hcju/proofwikips huggingface dataset:
Closure of Subset of Metric Space is Intersection of Closed Supersets
Let $M = \left({A, d}\right)$ be a metric space. Let $H \subseteq A$ be a subset of $A$. Let $H^-$ denote the closure of $H$. Then $H^-$ is the intersection of all closed sets of $M$ of which $H$ is a subset.
Rational Numbers form Metric Subspace of Real Numbers under Euclidean Metric
Let $\struct {\Q, d_\Q}$ be the set of rational numbers under the function $d_\Q: \Q \times \Q \to \R$ defined as: :$\forall x, y \in \Q: \map d {x, y} = \size {x - y}$ Let $\struct {\R, d}$ denote the real number line with the usual (Euclidean) metric. Then $\struct {\Q, d_\Q}$ is a metric subspace of $\struct {\R, d}$, where:
Euclidean Metric on Real Number Line is Metric
The Euclidean metric on the real number line $\R$ is a metric.
Definition:Unit n-Cube
Let $n \in \N$. The '''unit $n$-cube''' $I^n$ is the Cartesian product of $n$ instances of the closed real interval $\left\{ {x \in \R: 0 \le x \le 1}\right\}$: :$I^n = \left[{0 \,.\,.\, 1}\right]^n$
Unit n-Cube under Chebyshev Distance is Subspace of Real Vector Space
Let $n \in \N$. Let $I^n$ denote the unit $n$-cube: :$I^n = \left[{0 \,.\,.\, 1}\right]^n$ that is, the Cartesian product of $n$ instances of the closed real interval $\left\{ {x \in \R: 0 \le x \le 1}\right\}$. Let $d_c: I^n \times I^n \to \R$ be defined as: :$\displaystyle d_c \left({x, y}\right) = \max_{i \mathop = 1}^n \left\{ {\left\lvert{x_i - y_i}\right\rvert}\right\}$ where $x = \left({x_1, x_2, \ldots, x_n}\right), y = \left({x_1, x_2, \ldots, x_n}\right) \in I^n$. Then $\left({I_n, d_c}\right)$ is a metric subspace of $\left({\R^n, d_\infty}\right)$, where $d_\infty$ is the Chebyshev distance on the real vector space $\R_n$.
Unit n-Sphere under Euclidean Metric is Metric Subspace of Euclidean Real Vector Space
Let $\Bbb S^n$ be the unit $n$-sphere. Let $d_S: \Bbb S^n \times \Bbb S^n \to \R$ be the real-valued function defined as: :$\displaystyle \forall x, y \in \Bbb S^n: d_S \left({x, y}\right) = \sqrt {\sum_{i \mathop = 1}^{n + 1} \left({x_i - y_i}\right)^2}$ where $x = \left({x_1, x_2, \ldots, x_{n + 1} }\right), y = \left({y_1, y_2, \ldots, y_{n + 1} }\right)$. Then $\left({\Bbb S^n, d_S}\right)$ is a metric subspace of $\left({\R^{n + 1}, d}\right)$, where $d$ is the Euclidean metric on the real vector space $\R^{n + 1}$.
Vector Subspace of Real Vector Space under Chebyshev Metric is Metric Subspace
Let $n \in \N$. Let $A$ be the set of all ordered $n+1$-tuples $\tuple {x_1, x_2, \ldots, x_{n + 1} }$ of real numbers such that $x_{n + 1} = 0$. Let $d: A \times A \to \R$ be the function defined as: :$\displaystyle \forall x, y \in A: \map d {x, y} = \max_{i \mathop = 1}^n \set {\size {x_i - y_i} }$ where $x = \tuple {x_1, x_2, \ldots, x_{n + 1} }, y = \tuple {y_1, y_2, \ldots, y_{n + 1} }$. Then $\struct {A, d}$ is a metric subspace of $\struct {\R^{n + 1}, d_\infty}$ where $d_\infty$ is the Chebyshev distance on the real vector space $\R^{n + 1}$.
Bertrand-Chebyshev Theorem
For all $n \in \N_{>0}$, there exists a prime number $p$ with $n < p \le 2 n$.
