\langle{H, H^g}\right\rangle: \left\langle{H, H^k}\right\rangle = \left\langle{H, H^g}\right\rangle$ where: : $\left\langle{H, H^g}\right\rangle$ is the subgroup generated by $H$ and $H^g$ : $H^g$ is the conjugate of $H$ by $g$.


Definition:Polynormal Subgroup

Let $G$ be a group. Let $H$ be a subgroup of $G$. Then $H$ is a '''polynormal subgroup''' of $G$ {{iff}} for all $g \in G$, the conjugate closure of $H$ in $\langle H, g \rangle$ is equal to the conjugate closure of $H$ in $H^{\langle g \rangle}$. That is, $H$ is '''polynormal in $G$''' {{iff}}: : $\forall g \in G: H^{\left\langle g \right\rangle} = H^{H^{\left\langle g \right\rangle}}$ where: : $H^{\left\langle g \right\rangle}$ is the subgroup generated by the set of all elements of the form $g^nhg^{-n}$ where $h \in H, n \in \mathbb{Z}$, : $H^{H^{\left\langle g \right\rangle}}$ is the subgroup generated by the set of all elements of the form $khk^{-1}$ where $k \in H^{\left\langle g \right\rangle}$.


Definition:Underlying Set/Abstract Algebra

Let $\struct {S, \circ}$ be an algebraic structure. Then the '''underlying set''' of $\struct {S, \circ}$ is the set $S$.


Definition:Underlying Set/Metric Space

Let $M = \left({A, d}\right)$ be a metric space. Then the '''underlying set''' of $M$ is the set $A$.


Definition:Underlying Set/Topological Space

Let $T = \left({S, \tau}\right)$ be a topological space. Then the '''underlying set''' of $T$ is the set $S$.


Definition:Multiplication/Complex Numbers

The '''multiplication operation''' in the domain of complex numbers $\C$ is written $\times$. Let $z = a + i b, w = c + i d$ where $a, b, c, d \in \R$. Then $z \times w$ is defined as: :$\paren {a + i b} \times \paren {c + i d} = \paren {a c - b d} + i \paren {a d + b c}$


Definition:Coproduct

Let $\mathbf C$ be a metacategory. Let $A$ and $B$ be objects of $\mathbf C$. A '''coproduct diagram''' for $A$ and $B$ comprises an object $P$ and morphisms $i_1: A \to P$, $i_2: B \to P$: ::$\begin{xy} <-4em,0em>*+{A} = "A", <0em,0em>*+{P}  = "P", <4em,0em>*+{B}  = "B", "A";"P" **@{-} ?>*@{>} ?*!/^.8em/{i_1}, "B";"P" **@{-} ?>*@{>} ?*!/_.8em/{i_2}, \end{xy}$ subjected to the following universal mapping property: :For any object $X$ and morphisms $x_1, x_2$ like so: ::$\begin{xy} <-4em,0em>*+{A} = "A", <0em,0em>*+{X}  = "X", <4em,0em>*+{B}  = "B", "A";"X" **@{-} ?>*@{>} ?*!/^.8em/{x_1}, "B";"X" **@{-} ?>*@{>} ?*!/_.8em/{x_2}, \end{xy}$ :there is a unique morphism $u: P \to X$ such that: ::$\begin{xy} <0em,5em>*+{X}  = "X", <-5em,0em>*+{A} = "A", <0em,0em>*+{P}  = "P", <5em,0em>*+{B}  = "B", "A";"X" **@{-}  ?>*@{>} ?*!/^.8em/{x_1}, "B";"X" **@{-}  ?>*@{>} ?*!/_.8em/{x_2}, "P";"X" **@{--} ?>*@{>} ?*!/_.6em/{u}, "A";"P" **@{-}  ?>*@{>} ?*!/_.8em/{i_1}, "B";"P" **@{-}  ?>*@{>} ?*!/^.8em/{i_2}, \end{xy}$ :is a commutative diagram. That is: :$x_1 = u \circ i_i$ and $x_2 = u \circ i_2$ In this situation, $P$ is called a '''coproduct of $A$ and $B$''' and may be denoted $A + B$. We generally write $\sqbrk {x_1, x_2}$ for the unique morphism $u$ determined by above diagram. The morphisms $i_1$ and $i_2$ are often taken to be implicit. They are called '''injections'''; if necessary, $i_1$ can be called the '''first injection''' and $i_2$ the '''second injection'''. {{expand|the injection definition may merit its own, separate page}} {{definition wanted|general definition}} === Coproduct of Sets === When the objects $A$ and $B$ are sets, the definition of '''coproduct''' takes on the following form. {{:Definition:Coproduct/Sets}}


Definition:Variable

A '''variable''' is a label which is used to refer to an unspecified object. A '''variable''' can be identified by means of a symbol, for example: :$x, y, z, A, B, C, \phi, \psi, \aleph$ It is often convenient to append a subscript letter or number to distinguish between different objects of a similar type: :$a_0, a_1, a_2, \ldots, a_n; S_\phi, S_{\phi_x}, \ldots$ The type of symbol used to define a '''variable''' is purely conventional. Particular types of object, as they are introduced, frequently have a particular range of symbols specified to define them, but there are no strict rules on the subject.


A '''variable''' is a characteristic property of all individuals in a population or sample. It is a categorization of the population such that each individual can be unambiguously described with respect to said variable. === Quantitative Variable === {{:Definition:Variable/Descriptive Statistics/Quantitative Variable}} === Qualitative Variable === {{:Definition:Variable/Descriptive Statistics/Qualitative Variable|Qualitative Variable}}


Definition:Variable/Descriptive Statistics/Quantitative Variable

A '''quantitative variable''' is a variable such that: : The variable can be described by numbers : The performing of arithmetical operations on the data is meaningful.


Definition:Variable/Descriptive Statistics/Qualitative Variable

A '''qualitative variable''' is a variable such that: : The variable is not a quantitative variable : The variable describes each individual as either having, or not having, some specific property.


{{:Definition:Basis (Topology)/Analytic Basis/Definition 1}}


Let $S$ be a set. ==== Definition 1 ==== {{:Definition:Basis (Topology)/Synthetic Basis/Definition 1}} ==== Definition 2 ==== {{:Definition:Basis (Topology)/Synthetic Basis/Definition 2}}


Definition:Countable Basis

A '''countable (analytic) basis''' for a topology $\tau$ is an analytic basis for $\tau$ which is a countable set.


Definition:Sub-Basis/Analytic Sub-Basis

Let $\struct {S, \tau}$ be a topological space. Let $\SS \subseteq \tau$. Define: :$\displaystyle \BB = \set {\bigcap \FF: \FF \subseteq \SS, \FF \text{ is finite} }$ That is, $\BB$ is the set of all finite intersections of sets in $\SS$. Note that $\FF$ is allowed to be empty in the above definition. Define: :$\displaystyle \tau' = \set {\bigcup \AA: \AA \subseteq \BB}$ Suppose that $\tau \subseteq \tau'$. That is, suppose that every $U \in \tau$ is a union of finite intersections of sets in $\SS$, together with $\O$ and $S$ itself. Then $\SS$ is called an '''analytic sub-basis''' for $\tau$.


Definition:Sub-Basis/Synthetic Sub-Basis

Let $X$ be a set. A '''synthetic sub-basis on $X$''' is ''any'' subset $\mathcal S \subseteq \mathcal P \left({X}\right)$ of the power set of $X$.


Definition:Differential/Real Function

=== At a Point === {{:Definition:Differential/Real Function/Point}} === On an Open Set === {{:Definition:Differential/Real Function/Open Set}} Also: :$f \left({x + h}\right) - f \left({x}\right) - \mathrm d f \left({x; h}\right) = o \left({h}\right)$ as $h \to 0$. In the above, $o \left({h}\right)$ is interpreted as little-O of $h$. {{refactor|put the above somewhere else}}


{{:Definition:Differential/Real-Valued Function/Point}}


=== At a Point === {{:Definition:Differential/Vector-Valued Function/Point}} === On an Open Set === Let $O \subseteq \R^n$ be an open set. Let $f = \tuple {f_1, \ldots, f_m}^\intercal: O \to \R^m$ be a vector valued function, differentiable at $x \in O$. The '''differential''' $\d f$ is a function of two variables, defined as: :$\map {\d f} {x; h} = \map {J_f} x \cdot h$ where $\map {J_f} x$ be the Jacobian matrix of $f$ at $x$. That is, if $h = \tuple {h_1, \ldots, h_n}$: :$\map {\d f} {x; h} = \begin {pmatrix} \map {\dfrac {\partial f_1} {\partial x_1} } x & \cdots & \map {\dfrac {\partial f_1} {\partial x_n} } x \\ \vdots & \ddots & \vdots \\ \map {\dfrac {\partial f_m} {\partial x_1} } x & \cdots & \map {\dfrac {\partial f_m} {\partial x_n} } x \end {pmatrix} \begin {pmatrix} h_1 \\ \vdots \\ h_n \end {pmatrix}$ </onlyinclude>


Definition:Dependent Variable/Real Function

Let $f: \R \to \R$ be a real function. Let $\map f x = y$. Then $y$ is referred to as a '''dependent variable'''.


Definition:Independent Variable/Real Function

Let $f: \R \to \R$ be a real function. Let $\map f x = y$. Then $x$ is referred to as an '''independent variable'''.


Definition:Jacobian

Let $U$ be an open subset of $\R^n$. Let $\mathbf f = \paren {f_1, f_2, \ldots, f_m}^\intercal: U \to \R^m$ be a vector valued function, differentiable at $\mathbf x = \paren {x_1, x_2, \ldots, x_n}^\intercal \in U$.


Definition:Jacobian/Matrix

The '''Jacobian matrix''' of $\mathbf f$ at $\mathbf x$ is defined to be the matrix of partial derivatives: :$\displaystyle \mathbf J_{\mathbf f} := \begin{pmatrix}  \map {\dfrac {\partial f_1} {\partial x_1} } {\mathbf x} & \cdots & \map {\dfrac {\partial f_1} {\partial x_n} } {\mathbf x} \\  \vdots & \ddots & \vdots \\  \map {\dfrac {\partial f_m} {\partial x_1} } {\mathbf x} & \cdots & \map {\dfrac {\partial f_m} {\partial x_n} } {\mathbf x} \end{pmatrix}$


Definition:Jacobian/Determinant

The '''Jacobian determinant''' of $\mathbf f$ at $\mathbf x$ is defined to be the determinant of the Jacobian matrix: :$\displaystyle \map \det {\mathbf J_{\mathbf f} } := \begin{vmatrix}  \map {\dfrac {\partial f_1} {\partial x_1} } {\mathbf x} & \cdots & \map {\dfrac {\partial f_1} {\partial x_n} } {\mathbf x} \\  \vdots & \ddots & \vdots \\  \map {\dfrac {\partial f_m} {\partial x_1} } {\mathbf x} & \cdots & \map {\dfrac {\partial f_m} {\partial x_n} } {\mathbf x} \end{vmatrix}$


{{:Definition:Differentiable Mapping/Real Function/Point}}


{{:Definition:Differentiable Mapping/Complex Function/Point}}


{{:Definition:Differentiable Mapping/Real-Valued Function/Point}}


{{:Definition:Differentiable Mapping/Vector-Valued Function/Point}}


Let $f$ be a real function defined on an open interval $\openint a b$. Let $\xi$ be a point in $\openint a b$. ==== Definition 1 ==== {{:Definition:Differentiable Mapping/Real Function/Point/Definition 1}} ==== Definition 2 ==== {{:Definition:Differentiable Mapping/Real Function/Point/Definition 2}} These limits, if they exist, are called the derivative of $f$ at $\xi$.


Definition:Differentiable Mapping/Real Function/Interval/Closed Interval

Let $f$ be differentiable on the open interval $\openint a b$. If the following limit from the right exists: :$\displaystyle \lim_{x \mathop \to a^+} \frac {\map f x - \map f a} {x - a}$ as well as this limit from the left: :$\displaystyle \lim_{x \mathop \to b^-} \frac {\map f x - \map f b} {x - b}$ then $f$ is '''differentiable on the closed interval $\closedint a b$'''.


