Restriction of Non-Continuous Mapping on Metric Space to Subspace may be Continuous

Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be a metric spaces. Let $f: A_1 \to A_2$ be a mapping. Let $Y \subseteq A_1$. Let $f {\restriction_Y}: Y \to A_2$ be the restriction of $f$ to $Y$. Let $f {\restriction_Y}$ be $\left({d_Y, d_2}\right)$-continuous. Then it is not necessarily the case that $f$ is also $\left({d_1, d_2}\right)$-continuous.


Supremum Metric on Bounded Real Functions on Closed Interval is Metric

Let $\closedint a b \subseteq \R$ be a closed real interval. Let $A$ be the set of all bounded real functions $f: \closedint a b \to \R$. Let $d: A \times A \to \R$ be the supremum metric on $A$. Then $d$ is a metric.


Supremum Metric on Continuous Real Functions is Subspace of Bounded

Let $\left[{a \,.\,.\, b}\right] \subseteq \R$ be a closed real interval. Let $\mathscr C \left[{a \,.\,.\, b}\right]$ be the set of all continuous functions $f: \left[{a \,.\,.\, b}\right] \to \R$. Let $\mathscr B \left({\left[{a \,.\,.\, b}\right], \R}\right)$ be the set of all bounded real functions $f: \left[{a \,.\,.\, b}\right] \to \R$. Let $d$ be the supremum metric on $\mathscr B \left({\left[{a \,.\,.\, b}\right], \R}\right)$. Then $\left({\mathscr C \left[{a \,.\,.\, b}\right], d_{\mathscr C} }\right)$ is a subspace of $\left({\mathscr B \left({\left[{a \,.\,.\, b}\right], \R}\right), d}\right)$.


Zero Definite Integral of Nowhere Negative Function implies Zero Function

Let $\left[{a \,.\,.\, b}\right] \subseteq \R$ be a closed real interval. Let $h: \left[{a \,.\,.\, b}\right] \to \R$ be a continuous real function such that: :$\forall x \in \left[{a \,.\,.\, b}\right]: h \left({x}\right) \ge 0$ Let: :$\displaystyle \int_a^b h \left({x}\right) \rd x = 0$ Then: :$\forall x \in \left[{a \,.\,.\, b}\right]: h \left({x}\right) = 0$


L2 Metric on Closed Real Interval is Metric

Let $S$ be the set of all real functions which are continuous on the closed interval $\closedint a b$. Let $d: S \times S \to \R$ be the $L^2$ metric on $\closedint a b$: :$\displaystyle \forall f, g \in S: \map {d_2} {f, g} := \paren {\int_a^b \paren {\map f t - \map g t}^2 \rd t}^{\frac 1 2}$ Then $d_2$ is a metric.


Element in Bounded Metric Space has Bound

Let $M = \left({X, d}\right)$ be a metric space. Let $M' = \left({Y, d_Y}\right)$ be a subspace of $M$. Let $M'$ be bounded in $M$. Then: :$\forall a' \in X: \exists K' \in \R: \forall x \in Y: d \left({x, a}\right) \le K'$ That is, if there is one element of $X$ which satisfies the condition for $Y$ to be bounded in $M$, they ''all'' do.


Convergence of Square of Linear Combination of Sequences whose Squares Converge

Let $\sequence {x_i}$ and $\sequence {y_i}$ be real sequences such that the series $\ds \sum_{i \mathop \ge 0} x_i^2$ and $\ds \sum_{i \mathop \ge 0} y_i^2$ are convergent. Let $\lambda, \mu \in \R$ be real numbers. Then $\ds \sum_{i \mathop \ge 0} \paren {\lambda x_i + \mu y_i}^2$ is convergent.


Hilbert Sequence Space is Metric Space

Let $A$ be the set of all real sequences $\left\langle{x_i}\right\rangle$ such that the series $\displaystyle \sum_{i \mathop \ge 0} x_i^2$ is convergent. Let $\ell^2 = \left({A, d_2}\right)$ be the Hilbert sequence space on $\R$. Then $\ell^2$ is a metric space.


Supremum Metric on Bounded Real-Valued Functions is Metric

Let $X$ be a set. Let $A$ be the set of all bounded real-valued functions $f: X \to \R$. Let $d: A \times A \to \R$ be the supremum metric on $A$. Then $d$ is a metric.


Supremum Metric on Bounded Real Sequences is Metric

Let $A$ be the set of all bounded real sequences. Let $d: A \times A \to \R$ be the supremum metric on $A$. Then $d$ is a metric.


Supremum Metric is Metric

Let $S$ be a set. Let $M = \left({A', d'}\right)$ be a metric space. Let $A$ be the set of all bounded mappings $f: S \to M$. Let $d: A \times A \to \R$ be the supremum metric on $A$. Then $d$ is a metric.


Supremum Metric on Bounded Continuous Mappings is Metric

Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces. Let $A$ be the set of all continuous mappings $f: M_1 \to M_2$ which are also bounded. Let $d: A \times A \to \R$ be the supremum metric on $A$. Then $d$ is a metric.


Harmonic Number is not Integer

Let $H_n$ be the $n$th harmonic number. Then $H_n$ is not an integer for $n \ge 2$. That is, the only harmonic numbers that are integers are $H_0$ and $H_1$.


Supremum Metric on Differentiability Class is Metric

Let $\left[{a \,.\,.\, b}\right] \subseteq \R$ be a closed real interval. Let $r \in \N$ be a natural number. Let $A := \mathscr D^r \left[{a \,.\,.\, b}\right]$ be the set of all continuous functions $f: \left[{a \,.\,.\, b}\right] \to \R$ which are of differentiability class $r$. Let $d: A \times A \to \R$ be the supremum metric on $A$. Then $d$ is a metric.


Subspace of Metric Space is Metric Space

Let $M = \left({A, d}\right)$ be a metric space. Let $H \subseteq A$. Let $d_H: H \times H \to \R$ be the restriction $d \restriction_{H \times H}$ of $d$ to $H$. Let $\left({H, d_H}\right)$ be a metric subspace of $\left({A, d}\right)$. Then $d_H$ is a metric on $H$.


Lp Metric on Closed Real Interval is Metric

Let $S$ be the set of all real functions which are continuous on the closed interval $\left[{a \,.\,.\, b}\right]$. Let $p \in \R_{\ge 1}$. Let $d_p: S \times S \to \R$ be the $L^p$ metric on $\left[{a \,.\,.\, b}\right]$: :$\displaystyle \forall f, g \in S: d \left({f, g}\right) := \left({\int_a^b \left\vert{f \left({t}\right) - g \left({t}\right)}\right\vert^p \ \mathrm d t}\right)^{\frac 1 p}$ Then $d_p$ is a metric.


P-Sequence Space of Real Sequences is Metric Space

Let $A$ be the set of all real sequences $\left\langle{x_i}\right\rangle$ such that the series $\displaystyle \sum_{i \mathop \ge 0} x_i^2$ is convergent. Let $d_p$ be the $p$-sequence metric on $\R$. Then $\ell^p := \left({A, d_p}\right)$ is a metric space.


Peano Structure is Unique

Let $\left({P, s, 0}\right)$ and $\left({P', s', 0'}\right)$ be Peano structures. Then there is a unique bijection $f: P \to P'$ such that: {{begin-eqn}} {{eqn|l = f \left({0}\right)      |r = 0' }} {{eqn|l = \forall n \in P: f \left({s \left({n}\right)}\right)      |r = s' \left({f \left({n}\right)}\right) }} {{end-eqn}}


Constant Function is Continuous/Metric Space/Proof 2

Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces. Let $f_c: A_1 \to A_2$ be the constant mapping from $A_1$ to $A_2$: :$\exists c \in A_2: \forall a \in A_1: f_c \left({a}\right) = c$ That is, every point in $A_1$ maps to the same point $c$ in $A_2$. Then $f_c$ is continuous throughout $A_1$ with respect to $d_1$ and $d_2$.


Constant Function is Continuous/Metric Space/Proof 1


Constant Function is Continuous/Metric Space


Constant Function is Continuous/Real Function

Let $f_c: \R \to \R$ be the constant mapping: :$\exists c \in \R: \forall a \in \R: \map {f_c} a = c$ Then $f_c$ is continuous on $\R$.


Identity Mapping is Continuous/Metric Space

Let $M = \left({A, d}\right)$ be a metric space. The identity mapping $I_A: A \to A$ defined as: :$\forall x \in A: I_A \left({x}\right) = x$ is a continuous mapping.


Identity Mapping on Real Vector Space from Chebyshev to Euclidean Metric is Continuous

Let $\R^n$ be an $n$-dimensional real vector space. Let $d_2$ be the Euclidean metric on $\R^n$. Let $d_\infty$ be the Chebyshev distance on $\R^n$. Let $I: \R^n \to \R^n$ be the identity mapping from $\R^n$ to itself. Then the mapping: :$I: \struct {\R^n, d_\infty} \to \struct {\R^n, d_2}$ is $\tuple {d_\infty, d_2}$-continuous.


Identity Mapping on Real Vector Space from Euclidean to Chebyshev Distance is Continuous

Let $\R^n$ be an $n$-dimensional real vector space. Let $d_2$ be the Euclidean metric on $\R^n$. Let $d_\infty$ be the Chebyshev distance on $\R^n$. Let $I: \R^n \to \R^n$ be the identity mapping from $\R^n$ to itself. Then the mapping: :$I: \left({\R^n, d_2}\right) \to \left({\R^n, d_\infty}\right)$ is $\left({d_2, d_\infty}\right)$-continuous.


Composite of Continuous Mappings at Point between Metric Spaces is Continuous at Point

Let $M_1 = \left({X_1, d_1}\right), M_2 = \left({X_2, d_2}\right), M_3 = \left({X_3, d_3}\right)$ be metric spaces. Let $f: M_1 \to M_2$ be continuous at $a \in X_1$. Let $g: M_2 \to M_3$ be continuous at $f \left({a}\right) \in X_2$. Then their composite $g \circ f: M_1 \to M_3$ is continuous at $a \in X_1$.


Composite of Continuous Mappings between Metric Spaces is Continuous

Let $M_1 = \left({X_1, d_1}\right), M_2 = \left({X_2, d_2}\right), M_3 = \left({X_3, d_3}\right)$ be metric spaces. Let $f: M_1 \to M_2$ and $g: M_2 \to M_3$ be continuous mappings. Then their composite $g \circ f: M_1 \to M_3$ is continuous.


Mapping from L1 Space to Real Number Space is Continuous

Let $\left({\R, d}\right)$ be the real number line under the usual metric $d$. Let $X$ be the set of continuous real functions $f: \left[{a \,.\,.\, b}\right] \to \R$. Let $d_1$ be the $L^1$ metric on $X$. Let $I: X \to \R$ be the real-valued function defined as: :$\displaystyle \forall f \in X: I \left({f}\right) := \int_a^b f \left({t}\right) \ \mathop d t$ Then the mapping: :$I : \left({X, d_1}\right) \to \left({\R, d}\right)$ is continuous.