Sequence of Binomial Coefficients is Strictly Increasing to Half Upper Index
Let $n \in \Z_{>0}$ be a strictly positive integer. Let $\dbinom n k$ be the binomial coefficient of $n$ over $k$ for a positive integer $k \in \Z_{\ge 0}$. Let $S_n = \left\langle{x_k}\right\rangle$ be the sequence defined as: :$x_k = \dbinom n k$ Then $S_n$ is strictly increasing exactly where $0 \le k < \dfrac n 2$.
Isometry between Metric Spaces is Continuous
Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces. Let $\phi: M_1 \to M_2$ be an isometry. Then $\phi: M_1 \to M_2$ is a continuous mapping.
Isometric Metric Spaces are Homeomorphic
Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces. Let $M_1$ and $M_2$ be isometric. Then $M_1$ and $M_2$ are homeomorphic.
Homeomorphic Metric Spaces are not necessarily Isometric
Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces. Let $M_1$ and $M_2$ be homeomorphic. Then it is not necessarily the case that $M_1$ and $M_2$ are isometric.
Identity Mapping between Metrics separated by Scale Factor is Continuous
Let $M_1 = \struct {A, d_1}$ and $M_2 = \struct {A, d_2}$ be metric spaces on the same underlying set $A$. Let $d_1$ and $d_2$ be such that: :$\forall x, y \in A: \map {d_2} {x, y} \le K \map {d_2} {x, y}$ Let $I_A: A \to A$ be the identity mapping on $A$. Then $I_A$ is continuous from $M_1$ to $M_2$.
Metric Spaces on Topologically Equivalent Metrics on same Underlying Set are Homeomorphic
Let $M_1 = \left({A, d_1}\right)$ and $M_2 = \left({A, d_2}\right)$ be metric spaces on the same underlying set $A$. Let $d_1$ and $d_2$ be topologically equivalent. Then $M_1$ and $M_2$ are homeomorphic.
Real Vector Space under Chebyshev Distance is Homeomorphic to that under Euclidean Metric
Let $\R^n$ be an $n$-dimensional real vector space. Let $d_\infty: \R^n \times \R^n \to \R$ be the Chebyshev distance on $\R^n$. Let $d_2: \R^n \times \R^n \to \R$ be the Euclidean metric on $\R^n$. Let $M_1 = \left({\R^n, d_\infty}\right)$ and $M_2 = \left({\R^n, d_2}\right)$ be the corresponding metric spaces. Then $M_1$ and $M_2$ are homeomorphic.
Equivalence of Definitions of Homeomorphic Metric Spaces
{{TFAE|def = Homeomorphic Metric Spaces}} Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces. Let $f: A_1 \to A_2$ be a bijection.
Liouville's Theorem (Number Theory)
Let $x$ be an irrational number that is algebraic of degree $n$. Then there exists a constant $c > 0$ (which can depend on $x$) such that: :$\size {x - \dfrac p q} \ge \dfrac c {q^n}$ for every pair $p, q \in \Z$ with $q \ne 0$.
Liouville Numbers are Irrational
Liouville numbers are irrational.
Cardinality of Power Set of Natural Numbers Equals Cardinality of Real Numbers
The cardinality of the power set of the natural numbers is equal to the cardinality of the real numbers.
Liouville's Constant is Transcendental
Liouville's constant: {{begin-eqn}} {{eqn | l = \sum_{n \mathop \ge 1} \dfrac 1 {10^{n!} }       | r = \frac 1 {10^1} + \frac 1 {10^2} + \frac 1 {10^6} +  \frac 1 {10^{24} } + \cdots       | c =  }} {{eqn | r = 0.11000 \, 10000 \, 00000 \, 00000 \, 00010 \, 00 \ldots       | c =  }} {{end-eqn}} is transcendental.
Isometry of Metric Spaces is Equivalence Relation
Let $M_1$ and $M_2$ be metric spaces. Let $M_1 \sim M_2$ denote that $M_1$ and $M_2$ are isometric. The relation $\sim$ is an equivalence relation.
Homeomorphism of Metric Spaces is Equivalence Relation
Let $M_1$ and $M_2$ be metric spaces. Let $M_1 \sim M_2$ denote that $M_1$ and $M_2$ are homeomorphic. The relation $\sim$ is an equivalence relation.
