hcju/proofwikips huggingface dataset:
P-Product Metric on Real Vector Space is Metric/Proof 2
Let $\R^n$ be an $n$-dimensional real vector space. Let $p \in \R_{\ge 1}$. Let $d_p: \R^n \times \R^n \to \R$ be the $p$-product metric on $\R^n$: : $\displaystyle d_p \left({x, y}\right) :=  \left({\sum_{i \mathop = 1}^n \left \vert {x_i - y_i} \right \vert^p}\right)^{\frac 1 p}$ where $x = \left({x_1, x_2, \ldots, x_n}\right), y = \left({y_1, y_2, \ldots, y_n}\right) \in \R^n$. Then $d_p$ is a metric.
Surjection from Natural Numbers iff Right Inverse
Let $S$ be a set. Let $f: \N \to S$ be a mapping, where $\N$ denotes the set of natural numbers. Then $f$ is a surjection  $f$ admits a right inverse.
Sum of Squared Deviations from Mean
Let $x_1, x_2, \ldots, x_n$ be real data about some quantitative variable. Let $\overline x$ be the arithmetic mean of the above data. Then:  :$\displaystyle \sum_{i \mathop = 1}^n \paren {x_i - \overline x}^2 = \sum_{i \mathop = 1}^n \paren {x_i^2 - \overline x^2}$
Surjection from Natural Numbers iff Countable
Let $S$ be a non-empty set. Then $S$ is countable  there exists a surjection $f: \N \to S$.
Summation is Linear
Let $\left({x_1, \ldots, x_n}\right)$ and $\left({y_1, \ldots, y_n}\right)$ be finite sequences of numbers of equal length. Let $\lambda$ be a number. Then:
Sum of Identical Terms
Let $x$ be a number, and let $n \in \N$ be a natural number. Suppose that $n \ge 1$. Then: :$\displaystyle \sum_{i \mathop = 1}^n x = n x$
Synthetic Sub-Basis and Analytic Sub-Basis are Compatible
Let $\left({X, \tau}\right)$ be a topological space. Let $\mathcal S \subseteq \mathcal P \left({X}\right)$, where $\mathcal P \left({X}\right)$ denotes the power set of $X$. Then $\mathcal S$ is an analytic sub-basis for $\tau$  $\tau$ is the topology on $X$ generated by the synthetic sub-basis $\mathcal S$.
Analytic Basis is Analytic Sub-Basis
Let $\left({X, \tau}\right)$ be a topological space. Let $\mathcal B \subseteq \tau$ be an analytic basis for $\tau$. Then $\mathcal B$ is an analytic sub-basis for $\tau$.
Continuity Test using Sub-Basis
Let $\struct {X_1, \tau_1}$ and $\struct {X_2, \tau_2}$ be topological spaces. Let $f: X_1 \to X_2$ be a mapping. Let $\SS$ be an analytic sub-basis for $\tau_2$. Suppose that: :$\forall S \in \SS: f^{-1} \sqbrk S \in \tau_1$ where $f^{-1} \sqbrk S$ denotes the preimage of $S$ under $f$. Then $f$ is continuous.
Lagrange Interpolation Formula
Let $\tuple {x_0, \ldots, x_n}$ and $\tuple {a_0, \ldots, a_n}$ be ordered tuples of real numbers such that $x_i \ne x_j$ for $i \ne j$. Then there exists a unique polynomial $P \in \R \sqbrk X$ of degree at most $n$ such that: :$\map P {x_i} = a_i$ for all $i \in \set {0, 1, \ldots, n}$ Moreover $P$ is given by the formula: :$\displaystyle \map P X = \sum_{j \mathop = 0}^n a_i \map {L_j} X$ where $\map {L_j} X$ is the $j$th Lagrange basis polynomial associated to the $x_i$.
Product Formula for Sine
:$\displaystyle \map \sin {n z} = 2^{n - 1} \prod_{k \mathop = 0}^{n - 1} \map \sin {z + \frac{k \pi} n}$
Final Topology is Topology
Let $X$ be a set. Let $I$ be an indexing set. Let $\family {\struct {Y_i, \tau_i} }_{i \mathop \in I}$ be an $I$-indexed family of topological spaces. Let $\family {f_i: Y_i \to X}_{i \mathop \in I}$ be an $I$-indexed family of mappings. Let $\tau$ be the final topology on $X$ with respect to $\family {f_i}_{i \mathop \in I}$. Then $\tau$ is a topology on $X$.
