Real Function is Continuous at Isolated Point

Let $A \subseteq \R$ be any subset of the real numbers. Let $f: A \to \R$ be a real function. Let $x \in A$ be an isolated point of $A$. Then $f$ is continuous at $x$.


Metric Space Continuity by Epsilon-Delta

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:


Equalizer is Monomorphism

Let $\mathbf C$ be a metacategory. Let $e: E \to C$ be the equalizer of two morphisms $f, g: C \to D$. Then $e$ is a monomorphism.


Metric Space Continuity by Open Ball


Mapping from Standard Discrete Metric on Real Number Line is Continuous

Let $\R$ be the real number line. Let $\left({\R, d_1}\right)$ be the metric space such that $d_1$ be the Euclidean metric on $\R$. Let $\left({\R, d_2}\right)$ be the metric space such that $d_2$ be the standard discrete metric on $\R$. Let $f: \left({\R, d_2}\right) \to \left({\R, d_1}\right)$ be a real function. Then $f$ is $\left({d_2, d_1}\right)$-continuous on $\R$.


Properties of Affine Spaces

Let $\mathcal E$ be an affine space with difference space $V$. Let $0$ denote the zero element of $V$. Then the following hold for all $p,q,r \in \mathcal E$ and all $u,v \in V$: : $(1): \quad p - p = 0$ : $(2): \quad p + 0 = p$ : $(3): \quad p + u = p + v \iff u = v$ : $(4): \quad q - p = r - p \iff q = r$


P-Product Metric on Real Vector Space is Metric/Proof 1

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.


P-Product Metric on Real Vector Space is Metric/Proof 2


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 {{iff}} $f$ 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 {{iff}} 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 $\tau$ {{iff}} $\tau$ 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 $T'$ {{iff}} $V$ 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 $\R$ {{iff}} $S$ 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 $\tau$ {{iff}} 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 $\tau$ {{iff}} 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 $\tau$ {{iff}} 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 {{iff}} 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 {{iff}} $S$ 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 {{iff}} $z = 0$.


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 {{iff}} 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 {{iff}} 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.


Separable Metric Space is Second-Countable

Let $M = \struct {A, d}$ be a metric space. Let $M$ be separable. Then $M$ is second-countable.


Lipschitz Equivalence is Equivalence Relation

Let $A$ be a set. Let $\mathcal D$ be the set of all metrics on $A$. Let $\sim$ be the relation on $\mathcal D$ defined as: :$\forall d_1, d_2 \in \mathcal D: d_1 \sim d_2 \iff d_1$ is Lipschitz equivalent to $d_2$ Then $\sim$ is an equivalence relation.


Sequentially Compact Metric Space is Separable

A sequentially compact metric space is separable.


Sequentially Compact Metric Space is Second-Countable

A sequentially compact metric space is second-countable.


Lipschitz Equivalent Metrics are Topologically Equivalent

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 Lipschitz equivalent. Then $d_1$ and $d_2$ are topologically equivalent.


Compact Metric Space is Complete

A compact metric space is complete.


Pullback as Equalizer

Let $\mathbf C$ be a metacategory with binary products. Suppose that the following diagram commutes (possibly except for the square): ::$\begin{xy}\xymatrix@+1em{  P   \ar@/^/[rrd]^*+{p_1}   \ar@/_/[ddr]_*+{p_2}   \ar[rd]^*+{e} \\ &  A \times B   \ar[r]_*+{\pi_1}   \ar[d]^*+{\pi_2} &  A   \ar[d]^*+{f} \\ &  B   \ar[r]_*+{g} &  C }\end{xy}$ where $A \times B$ is the binary product of $A$ and $B$, and $\pi_1, \pi_2$ are the associated projections. Then the border of the diagram, i.e.: ::$\begin{xy}\xymatrix{  P   \ar[r]^*+{p_1}   \ar[d]_*+{p_2} &  A   \ar[d]^*+{f} \\  B   \ar[r]_*+{g} &  C }\end{xy}$ is a pullback of $f$ and $g$ iff $e: P \to A \times B$ is the equalizer of $f \circ \pi_1$ and $g \circ \pi_2$.