Isometry of Metric Spaces is Homeomorphism
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 an isometry. Then $f$ is a homeomorphism from $M_1$ to $M_2$.
Equivalence Class of Isometries is Subset of Equivalence Class of Homeomorphisms
Let $\mathcal C_I$ be an equivalence class of isometries on the set of metric spaces. Then $\mathcal C_I$ is a subset of an equivalence class of homeomorphisms on the set of metric spaces.
Equivalence of Local Uniform Convergence and Compact Convergence
Let $U \subseteq \C$ be an open set of the complex plane. Let $f_n: U \to \C$ be a sequence of functions which converges pointwise to $f: U \to \C$. Then $f_n$ converges to $f$ uniformly on all compact subsets of $U$  $f_n$ converges locally uniformly on $U$.
Frattini's Argument
Let $\struct {G, \circ}$ be a group. Let $K$ be a finite normal subgroup of $G$, and $p$ a prime which divides the order of $K$. Let $P$ be a Sylow $p$-subgroup of $K$, and $\map {N_G} P$ the normalizer of $P$ in $G$. Then: :$G = \map {N_G} P \circ K = K \circ \map {N_G} P$
Closure of Pointwise Operation on Algebraic Structure
Let $S$ be a set such that $S \ne \varnothing$. Let $\struct {T, \circ}$ be an algebraic structure. Let $T^S$ be the set of all mappings from $S$ to $T$. Let $f, g \in T^S$, that is, let $f: S \to T$ and $g: S \to T$ be mappings. Let $\oplus: T^S \to T^S$ be the pointwise operation on $T^S$ induced by $\circ$. Then $\oplus$ is closed on $T^S$  $\struct {T, \circ}$ is closed.
Pointwise Operation on Distributive Structure is Distributive
Let $S$ be a set. Let $\left({T, +, \circ}\right)$ be an algebraic structure with two operations $+$ and $\circ$. Let $T^S$ be the set of all mappings from $S$ to $T$. Let $\left({T^S, +', \circ'}\right)$ be the structure induced on $T^S$ by $+$ and $\circ$. Let $\circ$ be distributive over $+$ in $T$. Then $\circ'$ is distributive over $+'$ in $T$.
Identity Mapping on Metric Space is Homeomorphism
Let $M = \struct {A, d}$ be a metric space. The identity mapping $I_A: M \to M$ defined as: :$\forall x \in A: \map {I_A} x = x$ is a homeomorphism.
Inverse of Homeomorphism between Metric Spaces is Homeomorphism
Let $M = \left({A_1, d_1}\right), M_2 = \left({A_2, d_2}\right)$ be metric spaces. Let $f: M_1 \to M_2$ be a homeomorphism. Then $f^{-1}: M_2 \to M_1$ is also a homeomorphism.
Composite of Homeomorphisms between Metric Spaces is Homeomorphism
Let $M_1, M_2, M_3$ be metric spaces.  Let $f: M_1 \to M_2$ and $g: M_2 \to M_3$ be homeomorphisms. Then $g \circ f: M_1 \to M_3$ is also a homeomorphism.
Metrizable Space is Hausdorff
Let $T$ be a metrizable topological space. Then $T$ is a $T_2$ (Hausdorff) space.
Underlying Set of Topological Space is Closed
Let $T = \struct {S, \tau}$ be a topological space. Then the underlying set $S$ of $T$ is closed in $T$.
Basic Properties of Neighborhood in Topological Space
This page gathers together the basic properties of a neighborhood of a point in a topological space.
Point in Topological Space has Neighborhood
Let $T = \left({S, \tau}\right)$ be a topological space. Let $x \in S$. Then there exists in $T$ at least one neighborhood of $x$.
Point in Topological Space is Element of its Neighborhood
Let $T = \left({S, \tau}\right)$ be a topological space. Let $x \in S$. Let $N$ be a neighborhood of $x$ in $T$. Then $a \in N$.
Intersection of Neighborhoods in Topological Space is Neighborhood
Let $T = \left({S, \tau}\right)$ be a topological space. Let $x \in S$. Let $M, N$ be a neighborhoods of $x$ in $T$. Then $M \cap N$ is a neighborhood of $x$ in $T$.