Subspace Topology is Initial Topology with respect to Inclusion Mapping
Let $\struct {X, \tau}$ be a topological space. Let $Y$ be a non-empty subset of $X$. Let $\iota: Y \to X$ be the inclusion mapping. Let $\tau_Y$ be the initial topology on $Y$ with respect to $\iota$. Then $\struct {Y, \tau_Y}$ is a topological subspace of $\struct {X, \tau}$. That is: :$\tau_Y = \set {U \cap Y: U \in \tau}$
Topological Subspace is Topological Space
Let $\left({X, \tau}\right)$ be a topological space. Let $H \subseteq X$ be a non-empty subset of $X$. Let $\tau_H = \left\{{U \cap H: U \in \tau}\right\}$ be the subspace topology on $H$. Then the topological subspace $\left({H, \tau_H}\right)$ is a topological space.
Initial Topology with respect to Mapping equals Set of Preimages
Let $X$ be a set. Let $\struct {Y, \tau_Y}$ be a topological space. Let $f: X \to Y$ be a mapping. Let $\tau_X$ be the initial topology on $X$ with respect to $f$. Then: :$\tau_X = \set {f^{-1} \sqbrk U: U \in \tau_Y}$
Sommerfeld-Watson Transform
Let $f \left({z}\right)$ be a mapping with isolated poles. {{explain|Explain the context in which this theorem is placed. For example, what is the domain and range of $f$? One supposes $\C$ but it needs to be made clear.}} Let $f$ go to zero faster than $\dfrac 1 {\left|{z}\right|}$ as $\left|{z}\right| \to \infty$. {{explain|"faster"}} Let $C$ be a contour that is deformed such that all poles of $f \left({z}\right)$ are contained in $C$. Then: :$\displaystyle \sum \limits_{n \mathop = -\infty}^\infty \left({-1}\right)^n f \left({n}\right) = \frac 1 {2 i} \oint_C \frac {f \left({z}\right)} {\sin \pi z} \, \mathrm d z$
Linear Transformation of Arithmetic Mean
Let $D = \set {x_0, x_1, x_2, \ldots, x_n}$ be a set of real data describing a quantitative variable. Let $\overline x$ be the arithmetic mean of the data in $D$. Let $T: \R \to \R$ be a linear transformation such that: :$\forall i \in \set {0, 1, \ldots, n}: \map T {x_i} = \lambda x_i + \gamma$ Let $T \sqbrk D$ be the image of $D$ under $T$. Then the arithmetic mean of the data in $T \sqbrk D$ is given by: :$\map T {\overline x} = \lambda \overline x + \gamma$
Category of Pointed Sets is Category
Let $\mathbf{Set}_*$ be the category of pointed sets. Then $\mathbf{Set}_*$ is a metacategory.
Closed Set in Topological Subspace/Corollary
Let subspace $T'$ be closed in $T$. Then $V \subseteq T'$ is closed in $T'$  $V$ is closed in $T$.
Unit Interval is Path-Connected in Real Numbers
Let $\R$ be the real number line with the usual (Euclidean} metric. The closed unit interval $\mathbf I = \closedint 0 1$ is a path-connected metric subspace of $\R$.
Subset of Real Numbers is Path-Connected iff Interval
Let $\R$ be the real number line considered as an Euclidean space. Let $S \subseteq \R$ be a subset of $\R$. Then $S$ is a path-connected metric subspace of $\R$  $S$ is a real interval.
Size of Linearly Independent Subset is at Most Size of Finite Generator
Let $R$ be a division ring. Let $V$ be an $R$-vector space. Let $F \subseteq V$ be a finite generator of $V$ over $R$. Let $L \subseteq V$ be linearly independent over $R$. Then: :$\size L \le \size F$
Intermediate Value Theorem/Corollary
Let $0 \in \R$ lie between $f \left({a}\right)$ and $f \left({b}\right)$. That is, either: : $f \left({a}\right) < 0 < f \left({b}\right)$ or: : $f \left({b}\right) < 0 < f \left({a}\right)$ Then $f$ has a root in $\left({a \,.\,.\, b}\right)$.
