hcju/proofwikips huggingface dataset:
Dicyclic Group is Group
The dicyclic group $Q_n$ is a non-abelian group on two generators.
Existence of Interval of Convergence of Power Series
Let $\xi \in \R$ be a real number. Let $\displaystyle \map S x = \sum_{n \mathop = 0}^\infty a_n \paren {x - \xi}^n$ be a power series about $\xi$. Then the interval of convergence of $\map S x$ is a real interval whose midpoint is $\xi$.
Power Series is Differentiable on Interval of Convergence
Let $\xi \in \R$ be a real number. Let $\displaystyle \map f x = \sum_{n \mathop = 0}^\infty a_n \paren {x - \xi}^n$ be a power series about $\xi$. Let $\map f x$ have an interval of convergence $I$. Then $\map f x$ is continuous on $I$, and differentiable on $I$ except possibly at its endpoints. Also: :$\displaystyle \map {D_x} {\map f x} = \sum_{n \mathop = 1}^\infty n a_n \paren {x - \xi}^{n - 1}$
Power Series Expansion for Exponential Function
Let $\exp x$ be the exponential function. Then: {{begin-eqn}} {{eqn | ll= \forall x \in \R:       | l = \exp x       | r = \sum_{n \mathop = 0}^\infty \frac {x^n} {n!}       | c =  }} {{eqn | r = 1 + x + \frac {x^2} {2!} + \frac {x^3} {3!} + \cdots       | c =  }} {{end-eqn}}
Properties of Real Sine Function
Let $x \in \R$ be a real number. Let $\sin x$ be the sine of $x$. Then:
Properties of Real Cosine Function
Let $x \in \R$ be a real number. Let $\cos x$ be the cosine of $x$. Then:
Derivative of Cosine Function
:$\map {\dfrac \d {\d x} } {\cos x} = -\sin x$
Limit of Sine of X over X
:$\displaystyle \lim_{x \mathop \to 0} \frac {\sin x} x = 1$
Sum of Squares of Sine and Cosine
:$\cos^2 x + \sin^2 x = 1$
Cosine of Sum
:$\map \cos {a + b} = \cos a \cos b - \sin a \sin b$
Equivalence of Definitions of Sine and Cosine
The definitions for sine and cosine are equivalent. That is: : $\displaystyle \sin x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!} \iff \sin x = \frac {\text{Opposite}} {\text{Hypotenuse}}$ : $\displaystyle \cos x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!} \iff \cos x = \frac {\text{Adjacent}} {\text{Hypotenuse}}$
Real Cosine Function is Bounded
: $\size {\cos x} \le 1$
Boundedness of Sine X over X
Let $x \in \R$. Then: :$\size {\dfrac {\sin x} x} \le 1$
Sine and Cosine are Periodic on Reals
The sine and cosine functions are periodic on the set of real numbers $\R$: :$(1): \quad \map \cos {x + 2 \pi} = \cos x$ :$(2): \quad \map \sin {x + 2 \pi} = \sin x$ :800px
Differentiable Bounded Convex Real Function is Constant
Let $f$ be a real function which is: :$(1): \quad$ Differentiable on $\R$ :$(2): \quad$ Bounded on $\R$ :$(3): \quad$ Convex on $\R$. Then $f$ is constant.
Banach-Tarski Paradox
The unit ball $\mathbb D^3 \subset \R^3$ is equidecomposable to the union of two unit balls.
Equidecomposability is Equivalence Relation
The property of being equidecomposable is an equivalence relation on the power set $\mathcal P \left({\R^n}\right)$.
Equidecomposability Unaffected by Union
{{explain|I guess the sets should be disjoint, in the sense that $S_i \cap S_j$ is empty for $i \neq j$. Similar for $T$.}} Let $\left\{{S_1, \ldots, S_m}\right\}, \left\{{T_1, \ldots, T_m }\right\}$ be sets of sets in $\R^n$ such that: : for each $k \in \left\{{1, \dots, m}\right\}, S_k$ and $T_k$ are equidecomposable.  Then the set $\displaystyle S = \bigcup_{i \mathop = 1}^m S_i$ is equidecomposable with $\displaystyle T = \bigcup_{i \mathop = 1}^m T_i$.