Cartesian Product under Chebyshev Distance of Continuous Mappings between Metric Spaces is Continuous

Let $n \in \N_{>0}$. 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 $N_1 = \left({B_1, d'_1}\right), N_2 = \left({B_2, d'_2}\right), \ldots, N_n = \left({B_n, d'_n}\right)$ be metric spaces. Let $f_i: M_i \to N_i$ be continuous mappings for all $i \in \left\{{1, 2, \ldots, n}\right\}$. Let $\displaystyle \mathcal M = \prod_{i \mathop = 1}^n M_i$ be the cartesian product of $A_1, A_2, \ldots, A_n$. Let $\displaystyle \mathcal N = \prod_{i \mathop = 1}^n N_i$ be the cartesian product of $B_1, B_2, \ldots, B_n$. Let $d_\infty$ be the Chebyshev distance on $\displaystyle \mathcal A = \prod_{i \mathop = 1}^n A_i$, and $\displaystyle \mathcal B = \prod_{i \mathop = 1}^n B_i$, defined as: : $\displaystyle d_\infty \left({x, y}\right) = \max_{i \mathop = 1}^n \left\{ {d_i \left({x_i, y_i}\right)}\right\}$ : $\displaystyle d_\infty \left({x, y}\right) = \max_{i \mathop = 1}^n \left\{ {d'_i \left({x_i, y_i}\right)}\right\}$ where $x = \left({x_1, x_2, \ldots, x_n}\right), y = \left({y_1, y_2, \ldots, y_n}\right) \in \mathcal A$ or $\mathcal B$. Let $F: M \to N$ be the mapping defined as: :$\forall x \in \mathcal A: F \left({x_1, x_2, \ldots, x_n}\right) = \left({f \left({x_1}\right), \left({x_2}\right), \ldots,  \left({x_n}\right)}\right)$ Then $F$ is continuous.


Addition of Coordinates on Euclidean Plane is Continuous Function

Let $\struct {\R^2, d_2}$ be the real number plane with the usual (Euclidean) metric. Let $f: \R^2 \to \R$ be the real-valued function defined as: :$\forall \tuple {x_1, x_2} \in \R^2: \map f {x_1, x_2} = x_1 + x_2$ Then $f$ is continuous.


Metric Space Continuity by Inverse of Mapping between Open Balls

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 $a \in A_1$ be a point in $A_1$. $f$ is continuous at $a$ with respect to the metrics $d_1$ and $d_2$ iff: :$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: B_\delta \left({a; d_1}\right) \subseteq f^{-1} \left[{B_\epsilon \left({f \left({a}\right); d_2}\right)}\right]$ where $B_\epsilon \left({f \left({a}\right); d_2}\right)$ denotes the open $\epsilon$-ball of $f \left({a}\right)$ with respect to the metric $d_2$, and similarly for $B_\delta \left({a; d_1}\right)$.


Open Ball is Neighborhood of all Points Inside

Let $M = \left({A, d}\right)$ be a metric space. Let $a \in A$. Let $B_\epsilon \left({a}\right)$ be an open $\epsilon$-ball of $a$ in $M$. Let $x \in B_\epsilon \left({a}\right)$. Then $B_\epsilon \left({a}\right)$ is a neighborhoods of $x$ in $M$.


Metric Space Continuity by Neighborhood

Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces. Let $f: A_1 \to A_2$ be a mapping from $A_1$ to $A_2$. Let $a \in A_1$ be a point in $A_1$. Then the following definitions of continuity of $f$ at $a$ with respect to $d_1$ and $d_2$ are equivalent:


Sequence Characterization of Open Sets

Let $\left({X, d}\right)$ be a metric space. Let $G \subseteq X$. Then the following are equivalent: :$(1): \quad G \subseteq X$ is an open set of $\left({X, d}\right)$ :$(2): \quad \forall x \in G: \forall \left\langle{x_n}\right\rangle \in X: x_n \to x: \exists n_0 \in \N: \forall n \ge n_0: \left\langle{x_n}\right\rangle \in G$


Metric Space Continuity by Inverse of Mapping between Neighborhoods

Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces. Let $f: A_1 \to A_2$ be a mapping from $A_1$ to $A_2$. Let $a \in A_1$ be a point in $A_1$. $f$ is continuous at $a$ with respect to the metrics $d_1$ and $d_2$ {{iff}}: :for each neighborhood $N$ of $\map f a$ in $M_2$, $f^{-1} \sqbrk N$ is a neighborhood of $a$.


Point in Metric Space has Neighborhood

Let $M = \left({A, d}\right)$ be a metric space. Let $a \in A$ be a point in $M$. Then there exists some neighborhood of $a$ in $M$.


Point in Metric Space is Element of its Neighborhood

Let $N$ be a neighborhood of $a$ in $M$. Then $a \in N$.


Superset of Neighborhood in Metric Space is Neighborhood

Let $N$ be a neighborhood of $a$ in $M$. Let $N \subseteq N' \subseteq A$. Then $N'$ is a neighborhood of $a$ in $M$.


Intersection of Neighborhoods in Metric Space is Neighborhood

Let $N, N'$ be neighborhoods of $a$ in $M$. Then $N \cap N'$ is a neighborhood of $a$ in $M$.


Neighborhood in Metric Space has Subset Neighborhood

Let $N$ be a neighborhood of $a$ in $M$. Then there exists a neighborhood $N'$ of $a$ such that: :$(1): \quad N' \subseteq N$ :$(2): \quad N'$ is a neighborhood of each of its points.


Basic Properties of Neighborhood in Metric Space

This page gathers together the basic properties of a neighborhood in a metric space.


Basis for Element of Real Number Line

Let $M = \struct {\R, d}$ denote the real number line with the usual (Euclidean) metric. Let $a \in \R$ be a point in $M$. Then the set of all open intervals containing $a$ is a basis for the neighborhood system of $a$.


Open Ball in Real Number Line is Open Interval

Let $\struct {\R, d}$ denote the real number line $\R$ with the usual (Euclidean) metric $d$. Let $x \in \R$ be a point in $\R$. Let $\map {B_\epsilon} x$ be the open $\epsilon$-ball at $x$. Then $\map {B_\epsilon} x$ is the open interval $\openint {x - \epsilon} {x + \epsilon}$.


Open Real Interval is Open Ball

Let $\R$ denote the real number line with the usual (Euclidean) metric. Let $I := \openint a b \subseteq \R$ be an open real interval. Then $I$ is the open $\epsilon$-ball $\map {B_\epsilon} \alpha$ of some $\alpha \in \R$.


Neighborhoods in Standard Discrete Metric Space

Let $M = \left({A, d}\right)$ be a metric space where $d$ is the standard discrete metric. Let $a \in A$. Then $\left\{ {a}\right\}$ is a neighborhood of $a$ which forms a basis for the system of neighborhoods of $a$.


Subset of Standard Discrete Metric Space is Neighborhood of Each Point

Let $M = \left({A, d}\right)$ be a metric space where $d$ is the standard discrete metric. Let $S \subseteq A$. Let $a \in S$. Then $S$ is a neighborhood of $a$. That is, every subset of $A$ is a neighborhood of each of its points.


Continuity of Heaviside Step Function

Let $\mu_c: \R \to \R$ be the Heaviside step function: :$\mu_c \left({x}\right) = \begin{cases} 0 & : x < c \\ 1 & : x > c \\ \text{arbitrary} & : x = c \end{cases}$ Then $\mu_c$ is continuous at every point of $\R$ except at $c$.


Metric Space Continuity by Neighborhood Basis

Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces. Let $f: A_1 \to A_2$ be a mapping from $A_1$ to $A_2$. Let $a \in A_1$ be a point in $A_1$. Let $\BB_{\map f a}$ be a basis for the neighborhood system at $\map f a$. $f$ is continuous at $a$ with respect to the metrics $d_1$ and $d_2$ {{iff}}: :for each neighborhood $N$ in $\BB_{\map f a}$, $f^{-1} \sqbrk N$ is a neighborhood of $a$.


Closed Intervals form Neighborhood Basis in Real Number Line

Let $\R$ be the real number line with the usual (Euclidean) metric. Let $a \in \R$ be a point in $\R$. Let $\BB_a$ be defined as: :$\BB_a := \set {\closedint {a - \epsilon} {a + \epsilon}: \epsilon \in \R_{>0} }$ that is, the set of all closed intervals of $\R$ with $a$ as a midpoint. Then $\BB_a$ is a basis for the neighborhood system of $a$.


Open Reciprocal-N Balls form Neighborhood Basis in Real Number Line

Let $\R$ denote the real number line with the usual (Euclidean) metric. Let $a \in \R$ be a point in $\R$. Let $\BB_a$ be defined as: :$\BB_a := \set {\map {B_\epsilon} a: \epsilon \in \set {\dfrac 1 n: n \in \N} }$ that is, the set of all open $\epsilon$-balls of $a$ for $\epsilon$ which are reciprocals of integers. Then $\BB_a$ is a basis for the neighborhood system of $a$.


Subset of Open Reciprocal-N Balls forms Neighborhood Basis in Real Number Line

Let $\R$ denote the real number line with the usual (Euclidean) metric. Let $a \in \R$ be a point in $\R$. Let $k \in \Z$ be some fixed integer. Let $\BB_a$ be defined as: :$\BB_a := \set {\map {B_\epsilon} a: \epsilon \in \set {\dfrac 1 n: n \in \N, n > k} }$ that is, the set of all open $\epsilon$-balls of $a$ for $\epsilon$ which are reciprocals of integers greater than $k$. Then the $\BB_a$ is a basis for the neighborhood system of $a$.


Neighborhood Basis in Real Number Line is Infinite

Let $\R$ be the real number line with the usual (Euclidean) metric. Let $a \in \R$ be a point in $\R$. Let $\BB_a$ be a basis for the neighborhood system of $a$. Then $\BB_a$ is an infinite set.


Distinct Points in Metric Space have Disjoint Neighborhoods

Let $M = \left({A, d}\right)$ be a metric space. Let $x, y \in M: x \ne y$. Then there exist neighborhoods $N_x$ and $N_y$ of $x$ and $y$ respectively such that $N_x \cap N_y = \varnothing$, that is, that are disjoint.


