hcju/proofwikips huggingface dataset:
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 $d_2$ : :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 $d_2$ : :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 $\displaystyle \lim_{x \mathop \to c} \map f x = l$ : :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 $M$ : :$\forall x \in A: \map d {x, H} = 0 \implies x \in H$ 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 $a \in H^-$  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 $H^- \subseteq F$.