Definition:Differentiable Mapping/Real Function/Interval

Let $f$ be a real function defined on an open interval $\openint a b$. Then $f$ is '''differentiable on $\openint a b$''' {{iff}} $f$ is differentiable at each point of $\openint a b$.


Definition:Differentiable Mapping/Real Function/Real Number Line

Let $f$ be a real function defined on $\R$. By definition, $\R$ is an (unbounded) open interval. Let $f$ be differentiable on the open interval $\R$. That is, let $f$ be differentiable at every point of $\R$. Then $f$ is '''differentiable everywhere (on $\R$)'''.


Let $U$ be an open subset of $\R^n$.  Let $f: U \to \R$ be a real-valued function. Let $x \in U$. ==== Definition 1 ==== {{:Definition:Differentiable Mapping/Real-Valued Function/Point/Definition 1}} ==== Definition 2 ==== {{:Definition:Differentiable Mapping/Real-Valued Function/Point/Definition 2}}


Definition:Differentiable Mapping/Real-Valued Function/Open Set

Let $\mathbb X$ be an open subset of $\R^n$.  Let $f: \mathbb X \to \R$ be a real-valued function. Then $f$ is '''differentiable in the open set $\mathbb X$''' {{iff}} $f$ is differentiable at each point of $\mathbb X$.


Let $\mathbb X$ be an open subset of $\R^n$.  Let $f = \tuple {f_1, f_2, \ldots, f_m}^\intercal: \mathbb X \to \R^m$ be a vector valued function. ==== Definition 1 ==== {{:Definition:Differentiable Mapping/Vector-Valued Function/Point/Definition 1}} ==== Definition 2 ==== {{:Definition:Differentiable Mapping/Vector-Valued Function/Point/Definition 2}}


Definition:Differentiable Mapping/Vector-Valued Function/Region

Let $S \subseteq \mathbb X$. Then $f$ is '''differentiable in the open set $S$''' {{iff}} $f$ is differentiable at each $x$ in $S$. This can be denoted $f \in \map {\CC^1} {S, \R^m}$.


Definition:Vector-Valued Function/Component Function

Each $f_1, f_2, \ldots, f_n$ is a '''component function''' of $\mathbf r$.


Definition:Subdivision (Real Analysis)/Normal Subdivision

$P$ is a '''normal subdivision of $\closedint a b$''' {{iff}}: :the length of every interval of the form $\closedint {x_i} {x_{i + 1} }$ is the same as every other. That is, {{iff}}: :$\exists c \in \R_{> 0}: \forall i \in \N_{< n}: x_{i + 1} - x_i = c$


Let $S$ be a set. Let $\BB$ be a synthetic basis of $S$. === Definition 1 === {{:Definition:Topology Generated by Synthetic Basis/Definition 1}} === Definition 2 === {{:Definition:Topology Generated by Synthetic Basis/Definition 2}} === Definition 3 === {{:Definition:Topology Generated by Synthetic Basis/Definition 3}}


Definition:Complex Conjugate/Complex Conjugation

The operation of '''complex conjugation''' is the mapping: : $\overline \cdot: \C \to \C: z \mapsto \overline z$. where $\overline z$ is the complex conjugate of $z$. That is, it maps a complex number to its complex conjugate.


Definition:Addition/Complex Numbers

The '''addition operation''' in the domain of complex numbers $\C$ is written $+$. Let $z = a + i b, w = c + i d$ where $a, b, c, d \in \R, i^2 = -1$. Then $z + w$ is defined as: :$\paren {a + i b} + \paren {c + i d} = \paren {a + c} + i \paren {b + d}$


Definition:Category of Vector Spaces

The '''category of vector spaces''', denoted $\mathbf{Vect}$, is the metacategory with: {{DefineCategory |ob = vector spaces |mor = linear transformations }} {{DefinitionWanted|Explicate dependence on base field}}


Definition:Class (Descriptive Statistics)

Let $D$ be a finite collection of $n$ data regarding some quantitative variable.


Definition:Class (Descriptive Statistics)/Integer Data

Let the data in $D$ be described by natural numbers or by integers. Let $d_{\min}$ be the value of the smallest datum in $D$. Let $d_{\max}$ be the value of the largest datum in $D$.   Let $P = \left\{{x_0, x_1, x_2, \ldots, x_{n-1}, x_n}\right\} \subseteq \Z$ be a subdivision of $\left[{a \,.\,.\, b}\right]$, where $a \le x_0 \le x_n \le b$. The integer interval $\left[{a \,.\,.\, b}\right]$, where $a \le d_{\min} \le d_\max \le b$, is said to be divided into '''classes''' of integer intervals of the forms $\left[{x_i \,.\,.\, x_{i+1}}\right]$ or $\left[{x_i \,.\,.\, x_i}\right]$ {{iff}}: : Every datum is assigned into exactly one class : Every class is disjoint from every other : The union of all classes contains the entire integer interval $\left[{x_0 \,.\,.\, x_n}\right]$ By convention, the first and last classes are not empty classes.


Definition:Class (Descriptive Statistics)/Real Data

Let the data in $D$ be described by rational numbers or by real numbers. Let $d_{\min}$ be the value of the smallest datum in $D$. Let $d_{\max}$ be the value of the largest datum in $D$.   Let $P = \left\{{x_0, x_1, x_2, \ldots, x_{n-1}, x_n}\right\} \subseteq \R$ be a subdivision of $\left[{a \,.\,.\, b}\right]$, where $a \le x_0 \le x_n \le b$. The closed real interval $\left[{a \,.\,.\, b}\right]$, where $a \le d_{\text{min}} \le d_{\text{max}} \le b$, is said to be divided into '''classes''' of real intervals with endpoints $x_i$ and $x_{i+1}$ {{iff}}: : Every datum is assigned into exactly one class : Every class is disjoint from every other : The union of all classes contains the entire real interval $\left[{x_0 \,.\,.\, x_n}\right]$ The classes may be any combination of open, closed, or half-open intervals that fulfill the above criteria, but usually: : Every class except the last is of the form $\left[{x_i \,.\,.\, x_{i+1}}\right)$ : The last class is of the form  $\left[{x_{n-1} \,.\,.\, x_n}\right]$  By convention, the first and last classes are not empty classes.


Definition:Pointwise Addition

The (binary) operation of '''pointwise addition''' is defined on $\mathbb F^S$ as: :$+: \mathbb F^S \times \mathbb F^S \to \mathbb F^S: \forall f, g \in \mathbb F^S:$ ::$\forall s \in S: \map {\paren {f + g} } s := \map f s + \map g s$ where the $+$ on the {{RHS}} is conventional arithmetic addition.


Definition:Pointwise Addition of Complex-Valued Functions

Let $f, g: S \to \C$ be complex-valued functions. Then the '''pointwise sum of $f$ and $g$''' is defined as: :$f + g: S \to \C:$ ::$\forall s \in S: \map {\paren {f + g} } s := \map f s + \map g s$ where $+$ on the {{RHS}} is complex addition.


Definition:Pointwise Operation on Complex-Valued Functions

Let $\C^S$ be the set of all mappings $f: S \to \C$, where $\C$ is the set of complex numbers. Let $\oplus$ be a binary operation on $\C$. Define $\oplus: \C^S \times \C^S \to \C^S$, called '''pointwise $\oplus$''', by: :$\forall f, g \in \C^S: \forall s \in S: \map {\paren {f \oplus g} } s := \map f s \oplus \map g s$ In the above expression, the operator on the {{RHS}} is the given $\oplus$ on the complex numbers.


Definition:Closed under Mapping

Let $f: S \to T$ be a mapping. Let $S' \subseteq S$. Then $S'$ is '''closed under $f$''' {{iff}}: :$f \sqbrk {S'} \subseteq S'$ where $f \sqbrk {S'}$ is the image of $S'$ under $f$. That is: :$x \in S' \implies \map f x \in S'$ === Arbitrary Product === {{:Definition:Closed under Mapping/Arbitrary Product}}


Definition:Pointwise Operation on Number-Valued Functions

Let $S$ be a non-empty set. Let $\mathbb F$ be one of the standard number sets: $\Z, \Q, \R$ or $\C$. Let $\mathbb F^S$ be the set of all mappings $f: S \to \mathbb F$. Let $\oplus$ be a binary operation on $\mathbb F$. The (binary) operation '''pointwise $\oplus$''' is defined on $\mathbb F^S$ as: :$\oplus: \mathbb F^S \times \mathbb F^S \to \mathbb F^S: \forall f, g \in \mathbb F^S:$ ::$\forall s \in S: \map {\paren {f \oplus g} } s := \map f s \oplus \map g s$


Definition:Pointwise Operation on Integer-Valued Functions

Let $\Z^S$ be the set of all mappings $f: S \to \Z$, where $\Z$ is the set of integers. Let $\oplus$ be a binary operation on $\Z$. Define $\oplus: \Z^S \times \Z^S \to \Z^S$, called '''pointwise $\oplus$''', by: :$\forall f, g \in \Z^S: \forall s \in S: \map {\paren {f \oplus g} } s := \map f s \oplus \map g s$ In the above expression, the operator on the {{RHS}} is the given $\oplus$ on the integers.


Definition:Pointwise Operation on Rational-Valued Functions

Let $\Q^S$ be the set of all mappings $f: S \to \Q$, where $\Q$ is the set of rational numbers. Let $\oplus$ be a binary operation on $\Q$. Define $\oplus: \Q^S \times \Q^S \to \Q^S$, called '''pointwise $\oplus$''', by: :$\forall f, g \in \Q^S: \forall s \in S: \map {\paren {f \oplus g} } s := \map f s \oplus \map g s$ In the above expression, the operator on the {{RHS}} is the given $\oplus$ on the rational numbers.


Definition:Pointwise Addition of Integer-Valued Functions

Let $f, g: S \to \Z$ be integer-valued functions. Then the '''pointwise sum of $f$ and $g$''' is defined as: :$f + g: S \to \Z:$ ::$\forall s \in S: \map {\paren {f + g} } s := \map f s + \map g s$ where the $+$ on the {{RHS}} is integer addition.


Definition:Integer-Valued Function

Let $f: S \to T$ be a function. Let $S_1 \subseteq S$ such that $f \left({S_1}\right) \subseteq \Z$. Then $f$ is said to be '''integer-valued on $S_1$'''. That is, $f$ is defined as integer-valued on $S_1$ {{iff}} the image of $S_1$ under $f$ lies entirely within the set of integers $\Z$. An '''integer-valued function''' is a function $f: S \to \Z$ whose codomain is the set of integers $\Z$. That is, $f$ is '''integer-valued''' {{iff}} it is '''integer-valued''' over its entire domain. Category:Definitions/Integers rp9b74fhi6296cvp5f347ess83n70oq


Definition:Pointwise Addition of Rational-Valued Functions

Let $f, g: S \to \Q$ be rational-valued functions. Then the '''pointwise sum of $f$ and $g$''' is defined as: :$f + g: S \to \Q:$ ::$\forall s \in S: \map {\paren {f + g} } s := \map f s + \map g s$ where the $+$ on the {{RHS}} is integer addition.


Definition:Rational-Valued Function

Let $f: S \to T$ be a function. Let $S_1 \subseteq S$ such that $f \left({S_1}\right) \subseteq \Q$. Then $f$ is said to be '''rational-valued on $S_1$'''. That is, $f$ is defined as rational-valued on $S_1$ iff the image of $S_1$ under $f$ lies entirely within the set of rational numbers $\Q$. A '''rational-valued function''' is a function $f: S \to \Q$ whose codomain is the set of rational numbers $\Q$. That is, $f$ is '''rational-valued''' {{iff}} it is rational-valued over its entire domain. Category:Definitions/Rational Numbers 5lll7qnf2lmi2z6ki66v4o4j4uu7v4s


Definition:Pointwise Multiplication of Integer-Valued Functions

Let $f, g: S \to \Z$ be integer-valued functions. Then the '''pointwise product of $f$ and $g$''' is defined as: :$f \times g: S \to \Z:$ ::$\forall s \in S: \map {\paren {f \times g} } s := \map f s \times \map g s$ where the $\times$ on the {{RHS}} is integer multiplication.


