hcju/proofwikips huggingface dataset:
Definition:Ordered Product
Let $\left({S_1, \preceq_1}\right)$ and $\left({S_2, \preceq_2}\right)$ be tosets. Let: * the order type of $\left({S_1, \preceq_1}\right)$ be $\theta_1$; * the order type of $\left({S_2, \preceq_2}\right)$ be $\theta_2$. Let $T = S_1 \times S_2$ be the cartesian product of $S_1$ and $S_2$. Consider the relation $\preceq$ defined on $T$ as follows. Let $a_1$ and $a_2$ be arbitrary elements of $S_1$, and $b_1$ and $b_2$ be arbitrary elements of $S_2$. Then: * $b_1 \prec b_2 \implies \left({a_1, b_1}\right) \prec \left({a_2, b_2}\right)$ * $b_1 = b_2, a_1 \prec a_2 \implies \left({a_1, b_1}\right) \prec \left({a_2, b_2}\right)$ * $b_1 = b_2, a_1 = a_2 \implies \left({a_1, b_1}\right) = \left({a_2, b_2}\right)$ The ordered set $\left({S_1 \times S_2, \preceq}\right)$ is called the '''ordered product''' of $S_1$ and $S_2$, and is denoted $S_1 \cdot S_2$.
Definition:Euler Diagram
An '''Euler diagram''' is a graphical technique for illustrating the relationships between sets. It differs from a Venn diagram in that whereas the latter illustrates all possible intersections between a number of general sets, an '''Euler diagram''' depicts only those which are relevant for the situation being depicted.
Definition:Scope (Logic)
The '''scope''' of a logical connective is defined as the statements that it connects, whether this be simple or compound. In the case of a unary connective, there will be only one such statement. === Connective of Propositional Logic === {{:Definition:Scope (Logic)/Connective}} === Quantifier of Predicate Logic === {{:Definition:Scope (Logic)/Quantifier}}
Definition:Main Connective
In a compound statement, exactly '''one''' of its logical connectives has the largest scope. That connective is called the '''main connective'''. The scope of the '''main connective''' comprises the entire compound statement. {{transclude:Definition:Main Connective/Propositional Logic   |section = def   |title = Propositional Logic   |link = true   |header = 3   |increase = 1 }}
Definition:Codomain of Sequence
The codomain of a sequence can be elements of a set of any objects. If the codomain of a sequence $f$ is $S$, then the sequence is said to be a '''sequence of elements of $S$''', or a '''sequence in $S$'''.
Definition:Length of String
The '''length''' of a finite string in a formal language is the number of symbols it contains. The '''length''' of a string $S$ can be denoted $\map \len S$ or $\size S$.
Definition:Initial Part
A string $T$ is an '''initial part''' of a string $S$  $S$ can be formed by concatenating $T$ with another string $T'$: :$S = TT'$
Definition:Trivial Quotient
Let $\Delta_S$ be the diagonal relation on a set $S$. As $\Delta_S$ is an equivalence, we can form the quotient mapping: :$q_{\Delta_S}: S \to S / \Delta_S$. This quotient mapping is called the '''trivial quotient of $S$'''.
Definition:Conflation
A '''conflation''' is a mistake in which two or more separate but similar ideas become confused with one another.
Definition:Left Inverse Mapping
Let $S, T$ be sets where $S \ne \O$, i.e. $S$ is not empty. Let $f: S \to T$ be a mapping. Let $g: T \to S$ be a mapping such that: :$g \circ f = I_S$ where: :$g \circ f$ denotes the composite mapping $f$ followed by $g$; :$I_S$ is the identity mapping on $S$. Then $g: T \to S$ is called '''a left inverse (mapping)'''.
Definition:Right Inverse Mapping
Let $S, T$ be sets where $S \ne \O$, that is, $S$ is not empty. Let $f: S \to T$ be a mapping. Let $g: T \to S$ be a mapping such that: :$f \circ g = I_T$ where: :$f \circ g$ denotes the composite mapping $g$ followed by $f$ :$I_T$ is the identity mapping on $T$. Then $g: T \to S$ is called '''a right inverse (mapping) of $f$'''.