Subsets of Equidecomposable Subsets are Equidecomposable
Let $A, B \subseteq \R^n$ be equidecomposable. Let $S \subseteq A$. Then there exists $T \subseteq B$ such that $S$ and $T$ are equidecomposable.
Shape of Cosine Function
The cosine function is: :$(1): \quad$ strictly decreasing on the interval $\closedint 0 \pi$ :$(2): \quad$ strictly increasing on the interval $\closedint \pi {2 \pi}$ :$(3): \quad$ concave on the interval $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$ :$(4): \quad$ convex on the interval $\closedint {\dfrac \pi 2} {\dfrac {3 \pi} 2}$
Tangent Function is Periodic on Reals
The tangent function is periodic on the set of real numbers $\R$ with period $\pi$. This can be written: :$\tan x = \tan \left({x \bmod \pi}\right)$ where $x \bmod \pi$ denotes the modulo operation.
Derivative of Tangent Function
:$\map {\dfrac \d {\d x} } {\tan x} = \sec^2 x = \dfrac 1 {\cos^2 x}$ when $\cos x \ne 0$.
Stirling's Formula
The factorial function can be approximated by the formula: :$n! \sim \sqrt {2 \pi n} \paren {\dfrac n e}^n$ where $\sim$ denotes asymptotically equal.
Derivative of Cotangent Function
:$\map {\dfrac \d {\d x} } {\cot x} = -\csc^2 x = \dfrac {-1} {\sin^2 x}$ where $\sin x \ne 0$.
Shape of Sine Function
The sine function is: :$(1): \quad$ strictly increasing on the interval $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$ :$(2): \quad$ strictly decreasing on the interval $\closedint {\dfrac \pi 2} {\dfrac {3 \pi} 2}$ :$(3): \quad$ concave on the interval $\closedint 0 \pi$ :$(4): \quad$ convex on the interval $\closedint \pi {2 \pi}$
Derivative of Secant Function
:$\map {\dfrac \d {\d x} } {\sec x} = \sec x \tan x$ where $\cos x \ne 0$.
Derivative of Cosecant Function
:$\map {\dfrac \d {\d x} } {\csc x} = -\csc x \cot x$ where $\sin x \ne 0$.
Shape of Tangent Function
The nature of the tangent function on the set of real numbers $\R$ is as follows: : $\tan x$ is continuous and strictly increasing on the interval $\left({-\dfrac \pi 2 \,.\,.\, \dfrac \pi 2}\right)$ : $\tan x \to + \infty$ as $x \to \dfrac \pi 2 ^-$ : $\tan x \to - \infty$ as $x \to -\dfrac \pi 2 ^+$ : $\tan x$ is not defined on $\forall n \in \Z: x = \left({n + \dfrac 1 2}\right) \pi$, at which points it is discontinuous : $\forall n \in \Z: \tan \left({n \pi}\right) = 0$.
Derivative of Arcsine Function
:$\dfrac {\map \d {\arcsin x} } {\d x} = \dfrac 1 {\sqrt {1 - x^2} }$
Derivative of Arccosine Function
:$\map {D_x} {\arccos x} = \dfrac {-1} {\sqrt {1 - x^2}}$
Sum of Arcsine and Arccosine
Let $x \in \R$ be a real number such that $-1 \le x \le 1$. Then: : $\arcsin x + \arccos x = \dfrac \pi 2$ where $\arcsin$ and $\arccos$ denote arcsine and arccosine respectively.
Derivative of Arctangent Function
:$\dfrac {\map \d {\arctan x} } {\d x} = \dfrac 1 {1 + x^2}$
Complex Numbers form Field
Consider the algebraic structure $\struct {\C, +, \times}$, where: :$\C$ is the set of all complex numbers :$+$ is the operation of complex addition :$\times$ is the operation of complex multiplication Then $\struct {\C, +, \times}$ forms a field.