Definition:Pointwise Multiplication of Rational-Valued Functions

Let $f, g: S \to \Q$ be rational-valued functions. Then the '''pointwise product of $f$ and $g$''' is defined as: :$f \times g: S \to \Q:$ ::$\forall s \in S: \map {\paren {f \times g} } s := \map f s \times \map g s$ where the $\times$ on the {{RHS}} is rational multiplication.


Definition:Pointwise Multiplication of Real-Valued Functions

Let $f, g: S \to \R$ be real-valued functions. Then the '''pointwise product of $f$ and $g$''' is defined as: :$f \times g: S \to \R:$ ::$\forall s \in S: \map {\paren {f \times g} } s := \map f s \times \map g s$ where $\times$ on the {{RHS}} denotes real multiplication.


Definition:Pointwise Multiplication of Complex-Valued Functions

Let $f, g: S \to \Z$ be complex-valued functions. Then the '''pointwise product of $f$ and $g$''' is defined as: :$f \times g: S \to \Z:$ ::$\forall s \in S: \map {\paren {f \times g} } s := \map f s \times \map g s$ where the $\times$ on the {{RHS}} is complex multiplication.


Definition:Pointwise Scalar Multiplication of Number-Valued Function

When one of the functions is the constant mapping $f_\lambda: S \to \mathbb F: \map {f_\lambda} s = \lambda$, the following definition arises: The (binary) operation of '''pointwise scalar multiplication''' is defined on $\mathbb F \times \mathbb F^S$ as: :$\times: \mathbb F \times \mathbb F^S \to \mathbb F^S: \forall \lambda \in \mathbb F, f \in \mathbb F^S:$ ::$\forall s \in S: \map {\paren {\lambda \times f} } s := \lambda \times \map f s$ where the $\times$ on the {{RHS}} is conventional arithmetic multiplication.


Definition:Pointwise Scalar Multiplication of Integer-Valued Function

Let $f: S \to \Z$ be an integer-valued function. Let $\lambda \in \Z$ be an integer. Then the '''pointwise scalar product of $f$ by $\lambda$''' is defined as: :$\lambda \times f: S \to \Z:$ ::$\forall s \in S: \map {\paren {\lambda \times f} } s := \lambda \times \map f s$ where $\times$ on the {{RHS}} is integer multiplication.


Definition:Pointwise Scalar Multiplication of Rational-Valued Function

Let $f: S \to \Q$ be an rational-valued function. Let $\lambda \in \Q$ be an rational number. Then the '''pointwise scalar product of $f$ by $\lambda$''' is defined as: :$\lambda \times f: S \to \Q:$ ::$\forall s \in S: \map {\paren {\lambda \times f} } s := \lambda \times \map f s$ where the $\times $ on the {{RHS}} is rational multiplication.


Definition:Pointwise Scalar Multiplication of Real-Valued Function

Let $f: S \to \R$ be an real-valued function. Let $\lambda \in \R$ be an real number. Then the '''pointwise scalar product of $f$ by $\lambda$''' is defined as: :$\lambda \times f: S \to \R:$ ::$\forall s \in S: \map {\paren {\lambda \times f} } s := \lambda \times \map f s$ where the $\times $ on the {{RHS}} is real multiplication.


Definition:Pointwise Scalar Multiplication of Complex-Valued Function

Let $f: S \to \C$ be an complex-valued function. Let $\lambda \in \C$ be an complex number. Then the '''pointwise scalar product of $f$ by $\lambda$''' is defined as: :$\lambda \times f: S \to \C:$ ::$\forall s \in S: \map {\paren {\lambda \times f} } s := \lambda \times \map f s$ where $\times$ on the {{RHS}} denotes complex multiplication.


Definition:Addition/Rational Numbers

The addition operation in the domain of rational numbers $\Q$ is written $+$. Let: :$a = \dfrac p q, b = \dfrac r s$ where: :$p, q \in \Z$ :$r, s \in \Z_{\ne 0}$ Then $a + b$ is defined as: :$\dfrac p q + \dfrac r s = \dfrac {p s + r q} {q s}$ This definition follows from the definition of and proof of existence of the field of quotients of any integral domain, of which the set of integers is an example.


Definition:Quaternion/Addition

The '''sum''' of two quaternions $\mathbf x_1 = a_1 \mathbf 1 + b_1 \mathbf i + c_1 \mathbf j + d_1 \mathbf k$ and $\mathbf x_2 = a_2 \mathbf 1 + b_2 \mathbf i + c_2 \mathbf j + d_2 \mathbf k$ is defined as: :$\mathbf x_1 + \mathbf x_2 := \paren {a_1 + a_2} \mathbf 1 + \paren {b_1 + b_2} \mathbf i + \paren {c_1 + c_2} \mathbf j + \paren {d_1 + d_2} \mathbf k$


Definition:Quaternion/Multiplication

The '''product''' of two quaternions $\mathbf x_1 = a_1 \mathbf 1 + b_1 \mathbf i + c_1 \mathbf j + d_1 \mathbf k$ and $\mathbf x_2 = a_2 \mathbf 1 + b_2 \mathbf i + c_2 \mathbf j + d_2 \mathbf k$ is defined as: {{begin-eqn}}      {{eqn | l = \mathbf x_1 \mathbf x_2       | o = :=       | r = \paren {a_1 a_2 - b_1 b_2 - c_1 c_2 - d_1 d_2} \mathbf 1       | c =  }} {{eqn | o =        | ro= +       | r = \paren {a_1 b_2 + b_1 a_2 + c_1 d_2 - d_1 c_2} \mathbf i       | c =  }} {{eqn | o =        | ro= +       | r = \paren {a_1 c_2 - b_1 d_2 + c_1 a_2 + d_1 b_2} \mathbf j       | c =  }} {{eqn | o =        | ro= +       | r = \paren {a_1 d_2 + b_1 c_2 - c_1 b_2 + d_1 a_2} \mathbf k       | c =  }} {{end-eqn}}


Definition:Quaternion/Algebra over Field

An algebra of quaternions can be defined over any field as follows: Let $\mathbb K$ be a field, and $a$, $b \in \mathbb K$. Define the '''quaternion algebra''' $\left\langle{ a,b }\right\rangle_\mathbb K$ to be the $\mathbb K$-vector space with basis $\{1, i, j, k\}$ subject to: : $i^2 = a$ : $j^2 = b$ : $ij = k = -ji$ Formally this could be achieved as a multiplicative presentation of a suitable group, or as a linear subspace of a finite extension of $\mathbb K$. Taking $\mathbb K = \R$ and $a = b = -1$ we see that this generalises  Hamilton's quaternions.


Definition:Field Extension/Degree

Let $E / F$ be a field extension. The '''degree of $E / F$''', denoted $\index E F$, is the dimension of $E / F$ when $E$ is viewed as a vector space over $F$. === Finite === {{:Definition:Field Extension/Degree/Finite}} === Infinite === {{:Definition:Field Extension/Degree/Infinite}}


Definition:Class/Proper Class

A '''proper class''' is a class which is not a set. That is, $A$ is a '''proper class''' {{iff}}: : $\neg \exists x: x = A$ where $x$ is a set. A class which is not a proper class is a small class.


Definition:Normal Series/Factor Group

Let $\sequence {G_i}_{i \mathop \in \closedint 0 n}$ be a normal series for $G$: :$\sequence {G_i}_{i \mathop \in \closedint 0 n} = \tuple {\set e = G_0 \lhd G_1 \lhd \cdots \lhd G_{n - 1} \lhd G_n = G}$ The '''factor groups''' of $\sequence {G_i}_{i \mathop \in \closedint 0 n}$: :$\set e = G_0 \lhd G_1 \lhd \cdots \lhd G_n = G$ are the quotient groups: :$G_1 / G_0, G_2 / G_1, \ldots, G_i / G_{i - 1}, \ldots, G_n / G_{n-1}$


Definition:Normal Series/Sequence of Homomorphisms

Let $\sequence {G_i}_{i \mathop \in \closedint 0 n}$ be a normal series for $G$: :$\sequence {G_i}_{i \mathop \in \closedint 0 n} = \tuple {\set e = G_0 \lhd G_1 \lhd \cdots \lhd G_{n - 1} \lhd G_n = G}$ whose factor groups are: :$H_1 = G_1 / G_0, H_2 = G_2 / G_1, \ldots, H_i = G_i / G_{i - 1}, \ldots, H_n = G_n / G_{n - 1}$ By Kernel of Group Homomorphism Corresponds with Normal Subgroup of Domain, such a series can also be expressed as a sequence $\phi_1, \ldots, \phi_n$ of group homomorphisms: :$\set e \stackrel {\phi_1} {\to} H_1 \stackrel {\phi_2} {\to} H_2 \stackrel {\phi_3} {\to} \cdots \stackrel {\phi_n} {\to} H_n$


Definition:Normal Series/Infinite

A normal series may or may not terminate at either end: :$\cdots \stackrel {\phi_{i - 1} } {\longrightarrow} H_{i - 1} \stackrel {\phi_i} {\longrightarrow} H_i \stackrel {\phi_{i + 1} } {\longrightarrow} H_{i + 1} \stackrel {\phi_{i + 2} } {\longrightarrow} \cdots$ Such a series is referred to as an '''infinite normal series'''. The context will determine which end, if either, it terminates.


Definition:Normal Series/Length

Let $\sequence {G_i}_{i \mathop \in \closedint 0 n}$ be a normal series for $G$: :$\sequence {G_i}_{i \mathop \in \closedint 0 n} = \tuple {\set e = G_0 \lhd G_1 \lhd \cdots \lhd G_{n-1} \lhd G_n = G}$ The '''length''' of $\sequence {G_i}_{i \mathop \in \closedint 0 n}$ is the number of (normal) subgroups which make it. In this context, the '''length''' of $\sequence {G_i}_{i \mathop \in \closedint 0 n}$ is $n$. If such a normal series is infinite, then its '''length''' is not defined.


Definition:Non-Trivial Group

A '''non-trivial group''' is a group which is not the trivial group. It is often used in arguments to exclude the sometimes erratic behaviour of the trivial group.


Definition:Exact Normal Series

Let $\sequence {H_i}_{i \mathop \in I \mathop \subseteq \Z}$ be a normal series: :$\cdots \stackrel {\phi_{i - 1} } {\longrightarrow} H_{i - 1} \stackrel {\phi_i} {\longrightarrow} H_i \stackrel {\phi_{i + 1} } {\longrightarrow} H_{i + 1} \stackrel {\phi_{i + 2} } {\longrightarrow} \cdots$ Suppose that, for some $i \in I$: :$\Img {\phi_i} = \map \ker {\phi_{i + 1} }$ That is, the image of one homomorphism is the kernel of the next. Then $\sequence {H_i}$ is referred to as '''exact at $H_i$'''. If $\sequence {H_i}$ is '''exact''' for all $i \in I$, then $\sequence {H_i}$ itself is an '''exact normal series'''. </onlyinclude>


Definition:Equalizer

Let $\mathbf C$ be a metacategory. Let $f, g: C \to D$ be morphisms with common domain and codomain. An '''equalizer''' for $f$ and $g$ is the limit of the diagram $C \underset f{\overset g\rightrightarrows} D$, that is, a morphism $e: E \to C$ such that: :$f \circ e = g \circ e$ and subject to the following UMP: :For any $a: A \to C$ such that $f \circ a = g \circ a$, there is a unique $u: A \to E$ such that: :::$\begin{xy}\xymatrix{  E    \ar[r]^*{e} &  C   \ar[r]<2pt>^*{f}   \ar[r]<-2pt>_*{g}  &  D \\  A   \ar@{.>}[u]^*{u}   \ar[ur]_*{a} }\end{xy}$ :is a commutative diagram. I.e., $a = e \circ u$.