Minkowski's Inequality for Sums/Index 2

Let $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n \ge 0$ be non-negative real numbers. Then: :$\displaystyle \paren {\sum_{k \mathop = 1}^n \paren {a_k + b_k}^2}^{1/2} \le \paren {\sum_{k \mathop = 1}^n a_k^2}^{1/2} + \paren {\sum_{k \mathop = 1}^n b_k^2}^{1/2}$


Banach-Steinhaus Theorem

Let $X$ be a Banach space. Let $Y$ be a normed vector space with norm $\norm {\,\cdot\,}_Y$. Let $\family {T_\alpha: X \to Y}_{\alpha \mathop \in A}$ be an $A$-indexed family of bounded linear transformations from $X$ to $Y$. Suppose that: :$\displaystyle \forall x \in X: \sup_{\alpha \mathop \in A} \norm {T_\alpha x}_Y < \infty$ Then: :$\displaystyle \sup_{\alpha \mathop \in A} \norm {T_\alpha} < \infty$ where $\norm {T_\alpha}$ denotes the norm of the linear transformation $T_\alpha$. {{explain|We may need a definitive explanation for the intuitively understood $< \infty$.}}


Minkowski's Inequality for Sums/Index Greater than 1

Let $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n \in \R_{\ge 0}$ be non-negative real numbers. Let $p \in \R$ be a real number such that $p > 1$. Then: :$\displaystyle \left({\sum_{k \mathop = 1}^n \left({a_k + b_k}\right)^p}\right)^{1/p} \le \left({\sum_{k \mathop = 1}^n a_k^p}\right)^{1/p} + \left({\sum_{k \mathop = 1}^n b_k^p}\right)^{1/p}$


Minkowski's Inequality for Sums/Index Less than 1

Let $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n \in \R_{\ge 0}$ be non-negative real numbers. Let $p \in \R$ be a real number. If $p < 0$, then we require that $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n$ be strictly positive. If $p < 1$, $p \ne 0$, then: :$\displaystyle \left({\sum_{k \mathop = 1}^n \left({a_k + b_k}\right)^p}\right)^{1/p} \ge \left({\sum_{k \mathop = 1}^n a_k^p}\right)^{1/p} + \left({\sum_{k \mathop = 1}^n b_k^p}\right)^{1/p}$


Monotone Real Function is Darboux Integrable

Let $\closedint a b$ be a closed real interval, where $a < b$. Let $f: \closedint a b \to \R$ be a monotone real function. Then $f$ is Darboux integrable over $\closedint a b$.


Topologies Not Always Comparable by Coarseness

Let $S$ be a set with at least $2$ elements. Let $\mathbb T$ be the set of all topologies on $S$. For two topologies $\tau_a, \tau_b \in \mathbb T$, let $\tau_a \le \tau_b$ denote that $\tau_a$ is coarser than $\tau_b$. Then there exist $\tau_1, \tau_2 \in \mathbb T$ such that neither: :$\tau_1 \le \tau_2$ nor: :$\tau_2 \le \tau_1$ That is, there are always topologies on $S$ which are non-comparable.


Pullback of Subset Inclusion

Denote with $\mathbf{Set}$ the category of sets. Let $A, B$ be sets, and let $f: A \to B$ be a mapping. Let $V \subseteq B$ be a subset of $B$. Denote with $i: V \to B$ the inclusion mapping. Let $f^{-1} \left({V}\right) \subseteq A$ be the preimage of $V$ under $f$. Denote with $j: f^{-1} \left({V}\right) \to A$ the inclusion mapping. Denote with $\bar f = f \restriction_{f^{-1} \left({V}\right)}$ the restriction of $f$ to $f^{-1} \left({V}\right)$. Then: ::$\begin{xy}\xymatrix{  f^{-1} \left({V}\right)   \ar[r]^*+{\bar f}   \ar[d]_*+{j} &  V   \ar[d]^*+{i} \\  A   \ar[r]_*+{f} &  B }\end{xy}$ is a pullback diagram in $\mathbf{Set}$.


Poincaré Plane is Abstract Geometry

The Poincaré plane $\struct {\H, L_H}$ is an abstract geometry.


Pullback Lemma

Let $\mathbf C$ be a metacategory. Suppose that the following is a commutative diagram in $\mathbf C$: ::$\begin{xy}\xymatrix@+0.5em{  F   \ar[d]_*+{h''}   \ar[r]^*+{f'} &  E   \ar[d]^*+{h'}   \ar[r]^*+{g'} &  D   \ar[d]^*+{h} \\  A   \ar[r]_*+{f} &  B   \ar[r]_*+{g} &  C }\end{xy}$ Suppose furthermore that the right square is a pullback. Then the left square is a pullback iff the outer rectangle is.