Complex Multiplication is Commutative
The operation of multiplication on the set of complex numbers $\C$ is commutative: :$\forall z_1, z_2 \in \C: z_1 z_2 = z_2 z_1$
Complex Multiplication Distributes over Addition
The operation of multiplication on the set of complex numbers $\C$ is distributive over the operation of addition. :$\forall z_1, z_2, z_3 \in \C:$ ::$z_1 \paren {z_2 + z_3} = z_1 z_2 + z_1 z_3$ ::$\paren {z_2 + z_3} z_1 = z_2 z_1 + z_3 z_1$
Complex Modulus of Product of Complex Numbers
Let $z_1, z_2 \in \C$ be complex numbers. Let $\cmod z$ be the modulus of $z$. Then: :$\cmod {z_1 z_2} = \cmod {z_1} \cdot \cmod {z_2}$
Sum of Complex Conjugates
Let $z_1, z_2 \in \C$ be complex numbers. Let $\overline z$ denote the complex conjugate of the complex number $z$. Then: :$\overline {z_1 + z_2} = \overline {z_1} + \overline {z_2}$
Product of Complex Conjugates
Let $z_1, z_2 \in \C$ be complex numbers. Let $\overline z$ denote the complex conjugate of the complex number $z$. Then: :$\overline {z_1 z_2} = \overline {z_1} \cdot \overline {z_2}$
Sum of Complex Number with Conjugate
Let $z \in \C$ be a complex number. Let $\overline z$ be the complex conjugate of $z$. Let $\map \Re z$ be the real part of $z$. Then: :$z + \overline z = 2 \, \map \Re z$
Difference of Complex Number with Conjugate
Let $z \in \C$ be a complex number. Let $\overline z$ be the complex conjugate of $z$. Let $\map \Im z$ be the imaginary part of $z$. Then :$z - \overline z = 2 i \, \map \Im z$
Complex Number equals Conjugate iff Wholly Real
Let $z \in \C$ be a complex number. Let $\overline z$ be the complex conjugate of $z$. Then $z = \overline z$  $z$ is wholly real.
Complex Numbers cannot be Ordered Compatibly with Ring Structure
Let $\struct {\C, +, \times}$ be the field of complex numbers. There exists no total ordering on $\struct {\C, +, \times}$ which is compatible with the structure of $\struct {\C, +, \times}$.
Complex Plane is Metric Space
Let $\C$ be the set of all complex numbers. Let $d: \C \times \C \to \R$ be the function defined as: :$\map d {z_1, z_2} = \size {z_1 - z_2}$ where $\size z$ is the modulus of $z$. Then $d$ is a metric on $\C$ and so $\struct {\C, d}$ is a metric space.
Real Number Line is Metric Space
Let $\R$ be the real number line. Let $d: \R \times \R \to \R$ be defined as: :$\map d {x_1, x_2} = \size {x_1 - x_2}$ where $\size x$ is the absolute value of $x$. Then $d$ is a metric on $\R$ and so $\struct {\R, d}$ is a metric space.
P-Product Metric on Real Vector Space is Metric
Let $\R^n$ be an $n$-dimensional real vector space. Let $p \in \R_{\ge 1}$. Let $d_p: \R^n \times \R^n \to \R$ be the $p$-product metric on $\R^n$: : $\displaystyle d_p \left({x, y}\right) :=  \left({\sum_{i \mathop = 1}^n \left \vert {x_i - y_i} \right \vert^p}\right)^{\frac 1 p}$ where $x = \left({x_1, x_2, \ldots, x_n}\right), y = \left({y_1, y_2, \ldots, y_n}\right) \in \R^n$. Then $d_p$ is a metric.
Taxicab Metric is Metric
The taxicab metric is a metric.
Zero and One are the only Consecutive Perfect Squares
If $n$ is a perfect square other than $0$, then $n+1$ is not a perfect square.
ProofWiki:Jokes
If I cannot open these cans of food, I will die.