Open Ball in Cartesian Product under Chebyshev Distance

Let $M_1 = \struct {A_1, d_1}, M_2 = \struct {A_2, d_2}, \ldots, M_n = \struct {A_n, d_n}$ be metric spaces. Let $\displaystyle \AA = \prod_{i \mathop = 1}^n A_i$ be the cartesian product of $A_1, A_2, \ldots, A_n$. Let $d_\infty: \AA \times \AA \to \R$ be the Chebyshev distance on $\AA$: :$\displaystyle \map {d_\infty} {x, y} = \max_{i \mathop = 1}^n \set {\map {d_i} {x_i, y_i} }$ where $x = \tuple {x_1, x_2, \ldots, x_n}, y = \tuple {y_1, y_2, \ldots, y_n} \in \AA$. Let $a = \tuple {a_1, a_2, \ldots, a_n} \in \AA$. Let $\epsilon \in \R_{>0}$. Let $\map {B_\epsilon} {a; d_\infty}$ be the open $\epsilon$-ball of $a$ in $M = \struct {\AA, d_\infty}$. Then: :$\displaystyle \map {B_\epsilon} {a; d_\infty} = \prod_{i \mathop = 1}^n \map {B_\epsilon} {a_i; d_i}$


Neighborhood Basis in Cartesian Product under Chebyshev Distance

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 $d_\infty: \mathcal A \times \mathcal A \to \R$ be the Chebyshev distance on $\mathcal A$: : $\displaystyle d_\infty \left({x, y}\right) = \max_{i \mathop = 1}^n \left\{ {d_i \left({x_i, y_i}\right)}\right\}$ where $x = \left({x_1, x_2, \ldots, x_n}\right), y = \left({y_1, y_2, \ldots, y_n}\right) \in \mathcal A$. Let $a = \left({a_1, a_2, \ldots, a_n}\right) \in \mathcal A$. For all $i \in \left\{ {1, 2, \ldots, n}\right\}$, let $\mathcal B_{a_i}$ be a basis for the system of neighborhoods for $a_i$ in $M_i$. Let $\mathcal B_a$ be the cartesian product of $\mathcal B_{a_1}, \mathcal B_{a_2}, \ldots, \mathcal B_{a_n}$: :$\displaystyle \mathcal B_a = \prod_{i \mathop = 1}^n \mathcal B_{a_i}$ Then $\mathcal B_a$ is a basis for the system of neighborhoods for $a$ in $M$.


Projection from Cartesian Product under Chebyshev Distance is Continuous

Let $M_1 = \struct {A_1, d_1}, M_2 = \struct {A_2, d_2}, \ldots, M_n = \struct {A_n, d_n}$ be metric spaces. Let $\displaystyle \AA = \prod_{i \mathop = 1}^n A_i$ be the cartesian product of $A_1, A_2, \ldots, A_n$. Let $d_\infty: \AA \times \AA \to \R$ be the Chebyshev distance on $\AA$: :$\displaystyle \map {d_\infty} {x, y} = \max_{i \mathop = 1}^n \set {\map {d_i} {x_i, y_i} }$ where $x = \tuple {x_1, x_2, \ldots, x_n}, y = \tuple {y_1, y_2, \ldots, y_n} \in \AA$. For all $i \in \set {1, 2, \ldots, n}$, let $\pr_i: \AA \to A_i$ be the $i$th projection on $\AA$: :$\forall a \in \AA: \map {\pr_i} a = a_i$ where $a = \tuple {a_1, a_2, \ldots, a_n} \in \AA$. Then for all $i \in \set {1, 2, \ldots, n}$, $p_i$ is continuous on $\AA$.


Limit of Sequence in Metric Space in Neighborhood

Let $M = \left({A, d}\right)$ be a metric space. Let $\left\langle{a_n}\right\rangle$ be a sequence in $A$. Then $\displaystyle \lim_{n \mathop \to \infty} a_n = a$ iff for each neighborhood $V$ of $a$: :$\exists N \in \N: n > N \implies a_n \in V$


Square of Real Number is Non-Negative

Let $x \in \R$. Then: : $0 \le x^2$


Real Number between Zero and One is Greater than Square

Let $x \in \R$. Let $0 < x < 1$. Then: : $0 < x^2 < x$


Real Number Greater than One is Less than Square

Let $x \in \R$. Let $x > 1$. Then: : $x^2 > x$


Power of Real Number greater than One is Unbounded Above

Let $x \in \R$ be a real number such that $x > 1$. Let set $S = \set {x^n: n \in \N}$. Then $S$ is unbounded above.


Power of Real Number between Zero and One is Bounded

Let $x \in \R$ be a real number. Let $0 < x < 1$. Let set $S = \set {x^n: n \in \N}$. Then: :$\inf S = 0$ and: :$\sup S = 1$ where $\inf S$ and $\sup S$ are the infimum and supremum of $S$ respectively.


Limit of Function by Convergent Sequences/Real Number Line

Let $f$ be a real function defined on an open interval $\openint a b$, except possibly at the point $c \in \openint a b$. Then $\displaystyle \lim_{x \mathop \to c} \map f x = l$ {{iff}}: :for each sequence $\sequence {x_n}$ of points of $\openint a b$ such that $\forall n \in \N_{>0}: x_n \ne c$ and $\displaystyle \lim_{n \to \mathop \infty} x_n = c$ it is true that: :$\displaystyle \lim_{n \mathop \to \infty} \map f {x_n} = l$


Continuity of Mapping between Metric Spaces by Convergent Sequence

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. Then $f$ is continuous at $a \in X$ iff: : whenever $\displaystyle \lim_{n \mathop \to \infty} x_n = a$ for a sequence $\left \langle {x_n} \right \rangle$ of points of $A_1$   it is true that: : $\displaystyle \lim_{n \mathop \to \infty} f \left({x_n}\right) = f \left({a}\right)$


Infimum of Subset of Real Numbers is Arbitrarily Close

Let $A \subseteq \R$ be a subset of the real numbers. Let $b$ be an infimum of $A$. Let $\epsilon \in \R_{>0}$. Then: :$\exists x \in A: x - b < \epsilon$


Existence of Sequence in Set of Real Numbers whose Limit is Infimum

Let $A \subseteq \R$ be a non-empty subset of the real numbers. Let $b$ be an infimum of $A$. Then there exists a sequence $\sequence {a_n}$ in $\R$ such that: :$(1): \quad \forall n \in \N: a_n \in A$ :$(2): \quad \displaystyle \lim_{n \mathop \to \infty} a_n = b$


Existence of Sequence in Subset of Metric Space whose Limit is Infimum

Let $M = \struct {A, d}$ be a metric space. Let $a \in A$. Let $S \subseteq A$ be a non-empty subset of $A$. Then there exists a sequence $\sequence {a_n}$ of points of $S$ such that: :$\displaystyle \lim_{n \mathop \to \infty} \map d {a, a_n} = \map d {a, S}$


Limit of Sequence in Product of Metric Spaces under Chebyshev Distance

Let $M_1 = \struct {A_1, d_1}, M_2 = \struct {A_2, d_2}, \ldots, M_k = \struct {A_n, d_k}$ be metric spaces. Let $\displaystyle \AA = \prod_{i \mathop = 1}^k A_i$ be the cartesian product of $A_1, A_2, \ldots, A_k$. Let $d_\infty: \AA \times \AA \to \R$ be the Chebyshev distance on $\AA$: :$\displaystyle \map {d_\infty} {x, y} = \max_{i \mathop = 1}^k \set {\map {d_i} {x_i, y_i} }$ where $x = \tuple {x_1, x_2, \ldots, x_k}, y = \tuple {y_1, y_2, \ldots, y_k} \in \AA$. Let $\sequence {a_n}$ be a sequence of points of $\AA$: :$a_n = \tuple {a_{n 1}, a_{n 2}, \ldots, a_{n k} } \in \AA$ Let $c = \tuple {c_1, c_2, \ldots, c_k} \in \AA$. Then: :$\displaystyle \lim_{n \mathop \to \infty} a_n = c \iff \forall i \in \set {1, 2, \ldots, k}: \lim_{n \mathop \to \infty} a_{n i} = c_i$ where $\displaystyle \lim_{n \mathop \to \infty}$ denotes a limit.


Recurrence Formula for Sum of Sequence of Fibonacci Numbers

Let $g_n$ be the sum of the first $n$ Fibonacci numbers. Then: :$\forall n \ge 2: g_n = g_{n - 1} + g_{n - 2} + 1$


Limit of Subsequence equals Limit of Sequence/Metric Space

Let $M = \left({A, d}\right)$ be a metric space. Let $\left \langle {x_n} \right \rangle$ be a sequence in $A$. Let $l \in A$ such that: : $\displaystyle \lim_{n \mathop \to \infty} x_n = l$ Let $\left \langle {x_{n_r}} \right \rangle$ be a subsequence of $\left \langle {x_n} \right \rangle$. Then: : $\displaystyle \lim_{r \mathop \to \infty} x_{n_r} = l$


Convergent Real Sequence is Bounded

Let $\left \langle {x_n} \right \rangle$ be a sequence in $\R$. Let $l \in A$ such that $\displaystyle \lim_{n \mathop \to \infty} x_n = l$. Then $\left \langle {x_n} \right \rangle$ is bounded.


Monotone Convergence Theorem (Real Analysis)/Decreasing Sequence

Let $\sequence {x_n}$ be decreasing and bounded below. Then $\sequence {x_n}$ converges to its infimum.


Monotone Convergence Theorem (Real Analysis)/Increasing Sequence

Let $\sequence {x_n}$ be increasing and bounded above. Then $\sequence {x_n}$ converges to its supremum.


Triangle Inequality for Distance from Element to Set

Let $M = \struct {A, d}$ be a metric space. Let $S \subseteq A$ be a non-empty subset of $A$. Let $x, y \in A$. Then: :$\map d {x, S} \le \map d {x, y} + \map d {y, S}$ where $\map d {x, S}$ is the distance from $x$ to $S$.


Open Ball in Euclidean Plane is Interior of Circle

Let $\R^2$ be the real number plane with the usual (Euclidean) metric. Let $x = \tuple {x_1, x_2} \in \R^2$ be a point in $\R^2$. Let $\map {B_\epsilon} x$ be the open $\epsilon$-ball at $x$. Then $\map {B_\epsilon} x$ is the interior of the circle whose center is $x$ and whose radius is $\epsilon$.


Open Ball in Euclidean 3-Space is Interior of Sphere

Let $\R^3$ be the real Euclidean $3$-space considered as a metric space under the usual metric. Let $x = \tuple {x_1, x_2, x_3} \in \R^3$ be a point in $\R^3$. Let $\map {B_\epsilon} x$ be the open $\epsilon$-ball at $x$. Then $\map {B_\epsilon} x$ is the interior of the sphere whose center is $x$ and whose radius is $\epsilon$.


Open Ball in Standard Discrete Metric Space

Let $M = \struct {A, d}$ be a metric space. Let $d$ be the standard discrete metric on $M$. Let $a \in A$. Let $\map {B_\epsilon} {a; d}$ be an open $\epsilon$-ball of $a$ in $M$. Then: :$\map {B_\epsilon} {a; d} = \begin{cases} \set a & : \epsilon \le 1 \\ A & : \epsilon > 1 \end{cases}$


Distance from Point to Subset is Continuous Function

Let $M = \struct {X, d}$ be a metric space. Let $A \subseteq X$ be a non-empty subset of $X$. Let $f: X \to \R$ be the function defined as: :$\forall x \in X: \map f x = \map d {x, A}$ where $\map d {x, A}$ denotes the distance from $x$ to $A$. Then $f$ is continuous.