Carathéodory's Theorem (Convex Analysis)

Let $E \subset \R^\ell$. Let $\mathbb x \in \map {\operatorname{co} } E$. {{explain|Definition of $\map {\operatorname{co} } E$}} Then $\mathbf x$ is a convex combination of affinely independent points of $E$.  In particular, $\mathbf x $ is a convex combination of at most $\ell + 1$ points of $E$.


Limit of Sine of X over X/Corollary

:$\displaystyle \lim_{x \mathop \to 0} \frac x {\sin x} = 1$


Dicyclic Group is Group/Corollary

The quaternion group $Q_4$ is a non-abelian group.


Integral of Constant/Definite

:$\displaystyle \int_a^b c \ \ \mathrm dx = c \left({b-a}\right)$.


Primitive of Constant

:$\ds \int c \rd x = c x + C$ where $C$ is an arbitrary constant.


Unbounded Set of Real Numbers is not Compact

Let $\R$ be the set of real numbers considered as a Euclidean space. Let $S \subseteq \R$ be unbounded in $\R$. Then $S$ is not a compact subspace of $\R$.


Continuous Mapping is Sequentially Continuous

Let $X$ and $Y$ be topological spaces. Let $x \in X$. Let $f: X \to Y$ be a mapping that is continuous at $x$. Then $f$ is sequentially continuous at $x$.


Sequential Continuity is Equivalent to Continuity in Metric Space

Let $\left({X, d}\right)$ and $\left({Y, e}\right)$ be metric spaces. Let $f: X \to Y$ be a mapping. Let $x \in X$. Then $f$ is continuous at $x$ {{iff}} $f$ is sequentially continuous at $x$.


Pullback of Commutative Triangle

Let $\mathbf C$ be a metacategory. Suppose that the following is a commutative diagram in $\mathbf C$: ::$\begin{xy}\xymatrix@+1em@L+2px{  A'   \ar[rr]^*{h_\alpha}   \ar[dd]_*{\alpha'} & &  A   \ar[rd]^*{\gamma}   \ar[dd]^(.4)*{\alpha} \\ &  B'   \ar[ld]^*{\beta'}   \ar[rr] |{\hole} ^(.3)*{h_\beta} & &  B   \ar[ld]^*{\beta} \\  C'   \ar[rr]_*{h} & &  C }\end{xy}$ and that the two squares in it are pullback diagrams. Then there is a unique morphism $\gamma': A' \to B'$ making the following commute: ::$\begin{xy}\xymatrix@+1em@L+2px{  A'   \ar[rr]^*{h_\alpha}   \ar[dd]_*{\alpha'}   \ar@{-->}[rd]^*{\gamma'} & &  A   \ar[rd]^*{\gamma}   \ar[dd]^(.4)*{\alpha} \\ &  B'   \ar[ld]^*{\beta'}   \ar[rr] |{\hole} ^(.3)*{h_\beta} & &  B   \ar[ld]^*{\beta} \\  C'   \ar[rr]_*{h} & &  C }\end{xy}$ Furthermore, $\gamma'$ makes the following a pullback: ::$\begin{xy}\xymatrix@+.5em@L+2px{  A'   \ar[r]^*{h_\alpha}   \ar[d]_*{\gamma'} &  A   \ar[d]^*{\gamma} \\  B'   \ar[r]_*{h_\beta} &  B }\end{xy}$


Associative Commutative Idempotent Operation is Distributive over Itself

Let $\left({S, \circ}\right)$ be an algebraic structure, such that: :$(1): \quad \circ$ is associative :$(2): \quad \circ$ is commutative :$(3): \quad \circ$ is idempotent. Then $\circ$ is distributive over itself. That is: :$\forall a, b, c \in S: \left({a \circ b}\right) \circ \left({a \circ c}\right) = a \circ b \circ c = \left({a \circ c}\right) \circ \left({b \circ c}\right)$


Intersection Distributes over Intersection

Set intersection is distributive over itself: :$\forall A, B, C: \paren {A \cap B} \cap \paren {A \cap C} = A \cap B \cap C = \paren {A \cap C} \cap \paren {B \cap C}$ where $A, B, C$ are sets.


Inclusion Mapping is Restriction of Identity

Let $T$ be a set. Let $S \subseteq T$ be a subset of $T$. Let $i_S: S \to T$ be the inclusion mapping on $S$. Then $i_S$ is the restriction of the identity mapping $I_T: T \to T$ on $T$.