Definition:Commutative Diagram
A '''commutative diagram''' is a graphical technique designed to illustrate the construction of composite mappings. It is also a widespread tool in category theory, where it deals with morphisms instead of mappings. ::$\begin{xy} \xymatrix@L+2mu@+1em{  S_1 \ar[r]^*{f_1}      \ar@{-->}[rd]_*{f_2 \circ f_1} &  S_2 \ar[d]^*{f_2} \\ &  S_3 }\end{xy}$ It consists of: :$(1): \quad$ A collection of points representing the various domains and codomains of the mappings in question :$(2): \quad$ Arrows representing the mappings themselves. The diagram is properly referred to as '''commutative'''  all the various paths from the base of one arrow to the head of another represent equal mappings. A mapping which is uniquely determined by the rest of the diagram may be indicated by a dotted arrow. It is however generally advisable not to use more than one dotted arrow per diagram, so as to avoid confusion.
Definition:Extension of Mapping
As a mapping is, by definition, also a relation, the definition of an '''extension of a mapping''' is the same as that for an extension of a relation: Let: :$f_1 \subseteq X \times Y$ be a mapping on $X \times Y$ :$f_2 \subseteq S \times T$ be a mapping on $S \times T$ :$X \subseteq S$ :$Y \subseteq T$ :$f_2 \restriction_{X \times Y}$ be the restriction of $f_2$ to $X \times Y$. Let $f_2 \restriction_{X \times Y} = f_1$. Then $f_2$ '''extends''' or '''is an extension of''' $f_1$.
Definition:Extension of Operation
Let $\left({S, \circ}\right)$ be a magma. Let $\left({T, \circ \restriction_T}\right)$ be a submagma of $\left({S, \circ}\right)$, where $\circ \restriction_T$ denotes the restriction of $\circ$ to $T$. Then: : '''$\left({S, \circ}\right)$ is an extension of $\left({T, \circ \restriction_T}\right)$''' or : '''$\left({S, \circ}\right)$ extends $\left({T, \circ \restriction_T}\right)$''' We can use the term directly to the operation itself and say: : '''$\circ$ is an extension of $\circ \restriction_T$''' or: : '''$\circ$ extends $\circ \restriction_T$'''
Definition:Union Mapping
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 $T_2$ Let $f_1$ and $f_2$ be combinable, that is, that they agree on $S_1 \cap S_2$. Then the '''union mapping''' $f = f_1 \cup f_2$ of $f_1$ and $f_2$ is: :$f: S_1 \cup S_2 \to T_1 \cup T_2: \map f s = \begin{cases} \map {f_1} s : & s \in S_1 \\ \map {f_2} s : & s \in S_2 \end{cases}$
Definition:Increasing Union
Let $S_0, S_1, S_2, \ldots, S_i, \ldots$ be a nested sequence of sets, that is: :$S_0 \subseteq S_1 \subseteq S_2 \subseteq \ldots \subseteq S_i \subseteq \ldots$ Let $S$ be the set: :$\displaystyle S = \bigcup_{i \mathop \in \N} S_i$ where $\bigcup$ denotes set union. Then $S$ is called the '''increasing union''' of $S_0, S_1, S_2, \ldots, S_i, \ldots$
Definition:Bound Variable
A '''bound variable''' is a variable which, when it occurs in an expression, can be replaced with another variable without changing the meaning of the statement.
Definition:F-Sigma Set
Let $T = \struct {S, \tau}$ be a topological space. An '''$F_\sigma$ set ($F$-sigma set)''' is a set which can be written as a countable union of closed sets of $T$.
Definition:G-Delta Set
Let $T = \struct {S, \tau}$ be a topological space. A '''$G_\delta$ set ($G$-delta set)''' is a set which can be written as a countable intersection of open sets of $T$.