Mapping from Cartesian Product under Chebyshev Distance to Real Number Line is Continuous

Let $M = \struct {A, d'}$ be a metric space. Let $\displaystyle \AA = A \times A$ be the cartesian product of $A$ with itself. Let $d_\infty: \AA \times \AA \to \R$ be the Chebyshev distance on $\AA$: : $\displaystyle \map {d_\infty} {x, y} = \max \set {\map {d'} {x_1, y_1}, \map {d'} {x_2, y_2} }$ where $x = \tuple {x_1, x_2}, y = \tuple {y_1, y_2} \in \AA$. Then the mapping: :$d': \struct {A \times A, d_\infty} \to \struct {\R, d}$ where $d$ is the usual metric, is continuous.


Empty Set is Closed in Metric Space

Let $M = \struct {A, d}$ be a metric space. Then the empty set $\varnothing$ is closed in $M$.


Metric Space is Closed in Itself

Let $M = \struct {A, d}$ be a metric space. Then $A$ is closed in $M$.


Metric Space is Open and Closed in Itself

Let $M = \left({A, d}\right)$ be a metric space. Then $A$ is both open and closed in $M$.


Subset of Metric Space contains Limits of Sequences iff Closed

Let $M = \left({A, d}\right)$ be a metric space. Let $H \subseteq A$. Then $H$ is closed in $M$ iff: :for each sequence $\left\langle{a_n}\right\rangle$ of points of $H$ that converges to a point $a \in A$, it follows that $a \in H$.


Subset of Metric Space is Closed iff contains all Zero Distance Points

Let $M = \struct {A, d}$ be a metric space. Let $H \subseteq A$. Then $H$ is closed in $M$ {{iff}}: :$\forall x \in A: \map d {x, H} = 0 \implies x \in H$ where $\map d {x, H}$ denotes the distance between $x$ and $H$.


Continuity of Mapping between Metric Spaces by Closed Sets

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. Then $f$ is continuous iff: :for every $V \subseteq A_2$ which is closed in $M_2$, $f^{-1} \left({V}\right)$ is closed in $M_1$.


Metric Space defined by Closed Sets

Let $M = \struct {A, d}$ be a metric space. Then: {{begin-axiom}} {{axiom | n = C1         | lc= $A$ is closed in $M$ }} {{axiom | n = C2         | lc= $\O$ is closed in $M$ }} {{axiom | n = C3         | lc= The union of a finite number of closed sets of $M$ is a closed set of $M$ }} {{axiom | n = C4         | lc= The intersection of arbitrarily many closed sets of $M$ is a closed set of $M$ }} {{end-axiom}}


Infinite Union of Closed Sets of Metric Space may not be Closed

Let $M = \struct {A, d}$ be a metric space. Let $V_1, V_2, V_3, \ldots$ be an infinite set of closed sets of $M$. Then it is not necessarily the case that $\displaystyle \bigcup_{n \mathop \in \N} V_n$ is itself a closed set of $M$.


Open Sets of Cartesian Product of Metric Spaces under Chebyshev Distance

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 $d_\infty: \mathcal A \times \mathcal A \to \R$ be the Chebyshev distance on $\mathcal A$: : $\displaystyle d_\infty \left({x, y}\right) = \max_{i \mathop = 1}^n \left\{ {d_i \left({x_i, y_i}\right)}\right\}$ where $x = \left({x_1, x_2, \ldots, x_n}\right), y = \left({y_1, y_2, \ldots, y_n}\right) \in \mathcal A$. For $i \in \left\{ {1, 2, \ldots, n}\right\}$, let $U_i$ be open in $M_i$. Then $\displaystyle \prod_{i \mathop = 1}^n U_i$ is open in $M = \left({\mathcal A, d_\infty}\right)$.


Basis for Product of Metric Spaces under Chebyshev Distance

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 $d_\infty: \mathcal A \times \mathcal A \to \R$ be the Chebyshev distance on $\mathcal A$: : $\displaystyle d_\infty \left({x, y}\right) = \max_{i \mathop = 1}^n \left\{ {d_i \left({x_i, y_i}\right)}\right\}$ where $x = \left({x_1, x_2, \ldots, x_n}\right), y = \left({y_1, y_2, \ldots, y_n}\right) \in \mathcal A$. For $i \in \left\{ {1, 2, \ldots, n}\right\}$, let $U_i$ be open in $M_i$. Then $\left\{ {\displaystyle \prod_{i \mathop = 1}^n U_i}\right\}$ is a basis for the open sets of $M$.


Open Balls form Basis for Open Sets of Metric Space

Let $M = \left({A, d}\right)$ be a metric space. Let $\mathcal B$ be the set of all open balls of $M$. Then $\mathcal B$ is a basis for the open sets of $M$.


Graph of Continuous Mapping between Metric Spaces is Closed in Chebyshev Product

Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces. Let $\displaystyle \mathcal A = A_1 \times A_2$ be the cartesian product of $A_1$ and $A_2$. Let $d_\infty: \mathcal A \times \mathcal A \to \R$ be the Chebyshev distance on $\mathcal A$: : $\displaystyle d_\infty \left({x, y}\right) = \max \left\{ {d_1 \left({x_1, y_1}\right), d_2 \left({x_2, y_2}\right)}\right\}$ where $x = \left({x_1, x_2}\right), y = \left({y_1, y_2}\right) \in \mathcal A$. Let $\Gamma_f$ be the graph of $f$. Then $\Gamma_f$ is a closed set of $\left({\mathcal A, d_\infty}\right)$.


Point of Metric Space is Isolated iff not Limit Point

Let $M = \struct {A, d}$ be a metric space. Let $H \subseteq A$ be a subset of $A$. Let $H'$ be the set of limit points of $H$. Let $H^i$ be the set of isolated points of $H$. Then: :$A' \cap A^i = \O$


Subset of Metric Space is Subset of its Closure

Let $M = \left({A, d}\right)$ be a metric space. Let $H \subseteq A$ be a subset of $A$. Then: :$H \subseteq H^-$ where $H^-$ denotes the closure of $H$.


Point in Closure of Subset of Metric Space iff Limit of Sequence

Let $M = \struct {A, d}$ be a metric space. Let $H \subseteq A$ be a subset of $A$. Let $H^-$ denote the closure of $H$. Let $a \in A$. Then $a \in H^-$ {{iff}} there exists a sequence $\sequence {x_n}$ of points of $H$ which converges to the limit $a$.


Closure of Subset of Closed Set of Metric Space is Subset

Let $M = \struct {A, d}$ be a metric space. Let $F$ be a closed set of $M$. Let $H \subseteq F$ be a subset of $F$. Let $H^-$ denote the closure of $H$. Then $H^- \subseteq F$.


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 $U$ {{iff}} $f_n$ 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 $T^S$ {{iff}} $\struct {T, \circ}$ 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 $x$ {{iff}} 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 {{iff}} :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

:$\overline {T_1 \cap T_2} = \overline T_1 \cup \overline T_2$


De Morgan's Laws (Set Theory)/Set Complement/Complement of Union

:$\overline {T_1 \cup T_2} = \overline T_1 \cap \overline T_2$


De Morgan's Laws (Set Theory)/Proof by Induction/Difference with Intersection/Proof

Let $\mathbb T = \set {T_i: i \mathop \in I}$, where each $T_i$ is a set and $I$ is some finite indexing set. Then: :$\displaystyle S \setminus \bigcap_{i \mathop \in I} T_i = \bigcup_{i \mathop \in I} \paren {S \setminus T_i}$


De Morgan's Laws (Set Theory)/Set Difference/General Case/Difference with Intersection

:$\displaystyle S \setminus \bigcap \mathbb T = \bigcup_{T' \mathop \in \mathbb T} \paren {S \setminus T'}$ where: :$\displaystyle \bigcap \mathbb T := \set {x: \forall T' \in \mathbb T: x \in T'}$ that is, the intersection of $\mathbb T$


De Morgan's Laws (Set Theory)/Set Difference/General Case/Difference with Union

:$\displaystyle S \setminus \bigcup \mathbb T = \bigcap_{T' \mathop \in \mathbb T} \paren {S \setminus T'}$ where: :$\displaystyle \bigcup \mathbb T := \set {x: \exists T' \in \mathbb T: x \in T'}$ that is, the union of $\mathbb T$.


De Morgan's Laws (Set Theory)/Set Difference/Difference with Intersection

:$S \setminus \paren {T_1 \cap T_2} = \paren {S \setminus T_1} \cup \paren {S \setminus T_2}$


De Morgan's Laws (Set Theory)/Set Difference/Difference with Union

:$S \setminus \paren {T_1 \cup T_2} = \paren {S \setminus T_1} \cap \paren {S \setminus T_2}$


Isomorphism between Ring of Integers Modulo 2 and Parity Ring

The ring of integers modulo $2$ and the parity ring are isomorphic.


Parity Addition is Associative

Let $R := \struct {\set {\text{even}, \text{odd} }, +, \times}$ be the parity ring. The operation $+$ is associative: :$\forall a, b, c \in R: \paren {a + b} + c = a + \paren {b + c}$


Parity Multiplication is Commutative

Let $R := \struct {\set {\text{even}, \text{odd} }, +, \times}$ be the parity ring. The operation $\times$ is commutative: :$\forall a, b \in R: a \times b = b \times a$


Isomorphism between Roots of Unity under Multiplication and Integers under Modulo Addition

Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $\struct {R_n, \times}$ be the complex $n$th roots of unity under complex multiplication. Let $\struct {\Z_n, +_n}$ be the integers modulo $n$ under modulo addition. Then $\struct {R_n, \times}$ and $\struct {\Z_n, +_n}$ are isomorphic algebraic structures.


Isomorphism between Gaussian Integer Units and Integers Modulo 4 under Addition

Let $\struct {U_\C, \times}$ be the group of Gaussian integer units under complex multiplication. Let $\struct {\Z_n, +_4}$ be the integers modulo $4$ under modulo addition. Then $\struct {U_\C, \times}$ and $\struct {\Z_4, +_4}$ are isomorphic algebraic structures.


Gaussian Integer Units are 4th Roots of Unity

The units of the ring of Gaussian integers: :$\set {1, i, -1, -i}$ are the (complex) $4$th roots of $1$.


Isomorphism by Cayley Table

Let $\struct {S, \circ}$ and $\struct {T, *}$ be algebraic structures whose underlying sets are both finite. Then $\struct {S, \circ}$ and $\struct {T, *}$ are isomorphic {{iff}}: :a bijection $f: S \to T$ can be found such that: ::the Cayley table of $\struct {T, *}$ can be generated from the Cayley table of $\struct {S, \circ}$ by replacing each entry of $S$ with its image under $f$.


Bijection between Integers and Even Integers

Let $\Z$ be the set of integers. Let $2 \Z$ be the set of even integers. Then there exists a bijection $f: \Z \to 2 \Z$ between the two.