Preimage of Subset under Inclusion Mapping

Let $S$ be a set. Let $H \subseteq S$ be a subset of $S$. Let $i_H: H \to S$ be the inclusion mapping on $H$. Let $T \subseteq S$. Then: :$i_H^{-1} \left({T}\right) = T \cap H$ where $i_H^{-1} \left({T}\right)$ is the preimage of $T$ under $i_H$.


Inclusion Mapping is Continuous

Let $T = \struct {S, \tau}$ be a topological space. Let $T_H = \struct {H, \tau_H}$ be a topological subspace of $T$ where $H \subseteq S$. Let $i_H: H \to S$ be the inclusion mapping on $H$. Then $i_H$ is a $\struct {\tau_H, \tau}$-continuous mapping.


Continuity of Composite with Inclusion/Mapping on Inclusion

If $f$ is $\tuple {\tau, \tau'}$-continuous, then $f \circ i$ is $\tuple {\tau_H, \tau'}$-continuous


Continuity of Composite with Inclusion/Inclusion on Mapping

$g$ is $\left({\tau', \tau_H}\right)$-continuous iff $i \circ g$ is $\left({\tau', \tau}\right)$-continuous.


Continuity of Composite with Inclusion/Uniqueness of Induced Topology

The induced topology $\tau_H$ is the ''only'' topology on $H$ satisfying Continuity of Composite with Inclusion: Inclusion on Mapping for all possible $g$.


Products of Open Sets form Local Basis in Product Space

Let $T_1 = \struct{A_1, \tau_1}$ and $T_2 = \struct{A_2, \tau_2}$ be topological spaces. Let $\struct{T, \tau} = T_1 \times T_2$ be the product space of $T_1$ and $T_2$. Let $\tuple{x, y} \in A_1 \times A_2$. Let $W \in \tau$ be an open set of $T$ such that $\tuple{x, y} \in W$. Then: :$\exists U_1 \in \tau_1, U_2 \in \tau_2: \tuple{x, y} \in U_1 \times U_2 \subseteq W$ That is, products of open sets from $T_1$ and $T_2$ form a local basis at $\tuple{x, y}$.


Open Real Intervals are Homeomorphic

Consider the real numbers $\R$ as a metric space under the Euclidean metric. Let $I_1 := \left({a \,.\,.\, b}\right)$ and $I_2 := \left({c \,.\,.\, d}\right)$ be non-empty open real intervals. Then $I_1$ and $I_2$ are homeomorphic.


Half-Open Real Interval is not Closed in Real Number Line

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 not a closed set of $\R$. Similarly, the half-open interval $\left({a \,.\,.\, b}\right] \subset \R$ is not a closed set of $\R$.


Underlying Set of Topological Space is Clopen

Let $T = \left({S, \tau}\right)$ be a topological space. Then the underlying set $S$ of $T$ is both open and closed in $T$.


Empty Set is Closed in Topological Space

Let $T = \struct {S, \tau}$ be a topological space. Then $\O$ is closed in $T$.


Basis induces Local Basis

Let $\left({S, \tau}\right)$ be a topological space. Let $\mathcal B$ be an analytic basis for $\tau$. Let $x \in S$, and denote with $\mathcal B_x$ the set $\left\{{B \in \mathcal B: x \in B}\right\}$. Then $\mathcal B_x$ is a local basis at $x$.


Supremum of Bounded Above Set of Reals is in Closure

Let $\R$ be the real number line under the Euclidean metric. Let $H \subseteq \R$ be a bounded above subset of $\R$ such that $H \ne \varnothing$. Let $u = \map \sup H$ be the supremum of $H$. Then: :$u \in \map \cl H$ where $\map \cl H$ denotes the closure of $H$ in $\R$.


Topological Space is Open Neighborhood of Subset

Let $T = \left({S, \tau}\right)$ be a topological space. Let $H \subseteq S$ be a subset of $S$. Then $S$ is an open neighborhood of $H$.


Open Superset is Open Neighborhood

Let $T = \struct {S, \tau}$ be a topological space. Let $U \in \tau$ be an open set of $T$. Let $H \subseteq U$. Then $U$ is an open neighborhood of $H$.


Category has Products and Equalizers iff Pullbacks and Terminal Object

Let $\mathbf C$ be a metacategory. Then the following are equivalent: :$(1): \mathbf C$ has all finite products and equalizers. :$(2): \mathbf C$ has all pullbacks and a terminal object.