P-Product Metric is Metric
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 $p \in \R_{\ge 1}$. Let $d_p: \mathcal A \times \mathcal A \to \R$ be the $p$-product metric on $\mathcal A$: : $\displaystyle d_p \left({x, y}\right) := \left({\sum_{i \mathop = 1}^n \left({d_{i'} \left({x_i, y_i}\right)}\right)^p}\right)^{\frac 1 p}$ where $x = \left({x_1, x_2, \ldots, x_n}\right), y = \left({y_1, y_2, \ldots, y_n}\right) \in \mathcal A$. Then $d_p$ is a metric.
Euler Triangle Formula
Let $d$ be the distance between the incenter and the circumcenter of a triangle. Then: :$d^2 = R \left({R - 2 \rho}\right)$ where: :$R$ is the circumradius :$\rho$ is the inradius.
Open Ball of Point Inside Open Ball
Let $M = \left({A, d}\right)$ be a metric space. Let $B_\epsilon \left({x}\right)$ be an open $\epsilon$-ball in $M = \left({A, d}\right)$. Let $y \in B_\epsilon \left({x}\right)$. Then: : $\exists \delta \in \R: B_\delta \left({y}\right) \subseteq B_\epsilon \left({x}\right)$ That is, for every point in an open $\epsilon$-ball in a metric space, there exists an open $\delta$-ball of that point entirely contained within that open $\epsilon$-ball.
Open Sets in Metric Space
Let $M = \struct {A, d}$ be a metric space. Then $\O$ and $A$ are both open in $M$.
Equivalence of Definitions of Continuity on 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 mapping from $A_1$ to $A_2$. {{TFAE|def = Continuous on Metric Space|view = continuity|context = Metric Space|contextview = metric spaces}}
Finite Intersection of Open Sets of Metric Space is Open
Let $M = \struct {A, d}$ be a metric space. Let $U_1, U_2, \ldots, U_n$ be open in $M$. Then $\ds \bigcap_{i \mathop = 1}^n U_i$ is open in $M$. That is, a finite intersection of open subsets is open.
Union of Open Sets of Metric Space is Open
Let $M = \struct {A, d}$ be a metric space. The union of a set of open sets of $M$ is open in $M$.
Lipschitz Equivalent 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 Lipschitz equivalent. Then $M_1$ and $M_2$ are homeomorphic.
P-Product Metrics on Real Vector Space are Topologically Equivalent
For $n \in \N$, let $\R^n$ be an Euclidean space. Let $p \in \R_{\ge 1}$. Let $d_p$ be the $p$-product metric on $\R^n$. Let $d_\infty$ be the Chebyshev distance on $\R^n$. Then $d_p$ and $d_\infty$ are topologically equivalent.
Derivative of Function of Constant Multiple
Let $f$ be a real function which is differentiable on $\R$. Let $c \in \R$ be a constant. Then: :$\map {D_x} {\map f {c x} } = c \map {D_{c x} } {\map f {c x} }$
Cotangent Function is Periodic on Reals
The cotangent function is periodic on the set of real numbers $\R$ with period $\pi$.
Shape of Cotangent Function
The nature of the cotangent function on the set of real numbers $\R$ is as follows: : $\cot x$ is continuous and strictly decreasing on the interval $\left({0 \,.\,.\, \pi}\right)$ : $\cot x \to + \infty$ as $x \to 0^+$ : $\cot x \to - \infty$ as $x \to \pi^-$ : $\cot x$ is not defined on $\forall n \in \Z: x = n \pi$, at which points it is discontinuous : $\forall n \in \Z: \cot \left({n + \dfrac 1 2}\right) \pi = 0$
Change of Base of Logarithm
:$\log_b x = \dfrac {\log_a x} {\log_a b}$
Derivative of Function to Power of Function
Let $\map u x, \map v x$ be real functions which are differentiable on $\R$. Then: :$\map {\dfrac \d {\d x} } {u^v} = v u^{v - 1} \map {\dfrac \d {\d x} } u + u^v \paren {\ln u} \map {\dfrac \d {\d x} } v$
Liouville's Theorem (Complex Analysis)
Let $f: \C \to \C$ be a bounded entire function. Then $f$ is constant.