Definition:Metric Space/Distance Function

The mapping $d: A \times A \to \R$ is referred to as a '''distance function on $A$''' or simply '''distance'''.


Definition:Metric Space/Triangle Inequality

Axiom $\text M 2$ is referred to as the '''triangle inequality''', as it is a generalization of the Triangle Inequality which holds on the real number line and complex plane.


Definition:Metric Space/Point

The elements of $A$ are called the '''points''' of the space.


Definition:Quasimetric/Quasimetric Space

A '''quasimetric space''' $M = \left({A, d}\right)$ is an ordered pair consisting of a set $A \ne \varnothing$ followed by a quasimetric $d: A \times A \to \R$ which acts on that set.


Definition:Distance/Points/Real Numbers

Let $x, y \in \R$ be real numbers. Let $\size {x - y}$ be the absolute value of $x - y$. Then the function $d: \R^2 \to \R$: :$\map d {x, y} = \size {x - y}$ is called the '''distance between $x$ and $y$'''.


Definition:Pseudometric/Pseudometric Space

A '''pseudometric space''' $M = \left({A, d}\right)$ is an ordered pair consisting of a set $A \ne \varnothing$ followed by a pseudometric $d: A \times A \to \R$ which acts on that set.


Definition:Operation Induced by Injection

Let $\struct {T, \circ}$ be an algebraic structure. Let $f: S \to T$ be an injection. Then the '''operation induced on $S$ by $f$ and $\circ$''' is the binary operation $\circ_f$ on $S$ defined by: :$\circ_f: S \times S \to S: x \circ_f y := \map {f^{-1} } {\map f x \circ \map f y}$


Let $A \subseteq \R$ be any subset of the real numbers. Let $f: A \to \R$ be a real function. Let $x \in A$ be a point of $A$. === Definition by Epsilon-Delta === {{:Definition:Continuous Real Function/Point/Definition by Epsilon-Delta}} === Definition by Neighborhood === {{:Definition:Continuous Real Function/Point/Definition by Neighborhood}}


Definition:Continuous Real Function/Subset

Let $A \subseteq \R$ be any subset of the real numbers. Let $f: A \to \R$ be a real function. Then '''$f$ is continuous on $A$''' {{iff}} $f$ is continuous at every point of $A$.


Definition:Continuous Real Function/One Side

=== Continuity from the Left at a Point === {{Definition:Continuous Real Function/Left-Continuous/Point}} === Continuity from the Right at a Point === {{Definition:Continuous Real Function/Right-Continuous/Point}}


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


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


Definition:Frequency (Descriptive Statistics)

Let $S$ be a sample or a population. Let $\omega$ be a qualitative variable, or a class of a quantitative variable. The '''frequency''' of $\omega$ is the number of individuals in $S$ satisfying $\omega$.


Definition:Class (Descriptive Statistics)/Class Mark

The midpoint of a class is called the '''class mark'''.


Definition:Mean of Grouped Data

Let $\mathsf C_1, \mathsf C_2, \cdots, \mathsf C_n$ be classes of data. Let $m_i$ be the class mark of the $i$th class. Let $f_i$ be the frequency of the $i$th class. The '''mean of the grouped data''' for $\mathsf C_1, \mathsf C_2, \cdots, \mathsf C_n$ is defined as: :$\displaystyle \bar x_{\mathsf C} := \frac 1 n \sum_{i \mathop = 1}^n m_i f_i$


Definition:Class (Descriptive Statistics)/Empty Class

A class is '''empty''' if it is of frequency zero.


Definition:Euclidean Metric/Real Number Line

On the real number line, the Euclidean metric can be seen to degenerate to: :$\map d {x, y} := \sqrt {\paren {x - y}^2} = \size {x - y}$ where $\size {x - y}$ denotes the absolute value of $x - y$.


Definition:Open Ball/Radius

In $\map {B_\epsilon} a$, the value $\epsilon$ is referred to as the '''radius''' of the open $\epsilon$-ball.


Definition:Affine Space

=== Associativity Axioms === {{:Definition:Affine Space/Associativity Axioms}} === Group Action === {{:Definition:Affine Space/Group Action}} === Weyl's Axioms === {{:Definition:Affine Space/Weyl's Axioms}} === Addition === {{:Definition:Addition (Affine Geometry)}} === Subtraction === {{:Definition:Subtraction (Affine Geometry)}} === Tangent Space === {{:Definition:Tangent Space (Affine Geometry)}} === Vector === {{:Definition:Vector (Affine Geometry)}} === Point === {{:Definition:Point (Affine Geometry)}}


Definition:Sequentially Compact Space/In Itself

A subspace $H \subseteq S$ is '''sequentially compact in itself''' {{iff}} every infinite sequence in $H$ has a subsequence which converges to a point in $H$. This is understood to mean that $H$ is sequentially compact when we consider it as a topological space with the induced topology of $T$.


Definition:P-Product Metric/General Definition

Let $M_{1'} = \left({A_{1'}, d_{1'}}\right), M_{2'} = \left({A_{2'}, d_{2'}}\right), \ldots, M_{n'} = \left({A_{n'}, d_{n'}}\right)$ be metric spaces. Let $\displaystyle \mathcal A = \prod_{i \mathop = 1}^n A_{i'}$ be the cartesian product of $A_{1'}, A_{2'}, \ldots, A_{n'}$. Let $p \in \R_{\ge 1}$. The '''$p$-product metric''' on $\mathcal A$ is defined as: : $\displaystyle d_p \left({x, y}\right) := \left({\sum_{i \mathop = 1}^n \left({d_{i'} \left({x_i, y_i}\right)}\right)^p}\right)^{\frac 1 p}$ where $x = \left({x_1, x_2, \ldots, x_n}\right), y = \left({y_1, y_2, \ldots, y_n}\right) \in \mathcal A$. The metric space $\mathcal M_p := \left({\mathcal A, d_p}\right)$ is the '''$p$-product (space)''' of $M_{1'}, M_{2'}, \ldots, M_{n'}$.


Definition:Affine Geometry

'''Affine geometry''' is the study of the geometry of affine spaces. It provides a modern axiomatic approach to the study of configurations of lines, planes and hypersurfaces. In particular an affine space can be thought of as a finite dimensional vector space with no distinguished origin, and its affine transformations are those that preserve collinearity.


Definition:Path-Connected/Topology/Points

Let $T = \left({S, \tau}\right)$ be a topological space. Let $a, b \in S$ be such that there exists a path from $a$ to $b$. That is, there exists a continuous mapping $f: \left[{0 \,.\,.\, 1}\right] \to S$ such that $f \left({0}\right) = a$ and $f \left({1}\right) = b$. Then $a$ and $b$ are '''path-connected in $T$'''.


Definition:Path-Connected/Topology/Set

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


Definition:Path-Connected/Topology/Topological Space

Let $T = \struct {S, \tau}$ be a topological space. Then $T$ is a '''path-connected space''' {{iff}} $S$ is a path-connected set of $T$. That is, $T$ is a '''path-connected space''' {{iff}}: :for every $x, y \in S$, there exists a continuous mapping $f: \closedint 0 1 \to S$ such that $\map f 0 = x$ and $\map f 1 = y$.


Definition:Path-Connected/Topology

=== Points in Topological Space === {{:Definition:Path-Connected/Topology/Points}} === Set of Topological Space === {{:Definition:Path-Connected/Topology/Set}} === Topological Space === {{:Definition:Path-Connected/Topology/Topological Space}}


Definition:Population Parameter

A '''population parameter''' is a numerical description of a population.


Definition:Sample Statistic

A '''sample statistic''' is a numerical description of a sample.


Definition:Lagrange Basis Polynomial

Let $x_0, \ldots, x_n \in \R$ be real numbers. The '''Lagrange basis polynomials''' associated to the $x_i$ are the polynomials: :$\displaystyle \map {L_j} X := \prod_{\substack {0 \mathop \le i \mathop \le n \\ i \mathop \ne j} } \frac {X - x_i} {x_j - x_i} \in \R \sqbrk X$ {{explain|$\R \sqbrk X$}} {{mistake|Not sure if it's a mistake or a different way of defining it, but {{BookReference|Dictionary of Mathematics|1989|Ephraim J. Borowski|author2 = Jonathan M. Borwein|entry = Lagrange interpolation formula}} has this wrapped up in another product symbol}} {{NamedforDef|Joseph Louis Lagrange|cat = Lagrange}}


Definition:Final Topology

Let $X$ be a set. Let $I$ be an indexing set. Let $\family {\struct{Y_i, \tau_i}}_{i \mathop \in I}$ be an $I$-indexed family of topological spaces. Let $\family {f_i: Y_i \to X}_{i \mathop \in I}$ be an $I$-indexed family of mappings. === Definition 1 === {{:Definition:Final Topology/Definition 1}}


Definition:Convex Set (Order Theory)

=== Definition 1 === {{:Definition:Convex Set (Order Theory)/Definition 1}} === Definition 2 === {{:Definition:Convex Set (Order Theory)/Definition 2}}


Definition:Path-Connected/Metric Space

Let $M = \left({A, d}\right)$ be a metric space. $M$ is defined as '''path-connected''' iff: :$\forall m, n \in A: \exists f: \left[{0 \,.\,.\, 1}\right] \to A: f \left({0}\right) = m, f \left({1}\right) = n$ where $f$ is a continuous mapping. === Subset of Metric Space === {{:Definition:Path-Connected/Metric Space/Subset}}


Definition:Path-Connected/Metric Space/Subset

Let $M = \struct {A, d}$ be a metric space. Let $S \subseteq A$ be a subset of $M$. Then $S$ is '''path-connected (in $M$)''' {{iff}}: :$\forall m, n \in S: \exists f: \closedint 0 1 \to S: \map f 0 = m, \map f 1 = n$ where $f$ is a continuous mapping.


Definition:Percentile

Let $D$ be a finite collection of data regarding some quantitative variable. Let $D$ be divided into precisely $100$ classes. A $P$th '''percentile''' is a value in the interval defined by the $P$th class such that: :$\dfrac P {100}$ of the data in $D$ fall at or below the value chosen :$1 - \dfrac P {100}$ of the data in $D$ fall at or above the value chosen. Arbitrarily more precise definitions may be contrived as necessary to define a unique percentile for a given study.


Definition:Quartile

Let $D$ be a finite collection of data regarding some quantitative variable. Let $D$ be divided into precisely $4$ classes. A $Q$th '''quartile''' is a value in the interval defined by the $Q$th class such that: :$\dfrac Q 4$ of the data in $D$ fall at or below the value chosen; :$1 - \dfrac Q 4$ of the data in $D$ fall at or above the value chosen. Arbitrarily more precise definitions may be contrived as necessary to define a unique quartile for a given study. A common convention is: * The '''second quartile''', $Q_2$, is defined as the median of $D$ * The '''first quartile''', $Q_1$, is defined as the median of the data values below and not including $Q_2$  * The '''third quartile''', $Q_3$, is defined as the median of the data values above and not including $Q_2$


Definition:Interquartile Range

Let $Q_1$ and $Q_3$ be first and third quartiles. The '''interquartile range''' is defined and denoted as: :$\operatorname {IQR} := Q_3 - Q_1$


Definition:Coequalizer

Let $\mathbf C$ be a metacategory. Let $f, g: C \to D$ be morphisms with common domain and codomain. An '''equalizer''' for $f$ and $g$ is a morphism $q: D \to Q$ such that: :$q \circ f = q \circ g$ and subject to the following UMP: :For any $z: D \to Z$ such that $z \circ f = z \circ f$, there is a unique $u: Q \to Z$ such that: :::::$\begin{xy}\xymatrix{  C   \ar[r]<2pt>^*{f}   \ar[r]<-2pt>_*{g} &   D   \ar[r]^*{q}   \ar[rd]_*{z} &  Q   \ar@{.>}[d]^*{u} \\ & &  Z }\end{xy}$ :is a commutative diagram. I.e., $z = u \circ q$.