Definition:Equivalent Topological Bases
Let $X$ be a set. Let $\mathbb S_1$ and $\mathbb S_2$ be subsets of $\powerset X$, the power set of $X$. Let $\mathbb S_1$ and $\mathbb S_2$ be used as a synthetic basis or synthetic sub-basis to generate topologies for $X$. Let $\tau_1$ and $\tau_2$ be the topologies arising from $\mathbb S_1$ and $\mathbb S_2$ respectively. Then $\mathbb S_1$ and $\mathbb S_2$ are '''equivalent'''  $\tau_1 = \tau_2$.
Definition:Local Basis
Let $T = \struct {S, \tau}$ be a topological space. Let $x$ be an element of $S$. === Local Basis for Open Sets === {{:Definition:Local Basis/Local Basis for Open Sets}} === Neighborhood Basis of Open Sets === {{:Definition:Local Basis/Neighborhood Basis of Open Sets}}
Definition:Weakly Hereditary Property
Let $\xi$ be a property whose domain is the set of all topological spaces. Then $\xi$ is a '''weakly hereditary property''' : :$\map \xi X \implies \map \xi Y$ where $Y$ is any closed set of $X$ when considered as a subspace.
Definition:Hereditary Property (Topology)
Let $\xi$ be a property whose domain is the set of all topological spaces. Then $\xi$ is a '''hereditary property''' : :$\map \xi X \implies \map \xi Y$ where $Y$ is a subspace of $X$. That is, whenever a topological space has $\xi$, then so does any subspace.
Definition:Adherent Point
Let $T = \struct {S, \tau}$ be a topological space. Let $A \subseteq S$. === Definition from Neighborhood === {{:Definition:Adherent Point/Definition 3}} === Definition from Open Neighborhood === {{:Definition:Adherent Point/Definition 1}} === Definition from Closure === {{:Definition:Adherent Point/Definition 2}}
Definition:Omega-Accumulation Point
Let $T = \struct {S, \tau}$ be a topological space. Let $A \subseteq S$. An '''$\omega$-accumulation point''' of $A$ is a limit point $x$ of $A$ such that every open set containing $x$ also contains an infinite number of points of $A$.
Definition:Condensation Point
Let $T = \struct {X, \tau}$ be a topological space. Let $A \subseteq X$. A '''condensation point''' of $A$ is a limit point $x$ of $A$ such that every open set containing $x$ also contains an uncountable number of points of $A$.
Definition:Limit Point/Topology/Sequence
Let $T = \struct {S, \tau}$ be a topological space. Let $A \subseteq S$. Let $\sequence {x_n}$ be a sequence in $A$. Let $\sequence {x_n}$ converge to a value $\alpha \in A$. Then $\alpha$ is known as a '''limit (point) of $\sequence {x_n}$ (as $n$ tends to infinity)'''.
Definition:Accumulation Point/Sequence
Let $\sequence {x_n}_{n \mathop \in \N}$ be an infinite sequence in $A$. Let $x \in S$. Suppose that: :$\forall U \in \tau: x \in U \implies \set {n \in \N: x_n \in U}$ is infinite Then $x$ is an '''accumulation point''' of $\sequence {x_n}$.
Definition:Regular Open Set
Let $T$ be a topological space. Let $A \subseteq T$. Then $A$ is '''regular open in $T$''' : :$A = A^{- \circ}$
Definition:Regular Closed Set
Let $T$ be a topological space. Let $A \subseteq T$. Then $A$ is '''regular closed in $T$''' : :$A = A^{\circ -}$
Definition:Exterior (Topology)
Let $T$ be a topological space. Let $H \subseteq T$. === Definition 1 === {{:Definition:Exterior (Topology)/Definition 1}} === Definition 2 === {{:Definition:Exterior (Topology)/Definition 2}}
Definition:Dense-in-itself
Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$. Then $H$ is dense-in-itself  it contains no isolated points.
Definition:Meager Space
Let $T = \struct {S, \tau}$ be a topological space. Let $A \subseteq S$. $A$ is '''meager in $T$'''  it is a countable union of subsets of $S$ which are nowhere dense in $T$.