Riemann Removable Singularities Theorem
Let $U \subset \C$ be a domain, let $z_0 \in U$, and let $f: U \setminus \left\{ {z_0}\right\} \to \C$ be holomorphic. Then the following are equivalent: :$(1): \quad f$ extends to a holomorphic function $f: U \to \C$. :$(2): \quad f$ extends to a continuous function $f: U \to \C$. :$(3): \quad f$ is bounded in a neighborhood of $z_0$. :$(4): \quad f \left({z}\right) = o \left({\dfrac 1 {\left|{z - z_0}\right|} }\right)$ as $z \to z_0$.
Gaussian Integral
:$\displaystyle \int_{-\infty}^\infty e^{-x^2} \rd x = \sqrt \pi$
Indiscrete Topology is not Metrizable
Let $S$ be a set with more than one element. The indiscrete topology on $S$ is not metrizable.
Composite of Continuous Mappings is Continuous
Let $T_1, T_2, T_3$ be topological spaces. Let $f: T_1 \to T_2$ and $g: T_2 \to T_3$ be continuous mappings. Then the composite mapping $g \circ f: T_1 \to T_3$ is continuous.
Continuous Mapping is Continuous on Induced Topological Spaces
Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces. Let $\tau_{d_1}$ and $\tau_{d_2}$ be the topologies induced by the metrics $d_1$ and $d_2$. Let $T_1 = \left({A_1, \tau_{d_1}}\right)$ and $T_2 = \left({A_2, \tau_{d_2}}\right)$ be the resulting topological spaces. Let $f: A_1 \to A_2$ be a mapping. Then $f$ is $\left({d_1, d_2}\right)$-continuous  $f$ is $\left({\tau_{d_1}, \tau_{d_2}}\right)$-continuous.
Synthetic Basis and Analytic Basis are Compatible
Let $\struct {S, \tau}$ be a topological space. Then $\BB$ is an analytic basis for $\tau$  $\tau$ is the topology on $S$ generated by the synthetic basis $\BB$.
Picard's Existence Theorem
Let $\map f {x, y} : \R^2 \to \R$ be continuous in a region $D \subseteq \R^2$. Let $\exists M \in \R: \forall x, y \in D: \size {\map f {x, y} } < M$. Let $\map f {x, y}$ satisfy in $D$ the Lipschitz condition in $y$: :$\size{\map f {x, y_1} - \map f {x, y_2} } \le A \size {y_1 - y_2}$ where $A$ is independent of $x, y_1, y_2$. Let the rectangle $R$ be defined as $\set {\tuple {x, y} \in \R^2: \size {x - a} \le h, \size {y - b} \le k}$ such that $M h \le k$. Let $R \subseteq D$. Then $\forall x \in \R: \size {x - a} \le h$, the first order ordinary differential equation: :$y' = \map f {x, y}$ has one and only one solution $y = \map y x$ for which $b = \map y a$.
Continuity Test using Basis
Let $\left({X_1, \tau_1}\right)$ and $\left({X_2, \tau_2}\right)$ be topological spaces. Let $f: X_1 \to X_2$ be a mapping. Let $\mathcal B$ be an analytic basis for $\tau_2$. Suppose that: :$\forall B \in \mathcal B: f^{-1} \left({B}\right) \in \tau_1$ where $f^{-1} \left({B}\right)$ denotes the preimage of $B$ under $f$. Then $f$ is continuous.
Sub-Basis for Real Number Line
Let the real number line $\R$ be considered as a topology under the usual (Euclidean) metric. Then: :$\mathcal B := \left\{{\left({-\infty \,.\,.\, a}\right), \left({b \,.\,.\, \infty}\right): a, b \in \R}\right\}$ is a sub-basis for $\R$.
Synthetic Basis formed from Synthetic Sub-Basis
Let $X$ be a set. Let $\SS$ be a synthetic sub-basis on $X$. Define: :$\displaystyle \BB = \set {\bigcap \FF: \FF \subseteq \SS, \text{$\FF$ is finite} }$ Then $\BB$ is a synthetic basis on $X$.