Infimum of Bounded Below Set of Reals is in Closure

Let $\R$ be the real number line with the  usual (Euclidean} metric. Let $H \subseteq \R$ be a bounded below subset of $\R$ such that $H \ne \O$. Let $l = \map \inf H$ be the infimum of $H$. Then: :$l \in \map \cl H$ where $\map \cl H$ denotes the closure of $H$ in $\R$.


Nowhere Dense iff Complement of Closure is Everywhere Dense/Corollary

Let $H$ be a closed set of $T$. Then $H$ is nowhere dense in $T$ {{iff}} $S \setminus H$ is everywhere dense in $T$.


Equivalence of Definitions of Topology Generated by Synthetic Basis

Let $S$ be a set. Let $\BB$ be a synthetic basis on $S$. {{TFAE|def = Topology Generated by Synthetic Basis}}


Union from Synthetic Basis is Topology

Let $\BB$ be a synthetic basis on a set $X$. Let $\displaystyle \tau = \set {\bigcup \AA: \AA \subseteq \BB}$. Then $\tau$ is a topology on $X$.


Product as Limit

Let $\mathbf C$ be a metacategory. Let $C_1, C_2$ be objects of $\mathbf C$. Let their binary product $C_1 \times C_2$ exist in $\mathbf C$. Then $C_1 \times C_2$ is the limit of the diagram $D: 2 \to \mathbf C$ defined by: :$D_0 := C_1, D_1 := C_2$ where $2$ is the discrete category with two objects $0, 1$.


Equalizer as Limit

Let $\mathbf C$ be a metacategory. Let $f_1, f_2: C_1 \to C_2$ be morphisms of $\mathbf C$. Let their equalizer $e: E \to C_1$ exist in $\mathbf C$. Then $\left({E, e}\right)$ is the limit of the diagram $D: \mathbf J \to \mathbf C$ defined by: ::$\begin{xy}\xymatrix@+1em@L+3px{  \mathbf{J}: &  \ast   \ar[r]<2pt>   \ar[r]<-2pt> &  \star }\end{xy}$ ::$\begin{xy}\xymatrix@+1em@L+3px{  D: &  C_1   \ar[r]<2pt>^*+{f_1}   \ar[r]<-2pt>_*+{f_2} &  C_2 }\end{xy}$


Terminal Object as Limit

Let $\mathbf C$ be a metacategory. Let $\mathbf C$ have a terminal object $1$. Then $1$ is the limit of the unique diagram $D: \mathbf 0 \to \mathbf C$, where $\mathbf 0$ is the zero category.


Pullback as Limit

Let $\mathbf C$ be a metacategory. Let $f_1: A \to C$ and $f_2: B \to C$ be morphisms of $\mathbf C$. Let their pullback: ::$\begin{xy}\xymatrix@+1em@L+3px{  P   \ar[r]^*+{p_2}   \ar[d]_*+{p_1} &  A   \ar[d]^*+{f_1} \\  B   \ar[r]_*+{f_2} &  C }\end{xy}$ exist in $\mathbf C$. Then $\left({P, p_1, p_2}\right)$ is the limit of the diagram $D: \mathbf J \to \mathbf C$ defined by: ::$\begin{xy}\xymatrix@+1em@L+3px{  \save[]+<0em,-2em>*{\mathbf{J}:} \restore & &  \cdot   \ar[d] \\ &  \cdot   \ar[r] &  \cdot }\end{xy}$ ::$\begin{xy}\xymatrix@+1em@L+3px{  \save[]+<0em,-2em>*{D:} \restore & &  A   \ar[d]^*+{f_1} \\ &  B   \ar[r]_*+{f_2} &  C }\end{xy}$


Identification Topology is Topology

Let $T_1 = \struct {S_1, \tau_1}$ be a topological space. Let $S_2$ be a set. Let $f: S_1 \to S_2$ be a mapping. Let $\tau_2$ be the identification topology on $S_2$ with respect to $f$ and $\struct {S_1, \tau_1}$. Then $\tau_2$ is a topology on $S_2$.


Identification Mapping is Continuous

Let $T_1 = \struct {S_1, \tau_1}$ be a topological space. Let $S_2$ be a set. Let $f: S_1 \to S_2$ be a mapping. Let $\tau_2$ be the identification topology on $S_2$ with respect to $f$ and $\struct {S_1, \tau_1}$. Then the identification mapping $f$ is continuous.