Neighborhood in Topological Space has Subset Neighborhood
Let $T = \left({S, \tau}\right)$ be a topological space. Let $x \in S$. Let $N$ be a neighborhood of $x$ in $T$. Then there exists a neighborhood $N'$ of $x$ such that: :$(1): \quad N' \subseteq N$ :$(2): \quad N'$ is a neighborhood of each of its points.
Empty Set is Open in Neighborhood Space
Let $\struct {S, \mathcal N}$ be a neighborhood space. Then the empty set $\varnothing$ is an open set of $\struct {S, \mathcal N}$.
Whole Space is Open in Neighborhood Space
Let $\struct {S, \NN}$ be a neighborhood space. Then $S$ itself is an open set of $\struct {S, \NN}$.
Intersection of two Open Sets of Neighborhood Space is Open
Let $\left({S, \mathcal N}\right)$ be a neighborhood space. Let $U$ and $V$ be open sets of $\left({S, \mathcal N}\right)$. Then $U \cap V$ is an open set of $\left({S, \mathcal N}\right)$.
Union of Open Sets of Neighborhood Space is Open
Let $S$ be a neighborhood space. Let $I$ be an indexing set. Let $\family {U_\alpha}_{\alpha \mathop \in I}$ be a family of open sets of $\struct {S, \NN}$ indexed by $I$. Then their union $\ds \bigcup_{\alpha \mathop \in I} U_i$ is an open set of $\struct {S, \NN}$.
Neighborhood Space is Topological Space
Let $\left({S, \mathcal N}\right)$ be a neighborhood space. Let $\tau = \left\{{N: N \in \mathcal N}\right\}$ be the set of all open sets of $\left({S, \mathcal N}\right)$. Then $\left({S, \tau}\right)$ forms a topological space.
Subset in Neighborhood Space is Neighborhood iff it contains Open Set
Let $\left({S, \mathcal N}\right)$ be a neighborhood space. Let $x \in S$ be a point of $S$. Let $N \subseteq S$ be a subset of $S$. Then $N$ is a neighborhood of $x$  there exists an open set $U$ of $\left({S, \mathcal N}\right)$ such that $x \in U \subseteq N$.
Definition:Neighborhood Space Induced by Topological Space
Let $S$ be a set. Let $\tau$ be a topology on $S$, thus forming the topological space $\left({S, \tau}\right)$. For all $x \in S$, let $\mathcal N_x$ be the neighborhood filter of $x$ in $S$. Let $\mathcal N$ be the set of subsets of $S$ such that: :$\forall N \in \mathcal N: N$ is a neighborhood of each of its points Then $\left({S, \mathcal N}\right)$ is the '''neighborhood space induced by $\left({S, \tau}\right)$'''.
Topological Space Induced by Neighborhood Space is Topological Space
Let $\left({S, \mathcal N}\right)$ be a neighborhood space. Let $\left({S, \tau}\right)$ be the topological space induced by $\left({S, \mathcal N}\right)$. Then $\left({S, \tau}\right)$ is a topological space.
Topological Space induced by Neighborhood Space induced by Topological Space
Let $S$ be a set. Let $\tau$ be a topology on $S$, thus forming the topological space $\left({S, \tau}\right)$. Let $\left({S, \mathcal N}\right)$ be the neighborhood space induced by $\tau$ on $S$. Let $\left({S, \tau'}\right)$ be the topological space induced by $\mathcal N$ on $S$. Then $\tau = \tau'$.