Set of Integers under Addition is Isomorphic to Set of Even Integers under Addition

Let $\struct {\Z, +}$ be the algebraic structure formed by the set of integers under the operation of addition. Let $\struct {2 \Z, +}$ be the algebraic structure formed by the set of even integers under the operation of addition. Then $\struct {\Z, +}$ and $\struct {2 \Z, +}$ are isomorphic.


Natural Numbers under Addition do not form Group

The algebraic structure $\struct {\N, +}$ consisting of the set of natural numbers $\N$ under addition $+$ is not a group.


Natural Numbers under Multiplication do not form Group

The algebraic structure $\struct {\N, \times}$ consisting of the set of natural numbers $\N$ under multiplication $\times$ is not a group.


Symmetry Group of Square is Group

The symmetry group of the square is a non-abelian group.


Dihedral Group is Group

Let $D_n$ be the dihedral group of order $2 n$. Then $D_n$ is indeed a group.


Symmetric Group is Subgroup of Monoid of Self-Maps

Let $S$ be a set. Let $S^S$ be the set of all mappings from $S$ to itself Let $\struct {\Gamma \paren S, \circ}$ denote the symmetric group on $S$. Let $\struct {S^S, \circ}$ be the monoid of self-maps under composition of mappings. Then $\struct {\Gamma \paren S, \circ}$ is a subgroup of $\struct {S^S, \circ}$.


Cancellability by Cayley Table

Let $\left({S, \circ}\right)$ be a finite algebraic structure. Let $\mathcal T$ be the Cayley table for $\left({S, \circ}\right)$. Let $a \in S$ be an element of $S$. Then $a$ is cancellable for $\circ$ {{iff}}: :$(1): \quad$ no element of $S$ is repeated in $\mathcal T$ in the row headed by $a$ and: :$(2): \quad$ no element of $S$ is repeated in $\mathcal T$ in the column headed by $a$.


Piecewise Continuous Function does not necessarily have Improper Integrals

Let $f$ be a real function defined on a closed interval $\left[{a \,.\,.\, b}\right]$, $a < b$.  Let $f$ be a piecewise continuous function: {{:Definition:Piecewise Continuous Function}} Then it is not necessarily the case that $f$ is a piecewise continuous function with improper integrals: {{:Definition:Piecewise Continuous Function with Improper Integrals}}


Right Cancellable Element is Right Cancellable in Subset

Let $\left ({S, \circ}\right)$ be an algebraic structure. Let $\left ({T, \circ}\right) \subseteq \left ({S, \circ}\right)$. Let $x \in T$ be right cancellable in $S$. Then $x$ is also right cancellable in $T$.


Left Cancellable Element is Left Cancellable in Subset

Let $\struct {S, \circ}$ be an algebraic structure. Let $\struct {T, \circ} \subseteq \struct {S, \circ}$. Let $x \in T$ be left cancellable in $S$. Then $x$ is also left cancellable in $T$.


Left Cancellable Elements of Semigroup form Subsemigroup

Let $\struct {S, \circ}$ be a semigroup. Let $C_\lambda$ be the set of left cancellable elements of $\struct {S, \circ}$. Then $\struct {C_\lambda, \circ}$ is a subsemigroup of $\struct {S, \circ}$.


Right Cancellable Elements of Semigroup form Subsemigroup

Let $\struct {S, \circ}$ be a semigroup. Let $C_\rho$ be the set of right cancellable elements of $\struct {S, \circ}$. Then $\struct {C_\rho, \circ}$ is a subsemigroup of $\struct {S, \circ}$.


Surjection iff Right Cancellable/Necessary Condition

Let $f$ be a surjection. Then $f$ is right cancellable.


Surjection iff Right Cancellable/Sufficient Condition

Let $f$ be a mapping which is right cancellable. Then $f$ is a surjection.


Absolute Value Function on Integers induces Equivalence Relation

Let $\Z$ be the set of integers. Let $\mathcal R$ be the relation on $\Z$ defined as: :$\forall x, y \in \Z: \left({x, y}\right) \in \mathcal R \iff \left\vert{x}\right\vert = \left\vert{y}\right\vert$ where $\left\vert{x}\right\vert$ denotes the absolute value of $x$. Then $\mathcal R$ is an equivalence relation.


Absolute Value induces Equivalence not Compatible with Integer Addition

Let $\Z$ be the set of integers. Let $\RR$ be the relation on $\Z$ defined as: :$\forall x, y \in \Z: \tuple {x, y} \in \RR \iff \size x = \size y$ where $\size x$ denotes the absolute value of $x$. Then $\RR$ is not a congruence relation for integer addition.


Equivalence Relation is Congruence for Right Operation

Every equivalence is a congruence for the right operation.


Equivalence Relation is Congruence for Left Operation

Every equivalence is a congruence for the left operation.


Left Coset Space forms Partition

The left coset space of $H$ forms a partition of its group $G$, and hence: {{begin-eqn}} {{eqn | l = x \equiv^l y \pmod H       | o = \iff       | r = x H = y H }} {{eqn | l = \neg \paren {x \equiv^l y} \pmod H       | o = \iff       | r = x H \cap y H = \O }} {{end-eqn}}


Right Coset Space forms Partition

The right coset space of $H$ forms a partition of its group $G$: {{begin-eqn}} {{eqn | l = x \equiv^r y \pmod H       | o = \iff       | r = H x = H y }} {{eqn | l = \neg \paren {x \equiv^r y} \pmod H       | o = \iff       | r = H x \cap H y = \O }} {{end-eqn}}


Right Cosets are Equal iff Product with Inverse in Subgroup

Let $H x$ denote the right coset of $H$ by $x$. Then: :$H x = H y \iff x y^{-1} \in H$


Left Cosets are Equal iff Product with Inverse in Subgroup

Let $x H$ denote the left coset of $H$ by $x$. Then: :$x H = y H \iff x^{-1} y \in H$


Left Congruence Class Modulo Subgroup is Left Coset

Let $\mathcal R^l_H$ be the equivalence defined as left congruence modulo $H$. The equivalence class $\eqclass g {\mathcal R^l_H}$ of an element $g \in G$ is the left coset $g H$. This is known as the '''left congruence class of $g \bmod H$'''.


Left Congruence Modulo Subgroup is Equivalence Relation

Let $x \equiv^l y \pmod H$ denote the relation that $x$ is left congruent modulo $H$ to $y$. Then the relation $\equiv^l$ is an equivalence relation.


Right Congruence Modulo Subgroup is Equivalence Relation

Let $x \equiv^r y \pmod H$ denote the relation that $x$ is right congruent modulo $H$ to $y$ Then the relation $\equiv^r$ is an equivalence relation.


Group Epimorphism is Isomorphism iff Kernel is Trivial

Let $\struct {G, \oplus}$ and $\struct {H, \odot}$ be groups. Let $\phi: \struct {G, \oplus} \to \struct {H, \odot}$ be a group epimorphism. Let $e_G$ and $e_H$ be the identities of $G$ and $H$ respectively. Let $K = \map \ker \phi$ be the kernel of $\phi$. Then: :the epimorphism $\phi$ is an isomorphism {{iff}} :$K = \set {e_G}$


Quotient Group of Quotient Group is Isomorphic to Quotient Group by Preimage under Quotient Mapping

Let $G$ be a group. Let $H \lhd G$ where $\lhd$ denotes that $H$ is a normal subgroup of $G$. Let $K \lhd G/H$ and $L = q_H^{-1} \left[{K}\right]$, where: :$q_H: G \to G/H$ is the quotient epimorphism from $G$ to the quotient group $G/H$ :$q_H^{-1} \left[{K}\right]$ is the preimage of $K$ under $q_H$. Then there exists a group isomorphism $\phi: \left({G / H}\right) / K \to G / L$ defined as: ::$\phi \circ q_K \circ q_H = q_L$


Real Numbers are not Well-Ordered under Conventional Ordering

Let $\left({\R, \leqslant}\right)$ be the ordered structure consisting of the real numbers under the conventional ordering. Then $\left({\R, \leqslant}\right)$ is not a well-ordered set.


Supremum is Unique

Let $\struct {S, \preceq}$ be an ordered set. Let $T$ be a non-empty subset of $S$. Then $T$ has at most one supremum in $S$.


Infimum is Unique

Let $\struct {S, \preceq}$ be an ordered set. Let $T$ be a non-empty subset of $S$. Then $T$ has at most one infimum in $S$.


Infimum of Subgroups in Lattice

Then: :$\inf \set {H, K} = H \cap K$


Supremum of Subgroups in Lattice

Let either $H$ or $K$ be normal in $G$. Then: :$\sup \left\{{H, K}\right\} = H \circ K$ where $H \circ K$ denotes subset product.


Lexicographic Order on Totally Ordered Sets is Total Ordering

Let $\struct {S_1, \preceq_1}$ and $\struct {S_2, \preceq_2}$ be ordered sets. Let $\preccurlyeq$ be the lexicographic order on $S_1 \times S_2$''': :$\tuple {x_1, x_2} \preccurlyeq \tuple {y_1, y_2} \iff \paren {x_1 \prec_1 y_1} \lor \paren {x_1 = y_1 \land x_2 \preceq_2 y_2}$ Then: :$\preccurlyeq$ is a total ordering on $S_1 \times S_2$ {{iff}} :both $\preceq_1$ and $\preceq_2$ are total orderings.


Lexicographic Order is Ordering

Let $\left({S_1, \preceq_1}\right)$ and $\left({S_2, \preceq_2}\right)$ be ordered sets. Let $\preccurlyeq$ be the lexicographic order on $S_1 \times S_2$''': :$\left({x_1, x_2}\right) \preccurlyeq \left({y_1, y_2}\right) \iff \left({x_1 \prec_1 y_1}\right) \lor \left({x_1 = y_1 \land x_2 \preceq_2 y_2}\right)$ Then $\preccurlyeq$ is an ordering on $S_1 \times S_2$.


Powers of Commuting Elements of Monoid Commute

:$\forall m, n \in \N: \paren {\circ^m a} \circ \paren {\circ^n b} = \paren {\circ^n b} \circ \paren {\circ^m a}$


Power of Product of Commuting Elements in Monoid equals Product of Powers

:$\forall n \in \N: \circ^n \paren {a \circ b} = \paren {\circ^n a} \circ \paren {\circ^n b}$


Index Laws/Product of Indices/Monoid

Let $\struct {S, \circ}$ be a monoid whose identity element is $e$. For $a \in S$, let $\circ^n a = a^n$ be the $n$th power of $a$. Then: :$\forall m, n \in \N: a^{n m} = \paren {a^n}^m = \paren {a^m}^n$


Cardinality of Proper Subset of Finite Set

Let $A$ and $B$ be finite sets such that $A \subsetneqq B$. Let $\card B = n$, where $\card {\, \cdot \,}$ denotes cardinality. Then $\card A < n$.


First-Countable Space is Hausdorff iff All Convergent Sequences have Unique Limit

Let $T = \struct {S, \tau}$ be a first-countable topological space. Then $T$ is Hausdorff {{iff}} all convergent sequences on $T$ have a unique limit point.