Existence and Uniqueness of Identification Topology

Let $T_1 = \left({S_1, \tau_1}\right)$ be a topological space. Let $S_2$ be a set. Let $f: S_1 \to S_2$ be a mapping. Let $\tau_2$ be the identification topology on $S_2$ with respect to $f$ and $\left({S_1, \tau_1}\right)$. Then the identification topology on $S_2$ with respect to $f$ and $\left({S_1, \tau_1}\right)$ always exists and is unique.


Identification Topology is Finest Topology for Mapping to be Continuous

Let $T_1 := \struct {S_1, \tau_1}$ be a topological space. Let $S_2$ be a set. Let $f: S_1 \to S_2$ be a mapping. Let $\tau_2$ be the identification topology on $S_2$ with respect to $f$ and $\struct {S_1, \tau_1}$. Let $T_2 := \struct {S_2, \tau_2}$ be the resulting topological space. Then $\tau_2$ is the finest topology on $S_2$ such that $f: T_1 \to T_2$ is continuous.


Identification Topology equals Quotient Topology on Induced Equivalence

Let $T_1 = \struct {S_1, \tau_1}$ be a topological space. Let $S_2$ be a set. Let $f: S_1 \to S_2$ be a mapping. Let $\tau_2$ be the identification topology on $S_2$ with respect to $f$ and $\struct {S_1, \tau_1}$: :$\tau_2 = \set {V \in \powerset {S_2}: f^{-1} \sqbrk V \in \tau_1}$ Let $\RR_f \subseteq S_1 \times S_1$ be the equivalence on $S_1$ induced by $f$: :$\tuple {s_1, s_2} \in \RR_f \iff \map f {s_1} = \map f {s_2}$ Let $\tau_{\RR_f}$ be the quotient topology on $S / \RR_f$ by $q_{\RR_f}$: :$\tau_{\RR_f} := \set {U \subseteq S / \RR_f: q_{\RR_f}^{-1} \sqbrk U \in \tau_1}$ Then: :$\tau_2$ is homeomorphic to $\tau_{\RR_f}$.


Continuity of Composite with Identification Mapping

Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces. Let $f_1: T_1 \to T_2$ be a mapping. Let $S_3$ be a set. Let $p: S_1 \to S_3$ be a mapping. Let $\tau_3$ be the identification topology on $S_3$ with respect to $p$ and $T_1$. Let $T_3 = \struct {S_3, \tau_3}$ be the resulting topological space. Let $f_2: S_3 \to S_2$ be a mapping such that: :$f_1 = f_2 \circ p$ where $f_2 \circ p$ is the composition of $f_2$ with $p$. Then $f_1$ is a continuous mapping {{iff}} $f_2$ is a continuous mapping.


Young's Inequality for Increasing Functions

Let $a_0$ and $b_0$ be strictly positive real numbers. Let $f: \closedint 0 {a_0} \to \closedint 0 {b_0}$ be a strictly increasing bijection. Let $a$ and $b$ be real numbers such that $0 \le a \le a_0$ and $0 \le b \le b_0$. Then: :$\displaystyle ab \le \int_0^a \map f u \rd u + \int_0^b \map {f^{-1}} v \rd v$ where $\displaystyle \int$ denotes the definite integral.


Set is Closed iff Equals Topological Closure

Let $T$ be a topological space. Let $H \subseteq T$. Then $H$ is closed in $T$ {{iff}}: :$H = \map \cl H$


Equivalence of Definitions of Compact Topological Subspace

{{TFAE|def = Compact Subspace}} Let $T = \left({S, \tau}\right)$ be a topological space. Let $T_H = \left({H, \tau_H}\right)$ be a topological subspace of $T$, where $H \subseteq S$.


Covariant Hom Functor is Continuous

Let $\mathbf{Set}$ be the category of sets. Let $\mathbf C$ be a locally small category. Let $C$ be an object of $\mathbf C$, and let $\hom \paren {C, \cdot}: \mathbf C \to \mathbf{Set}$ be the covariant hom functor based at $C$. Then $\hom \paren {C, \cdot}$ is a continuous functor.


Contravariant Hom Functor maps Colimits to Limits

Let $\mathbf{Set}$ be the category of sets. Let $\mathbf C$ be a locally small category. Let $C$ be an object of $\mathbf C$. Let $\map \hom {\cdot, C}: \mathbf C \to \mathbf{Set}$ be the contravariant hom functor based at $C$. Then $\map \hom {\cdot, C}$ maps every colimit to a limit, in that: :$\map \hom { {\varinjlim \,}_j \,D_j, C} \cong {\varprojlim \,}_j \, \map \hom {D_j, C}$ for every diagram $D: \mathbf J \to \mathbf C$.