Euclidean Space is Path-Connected
Let $\R^n$ be the $n$-dimensional Euclidean space for $n \in \N$ a natural number. Then $\R^n$ is path-connected.
Coequalizer is Epimorphism
Let $\mathbf C$ be a metacategory. Let $q: D \to Q$ be the coequalizer of two morphisms $f, g: C \to D$. Then $q$ is an epimorphism.
Euclidean Space without Origin is Path-Connected
Let $n \in \Z: n \ge 2$. Let $\R^n$ be the $n$-dimensional Euclidean space. Let $\R^n \setminus \set {\mathbf 0}$ be $\R^n$ with the origin removed. Then $\R^n \setminus \set {\mathbf 0}$ is path-connected.
Composite of Continuous Mappings is Continuous/Corollary
Let $T_1, T_2, T_3$ each be one of: :metric spaces :the complex plane :the real number line Let $f: T_1 \to T_2$ and $g: T_2 \to T_3$ be continuous mappings. Then the composite mapping $g \circ f: T_1 \to T_3$ is continuous.
Binomial Theorem/Multiindex
Let $\alpha$ be a multiindex, indexed by $\left\{{1, \ldots, n}\right\}$ such that $\alpha_j \ge 0$ for $j = 1, \ldots, n$. Let $x = \left({x_1, \ldots, x_n}\right)$ and $y = \left({y_1, \ldots, y_n}\right)$ be ordered tuples of real numbers. Then: :$\displaystyle \left({x + y}\right)^\alpha = \sum_{0 \mathop \le \beta \mathop \le \alpha} {\alpha \choose \beta} x^\beta y^{\alpha - \beta}$ where $\displaystyle {n \choose k}$ is a binomial coefficient.
Leibniz's Rule/One Variable
Let $f$ and $g$ be real functions defined on the open interval $I$. Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $x \in I$ be a point in $I$ at which both $f$ and $g$ are $n$ times differentiable. Then: :$\displaystyle \paren {\map f x \, \map g x}^{\paren n} = \sum_{k \mathop = 0}^n \binom n k \map {f^{\paren k} } x \, \map {g^{\paren {n - k} } } x$ where $\paren n$ denotes the order of the derivative.
Open Sets of Double Pointed Topology
Let $\struct {S, \tau_S}$ be a topological space. Let $D$ be a doubleton endowed with the indiscrete topology. Let $\struct {S \times D, \tau}$ be the double pointed topology on $S$. Then $X \subseteq S \times D$ is open in $\tau$  for some $U \in \tau$: :$X = U \times D$
Open Sets of Double Pointed Topology/Corollary
A subset $X \subseteq S \times D$ is closed in $\tau$  for some closed set $C$ of $\tau$: :$X = C \times D$
Open Set Less One Point is Open/Corollary
Let $S = \left\{{\alpha_1, \alpha_2, \ldots, \alpha_n}\right\} \subseteq U$ be a finite set of points in $U$. Then $U \setminus S$ is open in $M$.
Closed Real Interval is Neighborhood Except at Endpoints
Let $\R$ be the real number line considered as an Euclidean space. Let $\left[{a \,.\,.\, b}\right] \subset \R$ be a closed interval of $\R$. Then $\left[{a \,.\,.\, b}\right]$ is a neighborhood of all of its points except $a$ and $b$.
Open Real Interval is Open Set
Let $\R$ be the real number line considered as an Euclidean space. Let $\openint a b \subset \R$ be an open interval of $\R$. Then $\openint a b$ is an open set of $\R$.
Closed Real Interval is not Open Set
Let $\R$ be the real number line considered as an Euclidean space. Let $\left[{a \,.\,.\, b}\right] \subset \R$ be a closed interval of $\R$. Then $\left[{a \,.\,.\, b}\right]$ is not an open set of $\R$.
Open Set may not be Open Ball
Let $M = \struct {A, d}$ be a metric space. Let $U \subseteq M$ be an open set of $M$. Then it is not necessarily the case that $U$ is an open ball of some $x \in A$.