Definition:Separable Space
A topological space $T = \struct {S, \tau}$ is '''separable'''  there exists a countable subset of $S$ which is everywhere dense in $T$.
Definition:Second-Countable Space
A topological space $T = \struct {S, \tau}$ is '''second-countable''' or '''satisfies the Second Axiom of Countability'''  its topology has a countable basis.
Definition:First-Countable Space
A topological space $T = \struct {S, \tau}$ is '''first-countable''' or '''satisfies the First Axiom of Countability'''  every point in $S$ has a countable local basis.
Definition:Neighborhood Basis
Let $\struct{X, \tau}$ be a topological space. Let $x \in X$. Let $\BB$ be a set of neighborhoods of $x$. Then $\BB$ is a '''neighborhood basis''' at $x$ : :For each neighborhood $N$ of $x$, there is an $M \in \BB$ such that $M \subseteq N$.
Definition:Closed Mapping
Let $X, Y$ be topological spaces. Let $f: X \to Y$ be a mapping.  If, for any closed set $V \subseteq X$, the image $\map f V$ is closed in $Y$, then $f$ is referred to as a '''closed mapping'''.
Definition:Continuity/Metric Subspace
Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces. Let $f: A_1 \to A_2$ be a mapping from $A_1$ to $A_2$. Let $Y \subseteq A_1$. By definition, $\left({Y, d_Y}\right)$ is a metric subspace of $A_1$. Let $a \in Y$ be a point in $Y$. Then $f$ is '''$\left({d_Y, d_2}\right)$-continuous at $a$''' : : $\forall \epsilon > 0: \exists \delta > 0: d_Y \left({x, a}\right) < \delta \implies d_2 \left({f \left({x}\right), f \left({a}\right)}\right) < \epsilon$ Similarly, $f$ is '''$\left({d_Y, d_2}\right)$-continuous''' : : $\forall a \in Y: f$ is $\left({d_Y, d_2}\right)$-continuous at $a$
Definition:Open Set/Topology
Let $T = \left({S, \tau}\right)$ be a topological space. Then the elements of $\tau$ are called the '''open sets of $T$'''. Thus: : '''$U \in \tau$''' and: : '''$U$ is open in $T$''' are equivalent statements.
Definition:Open Set/Metric Space
Let $M = \struct {A, d}$ be a metric space. Let $U \subseteq A$. Then $U$ is an '''open set in $M$'''  it is a neighborhood of each of its points. That is: :$\forall y \in U: \exists \epsilon \in \R_{>0}: \map {B_\epsilon} y \subseteq U$ where $\map {B_\epsilon} y$ is the open $\epsilon$-ball of $y$.
Definition:Open Set/Complex Analysis
{{:Definition:Open Set/Complex Analysis/Definition 1}}
Definition:Open Set/Real Analysis
=== Real Numbers === {{:Definition:Open Set/Real Analysis/Real Numbers}} === Real Euclidean Space === {{:Definition:Open Set/Real Analysis/Real Euclidean Space}}
Definition:Open Region
=== Complex Analysis === {{:Definition:Open Region/Complex}} === Open Region in the Plane === {{:Definition:Open Region/Plane}}
Definition:Neighborhood (Topology)
Let $T = \struct {S, \tau}$ be a topological space. === Neighborhood of a Set === {{:Definition:Neighborhood (Topology)/Set}} === Neighborhood of a Point === The set $A$ can be a singleton, in which case the definition is of the '''neighborhood of a point'''. {{:Definition:Neighborhood (Topology)/Point}}
Definition:Open Ball
Let $M = \struct {A, d}$ be a metric space or pseudometric space. Let $a \in A$. Let $\epsilon \in \R_{>0}$ be a strictly positive real number. The '''open $\epsilon$-ball of $a$ in $M$''' is defined as: :$\map {B_\epsilon} a := \set {x \in A: \map d {x, a} < \epsilon}$ If it is necessary to show the metric or pseudometric itself, then the notation $\map {B_\epsilon} {a; d}$ can be used.