Composite Mapping is Mapping

Let $S_1, S_2, S_3$ be sets. Let $f: S_1 \to S_2$ and $g: S_2 \to S_3$ be mappings. Then the composite mapping $g \circ f$ is also a mapping.


Normal Space is Regular Space

Let $\struct {S, \tau}$ be a normal space. Then $\struct {S, \tau}$ is also a regular space.


Product of Hausdorff Factor Spaces is Hausdorff/General Result

Let $\SS = \family {\struct {S_\alpha, \tau_\alpha} }$ be an indexed family of topological spaces for $\alpha$ in some indexing set $I$. Let $\displaystyle T = \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\SS$. Let each of $\struct {S_\alpha, \tau_\alpha}$ for $\alpha \in I$ be $T_2$ (Hausdorff) spaces. Then $T$ is a $T_2$ (Hausdorff) space.


Product of Hausdorff Factor Spaces is Hausdorff

Let $T_\alpha = \struct {S_\alpha, \tau_\alpha}$ and $T_\beta = \struct {S_\beta, \tau_\beta}$ be topological spaces. Let $T = T_\alpha \times T_\beta$ be the product space of $T_\alpha$ and $T_\beta$ Let $T_\alpha$ and $T_\beta$ both be $T_2$ (Hausdorff) spaces. Then $T$ is also a $T_2$ (Hausdorff) space.


Domain of Continuous Injection to Hausdorff Space is Hausdorff

Let $T_\alpha = \struct {S_\alpha, \tau_\alpha}$ and $T_\beta = \struct {S_\beta, \tau_\beta}$ be topological spaces. Let $f: S_\alpha \to S_\beta$ be a continuous mapping which is an injection. If $T_\beta$ is a $T_2$ (Hausdorff) space, then $T_\alpha$ is also a $T_2$ (Hausdorff) space.


Quotient Space of Hausdorff Space is not necessarily Hausdorff

Let $T = \struct {S, \tau}$ be a Hausdorff space. Let $\RR \subseteq S \times S$ be an equivalence relation on $S$. Let $T_\RR := \struct {S / \RR, \tau_\RR}$ be the quotient space of $S$ by $\RR$. Then $T_\RR$ is not necessarily also a Hausdorff space.


Topological Product of Compact Spaces/Finite Product

Let $T_1, T_2, \ldots, T_n$ be topological spaces. Let $\displaystyle \prod_{i \mathop = 1}^n T_i$ be the product space of $T_1, T_2, \ldots, T_n$. Then $\displaystyle \prod_{i \mathop = 1}^n T_i$ is compact {{iff}} all of $T_1, T_2, \ldots, T_n$ are compact.


Uniform Continuity on Metric Space does not imply Compactness

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 uniformly continuous mapping on $A_1$. Then $M_1$ does not necessarily have to be a compact metric space.


Uniformly Continuous Function is Continuous/Metric Space

Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_1, d_1}\right)$ be metric spaces. Let the mapping $f: M_1 \to M_2$ be uniformly continuous on $M_1$. Then $f$ is continuous on $M_1$.


Uniformly Continuous Function is Continuous/Real Function

Let $I$ be an interval of $\R$. Let $f: I \to \R$ be a uniformly continuous real function on $I$. Then $f$ is continuous on $I$.


Category of Finite Sets is Cartesian Closed

Let $\mathbf{Finset}$ be the category of finite sets. Then $\mathbf{Finset}$ is Cartesian closed.


Components of Separation are Clopen

Let $T = \left({S, \tau}\right)$ be a topological space. Let $A \mid B$ be a separation of $T$. Then both $A$ and $B$ are clopen in $T$.


Equivalence of Definitions of Connected Topological Space/No Separation iff No Union of Closed Sets

{{TFAE|def = Connected Topological Space}} Let $T = \left({S, \tau}\right)$ be a topological space.


Complements of Components of Two-Component Partition form Partition

Let $S$ be a set with at least two elements. Let $A, B \subseteq S$. Let $\complement_S$ denote the complement relative to $S$. $A \mid B$ is a partition of $S$ {{iff}} $\relcomp S A \mid \relcomp S B$ is a partition of $S$.