Closed Real Interval is Closed Set
Let $\R$ be the real number line considered as an Euclidean space. Let $\closedint a b \subset \R$ be a closed interval of $\R$. Then $\closedint a b$ is a closed set of $\R$.
Kernel of Linear Transformation is Null Space of Matrix Representation
Let $V$ and $W$ be finite dimensional vector spaces. Let $\phi: V \to W$ be a linear transformation from $V$ to $W$. Let $\left({e_1, \ldots, e_n}\right)$ and $\left({f_1, \ldots, f_m}\right)$ be ordered bases of $V$ and $W$ respectively. Let $A$ be the matrix of $\phi$ in these bases. Define $f: V \to \R^n$ by: :$\displaystyle \sum_{i \mathop = 1}^n a_i e_i \mapsto \left({a_1, \ldots, a_n}\right)$ and $g : W \to \R^m$ by: :$\displaystyle \sum_{i \mathop = 1}^m b_i f_i \mapsto \left({b_1, \ldots, b_m}\right)$ Let $N \left({A}\right) = \left\{ {x \in \R^n: A x = 0}\right\}$ be the null space of $A$. Let $\ker \phi = \left\{ {x \in V: \phi x = 0}\right\}$ be the kernel of $\phi$. Then: :$f \left[{\ker \phi}\right] = N \left({A}\right)$ and :$f^{-1} \left[{N \left({A}\right)}\right] = \ker \phi$ where $f \left[{X}\right]$ denotes the image set of a subset $X$ of the domain of $f$.
Half-Open Real Interval is neither Open nor Closed
Let $\R$ be the real number line considered as an Euclidean space. Let $\left[{a \,.\,.\, b}\right) \subset \R$ be a half-open interval of $\R$. Then $\left[{a \,.\,.\, b}\right)$ is neither an open set nor a closed set of $\R$. Similarly, the half-open interval $\left({a \,.\,.\, b}\right] \subset \R$ is neither an open set nor a closed set of $\R$.
Compact Sets of Double Pointed Topology
Let $\struct {S, \tau_S}$ be a topological space. Let $D$ be a doubleton endowed with the indiscrete topology. Let $\struct {S \times D, \tau}$ be the double pointed topology on $S$. Then $X \subseteq S \times D$ is compact in $\tau$  for some compact set $C$ of $\tau_S$: :$\map {\pr_1} X = C$ where $\pr_1$ denotes the first projection on $S \times D$.
Closure of Subset of Double Pointed Topological Space
Let $\struct {S, \tau_S}$ be a topological space. Let $D$ be a doubleton endowed with the indiscrete topology. Let $\struct {S \times D, \tau}$ be the double pointed topology on $S$. Let $X \subseteq S \times D$ be a subset of $S \times D$. Then the closure of $X$ in $\tau$ is: :$\map \cl X = \map \cl {\map {\pr_1} X} \times D$ where $\pr_1$ denotes the first projection on $S \times D$.
Interior of Subset of Double Pointed Topological Space
Let $\struct {S, \tau_S}$ be a topological space. Let $D$ be a doubleton endowed with the indiscrete topology. Let $\struct {S \times D, \tau}$ be the double pointed topology on $S$. Let $X \subseteq S \times D$ be a subset of $S \times D$. Define $A \subseteq S$ by: :$A := \set {s \in S: \paren {\forall d \in D: \tuple {s, d} \in X} }$ Then the interior of $X$ in $\tau$ is: :$X^\circ = A^\circ \times D$
Projection of Complement Contains Complement of Projection
Let $S$ and $T$ be non-empty sets. Let $X \subseteq S \times T$ be a subset of the Cartesian product $S \times T$. Denote with $\operatorname{pr}_1, \operatorname{pr}_2$ and $\complement$ the first and second projections, and the complement operation, respectively. Then: {{begin-eqn}} {{eqn|l = \complement \left({\operatorname{pr}_1 \left({X}\right)}\right)       |o = \subseteq      |r = \operatorname{pr}_1 \left({\complement \left({X}\right)}\right) }} {{eqn|l = \complement \left({\operatorname{pr}_2 \left({X}\right)}\right)       |o = \subseteq      |r = \operatorname{pr}_2 \left({\complement \left({X}\right)}\right) }} {{end-eqn}}
Interior of Subset
Let $\left({S, \tau}\right)$ be a topological space. Let $X$ and $Y$ be subsets of $S$, and suppose that $X \subseteq Y$. Then: :$X^\circ \subseteq Y^\circ$ where $X^\circ$ denotes the interior of $X$.