Definition:Neighborhood (Complex Analysis)
Let $z_0 \in \C$ be a complex number. Let $\epsilon \in \R_{>0}$ be a (strictly) positive real number. The '''$\epsilon$-neighborhood''' of $z_0$ is defined as: :$\map {N_\epsilon} {z_0} := \set {z \in \C: \cmod {z - z_0} < \epsilon}$
Definition:Neighborhood (Real Analysis)/Epsilon
On the real number line with the usual metric, the '''$\epsilon$-neighborhood''' of $\alpha$ is defined as the open interval: :$\map {N_\epsilon} \alpha := \openint {\alpha - \epsilon} {\alpha + \epsilon}$ where $\epsilon \in \R_{>0}$ is a (strictly) positive real number.
Definition:Limit of Real Function
Let $\openint a b$ be an open real interval. Let $c \in \openint a b$. Let $f: \openint a b \setminus \set c \to \R$ be a real function. Let $L \in \R$. Suppose that: :$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in \R: 0 < \size {x - c} < \delta \implies \size {\map f x - L} < \epsilon$ where $\R_{>0}$ denotes the set of strictly positive real numbers. That is: :For every (strictly) positive real number $\epsilon$, there exists a (strictly) positive real number $\delta$ such that ''every'' real number $x \ne c$ in the domain of $f$ within $\delta$ of $c$ has an image within $\epsilon$ of $L$. $\epsilon$ is usually considered as having the connotation of being "small" in magnitude, but this is a misunderstanding of its intent: the point is that (in this context) $\epsilon$ ''can be made'' arbitrarily small. :400px Then $\map f x$ is said to '''tend to the limit $L$ as $x$ tends to $c$''', and we write: :$\map f x \to L$ as $x \to c$ or :$\displaystyle \lim_{x \mathop \to c} \map f x = L$ This is voiced: :'''the limit of $\map f x$ as $x$ tends to $c$'''. It can directly be seen that this definition is the same as that for a general metric space.
Definition:Limit of Function (Metric Space)
Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces. Let $c$ be a limit point of $M_1$. Let $f: A_1 \to A_2$ be a mapping from $A_1$ to $A_2$ defined everywhere on $A_1$ ''except possibly'' at $c$. Let $L \in M_2$. $f \left({x}\right)$ is said to '''tend to the limit $L$ as $x$ tends to $c$''' and is written: :$f \left({x}\right) \to L$ as $x \to c$ or :$\displaystyle \lim_{x \mathop \to c} f \left({x}\right) = L$  the following equivalent conditions hold: === $\epsilon$-$\delta$ Condition === {{:Definition:Limit of Function (Metric Space)/Epsilon-Delta Condition}} === $\epsilon$-Ball Condition === {{:Definition:Limit of Function (Metric Space)/Epsilon-Ball Condition}}
Definition:Limit of Complex Function
The definition for the limit of a complex function is exactly the same as that for the general metric space.
Definition:Limit of Sequence/Metric Space
Let $M = \left({A, d}\right)$ be a metric space or pseudometric space. Let $\left \langle {x_n} \right \rangle$ be a sequence in $M$. Let $\left \langle {x_n} \right \rangle$ converge to a value $l \in A$. Then $l$ is a '''limit of $\left \langle {x_n} \right \rangle$ as $n$ tends to infinity'''. If $M$ is a metric space, this is usually written: :$\displaystyle l = \lim_{n \mathop \to \infty} x_n$
Definition:Limit of Sequence (Number Field)
=== Real Numbers === {{:Definition:Limit of Sequence/Real Numbers}} === Rational Numbers === {{:Definition:Limit of Sequence/Rational Numbers}} === Complex Numbers === {{:Definition:Limit of Sequence/Complex Numbers}}
Definition:Divergent Sequence
A sequence which is not convergent is '''divergent'''.
Definition:Divergent Series
A series which is not convergent is '''divergent'''.