Definition:Ordering on Multiindices

Let $Z$ be the set of multiindices. We define an ordering on $Z$ as follows. If $k = \left \langle {k_j}\right \rangle_{j \in J}$ and $\ell = \left \langle {\ell_j}\right \rangle_{j \in J}$ are multiindices, then we write $k \preceq \ell$ if: :$\left( \forall j \in J \right) \left( k_j \le \ell_j \right)$ By abuse of notation, it is standard to use $\leq$ to denote the ordering on the set of multiindices as well.


Definition:Addition/Multiindices

The addition operation in the domain of multiindices $Z$ is written $+$. Let $k = \sequence {k_j}_{j \mathop \in J}$ and $l = \sequence {l_j}_{j \mathop \in J}$ be multiindices. Then $k + l$ is defined as: :$\paren {\forall j \in J} \paren {\paren {k + l}_j = k_j + l_j}$


Definition:Multiindex/Modulus

Let $k = \sequence {k_j}_{j \mathop \in J}$ be a multiindex. The '''modulus''' of such a multiindex $k$  is defined by: :$\displaystyle \size k = \sum_{j \mathop \in J} k_j$


Definition:Binomial Coefficient/Multiindices

Let $k = \left \langle {k_j}\right \rangle_{j \mathop \in J}$ and $\ell = \left \langle {\ell_j}\right \rangle_{j \mathop \in J}$ be multiindices. Let $\ell \le k$. Then $\dbinom k \ell$ is defined as: :$\displaystyle \binom k \ell = \prod_{j \mathop \in J} \binom {k_j} {\ell_j}$ Note that since by definition only finitely many of the $k_j$ are non-zero, the product in the definition of $\dbinom k \ell$ is convergent.


Definition:Power (Algebra)/Multiindices

Let $k = \left \langle {k_j}\right \rangle_{j = 1, \ldots, n}$ be a multiindex indexed by $\left\{{1, \ldots, n}\right\}$. Let $x = \left({x_1, \ldots, x_n}\right) \in \R^n$ be an ordered tuple of real numbers. Then $x^k$ is defined as: :$\displaystyle x^k := \prod_{j \mathop = 1}^n x_j^{k_j}$ where the powers on the right hand side are integer powers.


Definition:Coordinate System/Origin

The '''origin''' of a coordinate system is the zero vector. In the $x y$-plane, it is the point: :$O = \tuple {0, 0}$ and in general, in the Euclidean space $\R^n$: :$O = \underbrace {\tuple {0, 0, \ldots, 0} }_{\text{$n$ coordinates} }$ Thus it is the point where the axes cross over each other.


Definition:Neighborhood (Metric Space)

Let $M = \struct {A, d}$ be a metric space. Let $S \subseteq A$ be a subset of $A$. Let $x \in S$. Let there exist $\epsilon \in \R_{>0}$ such that the open $\epsilon$-ball at $x$ lies completely in $S$, that is: :$\map {B_\epsilon} x \subseteq S$ Then $S$ is a '''neighborhood of $x$ in $M$'''.


Definition:Opposite Magma

$\struct {S, *}$ is the '''opposite magma of $\struct {S, \circ}$''' {{iff}}: :$\forall x_1, x_2, x_3 \in S: x_1 \circ x_2 = x_3 \iff x_2 * x_1 = x_3$


Definition:Partial Differential Operator

Let $U \subseteq \R^n$ be a open set. Let $\CC \subseteq \map {\CC^k} {U, \R}$ be a set of $k$-times continuously differentiable functions. Let $\displaystyle \partial_i = \frac {\partial} {\partial x_i}$ denote the partial derivative, $i = 1, \ldots, n$. For a multiindex $\alpha = \tuple {\alpha_1, \ldots, \alpha_n}$ indexed by $\set {1, \ldots, n}$ let $\partial^\alpha = \partial_1^{\alpha_1} \cdots \partial_n^{\alpha_n}$. A mapping $T : \CC \to \map {\CC^k} {U, \R}$ is a '''partial differential operator''' if there exist $r \in \N$ and functions $f_\alpha : \R^n \to \C$ for each multiindex $\alpha$ with $\cmod \alpha \le r$ such that for all $g \in \CC$: :$\displaystyle \map T g = \sum_{\cmod \alpha \mathop \le r} f_\alpha \partial^\alpha g$ Category:Definitions/Partial Differentiation ssux2u3lv8vz8e0zdzfrq79l0aq2xei


Definition:Region/Metric Space

Let $M = \left({A, d}\right)$ be a metric space. A '''region''' of $M$ is a subset $U$ of $M$ such that $U$ is: :$(1): \quad$ non-empty :$(2): \quad$ path-connected.


Definition:Open Set/Pseudometric Space

Let $P = \struct {A, d}$ be a pseudometric space. An '''open set''' in $P$ is defined in exactly the same way as for a metric space: $U$ is an '''open set in $P$''' {{iff}}: :$\forall y \in U: \exists \map \epsilon y > 0: \map {B_\epsilon} y \subseteq U$ where $\map {B_\epsilon} y$ is the open $\epsilon$-ball of $y$.


Definition:Relatively Closed Set

Let $T = \left({S, \tau}\right)$ be a topological space. Let $A \subseteq B \subseteq S$. === Definition 1 === {{:Definition:Relatively Closed Set/Definition 1}} === Definition 2 === {{:Definition:Relatively Closed Set/Definition 2}}


Definition:Closed Point

Let $a \in S$. Then '''$a$ is closed (in $T\,$)''' {{iff}} $\left\{{a}\right\}$ is a closed set (in $T\,$).


Definition:Closed Set/Metric Space

{{:Definition:Closed Set/Metric Space/Definition 1}}


Definition:Bounded Mapping/Metric Space

Let $M$ be a metric space. Let $f: X \to M$ be a mapping from any set $X$ into $M$. Then $f$ is a '''bounded mapping''' {{iff}} $\map f X$ is bounded in $M$.


Definition:Bounded Metric Space/Unbounded

Let $M = \left({X, d}\right)$ be a metric space. Let $M' = \left({Y, d_Y}\right)$ be a subspace of $M$. Then '''$M'$ is unbounded (in $M$)''' {{iff}} $M'$ is not bounded in $M$.


Definition:Number Field

A '''number field''' is a subfield of the field of complex numbers.


Definition:Real Interval/Endpoints

The numbers $a, b \in \R$ are known as the '''endpoints''' of the interval. $a$ is sometimes called the '''left hand endpoint''' and $b$ the '''right hand endpoint''' of the interval.


Definition:Real Interval/Length

The difference $b - a$ between the endpoints is called the '''length''' of the interval.


Definition:Real Interval/Midpoint

The '''midpoint''' of a real interval is the number: : $\dfrac {a + b} 2$


Definition:Real Interval/Open

The '''open (real) interval''' from $a$ to $b$ is defined as: :$\openint a b := \set {x \in \R: a < x < b}$


Definition:Real Interval/Half-Open

There are two '''half-open (real) intervals''' from $a$ to $b$. ==== Right half-open ==== {{Definition:Real Interval/Half-Open/Right}} ==== Left half-open ==== {{Definition:Real Interval/Half-Open/Left}}


Definition:Real Interval/Closed

The '''closed (real) interval''' from $a$ to $b$ is defined as: :$\closedint a b = \set {x \in \R: a \le x \le b}$


Definition:Real Interval/Unbounded Closed

There are two '''unbounded closed intervals''' involving a real number $a \in \R$, defined as: {{begin-eqn}} {{eqn | l = \hointr a \to       | o = :=       | r = \set {x \in \R: a \le x} }} {{eqn | l = \hointl \gets a       | o = :=       | r = \set {x \in \R: x \le a} }} {{end-eqn}}


Definition:Real Interval/Unbounded Open

There are two '''unbounded open intervals''' involving a real number $a \in \R$, defined as: {{begin-eqn}} {{eqn | l = \openint a \to       | o = :=       | r = \set {x \in \R: a < x} }} {{eqn | l = \openint \gets a       | o = :=       | r = \set {x \in \R: x < a} }} {{end-eqn}} Using the same symbology, the set $\R$ can be represented as an '''unbounded open interval''' with no endpoints: :$\openint \gets \to = \R$


Definition:Real Interval/Empty

When $a > b$: {{begin-eqn}} {{eqn | l = \left [{a \,.\,.\, b} \right]       | m = \left\{ {x \in \R: a \le x \le b}\right\}       | mo = =       | r = \varnothing }} {{eqn | l = \left [{a \,.\,.\, b} \right)       | m = \left\{ {x \in \R: a \le x < b}\right\}       | mo = =       | r = \varnothing }} {{eqn | l = \left ({a \,.\,.\, b} \right]       | m = \left\{ {x \in \R: a < x \le b}\right\}       | mo = =       | r = \varnothing }} {{eqn | l = \left ({a \,.\,.\, b} \right)       | m = \left\{ {x \in \R: a < x < b}\right\}       | mo = =       | r = \varnothing }} {{end-eqn}} When $a = b$: : $\left [{a \,.\,.\, b} \right) = \left [{a \,.\,.\, a} \right) = \left\{{x \in \R: a \le x < a}\right\} = \varnothing$ : $\left ({a \,.\,.\, b} \right] = \left ({a \,.\,.\, a} \right] = \left\{{x \in \R: a < x \le a}\right\} = \varnothing$ : $\left ({a \,.\,.\, b} \right) = \left ({a \,.\,.\, a} \right) = \left\{{x \in \R: a < x < a}\right\} = \varnothing$ Such empty sets are referred to as '''empty intervals'''.


Definition:Real Interval/Singleton

When $a = b$: :$\closedint a b = \closedint a a = \set {x \in \R: a \le x \le a} = \set a$


Definition:Interval/Notation/Wirth

{{begin-eqn}} {{eqn | l = \openint a b       | o = :=       | r = \set {s \in S: \paren {a \prec s} \land \paren {s \prec b} }       | c = Open Interval }} {{eqn | l = \hointr a b       | o = :=       | r = \set {s \in S: \paren {a \preccurlyeq s} \land \paren {s \prec b} }       | c = Right Half-Open Interval }} {{eqn | l = \hointl a b       | o = :=       | r = \set {s \in S: \paren {a \prec s} \land \paren {s \preccurlyeq b} }       | c = Left Half-Open Interval }} {{eqn | l = \closedint a b       | o = :=       | r = \set {s \in S: \paren {a \preccurlyeq s} \land \paren {s \preccurlyeq b} }       | c = Closed Interval }} {{end-eqn}}


Definition:Real Interval/Notation/Conventional

These are the notations usually seen for real intervals: {{begin-eqn}} {{eqn | l = \left ({a, b}\right)       | o = :=       | r = \set {x \in \R: a < x < b}       | c = Open real interval }} {{eqn | l = \left [{a, b}\right)       | o = :=       | r = \set {x \in \R: a \le x < b}       | c = Half-open real interval }} {{eqn | l = \left ({a, b}\right]       | o = :=       | r = \set {x \in \R: a < x \le b}       | c = Half-open real interval }} {{eqn | l = \left [{a, b}\right]       | o = :=       | r = \set {x \in \R: a \le x \le b}       | c = Closed real interval }} {{end-eqn}} but they can be confused with other usages for this notation. In particular, there exists the danger of taking $\paren {a, b}$ to mean an ordered pair.


Definition:Real Interval/Notation/Reverse-Bracket

In order to avoid the ambiguity problem arising from the conventional notation for intervals where an open real interval can be confused with an ordered pair, some authors use the '''reverse-bracket notation''' for open and half-open intervals: {{begin-eqn}} {{eqn | l = \left ] {\, a, b} \right [       | o = :=       | r = \set {x \in \R: a < x < b}       | c = Open real interval }} {{eqn | l = \left [ {a, b} \right [       | o = :=       | r = \set {x \in \R: a \le x < b}       | c = Half-open on the right }} {{eqn | l = \left ] {\, a, b} \right ]       | o = :=       | r = \set {x \in \R: a < x \le b}       | c = Half-open on the left }} {{end-eqn}} These are often considered to be both ugly ''and'' confusing, and hence are limited in popularity.