Real Interval is Bounded in Real Numbers
Let $\R$ be the real number line considered as an Euclidean space. Let $a, b \in \R$. Let $\mathcal I$ be one of the following real intervals: {{begin-eqn}} {{eqn | l = \openint a b       | o = :=       | r = \set {x \in \R: a < x < b}       | c = Open Real Interval }} {{eqn | l = \hointr a b       | o = :=       | r = \set {x \in \R: a \le x < b}       | c = Half-Open (to the right) Real Interval }} {{eqn | l = \hointl a b       | o = :=       | r = \set {x \in \R: a < x \le b}       | c = Half-Open (to the left) Real Interval }} {{eqn | l = \closedint a b       | o = :=       | r = \set {x \in \R: a \le x \le b}       | c = Closed Real Interval }} {{end-eqn}} where $b \ge a$. Then $\mathcal I$ is bounded in $\R$.
Set of Integers is not Bounded
Let $\R$ be the real number line considered as an Euclidean space. The set $\Z$ of integers is not bounded in $\R$.
Closed Real Interval is Compact
Let $\R$ be the real number line considered as an Euclidean space. Let $I = \closedint a b$ be a closed real interval. Then $I$ is compact.
Open Real Interval is not Closed Set
Let $\R$ be the real number line with the usual (Euclidean) metric. Let $I = \openint a b$ be an open real interval. Then $I$ is not a closed set of $\R$.
Open Real Interval is not Compact
Let $\R$ be the real number line considered as an Euclidean space. Let $I = \openint a b$ be an open real interval. Then $I$ is not compact.
Set of Integers is not Compact
Let $\R$ be the real number line considered as an Euclidean space. Let $\Z$ be the set of integers. Then $\Z$ is not compact.
Heine-Borel Theorem/Euclidean Space
Let $n \in \N_{> 0}$. Let $C$ be a subspace of the Euclidean space $\R^n$. Then $C$ is closed and bounded  it is compact.
Continuous Function on Compact Space is Bounded
Let $\struct {X, \tau}$ be a topological space. Let $\struct {Y, \norm {\, \cdot \, } }$ be a normed vector space. Let $f: X \to Y$ be continuous. Then $f$ is bounded.
Hausdorff Space is Hereditarily Compact iff Finite
Let $\left({S, \tau}\right)$ be a Hausdorff space. Then $\left({S, \tau}\right)$ is hereditarily compact  $S$ is finite.
Complete and Totally Bounded Metric Space is Sequentially Compact
Let $M = \struct {A, d}$ be a metric space. Let $M$ be complete and totally bounded. Then $M$ is sequentially compact.
Closed Bounded Subset of Real Numbers is Compact
Let $\R$ be the real number line considered as an Euclidean space. Let $S \subseteq \R$ be a closed and bounded subspace of $\R$. Then $S$ is compact in $\R$.
Non-Closed Set of Real Numbers is not Compact
Let $\R$ be the set of real numbers considered as an Euclidean space. Let $S \subseteq \R$ be non-closed in $\R$. Then $S$ is not a compact subspace of $\R$.
Compact Subspace of Real Numbers is Closed and Bounded
Let $\R$ be the real number line considered as a Euclidean space. Let $S \subseteq \R$ be compact subspace of $\R$. Then $S$ is closed and bounded in $\R$.
Category of Subobjects is Category
Let $\mathbf C$ be a metacategory. Let $C$ be an object of $\mathbf C$. Let $\mathbf{Sub}_{\mathbf C} \left({C}\right)$ be the category of subobjects of $C$. Then $\mathbf{Sub}_{\mathbf C} \left({C}\right)$ is a metacategory.