Correspondence between Neighborhood Space and Topological Space
Let $S$ be a set. Let $\left({S, \tau}\right)$ be a topological space. Let $\left({S, \mathcal N}\right)$ be the neighborhood space induced by $\tau$ on $S$. Let $\phi: \left({S, \tau}\right) \to \left({S, \mathcal N}\right)$ be the mapping defined as: :$\forall x \in S: \phi \left({x}\right) = x$ :$\forall U \in \tau: \phi \left({U}\right) = U \in \mathcal N$ Let $\left({T, \mathcal N'}\right)$ be a neighborhood space. Let $\left({T, \tau'}\right)$ be the topological space induced by $\mathcal N$ on $S$. Let $\psi: \left({T, \mathcal N'}\right) \to \left({T, \tau'}\right)$ be the mapping defined as: :$\forall y \in T: \psi \left({y}\right) = y$ :$\forall V \in \mathcal N': \psi \left({V}\right) = V \in \tau'$ Then: :$(1): \quad \phi^{-1} \left({\phi \left({S, \tau}\right)}\right) = \left({S, \tau}\right)$ :$(2): \quad \psi^{-1} \left({\psi \left({T, \mathcal N'}\right)}\right) = \left({T, \mathcal N'}\right)$
Neighborhood Space induced by Topological Space induced by Neighborhood Space
Let $\left({S, \mathcal N}\right)$ be a neighborhood space. Let $\left({S, \tau}\right)$ be the topological space induced by $\mathcal N$ on $S$. Let $\left({S, \mathcal N'}\right)$ be the neighborhood space induced by $\tau$ on $S$. Then $\mathcal N = \mathcal N'$.
Binomial Coefficient with Zero/Integer Coefficients
:$\forall n \in \N: \dbinom n 0 = 1$ where $\dbinom n 0$ denotes a binomial coefficient.
Form of Geometric Sequence of Integers in Lowest Terms
Let $G_n = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a geometric sequence of length $n$ consisting of positive integers only. Let $r$ be the common ratio of $G_n$. Let the elements of $G_n$ be the smallest positive integers such that $G_n$ has common ratio $r$. Then the $j$th term of $G_n$ is given by: :$a_j = p^{n - j} q^j$ where $r = \dfrac q p$. That is: :$G_n = \tuple {p^n, p^{n - 1} q, p^{n - 2} q^2, \ldots, p q^{n - 1}, q^n}$
Form of Geometric Sequence of Integers with Coprime Extremes
Let $Q_n = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a geometric sequence of length $n$ consisting of positive integers only. Let $a_1$ and $a_n$ be coprime. Then the $j$th term of $Q_n$ is given by: :$a_j = q^j p^{n - j}$
Naturally Ordered Semigroup forms Peano Structure
Let $\left({S, \circ, \preceq}\right)$ be a naturally ordered semigroup. Let $0 \in S$ be the zero of $S$. Let $1 \in S$ be the one of $S$. Let $s: S \to S$ be the mapping defined as: :$s \left({n}\right) := n \circ 1$ Then $\left({S, 0, s}\right)$ is a Peano structure.
Divisibility of Elements in Geometric Sequence of Integers
Let $Q_n = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a geometric sequence of integers. Let $j \ne k$. Then: :$\paren {\exists j \in \set {0, 1, \ldots, n - 1}: a_j \divides a_{j + 1} } \iff \paren {\forall j, k \in \set {0, 1, \ldots, n}, j < k: a_j \divides a_k}$ where $\divides$ denotes integer divisibility. That is: :One term of a geometric sequence of integers is the divisor of the next term  :All terms are divisors of all later terms.
Form of Geometric Sequence of Integers from One
Let $Q_n = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a geometric sequence consisting of integers only. Let $a_0 = 1$. Then the $j$th term of $Q_n$ is given by: :$a_j = a^j$ where: :the common ratio of $Q_n$ is $a$ :$a = a_1$. Thus: :$Q_n = \tuple {1, a, a^2, \ldots, a^n}$
Divisors of Power of Prime
Let $p$ be a prime number. Let $n \in \Z_{> 0}$ be a (strictly) positive integer. Then the only divisors of $p^n$ are $1, p, p^2, \ldots, p^{n - 1}, p^n$.
Piecewise Continuously Differentiable Function/Definition 2 is Continuous
Let $f$ be a real function defined on a closed interval $\closedint a b$. Let $f$ satisfy Piecewise Continuously Differentiable Function/Definition 2. Then $f$ is continuous.
Homomorphism of Chain Complexes induces Homomorphism of Homology
Let $A_\bullet$ and $B_\bullet$ be chain complexes of abelian groups. Let $f: A_\bullet \to B_\bullet$ be a homomorphism. Then for every $n$, $f$ induces a morphism $H_n \left({A_\bullet}\right) \to H_n \left({B_\bullet}\right)$ of homology groups. {{explain|Domain of $n$}}
Homotopic Chain Maps Induce Equal Maps on Homology
Let $A_\bullet$, $B_\bullet$ be chain complexes of abelian groups. Let $f, g: A_\bullet \to B_\bullet$ be chain maps which are homotopic. Then $f$ and $g$ induce equal maps on homology.