Epimorphism Preserves Distributivity
Let $\struct {R_1, +_1, \circ_1}$ and $\struct {R_2, +_2, \circ_2}$ be algebraic structures. Let $\phi: R_1 \to R_2$ be an epimorphism. :If $\circ_1$ is left distributive over $+_1$, then $\circ_2$ is left distributive over $+_2$. :If $\circ_1$ is right distributive over $+_1$, then $\circ_2$ is right distributive over $+_2$. Consequently, if $\circ_1$ is distributive over $+_1$, then $\circ_2$ is distributive over $+_2$. That is, epimorphism preserves distributivity.
Epimorphism Preserves Groups
Let $\struct {S, \circ}$ and $\struct {T, *}$ be algebraic structures. Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be an epimorphism. Let $\struct {S, \circ}$ be a group. Then $\struct {T, *}$ is also a group.
Continuity of Composite with Inclusion
Let $T = \left({A, \tau}\right)$ and $T' = \left({A', \tau'}\right)$ be topological spaces. Let $H \subseteq A$. Let $T_H = \left({H, \tau_H}\right)$ be a topological subspace of $T$. Let $i: H \to A$ be the inclusion mapping. Let $f: A \to A'$ and $g: A' \to H$ be mappings. Then the following apply:
Totally Bounded Metric Space is Bounded
Let $M = \struct {A, d}$ be a totally bounded metric space. Then $M$ is bounded.
Convergent Subsequence of Cauchy Sequence
Let $\left({A, d}\right)$ be a metric space. Let $\left\langle{x_n}\right\rangle_{n \in \N}$ be a Cauchy sequence in $A$.  Let $x \in A$. Then $\left\langle{x_n}\right\rangle$ converges to $x$  it has a subsequence that converges to $x$.
Existence and Uniqueness of Sigma-Algebra Generated by Collection of Subsets
Let $X$ be a set. Let $\mathcal G \subseteq \powerset X$ be a collection of subsets of $X$. Then $\map \sigma {\mathcal G}$, the $\sigma$-algebra generated by $\mathcal G$, exists and is unique.
Projection from Product Topology is Continuous
Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces. Let $T = \struct {T_1 \times T_2, \tau}$ be the product space of $T_1$ and $T_2$, where $\tau$ is the Tychonoff topology on $S$. Let $\pr_1: T \to T_1$ and $\pr_2: T \to T_2$ be the first and second projections from $T$ onto its factors. Then both $\pr_1$ and $\pr_2$ are are continuous.
Continuous Mapping to Topological Product
Let $T = T_1 \times T_2$ be a product space of two topological spaces $T_1$ and $T_2$. Let $\pr_1: T \to T_1$ and $\pr_2: T \to T_2$ be the first and second projections from $T$ onto its factors. Let $T'$ be a topological space. Let $f: T' \to T$ be a mapping. Then $f$ is continuous  $\pr_1 \circ f$ and $\pr_2 \circ f$ are continuous.
Open and Closed Sets in Topological Space
Let $T = \struct {S, \tau}$ be a topological space. Then $S$ and $\O$ are both both open and closed in $T$.
Absolutely Continuous Real Function is Continuous
Let $I \subseteq \R$ be a real interval. Let $f : I \to \R$ be an absolutely continuous function. Then $f$ is continuous.
Closed Set in Topological Subspace
Let $T$ be a topological space. Let $T' \subseteq T$ be a subspace of $T$. Then $V \subseteq T'$ is closed in $T'$  $V = T' \cap W$ for some $W$ closed in $T$.
Continuity Defined from Closed Sets
Let $T_1$ and $T_2$ be topological spaces. Let $f: T_1 \to T_2$ be a mapping. Then $f$ is continuous  for all $V$ closed in $T_2$, $f^{-1} \sqbrk V$ is closed in $T_1$.
Real Numbers are Uncountable
The set of real numbers $\R$ is uncountably infinite.
Condition for Point being in Closure
Let $T$ be a topological space. Let $H \subseteq T$. Let $x \in T$. Then $x \in H^-$  every open set of $T$ which contains $x$ contains a point in $H$.
Topological Closure of Subset is Subset of Topological Closure
Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq K$ and $K \subseteq S$. Then: :$\map \cl H \subseteq \map \cl K$ where $\map \cl H$ denotes the closure of $H$.