Definition:Interval/Notation/Unbounded Intervals

In Wirth interval notation, unbounded intervals of an ordered set $\struct {S, \preccurlyeq}$ are written as follows: {{begin-eqn}} {{eqn | l = \hointr a \to       | o = :=       | r = \set {x \in S: a \preccurlyeq x} }} {{eqn | l = \hointl \gets a       | o = :=       | r = \set {x \in S: x \preccurlyeq a} }} {{eqn | l = \openint a \to       | o = :=       | r = \set {x \in S: a \prec x} }} {{eqn | l = \openint \gets a       | o = :=       | r = \set {x \in S: x \prec a} }} {{eqn | l = \openint \gets \to       | o = :=       | r = \set {x \in S} = S }} {{end-eqn}}


Definition:Real Interval/Notation

An arbitrary (real) interval is frequently denoted $\mathbb I$. Sources which use the $\textbf {boldface}$ font for the number sets $\N, \Z, \Q, \R, \C$ tend also to use $\mathbf I$ for this entity. Some sources merely use the ordinary $\textit {italic}$ font $I$. === Wirth Interval Notation === {{:Definition:Real Interval/Notation/Wirth}}


Definition:Hereditarily Compact Space

Let $T = \left({S, \tau}\right)$ be a topological space. === Definition 1 === {{:Definition:Hereditarily Compact Space/Definition 1}} === Definition 2 === {{:Definition:Hereditarily Compact Space/Definition 2}}


{{:Definition:Compact Space/Topology/Subspace/Definition 2}}


Definition:Compact Space/Euclidean Space

Let $\R^n$ denote Euclidean $n$-space. Let $H \subseteq \R^n$. Then $H$ is '''compact in $\R^n$''' {{iff}} $H$ is closed and bounded. === Real Analysis === The same definition applies when $n = 1$, that is, for the real number line: {{:Definition:Compact Space/Real Analysis}} === Complex Analysis === {{:Definition:Compact Space/Complex Analysis}}


Definition:Subobject

Let $\mathbf C$ be a metacategory. Let $C$ be an object of $\mathbf C$. A '''subobject''' of $C$ is a monomorphism $m: B \rightarrowtail C$ with codomain $C$.


Definition:Category of Subobjects

Let $\mathbf C$ be a metacategory. Let $C$ be an object of $\mathbf C$. The '''category of subobjects of $C$''', denoted $\mathbf{Sub}_{\mathbf C} \left({C}\right)$, is defined as follows: {{DefineCategory |ob = Subobjects $m: B \to C$ of $C$ |mor = Morphisms $f: \operatorname{dom} m \to \operatorname{dom} m'$ of $\mathbf C$ such that $m' \circ f = m$ in $\mathbf C$ |comp = Inherited from $\mathbf C$ |id = $\operatorname{id}_m := \operatorname{id}_{\operatorname{dom} m}$, the identity morphism in $\mathbf C$ of the domain of $m$ }} The behaviour of the morphisms is shown in the following commutative diagram in $\mathbf C$: ::$\begin{xy}\xymatrix@+1em{  B   \ar[r]^*+{f}   \ar[rd]_*+{m} &  B'   \ar[d]^*+{m'} \\ &  C }\end{xy}$


Definition:Sphere/Geometry

A '''sphere''' is a surface in solid geometry such that all straight lines falling upon it from one particular point inside it are equal. {{EuclidDefinition|book = XI|def = 14|name = Sphere}}


Definition:Sphere/Geometry/Center

That point is called the '''center''' of the sphere.


Definition:Sphere/Geometry/Radius

A '''radius''' of a sphere is a straight line segment whose endpoints are the center and the surface of the sphere. '''The radius''' of a sphere is the length of one such radius.


Definition:Sphere/Topology

The $n$-dimensional '''sphere''', or '''$n$-sphere''', is the set: :$\Bbb S^n = \set {x \in \R^{n + 1} : \size {x - y} = r}$ where $\size {\, \cdot \, }$ denotes the Euclidean distance.


Definition:Sphere/Topology/Unit Sphere

The $n$-dimensional '''unit sphere''', or '''unit $n$-sphere''', is the $n$-sphere of radius $1$ and center the origin: :$\Bbb S^n = \left\{{x \in \R^{n+1} : \left\lvert{x}\right\rvert = 1}\right\}$


Definition:Inclusion Relation on Subobjects

Let $\mathbf C$ be a metacategory. Let $C$ be an object of $\mathbf C$. Let $\mathbf{Sub}_{\mathbf C} \left({C}\right)$ be the category of subobjects of $C$. The '''inclusion relation $\subseteq$''' on subobjects of $C$ is defined as follows: :$m \subseteq m'$ iff there exists a morphism $f: m \to m'$


Definition:Homotopy/Relative

Let $K \subseteq X$ be a subset of $X$. We say that $f$ and $g$ are '''homotopic relative to $K$''' if there exists a free homotopy $H$ between $f$ and $g$, and: :$(1): \quad \forall x \in K: f \left({x}\right) = g \left({x}\right)$ :$(2): \quad \forall x \in K, t \in \left[{0 \,.\,.\, 1}\right]: H \left({x, t}\right) = f \left({x}\right)$


Definition:Equivalent Subobjects

Let $\mathbf C$ be a metacategory. Let $C$ be an object of $\mathbf C$. Let $\mathbf{Sub}_{\mathbf C} \left({C}\right)$ be the category of subobjects of $C$. Two subobjects $m, m'$ of $C$ are said to be '''equivalent''' iff: :$m \subseteq m'$ and $m' \subseteq m$ where $\subseteq$ denotes the inclusion relation on subobjects.


Definition:Subobject Class

Let $\mathbf C$ be a metacategory. Let $C$ be an object of $\mathbf C$. Let $\mathbf{Sub}_{\mathbf C} \left({C}\right)$ be the category of subobjects of $C$. A '''subobject class of $C$''' is an equivalence class of subobjects of $C$ under the equivalence of subobjects. If $m$ is a subobject, its associated '''subobject class''' may be denoted by $\overline m$ or $\left[\!\left[{m}\right]\!\right]$. === Morphism Class === {{:Definition:Subobject Class/Morphism Class}}


Definition:Homotopy/Free

Let $X$ and $Y$ be topological spaces. Let $f: X \to Y$, $g: X \to Y$ be continuous mappings. Then $f$ and $g$ are '''(freely) homotopic''' {{iff}} there exists a continuous mapping: : $H: X \times \left[{0 \,.\,.\, 1}\right] \to Y$ such that, for all $x \in X$: : $H \left({x, 0}\right) = f \left({x}\right)$ and: : $H \left({x, 1}\right) = g \left({x}\right)$ $H$ is called a '''(free) homotopy between $f$ and $g$'''.


Definition:Category of Subobject Classes

Let $\mathbf C$ be a metacategory. Let $C$ be an object of $\mathbf C$. Let $\mathbf{Sub}_{\mathbf C} \left({C}\right)$ be the category of subobjects of $C$. The '''category of subobject classes of $C$''', denoted $\overline{\mathbf{Sub}}_{\mathbf C} \left({C}\right)$, is defined as follows: {{DefineCategory |ob = Subobject classes $\left[\!\left[{m}\right]\!\right]$ of $C$ |mor = Morphism classes $\left[\!\left[{f}\right]\!\right]: \left[\!\left[{m}\right]\!\right] \to \left[\!\left[{m'}\right]\!\right]$ |comp = Inherited from $\mathbf{Sub}_{\mathbf C} \left({C}\right)$, i.e. $\left[\!\left[{g}\right]\!\right] \circ \left[\!\left[{f}\right]\!\right] := \left[\!\left[{g \circ f}\right]\!\right]$ |id = $\operatorname{id}_{\left[\!\left[{m}\right]\!\right]} := \left[\!\left[{\operatorname{id}_m}\right]\!\right]$, the morphism class of the identity morphism of $m$ in $\mathbf{Sub}_{\mathbf C} \left({C}\right)$ }}


Definition:Subobject Class/Morphism Class

Define the equivalence $\sim$ on the morphisms of $\mathbf{Sub}_{\mathbf C} \left({C}\right)$ as follows. For morphisms $f: m \to n$ and $g: m' \to n'$ of $\mathbf{Sub}_{\mathbf C} \left({C}\right)$: :$f \sim g$ iff $m \sim m'$ and $n \sim n'$ where $m \sim m'$ signifies equivalence of subobjects. That $\sim$ in fact is an equivalence is shown on Morphism Class Equivalence is Equivalence. A '''morphism class''' is an equivalence class $\left[\!\left[{f}\right]\!\right]$ under $\sim$ of a morphism $f: m \to m'$. The domain and codomain of $\left[\!\left[{f}\right]\!\right]$ are taken to be $\left[\!\left[{m}\right]\!\right]$ and $\left[\!\left[{m'}\right]\!\right]$, respectively.


Definition:Inclusion Relation on Subobject Classes

Let $\mathbf C$ be a metacategory. Let $C$ be an object of $\mathbf C$. The '''inclusion relation $\subseteq$''' on subobject classes of $C$ is defined as follows: :$\left[\!\left[{m}\right]\!\right] \subseteq \left[\!\left[{m'}\right]\!\right]$ iff there exists a morphism $\left[\!\left[{f}\right]\!\right]: \left[\!\left[{m}\right]\!\right] \to \left[\!\left[{m'}\right]\!\right]$


Definition:Linear Continuum

A totally ordered set $\struct {S, \preceq}$ is a '''linear continuum''' {{iff}}: :$(1): \quad \struct {S, \preceq}$ is densely ordered :$(2): \quad \struct {S, \preceq}$ is Dedekind complete


Definition:Close Packed/Subset

A subset $T \subseteq S$ is said to be '''close packed in $\left({S, \preceq}\right)$''' {{iff}}: :$\forall a, b \in S: a \prec b \implies \exists c \in T: a \prec c \prec b$


Definition:Increasing/Sequence/Real Sequence

Let $\sequence {x_n}$ be a sequence in $\R$. Then $\sequence {x_n}$ is '''increasing''' {{iff}}: :$\forall n \in \N: x_n \le x_{n + 1}$


Definition:Decreasing/Sequence/Real Sequence

Let $\sequence {x_n}$ be a sequence in $\R$. Then $\sequence {x_n}$ is '''decreasing''' {{iff}}: :$\forall n \in \N: x_{n + 1} \le x_n$


Definition:Strictly Increasing/Sequence/Real Sequence

Let $\sequence {x_n}$ be a sequence in $\R$. Then $\sequence {x_n}$ is '''strictly increasing''' {{iff}}: :$\forall n \in \N: x_n < x_{n + 1}$


Definition:Strictly Decreasing/Sequence/Real Sequence

Let $\sequence {x_n}$ be a sequence in $\R$. Then $\sequence {x_n}$ is '''strictly decreasing''' {{iff}}: :$\forall n \in \N: x_{n + 1} < x_n$


Definition:Monotone (Order Theory)/Sequence/Real Sequence

Let $\sequence {x_n}$ be a sequence in $\R$. Then $\sequence {x_n}$ is '''monotone''' {{iff}} it is either increasing or decreasing.


Definition:Strictly Monotone/Sequence/Real Sequence

Let $\sequence {x_n}$ be a sequence in $\R$. Then $\sequence {x_n}$ is '''strictly monotone''' {{iff}} it is either strictly increasing or strictly decreasing.