Equivalence of Definitions of Piecewise Continuously Differentiable Function
{{TFAE|def = Piecewise Continuously Differentiable Function}} Let $f$ be a real function defined on a closed interval $\closedint a b$.
Bounded Function Continuous on Open Interval is Darboux Integrable
Let $f$ be a real function defined on an interval $\closedint a b$ such that $a < b$. Let $f$ be continuous on $\closedint a b$. Let $f$ be bounded on $\closedint a b$. Then $f$ is Darboux integrable on $\closedint a b$.
Square Inscribed in Circle is greater than Half Area of Circle
A square inscribed in a circle has an area greater than half that of the circle.
Bounded Piecewise Continuous Function is Darboux Integrable
Let $f$ be a real function defined on the closed interval $\closedint a b$. Let $f$ be piecewise continuous and bounded on $\closedint a b$. Then $f$ is Darboux integrable on $\closedint a b$.
Rational Numbers form Subset of Real Numbers
The set $\Q$ of rational numbers forms a subset of the real numbers $\R$.
Real Number is not necessarily Rational Number
Let $x$ be a real number. Then it is not necessarily the case that $x$ is also a rational number.
Zero is both Positive and Negative
The number $0$ (zero) is the only (real) number which is both: :a positive (real) number and :a negative (real) number.
Binary Operation on Subset is Binary Operation
Let $S$ be a set. Let $\circ$ be a binary operation on $S$. Let $T \subseteq S$. Let $\circ {\restriction}_T$ be the restriction of $\circ$ to $T$. Then $\circ {\restriction}_T$ is a binary operation on $T$.
Matrix Entrywise Addition over Ring is Associative
Let $\struct {R, +, \circ}$ be a ring. Let $\map {\MM_R} {m, n}$ be a $m \times n$ matrix space over $R$. For $\mathbf A, \mathbf B \in \map {\MM_R} {m, n}$, let $\mathbf A + \mathbf B$ be defined as the matrix entrywise sum of $\mathbf A$ and $\mathbf B$. The operation $+$ is associative on $\map {\MM_R} {m, n}$. That is: :$\paren {\mathbf A + \mathbf B} + \mathbf C = \mathbf A + \paren {\mathbf B + \mathbf C}$ for all $\mathbf A$, $\mathbf B$ and $\mathbf C$ in $\map {\MM_R} {m, n}$.
Final Topology with respect to Mapping
Let $\left({X, \tau_X}\right)$ be a topological space. Let $Y$ be a set. Let $f: X \to Y$ be a mapping. Let $\tau_Y$ be the final topology on $Y$ with respect to $f$. Then: :$\tau_Y = \left\{ U \subseteq Y : f^{-1} \left({U}\right) \in \tau_X \right\}$ Observe that the set on the right-hand side of the equality is sometimes denoted $f\tau$. Further, the following holds true: # $\forall \left(Z, \tau_Z \right)$ topological space $: \forall g: \left({Y, f\tau}\right) \to \left({Z, \tau_Z}\right)$ mapping $: g$ is continuous $\iff g \circ f$ is continuous (where $f$ is defined from $\left({X, \tau_X}\right)$ to $\left(Y, f\tau \right)$). # $\forall \tau_Y$ topology in $Y : \forall \left(Z, \tau_Z \right)$ topological space, the following are equivalent: ##$\tau_Y = f\tau$ ##$\forall g: \left({Y, \tau_Y}\right) \to \left({Z, \tau_Z}\right) : g$ is continuous $\iff g \circ f$ is continuous (where $f$ is defined from $\left({X, \tau_X}\right)$ to $\left(Y, \tau_Y \right)$). That is, the topology $f\tau$ is ''characterised'' by the fact that it is the only one which makes all mappings defined from it to any topological space continuous.
Composition of Symmetries is Symmetry
Let $P$ be a geometric figure. Let $S_P$ be the set of all symmetries of $P$. Let $\circ$ denote composition of mappings. Let $\phi$ and $\psi$ be symmetries of $P$. Then $\phi \circ \psi$ is also a symmetry of $P$.