Topologically Distinguishable Points are Distinct

Let $T = \left({X, \tau}\right)$ be a topological space. Let $x, y \in X$ be topologically distinguishable. Then the singleton sets $\left\{{x}\right\}$ and $\left\{{y}\right\}$ are disjoint and so: : $x \ne y$


Restriction of Strict Total Ordering is Strict Total Ordering

Let $\left({S, \prec}\right)$ be a strict total ordering. Let $T \subseteq S$. Let $\prec \restriction_T$ be the restriction of $\prec$ to $T$. Then $\prec \restriction_T$ is a strict total ordering of $T$.


Principle of Recursive Definition

Let $\N$ be the natural numbers. Let $T$ be a set. Let $a \in T$. Let $g: T \to T$ be a mapping. Then there exists exactly one mapping $f: \N \to T$ such that: :$\forall x \in \N: \map f x = \begin{cases} a & : x = 0 \\ \map g {\map f n} & : x = n + 1 \end{cases}$


Principle of Recursive Definition/General Result

Let $p \in \N$. Let $p^\ge$ be the upper closure of $p$ in $\N$: :$p^\ge := \set {x \in \N: x \ge p} = \set {p, p + 1, p + 2, \ldots}$ Then there exists exactly one mapping $f: p^\ge \to T$ such that: :$\forall x \in p^\ge: \map f x = \begin{cases} a & : x = p \\ \map g {\map f n} & : x = n + 1 \end{cases}$


Real Zero is Zero Element

:$\forall x \in \R: 0 \times x = 0$


Negative of Real Zero equals Zero

Let $0$ denote the identity for addition in the real numbers $\R$. Then: :$-0 = 0$


Negative of Negative Real Number

:$\forall x \in \R: -\paren {-x} = x$


Multiplication by Negative Real Number

:$\forall x, y \in \R: x \times \paren {-y} = -\paren {x \times y} = \paren {-x} \times y$


Negative of Sum of Real Numbers

:$\forall x, y \in \R: -\paren {x + y} = -x - y$


Real Multiplication Identity is One/Corollary

:$\forall x \in \R_{\ne 0}: x \times y = x \implies y = 1$


Real Number Divided by Itself

:$\forall x \in \R_{\ne 0}: \dfrac x x = 1$


Product of Reciprocals of Real Numbers

:$\forall x, y \in \R_{\ne 0}: \dfrac 1 x \times \dfrac 1 y = \dfrac 1 {x \times y}$


Product of Quotients of Real Numbers

:$\forall x, w \in \R, y, z \in \R_{\ne 0}: \dfrac x y \times \dfrac w z = \dfrac {x \times w} {y \times z}$


Sum of Quotients of Real Numbers

:$\forall x, w \in \R, y, z \in \R_{\ne 0}: \dfrac x y + \dfrac w z = \dfrac {\paren {x \times z} + \paren {y \times w} } {y \times z}$


Reciprocal of Quotient of Real Numbers

:$\forall x, y \in \R_{\ne 0}: \dfrac 1 {x / y} = \dfrac y x$


Quotient of Quotients of Real Numbers

:$\forall x \in \R, y, w, z \in \R_{\ne 0}: \dfrac {x / y} {w / z} = \dfrac {x \times z} {y \times w}$


Product of Real Number with Quotient

:$\forall a, x \in \R, y \in \R_{\ne 0}: \dfrac {a \times x} y = a \times \dfrac x y$


Neighborhood in Topological Subspace

Let $\struct {X, \tau}$ be a topological space. Let $S \subseteq X$ be a subset of $X$. Let $\tau_S$ denote the subspace topology on $S$. Let $x \in S$ be an arbitrary point of $S$. Let $E \subseteq S$. Then: :$E$ is a neighborhood of $x$ in $\struct {S, \tau_S}$ {{iff}}: :$\exists D \subseteq X$ such that: ::$D$ is a neighborhood of $x$ in $X$ ::$E = D \cap S$.


Negative of Quotient of Real Numbers

:$\forall x \in \R, y \in \R_{\ne 0}: \dfrac {-x} y = -\dfrac x y = \dfrac x {-y}$


Product of Strictly Positive Real Numbers is Strictly Positive

:$x, y \in \R_{>0} \implies x \times y \in \R_{>0}$


Real Number is Greater than Zero iff its Negative is Less than Zero

:$\forall x \in \R: x > 0 \iff \paren {-x} < 0$


Order of Real Numbers is Dual of Order of their Negatives

:$\forall x, y \in \R: x > y \iff \paren {-x}< \paren {-y}$


Square of Non-Zero Real Number is Strictly Positive

:$\forall x \in \R: x \ne 0 \implies x^2 > 0$


Real Zero is Less than Real One

The real number $0$ is less than the real number $1$: :$0 < 1$


Product of Real Numbers is Positive iff Numbers have Same Sign

The product of two real numbers is greater than $0$ {{iff}} either both are greater than $0$ or both are less than $0$. :$\forall x, y \in \R: x \times y > 0 \iff \paren {x, y \in \R_{>0} } \lor \paren {x, y \in \R_{<0} }$


Reciprocal of Strictly Positive Real Number is Strictly Positive

:$\forall x \in \R: x > 0 \implies \dfrac 1 x > 0$


Order of Strictly Positive Real Numbers is Dual of Order of their Reciprocals

:$\forall x, y \in \R: x > y > 0 \implies \dfrac 1 x < \dfrac 1 y$


Mean of Unequal Real Numbers is Between them

:$\forall x, y \in \R: x < y \implies x < \dfrac {x + y} 2 < y$


Intersection of Inductive Set as Subset of Real Numbers is Inductive Set

Let $\AA$ be a set of inductive sets defined as subsets of real numbers. Then their intersection is an inductive set.


Non-Empty Subset of Initial Segment of Natural Numbers has Greatest Element

Let $n \in \N_{>0}$ be a non-zero natural number. Let $\N^*_n$ denote the Initial segment $\set {1, 2, \ldots, n}$ of the non-zero natural numbers. Then every non-empty subset of $\N^*_n$ has a greatest element.


Not every Non-Empty Subset of Natural Numbers has Greatest Element

Let $S \subseteq \N_{>0}$. Then, despite Non-Empty Subset of Initial Segment of Natural Numbers has Greatest Element, it is not necessarily the case that $S$ has a greatest element.


Existence of Integral on Union of Adjacent Intervals

Let $f$ be a real function defined on a closed interval $\closedint a b$ where $a < b$. Let $c$ be a point in $\openint a b$. Then: :$f$ is Darboux integrable on $\closedint a c$ and $\closedint c b$ {{iff}}: :$f$ is Darboux integrable on $\closedint a b$.


Cartesian Product of Bijections is Bijection

Let $S_1 \times S_2$ be the Cartesian product of two sets $S_1$ and $S_2$. Let $T_1 \times T_2$ be the Cartesian product of two sets $T_1$ and $T_2$. Let $f_1: S_1 \to T_1$ and $f_2: S_2 \to T_2$ be bijections. Let $f_1 \times f_2: S_1 \times S_2 \to T_1 \times T_2$ be the Cartesian product of $f_1$ and $f_2$ defined as: :$\forall \left({s_1, s_2}\right) \in S_1 \times S_2: f_1 \times f_2 \left({s_1, s_2}\right) := \left({f_1 \left({s_1}\right), f_2 \left({s_2}\right)}\right)$ Then $f_1 \times f_2$ is a bijection.


First Order ODE/y' - f (y) phi' (x) over f' (y) = phi (x) phi' (x) over f' (y)

Let $\map f y$ and $\map \phi x$ be known real functions of $y$ and $x$ respectively. The general solution of: :$(1): \quad \dfrac {\d y} {\d x} - \dfrac {\map f y} {\map {f'} y} \map {\phi'} x = \dfrac {\map \phi x \, \map {\phi'} x} {\map {f'} y}$ is: :$\map f y = C e^{\map \phi x} - \map \phi x - 1$


Cardinality of Set of All Mappings/Finite Sets

Let $S$ and $T$ be finite sets. The cardinality of the set of all mappings from $S$ to $T$ (that is, the total number of mappings from $S$ to $T$) is: :$\card {T^S} = \card T^{\card S}$


Cardinality of Set of All Mappings/Infinite Sets

Let $S$ and $T$ be sets such that either $S$ or $T$ is infinite. The cardinality of the set of all mappings from $S$ to $T$ (that is, the total number of mappings from $S$ to $T$) is: :$\card {T^S} = \card T^{\card S}$


Cardinality of Set Union/General Case

Let $S_1, S_2, \ldots$ be sets. Then: {{begin-eqn}}      {{eqn | l = \card {\bigcup_{i \mathop = 1}^n S_i}       | r = \sum_{i \mathop = 1}^n \card {S_i}       | c = }} {{eqn | o = -       | r = \sum_{1 \mathop \le i \mathop < j \mathop \le n} \card {S_i \cap S_j}       | c = }} {{eqn | o = +       | r = \sum_{1 \mathop \le i \mathop < j \mathop < k \mathop \le n} \card {S_i \cap S_j \cap S_k}       | c = }} {{eqn | o = \cdots       | c = }} {{eqn | o = +       | r = \paren {-1}^{n - 1} \card {\bigcap_{i \mathop = 1}^n S_i}       | c = }} {{end-eqn}}


Structure of Inverse Completion of Commutative Semigroup

Let $\left({S, \circ}\right)$ be a commutative semigroup. Let $\left ({C, \circ}\right) \subseteq \left({S, \circ}\right)$ be the subsemigroup of cancellable elements of $\left({S, \circ}\right)$. Let $\left({T, \circ'}\right)$ be an inverse completion of $\left({S, \circ}\right)$. Then: :$T = S \circ' C^{-1}$ where: :$C^{-1}$ is the inverse of $C$ in $T$ :$S \circ' C^{-1}$ is the subset product of $S$ with $C^{-1}$.


Inverse in Monoid is Unique

Let $\struct {S, \circ}$ be a monoid. Then an element $x \in S$ can have at most one inverse for $\circ$.


Inverse of Inverse/Monoid

Let $\struct {S, \circ}$ be a monoid. Let $x \in S$ be invertible, and let its inverse be $x^{-1}$. Then $x^{-1}$ is also invertible, and: :$\paren {x^{-1} }^{-1} = x$


Inverse of Product/Monoid

Let $\left({S, \circ}\right)$ be a monoid whose identity is $e$. Let $a, b \in S$ be invertible for $\circ$, with inverses $a^{-1}, b^{-1}$. Then $a \circ b$ is invertible for $\circ$, and: : $\left({a \circ b}\right)^{-1} = b^{-1} \circ a^{-1}$


Right Identity Element is Idempotent

Let $\struct {S, \circ}$ be an algebraic structure. Let $e_R \in S$ be a right identity with respect to $\circ$. Then $e_R$ is idempotent under $\circ$.


More than one Right Identity then no Left Identity

Let $\left({S, \circ}\right)$ be an algebraic structure. If $\left({S, \circ}\right)$ has more than one right identity, then it has no left identity.