Category of Subobjects is Preorder Category
Let $\mathbf C$ be a metacategory. Let $C$ be an object of $\mathbf C$. Let $\mathbf{Sub}_{\mathbf C} \left({C}\right)$ be the category of subobjects of $C$. Then $\mathbf{Sub}_{\mathbf C} \left({C}\right)$ is a preorder category.
Integral of Power/Fermat's Proof
:$\displaystyle \forall n \in \Q_{>0}: \int_0^b x^n \rd x = \frac {b^{n + 1} } {n + 1}$
Composition of Mapping with Inclusion is Restriction
Let $S$ and $T$ be sets. Let $f: S \to T$ be a mapping. Let $A \subseteq S$ be a subset of the domain of $S$. Let $i_A: A \to S$ be the inclusion mapping from $A$ to $S$. Then: :$f \circ i_A = f \restriction_A$ where $f \restriction_A$ denotes the restriction of $f$ to $A$.
Equivalence of Subobjects is Equivalence
Let $\mathbf C$ be a metacategory. Let $C$ be an object of $\mathbf C$. Let $\mathbf{Sub}_{\mathbf C} \left({C}\right)$ be the category of subobjects of $C$. The relation $\sim$ on $\mathbf{Sub}_{\mathbf C} \left({C}\right)$ defined by: :$m \sim m'$ iff $m$ and $m'$ are equivalent is an equivalence. {{improve|That's not a relation since we can't know we're dealing with sets; cover this}}
Morphism Class Equivalence is Equivalence
Let $\mathbf C$ be a metacategory. Let $C$ be an object of $\mathbf C$. Let $\mathbf{Sub}_{\mathbf C} \left({C}\right)$ be the category of subobjects of $C$. Let $\sim$ denote morphism class equivalence on the morphisms of $\mathbf{Sub}_{\mathbf C} \left({C}\right)$. Then $\sim$ is an equivalence.
Subobject Class in Category of Sets
Let $\mathbf{Set}$ be the category of sets. Let $S$ be a set. Let $\overline{\mathbf{Sub}}_{\mathbf{Set}} \left({S}\right)$ be the category of subobject classes of $S$. Let $\mathcal P \left({S}\right)$ be the order category on the power set of $S$ induced by Subset Relation on Power Set is Partial Ordering. Then $\overline{\mathbf{Sub}}_{\mathbf{Set}} \left({S}\right) \cong \mathcal P \left({S}\right)$.
Category of Subobject Classes is Category
Let $\mathbf C$ be a metacategory. Let $C$ be an object of $\mathbf C$. Let $\map {\overline {\mathbf {Sub} }_{\mathbf C} } C$ be the category of subobject classes of $C$. Then $\map {\overline {\mathbf {Sub} }_{\mathbf C} } C$ is a metacategory.
Category of Subobject Classes is Order Category
Let $\mathbf C$ be a metacategory. Let $C$ be an object of $\mathbf C$. Let $\overline{\mathbf{Sub}}_{\mathbf C} \left({C}\right)$ be the category of subobject classes of $C$. Then $\overline{\mathbf{Sub}}_{\mathbf C} \left({C}\right)$ is an order category.
Reverse Triangle Inequality/Real and Complex Fields
Let $x$ and $y$ be elements of either the real numbers $\R$ or the complex numbers $\C$. Then: :$\cmod {x - y} \ge \size {\cmod x - \cmod y}$
Triangle Inequality/Real Numbers
Let $x, y \in \R$ be real numbers. Let $\size x$ denote the absolute value of $x$. Then: :$\size {x + y} \le \size x + \size y$
Triangle Inequality/Complex Numbers
Let $z_1, z_2 \in \C$ be complex numbers. Let $\cmod z$ denote the modulus of $z$. Then: :$\cmod {z_1 + z_2} \le \cmod {z_1} + \cmod {z_2}$
Triangle Inequality/Vectors in Euclidean Space
Let $\mathbf x, \mathbf y$ be vectors in $\R^n$. Let $\norm {\, \cdot \,}$ denote vector length. Then: :$\norm {\mathbf x + \mathbf y} \le \norm {\mathbf x} + \norm {\mathbf y}$ If the two vectors are scalar multiples where said scalar is non-negative, an equality holds: :$\exists \lambda \in \R, \lambda \ge 0: \mathbf x = \lambda \mathbf y \iff \norm {\mathbf x + \mathbf y} = \norm {\mathbf x} + \norm {\mathbf y}$
Reverse Triangle Inequality/Real and Complex Fields/Corollary/Proof 2
Let $x$ and $y$ be elements of either the real numbers $\R$ or the complex numbers $\C$. Then: :$\cmod {x - y} \ge \cmod x - \cmod y$ where $\cmod x$ denotes either the absolute value of a real number or the complex modulus of a complex number.