Definition:Archimedean Property/Norm

Let $n: S \to \R$ be a norm on $S$. {{explain|What is a norm on a general algebraic structure?}} {{Disambiguate|Definition:Norm}} Then $n$ satisfies the '''Archimedean property on $S$''' {{iff}}: :$\forall a, b \in S: n \paren a < n \paren b \implies \exists m \in \N: n \paren {m \cdot a} > n \paren b$ Using the more common symbology for a norm: :$\forall a, b \in S: \norm a < \norm b \implies \exists m \in \Z_{>0}: \norm {m \cdot a} > \norm b$


Definition:Archimedean Property/Ordering

Let $\preceq$ be an ordering on $S$. Then $\preceq$ satisfies the '''Archimedean property on $S$''' {{iff}}: :$\forall a, b \in S: a \prec b \implies \exists m \in \Z_{>0}: b \prec m \cdot a$


Definition:Cauchy Sequence/Metric Space

Let $M = \struct {A, d}$ be a metric space. Let $\sequence {x_n}$ be a sequence in $M$. Then $\sequence {x_n}$ is a '''Cauchy sequence''' {{iff}}: :$\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall m, n \in \N: m, n \ge N: \map d {x_n, x_m} < \epsilon$


Definition:Net (Metric Space)/Finite Net

A '''finite $\epsilon$-net for $M$''' is an $\epsilon$-net for $M$ which is a finite set.


Definition:Cauchy Sequence/Real Numbers

Let $\sequence {x_n}$ be a sequence in $\R$. Then $\sequence {x_n}$ is a '''Cauchy sequence''' {{iff}}: : $\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall m, n \in \N: m, n \ge N: \size {x_n - x_m} < \epsilon$


Definition:Cauchy Sequence/Rational Numbers

Let $\left \langle {x_n} \right \rangle$ be a rational sequence. Then $\left \langle {x_n} \right \rangle$ is a '''Cauchy sequence''' {{iff}}: : $\forall \epsilon \in \Q_{>0}: \exists N \in \N: \forall m, n \in \N: m, n \ge N: \left|{x_n - x_m}\right| < \epsilon$ where $\Q_{>0}$ denotes the set of all strictly positive rational numbers.


Definition:Euclidean Space/Real

Let $\R^n$ be an $n$-dimensional real vector space. Let the Euclidean metric $d$ be applied to $\R^n$. Then $\struct {\R^n, d}$ is a '''Euclidean $n$-space'''.


Definition:Euclidean Space/Rational

Let $\Q^n$ be an $n$-dimensional vector space of rational numbers. Let the Euclidean Metric $d$ be applied to $\Q^n$. Then $\left({\Q^n, d}\right)$ is a Euclidean $n$-space.


Definition:Euclidean Space/Complex

Let $\C$ be the complex plane. Let $d$ be the Euclidean metric on $\C$. Then $\left({\C, d}\right)$ is a Euclidean space.


Definition:Euclidean Space/Euclidean Topology

Let $S$ be one of the standard number fields $\Q$, $\R$, $\C$. Let $S^n$ be a cartesian space for $n \in \N_{\ge 1}$. Let $M = \left({S^n, d}\right)$ be a Euclidean space. The topology $\tau_d$ induced by the Euclidean metric $d$ is called the '''Euclidean topology'''.


Definition:Discriminant of Polynomial/Quadratic Equation

The expression $b^2 - 4 a c$ is called the '''discriminant''' of the equation.


Definition:Linearly Ordered Space

Let $\struct {S, \preceq}$ be a linearly ordered set. Let $\tau$ be the order topology on $S$. Then $\struct {S, \preceq, \tau}$ is a '''linearly ordered space'''.


Definition:Bounded Mapping/Real-Valued/Attaining its Bounds

Let $f: S \to \R$ be a bounded real-valued function. Let $T$ be a subset of $S$. Suppose that: :$\exists a, b \in T: \forall x \in S: f \left({a}\right) \le f \left({x}\right) \le f \left({b}\right)$ Then $f$ '''attains its bounds on $T$'''.


Definition:Bounded Mapping/Real-Valued

{{:Definition:Bounded Mapping/Real-Valued/Definition 1}}


Definition:Bounded Mapping/Complex-Valued

Let $f: S \to \C$ be a complex-valued function. Then $f$ is '''bounded''' {{iff}} the real-valued function $\cmod f: S \to \R$ is bounded, where $\cmod f$ is the modulus of $f$. That is, $f$ is '''bounded''' if there is a constant $K \ge 0$ such that $\cmod {f \paren z} \le K$ for all $z \in S$.


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

$f$ is '''continuous from $\left({A_1, d_1}\right)$ to $\left({A_2, d_2}\right)$''' {{iff}}: :for every $U \subseteq A_2$ which is open in $M_2$, $f^{-1} \left[{U}\right]$ is open in $M_1$.


Definition:Composition Functor on Categories of Subobjects

Let $\mathbf C$ be a metacategory. Let $C$ and $D$ be objects of $\mathbf C$. Let $\mathbf{Sub}_{\mathbf C} \left({C}\right)$ and $\mathbf{Sub}_{\mathbf C} \left({D}\right)$ be the associated categories of subobjects. Let $g: C \to D$ be a monomorphism of $\mathbf C$. Then $g$ defines a '''composition functor''' $g_* : \mathbf{Sub}_{\mathbf C} \left({C}\right) \to \mathbf{Sub}_{\mathbf C} \left({D}\right)$: {{begin-axiom}} {{axiom|lc= Object functor:        |m = g_* f := g \circ f        |rc= The composition $\circ$ is taken in $\mathbf C$ }} {{axiom|lc= Morphism functor:        |m = g_* a := a }} {{end-axiom}} That it is in fact a functor is shown on Composition Functor on Categories of Subobjects is Functor. The effect of $g_*$ is captured in the following commutative diagram: ::$\begin{xy} <-3em,0em>*+{X} = "X", <3em,0em>*+{X'} = "X2", <0em,-4em>*+{C} = "C", <0em,-8em>*+{D} = "D", "X";"X2" **@{-} ?>*@{>} ?*!/_1em/{a}, "X";"C" **@{-} ?>*@{>} ?<>(.4)*!/^.6em/{f}, "X2";"C" **@{-} ?>*@{>} ?<>(.4)*!/_.6em/{f'}, "C";"D" **@{-} ?>*@{>} ?<>(.4)*!/^.6em/{g}, "X";"D" **\crv{<-5em,-4em>} ?>*@{>} ?*!/^1.6em/{g_* f = \\ g \circ f}, "X2";"D" **\crv{<5em,-4em>} ?>*@{>} ?*!/_1.6em/{g_* f' = \\ g \circ f'}, \end{xy}$


Definition:Local Membership Relation

Let $\mathbf C$ be a metacategory. Let $C$ be an object of $\mathbf C$. The '''local membership relation''' $\in_C$ between variable elements $z: Z \to C$ and subobjects $m: M \to C$ of $C$ is defined by: :$z \in_C m$ iff there exists an $f: Z \to M$: $z = m \circ f$. If $z \in_C m$, one says that $z$ is a '''local member''' of $m$.


Definition:Lipschitz Continuity

Let $M = \struct {A, d}$ and $M' = \struct {A', d'}$ be metric spaces. Let Let $f: A \to A'$ be a mapping. Then $f$ is a '''Lipschitz continuous mapping''' {{iff}} there exists a positive real number $K \in \R_{\ge 0}$ such that: :$\forall x, y \in A: \map {d'} {\map f x, \map f y} \le K \map d {x, y}$ That is, the distance between the images of two points lies within a fixed multiple of the distance between the points. === At a Point === {{:Definition:Lipschitz Continuity/Point}} === Lipschitz Constant === {{:Definition:Lipschitz Continuity/Lipschitz Constant}} === Real Function === The concept can be directly applied to the real numbers considered as a metric space under the usual topology: {{:Definition:Lipschitz Continuity/Real Function}}


Definition:Lipschitz Continuity/Lipschitz Constant

Let $f: A \to A'$ be a (Lipschitz continuous) mapping such that: : $\forall x, y \in A: d\,' \left({f \left({x}\right), f \left({y}\right)}\right) \le K d \left({x, y}\right)$ where $K \in \R_{\ge 0}$ is a positive real number. Then $K$ is '''a Lipschitz constant for $f$'''.


Definition:Lipschitz Continuity/Real Function

Let $A \subseteq \R$. Let $f: A \to \R$ be a real function. Let $I \subseteq A$ be a real interval on which: :$\exists K \in \R_{\ge 0}: \forall x, y \in I: \size {\map f x - \map f y} \le K \size {x - y}$ Then $f$ is '''Lipschitz continuous on $I$'''.


=== Definition 1 === {{:Definition:Lipschitz Equivalence/Metrics/Definition 1}} === Definition 2 === {{:Definition:Lipschitz Equivalence/Metrics/Definition 2}}


Definition:Pullback (Category Theory)

Let $\mathbf C$ be a metacategory. Let $f: A \to C$ and $g: B \to C$ be morphisms with common codomain. A '''pullback''' of $f$ and $g$ is a commutative diagram: ::$\begin{xy}\xymatrix{  P   \ar[r]^*+{p_1}   \ar[d]_*+{p_2} &  A   \ar[d]^*+{f} \\  B   \ar[r]_*+{g} &  C }\end{xy}$ such that $f \circ p_1 = g \circ p_2$, subject to the following UMP: :For any commutative diagram: :::$\begin{xy}\xymatrix{  Q   \ar[r]^*+{q_1}   \ar[d]_*+{q_2} &  A   \ar[d]^*+{f} \\  B   \ar[r]_*+{g} &  C }\end{xy}$ :there is a unique morphism $u: Q \to P$ making the following diagram commute: :::$\begin{xy}\xymatrix@+1em{  Q   \ar@/^/[rrd]^*+{q_1}   \ar@/_/[ddr]_*+{q_2}   \ar@{-->}[rd]^*+{u} \\ &  P   \ar[r]_*+{p_1}   \ar[d]^*+{p_2} &  A   \ar[d]^*+{f} \\ &  B   \ar[r]_*+{g} &  C }\end{xy}$ In this situation, $p_1$ is called the '''pullback of $f$ along $g$''' and may be denoted as $g^* f$. Similarly, $p_2$ is called the '''pullback of $g$ along $f$''' and may be denoted $f^* g$.


Let $T_\alpha = \left({S_\alpha, \tau_\alpha}\right)$ and $T_\beta = \left({S_\beta, \tau_\beta}\right)$ be topological spaces. Let $f: T_\alpha \to T_\beta$ be a bijection. === Definition 1 === {{:Definition:Homeomorphism/Topological Spaces/Definition 1}} === Definition 2 === {{:Definition:Homeomorphism/Topological Spaces/Definition 2}} === Definition 3 === {{:Definition:Homeomorphism/Topological Spaces/Definition 3}} === Definition 4 === {{:Definition:Homeomorphism/Topological Spaces/Definition 4}}


Definition:Homeomorphism/Manifolds

Let $X$ and $Y$ be manifolds. A '''homeomorphism''' of $X$ to $Y$ is a continuous bijection such that the inverse is also continuous.


Definition:Lipschitz Equivalence/Metric Spaces

Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces. Let $f: M_1 \to M_2$ be a mapping such that $\exists h, k \in \R_{>0}$ such that: :$\forall x, y \in A: h d_2 \left({f \left({x}\right), f \left({y}\right)}\right) \le d_1 \left({x, y}\right) \le k d_2 \left({f \left({x}\right), f \left({y}\right)}\right)$ Then $f$ is a '''Lipschitz equivalence''', and $M_1$ and $M_2$ are  described as '''Lipschitz equivalent'''.


Definition:Open Set Axioms

{{begin-axiom}} {{axiom | n = \text O 1         | t = The union of an arbitrary subset of $\tau$ is an element of $\tau$. }} {{axiom | n = \text O 2         | t = The intersection of any two elements of $\tau$ is an element of $\tau$. }} {{axiom | n = \text O 3         | t = $S$ is an element of $\tau$. }} {{end-axiom}}


Let $x \in S_1$. === Definition using Open Sets === {{Definition:Continuous Mapping (Topology)/Point/Open Sets}} === Definition using Filters === {{Definition:Continuous Mapping (Topology)/Point/Filters}} If necessary, we can say that '''$f$ is $\left({\tau_1, \tau_2}\right)$-continuous at $x$'''.


Definition:Continuous Mapping (Topology)/Set

Let $S$ be a subset of $S_1$. The mapping $f$ is '''continuous on $S$''' {{iff}} $f$ is continuous at every point $x \in S$.