Inverse Completion is Commutative Monoid

Let $\left({S, \circ}\right)$ be a commutative semigroup. Let $\left ({C, \circ}\right) \subseteq \left({S, \circ}\right)$ be the subsemigroup of cancellable elements of $\left({S, \circ}\right)$. Let $\left({T, \circ'}\right)$ be an inverse completion of $\left({S, \circ}\right)$. Then $\left({T, \circ'}\right)$ is a commutative monoid.


Inverse Completion is Commutative Semigroup

Let $\left({S, \circ}\right)$ be a commutative semigroup. Let $\left ({C, \circ}\right) \subseteq \left({S, \circ}\right)$ be the subsemigroup of cancellable elements of $\left({S, \circ}\right)$. Let $\left({T, \circ'}\right)$ be an inverse completion of $\left({S, \circ}\right)$. Then $T = S \circ' C^{-1}$, and is a commutative semigroup.


Complement of G-Delta Set is F-Sigma Set

Let $T = \struct {S, \tau}$ be a topological space. Let $X$ be an $G_\delta$ set of $T$. Then its complement $S \setminus X$ is an $F_\sigma$ set of $T$.


Bounded Piecewise Continuous Function has Improper Integrals

Let $f$ be a real function defined on a closed interval $\closedint a b$, $a < b$.  Let $f$ be piecewise continuous and bounded on $\closedint a b$. Then $f$ is a piecewise continuous function with improper integrals.


F-Sigma Set is not necessarily Closed Set

Let $T = \struct {S, \tau}$ be a topological space. Let $X$ be an $F_\sigma$ set of $T$. Then it is not necessarily the case that $X$ is a closed set of $T$.


Not every Closed Set is G-Delta Set

Let $T = \struct {S, \tau}$ be a topological space. Let $V$ be a closed set of $T$. Then it is not necessarily the case that $V$ is a $G_\delta$ set of $T$.


Open Set of Uncountable Finite Complement Topology is not F-Sigma

Let $T = \struct {S, \tau}$ be a finite complement topology on an uncountable set $S$. Let $U \in \tau$ be an open set of $T$. Then $U$ is not an $F_\sigma$ set.


Closed Set of Uncountable Finite Complement Topology is not G-Delta

Let $T = \struct {S, \tau}$ be a finite complement topology on an uncountable set $S$. Let $V \in \tau$ be a closed set of $T$. Then $V$ is not a $G_\delta$ set.


G-Delta Set is not necessarily Open Set

Let $T = \struct {S, \tau}$ be a topological space. Let $X$ be a $G_\delta$ set of $T$. Then it is not necessarily the case that $X$ is a open set of $T$.


Limit Point of Sequence may only be Adherent Point of Range

Let $T = \struct {S, \tau}$ be a topological space. Let $A \subseteq S$. Let $\sequence {x_n}$ be a sequence in $A$. Let $\alpha$ be a limit point of $\sequence {x_n}$. Then $\alpha$ may be only an adherent point of $A$ and not a limit point of $A$.


Limit Point of Sequence in Discrete Space not always Limit Point of Open Set

Let $T = \struct {S, \tau}$ be a discrete topological space. Let $U \in \tau$ be an open set of $T$. Let $\sequence {x_n}$ be a sequence in $U$. Let $x$ be the limit point of $\sequence {x_n}$. Then $x$ is not always a limit point of $U$.


Accumulation Point of Sequence of Distinct Terms is Omega-Accumulation Point of Range

Let $T = \struct {S, \tau}$ be a topological space. Let $\sequence {x_n}$ be a sequence of distinct terms of $S$. Let $\alpha$ be an accumulation point of $\sequence {x_n}$. Then $\alpha$ is also an $\omega$-accumulation point of $\set {x_n: n \in \N}$.


Limit Point of Sequence is Adherent Point of Range

Let $T = \left({S, \tau}\right)$ be a topological space. Let $\left \langle {x_n} \right \rangle$ be a sequence in $S$. Let $\alpha$ be a limit point of $\left \langle {x_n} \right \rangle$. Then $\alpha$ is an adherent point of $\left\{ {x_n: n \in \N}\right\}$.


Limit Point of Sequence of Distinct Terms is Omega-Accumulation Point of Range

Let $T = \struct {S, \tau}$ be a topological space. Let $\sequence {x_n}$ be a sequence of distinct terms of $S$. Let $\alpha$ be a limit point of $\sequence {x_n}$. Then $\alpha$ is also an $\omega$-accumulation point of $\set {x_n: n \in \N}$.


Equivalence of Definitions of Isolated Point

{{TFAE|def = Isolated Point of Subset|view = isolated point}} Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$ be a subset of $S$.


Equivalence of Definitions of Closed Set

{{TFAE|def = Closed Set (Topology)|view = Closed Set|context = Topology (Mathematical Branch)|contextview = topology}} Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$.


Kuratowski's Closure-Complement Problem

Let $T = \struct {S, \tau}$ be a topological space. Let $A \subseteq S$ be a subset of $T$. By successive applications of the operations of complement relative to $S$ and the closure, there can be as many as $14$ distinct subsets of $S$ (including $A$ itself).


Kuratowski's Closure-Complement Problem/Complement

The complement of $A$ in $\R$ is given by: {{begin-eqn}} {{eqn | l = A'       | r = \left({\gets \,.\,.\, 0}\right]       | c = {{Defof|Unbounded Closed Real Interval}} }} {{eqn | o =       | ro= \cup       | r = \left\{ {1} \right\}       | c = {{Defof|Singleton}} }} {{eqn | o =       | ro= \cup       | r = \left[{2 \,.\,.\, 3}\right)       | c = {{Defof|Half-Open Real Interval}} }} {{eqn | o =       | ro= \cup       | r = \left({3 \,.\,.\, 4}\right]       | c = ... adjacent to Half-Open Real Interval }} {{eqn | o =       | ro= \cup       | r = \left({\R \setminus \Q \cap \left[{4 \,.\,.\, 5}\right]}\right)       | c = Irrational Numbers from $4$ to $5$ }} {{eqn | o =       | ro= \cup       | r = \left[{5 \,.\,.\, \to}\right)       | c = {{Defof|Unbounded Closed Real Interval}} }} {{end-eqn}} :500px


Kuratowski's Closure-Complement Problem/Interior

The interior of $A$ in $\R$ is given by: {{begin-eqn}} {{eqn | l = A^\circ       | r = \left({0 \,.\,.\, 1}\right) \cup \left({1 \,.\,.\, 2}\right)       | c = Union of Adjacent Open Intervals }} {{end-eqn}} :500px


Kuratowski's Closure-Complement Problem/Closure

The closure of $A$ in $\R$ is given by: {{begin-eqn}} {{eqn | l = A^-       | r = \closedint 0 2       | c = {{Defof|Closed Real Interval}} }} {{eqn | o =       | ro= \cup       | r = \set 3       | c = {{Defof|Singleton}} }} {{eqn | o =       | ro= \cup       | r = \closedint 4 5       | c = {{Defof|Closed Real Interval}} }} {{end-eqn}} :500px


Kuratowski's Closure-Complement Problem/Exterior

The exterior of $A$ in $\R$ is given by: {{begin-eqn}} {{eqn | l = A^e       | r = \openint \gets 0       | c = {{Defof|Unbounded Open Real Interval}} }} {{eqn | o =       | ro= \cup       | r = \openint 2 3 \cup \openint 3 4       | c = {{Defof|Union of Adjacent Open Intervals}} }} {{eqn | o =       | ro= \cup       | r = \openint 5 \to       | c = {{Defof|Unbounded Open Real Interval}} }} {{end-eqn}} :500px


Kuratowski's Closure-Complement Problem/Closure of Complement

The closure of the complement of $A$ in $\R$ is given by: {{begin-eqn}} {{eqn | l = A^{\prime \, -}       | r = \hointl \gets 0       | c = {{Defof|Unbounded Closed Real Interval}} }} {{eqn | o =       | ro= \cup       | r = \set 1       | c = {{Defof|Singleton}} }} {{eqn | o =       | ro= \cup       | r = \hointr 2 \to       | c = {{Defof|Unbounded Closed Real Interval}} }} {{end-eqn}} :500px


Kuratowski's Closure-Complement Problem/Closure of Interior

The closure of the interior of $A$ in $\R$ is given by: {{begin-eqn}} {{eqn | l = A^{\circ \, -}       | r = \closedint 0 2       | c = {{Defof|Closed Real Interval}} }} {{end-eqn}} :500px


Kuratowski's Closure-Complement Problem/Interior of Closure

The interior of the closure of $A$ in $\R$ is given by: {{begin-eqn}} {{eqn | l = A^{- \, \circ}       | r = \openint 0 2       | c = {{Defof|Open Real Interval}} }} {{eqn | o =       | ro= \cup       | r = \openint 4 5       | c = {{Defof|Open Real Interval}} }} {{end-eqn}} :500px


Kuratowski's Closure-Complement Problem/Interior of Closure of Interior

The interior of the closure of the interior of $A$ in $\R$ is given by: {{begin-eqn}} {{eqn | l = A^{\circ \, - \, \circ}       | r = \left({0 \,.\,.\, 2}\right)       | c = {{Defof|Open Real Interval}} }} {{end-eqn}} :500px


Kuratowski's Closure-Complement Problem/Interior of Complement of Interior

The interior of the complement of the interior of $A$ in $\R$ is given by: {{begin-eqn}} {{eqn | l = A^{\circ \, \prime \, \circ}       | r = \openint \gets 0       | c = {{Defof|Unbounded Open Real Interval}} }} {{eqn | o =       | ro= \cup       | r = \openint 2 \to       | c = {{Defof|Unbounded Open Real Interval}} }} {{end-eqn}} :500px


Kuratowski's Closure-Complement Problem/Closure of Interior of Complement

The closure of the interior of the complement of $A$ in $\R$ is given by: {{begin-eqn}} {{eqn | l = A^{\prime \, \circ \, -}       | r = \hointl \gets 0       | c = {{Defof|Unbounded Closed Real Interval}} }} {{eqn | o =       | ro= \cup       | r = \closedint 2 4       | c = {{Defof|Closed Real Interval}} }} {{eqn | o =       | ro= \cup       | r = \hointr 5 \to       | c = {{Defof|Unbounded Closed Real Interval}} }} {{end-eqn}} :500px


Kuratowski's Closure-Complement Problem/Closure of Interior of Closure

The closure of the interior of the closure of $A$ in $\R$ is given by: {{begin-eqn}} {{eqn | l = A^{- \, \circ \, -}       | r = \left[{0 \,.\,.\, 2}\right]       | c = {{Defof|Closed Real Interval}} }} {{eqn | o =       | ro= \cup       | r = \left[{4 \,.\,.\, 5}\right]       | c = {{Defof|Closed Real Interval}} }} {{end-eqn}} :500px


Kuratowski's Closure-Complement Problem/Interior of Complement of Interior of Closure