Composition of Symmetries is Associative
Let $P$ be a geometric figure. Let $S_P$ be the set of all symmetries of $P$. Let $\circ$ denote composition of mappings. Let $\phi, \psi, \chi$ be symmetries of $P$. Then: : $\paren {\phi \circ \psi} \circ \chi = \phi \circ \paren {\psi \circ \chi}$ That is, composition of symmetries is associative.
Square of Modulo less One equals One
Let $m \in \Z$ be an integer. Let $\Z_m$ be the set of integers modulo $m$: :$\Z_m = \set {\eqclass 0 m, \eqclass 1 m, \ldots, \eqclass {m - 1} m}$ Then: :$\eqclass {m - 1} m \times_m \eqclass {m - 1} m = \eqclass 1 m$ where $\times_m$ denotes multiplication modulo $m$.
Symmetry Group of Equilateral Triangle is Symmetric Group
Let $D_3$ denote the symmetry group of the equilateral triangle. Let $S_3$ denote the symmetric group on $3$ letters. Then $D_3$ is isomorphic to $S_3$.
De Morgan's Laws (Set Theory)/Relative Complement/General Case/Complement of Intersection
:$\ds \relcomp S {\bigcap \mathbb T} = \bigcup_{H \mathop \in \mathbb T} \relcomp S H$
De Morgan's Laws (Set Theory)/Relative Complement/General Case/Complement of Union
:$\displaystyle \relcomp S {\bigcup \mathbb T} = \bigcap_{H \mathop \in \mathbb T} \relcomp S H$
De Morgan's Laws (Set Theory)/Relative Complement/Family of Sets/Complement of Union
:$\displaystyle \relcomp S {\bigcup_{i \mathop \in I} \mathbb S_i} = \bigcap_{i \mathop \in I} \relcomp S {S_i}$
De Morgan's Laws (Set Theory)/Relative Complement/Family of Sets/Complement of Intersection
:$\displaystyle \relcomp S {\bigcap_{i \mathop \in I} \mathbb S_i} = \bigcup_{i \mathop \in I} \relcomp S {S_i}$
De Morgan's Laws (Set Theory)/Relative Complement/Complement of Union
:$\relcomp S {T_1 \cup T_2} = \relcomp S {T_1} \cap \relcomp S {T_2}$
De Morgan's Laws (Set Theory)/Relative Complement/Complement of Intersection
:$\relcomp S {T_1 \cap T_2} = \relcomp S {T_1} \cup \relcomp S {T_2}$
De Morgan's Laws (Set Theory)/Set Difference/Family of Sets/Difference with Intersection
:$\displaystyle S \setminus \bigcap_{i \mathop \in I} T_i = \bigcup_{i \mathop \in I} \paren {S \setminus T_i}$ where: :$\displaystyle \bigcup_{i \mathop \in I} T_i := \set {x: \exists i \in I: x \in T_i}$ that is, the union of $\family {T_i}_{i \mathop \in I}$.
De Morgan's Laws (Set Theory)/Set Difference/Family of Sets/Difference with Union
:$\displaystyle S \setminus \bigcup_{i \mathop \in I} T_i = \bigcap_{i \mathop \in I} \paren {S \setminus T_i}$ where: :$\displaystyle \bigcap_{i \mathop \in I} T_i := \set {x: \forall i \in I: x \in T_i}$ that is, the intersection of $\family {T_i}_{i \mathop \in I}$.
De Morgan's Laws (Set Theory)/Set Complement/General Case/Complement of Intersection
:$\displaystyle \complement \paren {\bigcap \mathbb T} = \bigcup_{H \mathop \in \mathbb T} \complement \paren H$
De Morgan's Laws (Set Theory)/Set Complement/General Case/Complement of Union
:$\displaystyle \complement \paren {\bigcup \mathbb T} = \bigcap_{H \mathop \in \mathbb T} \complement \paren H$
De Morgan's Laws (Set Theory)/Set Complement/Complement of Intersection
:$\overline {T_1 \cap T_2} = \overline T_1 \cup \overline T_2$