=== Definition by Pointwise Continuity === {{Definition:Continuous Mapping (Topology)/Everywhere/Pointwise}} === Definition by Open Sets === {{Definition:Continuous Mapping (Topology)/Everywhere/Open Sets}}


Definition:Continuous Mapping (Topology)/Everywhere/Open Sets

The mapping $f$ is '''continuous on $S_1$''' {{iff}}: :$U \in \tau_2 \implies f^{-1} \sqbrk U \in \tau_1$ where $f^{-1} \sqbrk U$ denotes the preimage of $U$ under $f$.


Definition:Discrete Topology/Finite

Let $S$ be a finite set. Then $\tau = \powerset S$ is a '''finite discrete topology''', and $\struct {S, \tau} = \struct {S, \powerset S}$ is a '''finite discrete space'''.


Definition:Discrete Topology/Countable

Let $S$ be a countably infinite set. Then $\tau = \powerset S$ is a '''countable discrete topology''', and $\struct {S, \tau} = \struct {S, \powerset S}$ is a '''countable discrete space'''.


Definition:Discrete Topology/Uncountable

Let $S$ be an uncountably infinite set. Then $\tau = \powerset S$ is an '''uncountable discrete topology''', and $\struct {S, \tau} = \struct {S, \powerset S}$ is an '''uncountable discrete space'''.


Definition:Discrete Topology/Infinite

Let $S$ be an infinite set. Then $\tau = \powerset S$ is an '''infinite discrete topology''', and $\struct {S, \tau} = \struct {S, \powerset S}$ is an '''infinite discrete space'''.


Definition:Finer Topology/Strictly Finer

Let $\tau_1 \supsetneq \tau_2$. $\tau_1$ is said to be '''strictly finer''' than $\tau_2$. This can be expressed as: :$\tau_1 > \tau_2 := \tau_1 \supsetneq \tau_2$


Definition:Coarser Topology/Strictly Coarser

Let $\tau_1 \subsetneq \tau_2$. Then $\tau_1$ is said to be '''strictly coarser''' than $\tau_2$. This can be expressed as: :$\tau_1 < \tau_2 := \tau_1 \subsetneq \tau_2$


Definition:Uniform Continuity/Metric Space

Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces. Then a mapping $f: A_1 \to A_2$ is '''uniformly continuous on $A_1$''' {{iff}}: :$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x, y \in A_1: \map {d_1} {x, y} < \delta \implies \map {d_2} {\map f x, \map f y} < \epsilon$ where $\R_{>0}$ denotes the set of all strictly positive real numbers.


Definition:Uniform Continuity/Real Numbers

Let $I \subseteq \R$ be a real interval. A real function $f: I \to \R$ is said to be '''uniformly continuous''' on $I$ {{iff}}: :for every $\epsilon > 0$ there exists $\delta > 0$ such that the following property holds: ::for every $x, y \in I$ such that $\size {x - y} < \delta$ it happens that $\size {\map f x - \map f y} < \epsilon$. Formally: $f: I \to \R$ is '''uniformly continuous''' {{iff}} the following property holds: :$\forall \epsilon > 0: \exists \delta > 0: \paren {x, y \in I, \size {x - y} < \delta \implies \size {\map f x - \map f y} < \epsilon}$


Definition:Compact Space/Metric Space

Let $M = \left({A, d}\right)$ be a metric space. Let $\tau$ denote the topology on $A$ induced by $d$. Then $M$ is '''compact''' {{iff}} $\left({A, \tau}\right)$ is a compact topological space.


Definition:Upper Integral

Let $\closedint a b$ be a closed real interval. Let $f: \closedint a b \to \R$ be a bounded real function. The '''upper integral of $f$ over $\closedint a b$''' is defined as: :$\displaystyle \overline {\int_a^b} \map f x \rd x = \inf_P \map U P$ where: :the infimum is taken over all subdivisions $P$ of $\closedint a b$ :$\map U P$ denotes the upper sum of $f$ on $\closedint a b$ belonging to $P$.


Definition:Lower Integral

Let $\closedint a b$ be a closed real interval. Let $f: \closedint a b \to \R$ be a bounded real function. The '''lower integral of $f$ over $\closedint a b$''' is defined as: :$\displaystyle \underline {\int_a^b} \map f x \rd x = \sup_P \map L P$ where: :the supremum is taken over all subdivisions $P$ of $\closedint a b$ :$\map L P$ denotes the lower sum of $f$ on $\closedint a b$ belonging to $P$.


Definition:Abstract Geometry

Let $P$ be a set and $L$ be a set of subsets of $P$. Then $\left({P, L}\right)$ is an '''abstract geometry''' {{iff}}: {{begin-axiom}} {{axiom | n = 1         | q = \forall A, B \in P         | m = \exists l \in L: A, B \in l }} {{axiom | n = 2         | q = \forall l \in L         | m = \exists A, B \in P: A, B \in l \land A \ne B }} {{end-axiom}}


Definition:Euclidean Plane

For any real number $a$ let: :$L_a = \set {\tuple {x, y} \in \R^2: x = a}$ Furthermore, define: :$L_A = \set {L_a: a \in \R}$ For any two real numbers $m$ and $b$ let: :$L_{m, b} = \set {\tuple {x, y} \in \R^2: y = m x + b}$ Furthermore, define: :$L_{M, B} = \set {L_{m, b}: m, b \in \R}$ Finally let: :$L_E = L_A \cup L_{M, B}$ The abstract geometry $\struct {\R^2, L_E}$ is called the '''Euclidean plane'''.


Definition:Multiple Edge/Multiplicity

The '''multiplicity''' of an edge is the number of edges having the same pair of endvertices.


Definition:Poincaré Plane

Let: :$\H = \set {\tuple {x, y} \in \R^2: y > 0}$ Let $a \in \R$ be a real number. Let: :${}_a L := \set {\tuple {x, y} \in \H: x = a}$ Define: :${}_A L := \set{ {}_a L: a \in \R}$ Let $c \in \R$ be a real number and $r \in \R_{>0}$ be a strictly positive real number. Let: :${}_c L_r := \set {\tuple {x, y} \in \H: \paren {x - c}^2 + y^2 = r^2}$ Define: :${}_C L_R := \set { {}_c L_r: c \in \R \land r \in \R_{>0} }$ Finally let: :$L_H = {}_A L \cup {}_C L_R$ The abstract geometry $\struct {\H, L_H}$ is called the '''Poincaré plane'''. This is shown to be an abstract geometry in Poincaré Plane is Abstract Geometry.


Definition:Topology Generated by Synthetic Sub-Basis/Definition 1

Define: :$\displaystyle \BB = \set {\bigcap \FF: \FF \subseteq \SS, \FF \text{ is finite} }$ That is, $\BB$ is the set of all finite intersections of sets in $\SS$. Note that $\FF$ is allowed to be empty in the above definition. The '''topology generated by $\SS$''', denoted $\map \tau \SS$, is defined as: :$\displaystyle \map \tau \SS = \set {\bigcup \AA: \AA \subseteq \BB}$


Definition:Topology Generated by Synthetic Sub-Basis/Definition 2

The '''topology generated by $\mathcal S$''', denoted $\tau \left({\mathcal S}\right)$, is defined as the unique topology on $X$ that satisfies the following axioms: :$\left({1}\right): \quad \mathcal S \subseteq \tau \left({\mathcal S}\right)$ :$\left({2}\right): \quad$ For any topology $\mathcal T$ on $X$, the implication $\mathcal S \subseteq \mathcal T \implies \tau \left({\mathcal S}\right) \subseteq \mathcal T$ holds. That is, $\tau \left({\mathcal S}\right)$ is the coarsest topology on $X$ for which every element of $\mathcal S$ is open.


Definition:Balanced String

Let $S$ be a string in an alphabet containing the left and right brackets $($ and $)$. Then $S$ is said to be '''balanced''' iff it contains equally many left and right brackets.


Definition:Sequential Continuity

Let $X$ and $Y$ be topological spaces. Let $f: X \to Y$ be a mapping. === At a Point === {{:Definition:Sequential Continuity/Point}} === On a Domain === {{:Definition:Sequential Continuity/Domain}} Category:Definitions/Continuity Category:Definitions/Topology fc8tazwc9qttrg0qldxdvgcg2yxtqr9


Definition:Initial Topology/Definition 1

Let: :$\mathcal S = \left\{{f_i^{-1} \left[{U}\right]: i \in I, U \in \tau_i}\right\} \subseteq \mathcal P \left({X}\right)$ where $f_i^{-1} \left[{U}\right]$ denotes the preimage of $U$ under $f_i$. The topology $\tau$ on $X$ generated by $\mathcal S$ is called the '''initial topology on $X$ with respect to $\left \langle {f_i}\right \rangle_{i \mathop \in I}$'''.


Definition:Initial Topology/Definition 2

Let $\tau$ be the coarsest topology on $X$ such that each $f_i: X \to Y_i$ is $\left({\tau, \tau_i}\right)$-continuous. Then $\tau$ is known as the '''initial topology on $X$ with respect to $\left \langle {f_i} \right \rangle_{i \mathop \in I}$'''.


Definition:Product Space (Topology)/General Definition

Let $\family {\struct {S_i, \tau_i} }_{i \mathop \in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set. Let $S$ be the cartesian product of $\family {S_i}_{i \mathop \in I}$: :$\displaystyle S := \prod_{i \mathop \in I} S_i$ Let $\tau$ be the Tychonoff topology on $S$. From Natural Basis of Tychonoff Topology, $\tau$ is generated from: :the basis $\BB$ of cartesian products of the form $\displaystyle \prod_{i \mathop \in I} U_i$ where: ::for all $i \in I : U_i \in \tau_i$ ::for all but finitely many indices $i : U_i = S_i$ The topological space $\struct{X, \tau}$ is called the '''product space''' of $\family {\struct {S_i, \tau_i} }_{i \mathop \in I}$.


Definition:Neighborhood (Topology)/Set

Let $A \subseteq S$ be a subset of $S$. A '''neighborhood''' of $A$, which can be denoted $N_A$, is any subset of $S$ containing an open set of $T$ which itself contains $A$. That is: :$\exists U \in \tau: A \subseteq U \subseteq N_A \subseteq S$


Definition:Neighborhood (Topology)/Point

Let $z \in S$ be a point in a $S$. Let $N_z$ be a subset of $S$ which contains (as a subset) an open set of $T$ which itself contains (as an element) $z$. Then $N_z$ is a '''neighborhood''' of $z$. That is: :$\exists U \in \tau: z \in U \subseteq N_z \subseteq S$


Definition:Closed Neighborhood

If $\relcomp S {N_A} \in \tau$, that is if $N_A$ is closed in $S$, then $N_A$ is called a '''closed neighborhood'''.


Definition:Limit Point/Topology/Set/Definition 1

A point $x \in S$ is a '''limit point of $A$''' {{iff}} every open neighborhood $U$ of $x$ satisfies: :$A \cap \paren {U \setminus \set x} \ne \O$ That is, {{iff}} every open set $U \in \tau$ such that $x \in U$ contains some point of $A$ distinct from $x$.


Definition:Limit Point/Topology/Set/Definition 2

A point $x \in S$ is a '''limit point of $A$''' {{iff}} : $x$ belongs to the closure of $A$ but is not an isolated point of $A$.


Definition:Limit Point/Topology/Set/Definition 3

A point $x \in S$ is a '''limit point of $A$''' {{iff}} $x$ is an adherent point of $A$ but is not an isolated point of $A$.


Definition:Limit Point/Topology/Set/Definition 4

A point $x \in S$ is a '''limit point of $A$''' {{iff}} $\left({S \setminus A}\right) \cup \left\{{x}\right\}$ is ''not'' a neighborhood of $x$.


Definition:Adherent Point/Definition 1

A point $x \in S$ is an '''adherent point of $A$''' {{iff}} every open neighborhood $U$ of $x$ satisfies: :$A \cap U \ne \O$


Definition:Adherent Point/Definition 2

A point $x \in S$ is an '''adherent point of $A$''' {{iff}} $x$ is an element of the closure of $A$.