The interior of the complement of the interior of the closure of $A$ in $\R$ is given by: {{begin-eqn}} {{eqn | l = A^{- \, \circ \, \prime \, \circ}       | r = \left({\gets \,.\,.\, 0}\right)       | c = {{Defof|Unbounded Open Real Interval}} }} {{eqn | o =       | ro= \cup       | r = \left({2 \,.\,.\, 4}\right)       | c = {{Defof|Open Real Interval}} }} {{eqn | o =       | ro= \cup       | r = \left({5 \,.\,.\, \to}\right)       | c = {{Defof|Unbounded Open Real Interval}} }} {{end-eqn}} :500px


Kuratowski's Closure-Complement Problem/Complement of Interior of Closure of Interior

The complement of the interior of the closure of the interior of $A$ in $\R$ is given by: {{begin-eqn}} {{eqn | l = A^{\circ \, - \, \circ \, \prime}       | r = \left({\gets \,.\,.\, 0}\right]       | c = {{Defof|Unbounded Closed Real Interval}} }} {{eqn | o =       | ro= \cup       | r = \left[{2 \,.\,.\, \to}\right)       | c = {{Defof|Unbounded Closed Real Interval}} }} {{end-eqn}} :500px


Sum of Integrals on Adjacent Intervals for Integrable Functions

Let $f$ be a real function which is Darboux integrable on any closed interval $\mathbb I$. Let $a, b, c \in \mathbb I$. Then: :$\displaystyle \int_a^c \map f t \rd t + \int_c^b \map f t \rd t = \int_a^b \map f t \rd t$


Closure of Irrational Numbers is Real Numbers

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space under the same topology. Then: :$\paren {\R \setminus \Q}^- = \R$ where $\paren {\R \setminus \Q}-$ denotes the closure of $\R \setminus \Q$.


Closure of Union of Adjacent Open Intervals

Let $a, b, c \in R$ where $a < b < c$. Let $A$ be the union of the two adjacent open intervals: :$A := \left({a \,.\,.\, b}\right) \cup \left({b \,.\,.\, c}\right)$ Then: :$A^- = \left[{a \,.\,.\, c}\right]$ where: : $A^-$ is the closure of $A$.


Interior equals Complement of Closure of Complement

Let $T$ be a topological space. Let $H \subseteq T$. Let $H^-$ denote the closure of $H$ and $H^\circ$ denote the interior of $H$. Let $H^\prime$ denote the complement of $H$ in $T$: :$H^\prime = T \setminus H$ Then: :$H^\circ = H^{\prime \, - \, \prime}$


Closure of Half-Open Real Interval is Closed Real Interval

Let $\struct {\R, \tau_d}$ be the real number line under the usual (Euclidean) topology.   Let $H_1 = \hointl a b$ and $H_2 = \hointr a b$ be half-open intervals of $\R$. Then the closure of both $H_1$ and $H_2$ in $\R$ are the closed interval $\closedint a b$.


Closure of Irrational Interval is Closed Real Interval

Let $\struct {\R, \tau_d}$ be the real numbers under the usual (Euclidean) topology. Let $\struct{\R \setminus \Q, \tau_d}$ be the irrational number space under the same topology. Let $a, b \in \R$ such that $a < b$. Let $\Bbb I \subseteq \R$ be an interval of $\R$ Then the closure of the set: :$\Bbb I \cap \paren {\R \setminus \Q}$ is the closed real interval $\closedint a b$.


Closure of Rational Interval is Closed Real Interval

Let $\struct {\R, \tau_d}$ be the real numbers under the usual (Euclidean) topology. Let $\struct {\Q, \tau_d}$ be the rational number space under the same topology. Let $a, b \in \R$ such that $a < b$. Let $\Bbb I \subseteq \R$ be an interval of $\R$ Then the closure of the set : :$\Bbb I \cap \Q$ is the closed real interval $\closedint a b$.


Interior of Singleton in Real Number Line is Empty

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Let $a \in \R$ be a real number. Then: :$\set a^\circ = \O$ where $\set a^\circ$ denotes the interior of $\set a$ in $\R$.


Kuratowski's Closure-Complement Problem/Proof of Maximum

Let $T = \left({S, \tau}\right)$ be a topological space. Let $A \subseteq S$ be a subset of $T$. By successive applications of the operations of complement relative to $S$ and the closure, there can be no more than $14$ distinct subsets of $S$ (including $A$ itself).


Closure of Complement of Closure is Regular Closed

Let $T = \struct {S, \tau}$ be a topological space. Let $A \subseteq S$ be a subset of $T$. Let $A^-$ denote the closure of $A$ in $T$. Let $A'$ denote the complement of $A$ in $S$: $A' = S \setminus A$. Then $A^{- ' -}$ is regular closed.


Topology induced by Usual Metric on Positive Integers is Discrete

Let $\Z_{>0}$ be the set of (strictly) positive integers. Let $d: \Z_{>0} \times \Z_{>0} \to \R$ be the usual (Euclidean) metric on $\Z_{>0}$. Then the metric topology for $d$ is a discrete topology.


Topology induced by Scaled Euclidean Metric on Positive Integers is Discrete

Let $\Z_{>0}$ be the set of (strictly) positive integers. Let $\delta: \Z_{>0} \times \Z_{>0} \to \R$ be the metric on $\Z_{>0}$ defined as: :$\forall x, y \in \Z_{>0}: \map \delta {x, y} = \dfrac {\size {x - y} } {x y}$ Then the metric topology for $\delta$ is a discrete topology.


Scaled Euclidean Metric is Metric

Let $\R_{>0}$ be the set of (strictly) positive integers. Let $\delta: \R_{>0} \times \R_{>0} \to \R$ be the metric on $\R_{>0}$ defined as: :$\forall x, y \in \R_{>0}: \delta \left({x, y}\right) = \dfrac {\left\lvert{x - y}\right\rvert} {x y}$ Then $\delta$ is a metric.


Topologies induced by Usual Metric and Scaled Euclidean Metric on Positive Integers are Homeomorphic

Let $\Z_{>0}$ be the set of (strictly) positive integers. Let $d: \Z_{>0} \times \Z_{>0} \to \R$ be the usual (Euclidean) metric on $\Z_{>0}$. Let $\delta: \Z_{>0} \times \Z_{>0} \to \R$ be the metric on $\Z_{>0}$ defined as: :$\forall x, y \in \Z_{>0}: \map \delta {x, y} = \dfrac {\size {x - y} } {x y}$ Let $\tau_d$ denote the metric topology for $d$. Let $\tau_\delta$ denote the metric topology for $\delta$. Then $\struct {\Z_{>0}, \tau_d}$ and $\struct {\Z_{>0}, \tau_\delta}$ are homeomorphic.


Cauchy Sequence in Positive Integers under Usual Metric is eventually Constant

Let $\Z_{>0}$ be the set of (strictly) positive integers. Let $d: \Z_{>0} \times \Z_{>0} \to \R$ be the usual (Euclidean) metric on $\Z_{>0}$. Let $\sequence {x_n}$ be a Cauchy sequence in $\struct {\Z_{>0}, d}$. Then: :$\exists m, n \in \Z_{>0}: \forall r > n: x_r = m$ That is, $\sequence {x_n}$ is eventually constant.


Positive Integers under Usual Metric is Complete Metric Space

Let $\Z_{>0}$ be the set of (strictly) positive integers. Let $d: \Z_{>0} \times \Z_{>0} \to \R$ be the usual (Euclidean) metric on $\Z_{>0}$. Then $\struct {\Z_{>0}, d}$ is a complete metric space.


Cauchy Sequence in Positive Integers under Scaled Euclidean Metric

Let $\Z_{>0}$ be the set of (strictly) positive integers. Let $\delta: \Z_{>0} \times \Z_{>0} \to \R$ be the scaled Euclidean metric on $\Z_{>0}$ defined as: :$\forall x, y \in \Z_{>0}: \map \delta {x, y} = \dfrac {\size {x - y} } {x y}$ The sequence $\sequence {x_n}$ in $\Z_{>0}$ defined as: :$\forall n \in \N: x_n = n$ is a Cauchy sequence in $\struct {\Z_{>0}, \delta}$.


Positive Integers under Scaled Euclidean Metric is not Complete Metric Space

Let $\Z_{>0}$ be the set of (strictly) positive integers. Let $\delta: \Z_{>0} \times \Z_{>0} \to \R$ be the scaled Euclidean metric on $\Z_{>0}$ defined as: :$\forall x, y \in \Z_{>0}: \map \delta {x, y} = \dfrac {\size {x - y} } {x y}$ Then $\struct {\Z_{>0}, \delta}$ is not a complete metric space.


Subset of Euclidean Plane whose Product of Coordinates are Greater Than or Equal to 1 is Closed

Let $\struct {\R^2, \tau_d}$ be the real number plane with the usual (Euclidean) topology. Let $A \subseteq R^2$ be the set of all points defined as: :$A := \set {\tuple {x, y} \in \R^2: x y \ge 1}$ Then $A$ is a closed set in $\struct {\R^2, d}$.


Projection on Real Euclidean Plane is Open Mapping

Let $\struct {\R^2, d}$ be the real number plane with the usual (Euclidean) topology. Let $\rho: \R^2 \to \R$ be the first projection on $\R^2$ defined as: :$\forall \tuple{x, y} \in \R^2: \map \rho {x, y} = x$ Then $\rho$ is an open mapping. The same applies with the second projection on $\R^2$.


Projection on Real Euclidean Plane is not Closed Mapping

Let $\struct {\R^2, d}$ be the real number plane with the usual (Euclidean) topology. Let $\rho: \R^2 \to \R$ be the first projection on $\R^2$ defined as: :$\forall \tuple {x, y} \in \R^2: \map \rho {x, y} = x$ Then $\rho$ is not a closed mapping. The same applies with the second projection on $\R^2$.


Complement of Set of Rational Pairs in Real Euclidean Plane is Arc-Connected

Let $\struct {\R^2, d}$ be the real number plane with the usual (Euclidean) topology. Let $S \subseteq \R^2$ be the subset of $\R^2$ defined as: :$\forall x, y \in \R^2: \tuple {x, y} \in S \iff x, y \in \Q$ Hence let $A := \R^2 \setminus S$: :$\tuple {x, y} \in A$ {{iff}} either $x$ or $y$ or both is irrational. Then $A$ is arc-connected.


Empty Set is Compact Space

Let $T = \left({S, \tau}\right)$ be a topological space. Then the empty set $\varnothing$ is a compact subspace of $T$.


Alexandroff Extension is Topology

Let $T = \struct {S, \tau}$ be a non-empty topological space. Let $p$ be a new element not in $S$. Let $S^* := S \cup \set p$. Let $T^* = \struct {S^*, \tau^*}$ be the Alexandroff extension on $S$. Then $\tau^*$ is a topology on $S^*$.


Intersection of Closed Set with Compact Subspace is Compact

Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$ be closed in $T$. Let $K \subseteq S$ be compact in $T$. Then $H \cap K$ is compact in $T$.