Closure of Topological Closure equals Closure
Let $T$ be a topological space. Let $H \subseteq T$. Then: : $\left({H^-}\right)^- = H^-$ where $H^-$ denotes the closure of $H$.
Topological Closure is Closed
Let $T$ be a topological space. Let $H \subseteq T$. Then the closure $\map \cl H$ of $H$ is closed in $T$.
Equivalence of Definitions of Closure of Topological Subspace
{{TFAE|def = Closure (Topology)|view = Closure|context = Topology (Mathematical Branch)|contextview = Topology}} Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$.
Closure of Intersection is Subset of Intersection of Closures
Let $T$ be a topological space. Let $I$ be an indexing set. Let $\forall i \in I: H_i \subseteq T$. Then: : $\displaystyle \left({\bigcap_I H_i}\right)^- \subseteq \bigcap_I H_i^-$ where $H_i^-$ denotes the closure of $H_i$.
Closure of Finite Union equals Union of Closures
Let $T$ be a topological space. Let $n \in \N$. Let: :$\forall i \in \set {1, 2, \ldots, n}: H_i \subseteq T$ Then: :$\displaystyle \bigcup_{i \mathop = 1}^n \map \cl {H_i} = \map \cl {\bigcup_{i \mathop = 1}^n H_i}$
Closure of Open Ball in Metric Space
Let $M = \struct {A, d}$ be a metric space. Let $\map {B_\epsilon} x$ be an open $\epsilon$-ball in $M$. Let $y \in \map \cl {\map {B_\epsilon} x}$, where $\cl$ denotes the closure of $\map {B_\epsilon} x$. Then $\map d {x, y} \le \epsilon$.
Nowhere Dense iff Complement of Closure is Everywhere Dense
Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$. Then $H$ is nowhere dense in $T$  $S \setminus H^-$ is everywhere dense in $T$, where $H^-$ denotes the closure of $H$.
Totally Bounded Metric Space is Separable
A totally bounded metric space is separable.
Convergent Sequence in Hausdorff Space has Unique Limit
Let $T = \struct {S, \tau}$ be a Hausdorff space. Let $\sequence {x_n}$ be a convergent sequence in $T$. Then $\sequence {x_n}$ has exactly one limit.
Metric Space is Hausdorff
Let $M = \struct {A, d}$ be a metric space. Then $M$ is a Hausdorff space.
Convergent Real Sequence has Unique Limit
Let $\sequence {s_n}$ be a real sequence. Then $\sequence {s_n}$ can have at most one limit.
Distinct Points in Metric Space have Disjoint Open Balls
Let $M = \left({A, d}\right)$ be a metric space. Let $x, y \in M: x \ne y$. Then there exist disjoint open $\epsilon$-balls $B_\epsilon \left({x}\right)$ and $B_\epsilon \left({y}\right)$ containing $x$ and $y$ respectively.
Sequentially Compact Metric Space is Lindelöf
Let $M$ be a metric space. Let $M$ be sequentially compact. Then $M$ is also a Lindelöf space. That is, from every open cover of $M$, it is possible to extract a countable subcover.
Sequentially Compact Metric Space is Compact
A sequentially compact metric space is compact.
Compact Subspace of Metric Space is Bounded
Let $M = \left({A, d}\right)$ be a metric space. Let $C$ be a subspace of $M$. If $C$ is compact, then it is bounded.
Compact Subspace of Hausdorff Space is Closed
Let $H = \struct {A, \tau}$ be a Hausdorff space. Let $C$ be a compact subspace of $H$. Then $C$ is closed in $H$.
Continuous Image of Compact Space is Compact
Let $T_1$ and $T_2$ be topological spaces. Let $f: T_1 \to T_2$ be a continuous mapping. If $T_1$ is compact then so is its image $f \sqbrk {T_1}$ under $f$. That is, compactness is a continuous invariant.
Closed Subspace of Compact Space is Compact
A closed subspace of a compact space is compact. That is, the property of being compact is weakly hereditary.