Negative of Complex Modulus
Let $z \in \C$ be a complex number. Then: :$-\cmod z \le \cmod z$ where $\cmod z$ denotes the complex modulus of $z$. The equality holds  $z = 0$.
Limit of Function by Convergent Sequences/Corollary
Let $\openint a b$ be an open real interval. Let $f: \openint a b \to \R$ be a real function. Let $l \in \R$. Then: :$(1): \quad \displaystyle \lim_{x \mathop \to a^+} \map f x = l \iff \forall \sequence {x_n} \subseteq \openint a b: \lim_{n \mathop \to \infty} x_n = a \implies \lim_{n \mathop \to \infty} \map f {x_n} = l$ :$(2): \quad \displaystyle \lim_{x \mathop \to b^-} \map f x = l \iff \forall \sequence {x_n} \subseteq \openint a b: \lim_{n \mathop \to \infty} x_n = b \implies \lim_{n \mathop \to \infty} \map f {x_n} = l$ where: :$\displaystyle \lim_{x \mathop \to a^+} \map f x$ denotes the limit of $f$ from the right :$\displaystyle \lim_{x \mathop \to b^-} \map f x$ denotes the limit of $f$ from the left.
Half-Open Real Interval is not Open Set
Let $\R$ be the real number line considered as an Euclidean space. Let $\hointr a b \subset \R$ be a half-open interval of $\R$. Then $\hointr a b$ is not an open set of $\R$. Similarly, the half-open interval $\hointl a b \subset \R$ is not an open set of $\R$.
Empty Set is Open in Metric Space
Let $M = \struct {A, d}$ be a metric space. Then the empty set $\varnothing$ is an open set of $M$.
Metric Space is Open in Itself
Let $M = \struct {A, d}$ be a metric space. Then the set $A$ is an open set of $M$.
Subset of Standard Discrete Metric Space is Open
Let $M = \left({A, d}\right)$ be a standard discrete metric space. Let $S \subseteq A$ be a subset of $A$. Then $S$ is an open set of $M$.
Image of Open Set under Continuous Mapping in Metric Space may not be Open
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 $\left({d_1, d_2}\right)$-continuous mapping from $A_1$ to $A_2$. Let $U \subseteq A_1$ be an open set of $M_1$. Then it is not necessarily the case that $f \left({U}\right)$ is an open set of $M_2$.
Equivalence of Definitions of Adherent Point
Let $T = \struct {S, \tau}$ be a topological space. Let $A \subseteq S$. {{TFAE|def = Adherent Point|view = adherent point of $A$}}
Metric Space is Compact iff Countably Compact
A metric space is compact  it is countably compact.
Countably Compact Metric Space is Sequentially Compact
Let $M$ be a countably compact metric space. Then $M$ is sequentially compact.
Countably Infinite Set in Countably Compact Space has Omega-Accumulation Point
Let $\struct {X, \tau}$ be a countably compact topological space. Let $A \subseteq X$ be countably infinite. Then $A$ has an $\omega$-accumulation point in $X$.
First-Countable Space is Sequentially Compact iff Countably Compact
A first-countable topological space is sequentially compact  it is countably compact.
Accumulation Point of Infinite Sequence in First-Countable Space is Subsequential Limit
Let $\struct {X, \tau}$ be a first-countable topological space. Let $\sequence {x_n}_{n \mathop \in \N}$ be an infinite sequence in $X$. Let $x$ be an accumulation point of $\sequence {x_n}$. Then $x$ is a subsequential limit of $\sequence {x_n}$.
Sequentially Compact Metric Space is Totally Bounded
Let $M = \struct {A, d}$ be a metric space. Let $M$ be sequentially compact. Then $M$ is totally bounded.