Definition:Divergent Function
A function which is not convergent is '''divergent'''.
Definition:Convergent Mapping
=== Metric Space === {{:Definition:Convergent Mapping/Metric Space}} === Real Function === As the real number line $\R$ under the usual (Euclidean) metric forms a metric space, the definition also holds for real functions: {{:Definition:Convergent Mapping/Real Function}} === Complex Function === As the complex plane $\C$ under the usual (Euclidean) metric forms a metric space, the definition also holds for complex functions: {{:Definition:Convergent Mapping/Complex Function}}
Definition:Filter
Let $\left({S, \preccurlyeq}\right)$ be an ordered set. A '''filter of $\left({S, \preccurlyeq}\right)$''' (or a '''filter on $\left({S, \preccurlyeq}\right)$''') is a subset $\mathcal F \subseteq S$ which satisfies the following conditions: {{begin-axiom}} {{axiom | n = 1         | m = \mathcal F \ne \varnothing }} {{axiom | n = 2         | m = x, y \in \mathcal F \implies \exists z \in \mathcal F: z \preccurlyeq x, z \preccurlyeq y }} {{axiom | n = 3         | m = \forall x \in \mathcal F: \forall y \in S: x \preccurlyeq y \implies y \in \mathcal F }} {{end-axiom}} === Proper Filter === {{:Definition:Filter/Proper Filter}}
Definition:Convergent Filter
Let $\struct {S, \tau}$ be a topological space. Let $\FF$ be a filter on $S$. Then $\FF$ '''converges''' to a point $x \in S$ : :$\forall N_x \subseteq S: N_x \in \FF$ where $N_x$ is a neighborhood of $x$. That is, a filter '''converges''' to a point $x$  every neighborhood of $x$ is an element of that filter. If there is a point $x \in S$ such that $\FF$ '''converges''' to $x$, then $\FF$ is '''convergent'''.
Definition:Standard Number Field
The '''standard number fields''' are the following sets of numbers: :The rational numbers: $\Q = \set {p / q: p, q \in \Z, q \ne 0}$ :The real numbers: $\R = \set {x: x = \sequence {s_n} }$ where $\sequence {s_n}$ is a Cauchy sequence in $\Q$ :The complex numbers: $\C = \set {a + i b: a, b \in \R, i^2 = -1}$. These sets are indeed fields: :$\struct {\Q, +, \times, \le}$ is an ordered field, and also a metric space. :$\struct {\R, +, \times, \le}$ is an ordered field, and also a complete metric space. :$\struct {\C, +, \times}$ is a field, but cannot be ordered compatibly with $+$ and $\times$. However, it can be treated as a metric space.
Definition:Continuous Invariant
Let $P$ be a property whose domain is the set of all topological spaces. Suppose that whenever $\map P T$ holds, then so does $\map P {T'}$, where: :$T$ and $T'$ are topological spaces :$\phi: T \to T'$ is a continuous mapping from $T$ to $T'$ :$\phi \sqbrk T = T'$, where $\phi \sqbrk T$ denotes the image of $\phi$. Then $P$ is a '''continuous invariant'''.
Definition:Image (Set Theory)/Relation/Subset
Let $X \subseteq S$ be a subset of $S$. Then the '''image set (of $X$ by $\RR$)''' is defined as: :$\RR \sqbrk X := \set {t \in T: \exists s \in X: \tuple {s, t} \in \RR}$
Definition:Open Invariant
Let $P$ be a property whose domain is the set of all topological spaces. Suppose that whenever $\map P T$ holds, then so does $\map P {T'}$, where: :$T$ and $T'$ are topological spaces :$\phi: T \to T'$ is a mapping from $T$ to $T'$ :$\phi \sqbrk T = T'$, where $\phi \sqbrk T$ denotes the image of $\phi$ :$T'$ is an open set. Then $P$ is an '''open invariant'''.
Definition:Closed Invariant
Let $P$ be a property whose domain is the set of all topological spaces. Suppose that whenever $\map P T$ holds, then so does $\map P {T'}$, where: :$T$ and $T'$ are topological spaces :$\phi: T \to T'$ is a mapping from $T$ to $T'$ :$\phi \sqbrk T = T'$, where $\phi \sqbrk T$ denotes the image of $\phi$ :$T'$ is a closed set. Then $P$ is a '''closed invariant'''.
Definition:Product Space (Topology)
Let $\struct {S_1, \tau_1}$ and $\struct {S_2, \tau_2}$ be topological spaces. Let $S_1 \times S_2$ be the cartesian product of $S_1$ and $S_2$. Let $\tau$ be the Tychonoff topology on $S_1 \times S_2$. From Natural Basis of Tychonoff Topology of Finite Product, $\tau$ is the topology generated by the natural basis: :$\BB = \set {U_1 \times U_2: U_1 \in \tau_1, U_2 \in \tau_2}$ The topological space $\struct {S_1 \times S_2, \tau}$ is called  the '''product space''' of $\struct {S_1, \tau_1}$ and $\struct {S_2, \tau_2}$.
Definition:P-Product Metric
Let $M_{1'} = \struct {A_{1'}, d_{1'} }$ and $M_{2'} = \struct {A_{2'}, d_{2'} }$ be metric spaces. Let $A_{1'} \times A_{2'}$ be the cartesian product of $A_{1'}$ and $A_{2'}$. Let $p \in \R_{\ge 1}$. The '''$p$-product metric''' on $A_{1'} \times A_{2'}$ is defined as: :$\map {d_p} {x, y} := \paren {\paren {\map {d_{1'} } {x_1, y_1} }^p + \paren {\map {d_{2'} } {x_2, y_2} }^p}^{1/p}$ where $x = \tuple {x_1, x_2}, y = \tuple {y_1, y_2} \in A_{1'} \times A_{2'}$. The metric space $\mathcal M_p := \struct {A_{1'} \times A_{2'}, d_p}$ is the '''$p$-product (space)''' of $M_{1'}$ and $M_{2'}$.
Definition:Identification Topology
Let $\struct {S_1, \tau_1}$ be a topological space. Let $S_2$ be a set. Let $f: S_1 \to S_2$ be a mapping. The '''identification topology on $S_2$ with respect to $f$ and $\struct {S_1, \tau_1}$''' is defined as: :$\tau_2 = \set {V \in \powerset {S_2}: f^{-1} \sqbrk V \in \tau_1}$
Definition:Topological Sum
Let $\struct {X, \tau_1}$ and $\struct {Y, \tau_2}$ be topological spaces. The '''topological sum''' $\struct {Z, \tau_3}$ of $X$ and $Y$ is defined as: :$Z = X \sqcup Y$ where: :$X \sqcup Y$ denotes the disjoint union of $X$ and $Y$ :$\tau_3$ is the topology generated by $\tau_1$ and $\tau_2$.
Definition:Finite Intersection Property
Let $\Bbb S$ be a set of sets. Let $\Bbb S$ have the property that: :the intersection of any finite number of sets in $\Bbb S$ is not empty. Then $\Bbb S$ satisfies the '''finite intersection property'''.
Definition:Filter Sub-Basis
Let $S$ be a set. Let $\powerset S$ denote the power set of $S$. Let $\BB \subset \powerset S$ be a set of subsets of $\powerset S$ which satisfies the finite intersection property. That is, the intersection of any finite number of sets in $\BB$ is not empty. Then $\BB$, together with the finite intersections of all its elements, is a basis for a filter $\FF$ on $S$. Thus $\BB$ is a '''sub-basis''' for $\FF$.
Definition:Cluster Point of Filter
Let $S$ be a set. Let $\powerset S$ denote the power set of $S$. Let $\FF \subset \powerset X$ be a filter on $S$. Let $x \in S$ be an element of every set in $\FF$: :$x \in X: \forall U \in \FF: x \in U$ Then $x$ is a '''cluster point of $\FF$'''.