Sigma-Algebra Closed under Finite Intersection

Let $A_1, \ldots, A_n \in \Sigma$. Then $\displaystyle \bigcap_{k \mathop = 1}^n A_k \in \Sigma$.


Equivalence of Definitions of Connected Topological Space/No Union of Closed Sets implies No Subsets with Empty Boundary

Let $T = \left({S, \tau}\right)$ be a topological space. Let $T$ have no two disjoint non-empty closed sets whose union is $S$. Then the only subsets of $S$ whose boundary is empty are $S$ and $\varnothing$.


Equivalence of Definitions of Connected Topological Space/No Subsets with Empty Boundary implies No Clopen Sets

Let $T = \left({S, \tau}\right)$ be a topological space. Let $T$ be such that the only subsets of $S$ whose boundary is empty are $S$ and $\varnothing$. Then the only clopen sets of $T$ are $S$ and $\varnothing$.


Equivalence of Definitions of Connected Topological Space/No Clopen Sets implies No Union of Separated Sets

Let $T = \struct {S, \tau}$ be a topological space. Let the only clopen sets of $T$ be $S$ and $\O$. Then there are no two non-empty separated sets of $T$ whose union is $S$.


Equivalence of Definitions of Connected Topological Space/No Union of Separated Sets implies No Continuous Surjection to Discrete Two-Point Space

Let $T = \struct {S, \tau}$ be a topological space. Let $T$ be such that there are no two non-empty separated sets whose union is $S$. Then there exists no continuous surjection from $T$ onto a discrete two-point space.


Equivalence of Definitions of Connected Topological Space/No Continuous Surjection to Discrete Two-Point Space implies No Separation

Let $T = \left({S, \tau}\right)$ be a topological space. Let $T$ be such that there exists no continuous surjection from $T$ onto a discrete two-point space. Then there exist no open sets $A, B \in \tau$ such that $A, B \ne \varnothing$, $A \cup B = S$ and $A \cap B = \varnothing$.


Existence of Vector Space Bases implies Axiom of Choice

The supposition that every vector space has a basis, along with the Zermelo-Fraenkel axioms, implies that the axiom of choice holds.


Category of Posets is Cartesian Closed

Let $\mathbf{OrdSet}$ be the category of ordered sets. Then $\mathbf{OrdSet}$ is Cartesian closed.


Definition:Exponentiation Functor

Let $\mathbf C$ be a Cartesian closed metacategory. Let $A$ be an object of $\mathbf C$. Then '''exponentiation by $A$''', denoted $\left({-}\right)^A: \mathbf C \to \mathbf C$, is the functor defined by: {{begin-axiom}} {{axiom|lc= Object functor:        |m = C^A := C^A        |rc= $C^A$ is the exponential of $C$ by $A$ }} {{axiom|lc= Morphism functor:        |m = f^A := \widetilde{\left({f \circ \epsilon}\right)}        |rc= $f: B \to C$ is a morphism of $\mathbf C$ }} {{end-axiom}} Here $\epsilon: B^A \times A \to B$ denotes the evaluation morphism, and $\widetilde{\left({f \circ \epsilon}\right)}: B^A \to C^A$ is the exponential transpose of $f \circ \epsilon$. That it is in fact a functor is shown on Exponentiation Functor is Functor.


Exponentiation Functor is Functor

Let $\mathbf C$ be a Cartesian closed metacategory. Let $A$ be an object of $\mathbf C$. Let $\left({-}\right)^A: \mathbf C \to \mathbf C$ be the exponentiation functor. Then $\left({-}\right)^A$ is a functor.


Cosine Exponential Formulation

:$\cos z = \dfrac {\map \exp {i z} + \map \exp {-i z} } 2$


Reciprocal of i

Let $i$ be the imaginary unit such that $i^2 = -1$. Then $\dfrac 1 i = -i$.


Squeeze Theorem/Functions/Proof 3

Let $a$ be a point on an open real interval $I$. Also let $f$, $g$ and $h$ be real functions defined at all points of $I$ except for possibly at point $a$. Suppose that: : $\forall x \ne a \in I: g \left({x}\right) \le f \left({x}\right) \le h \left({x}\right)$ : $\displaystyle \lim_{x \mathop \to a} \ g \left({x}\right) = \lim_{x \mathop \to a} \ h \left({x}\right) = L$ Then: : $\displaystyle \lim_{x \mathop \to a} \ f \left({x}\right) = L$


Rational Numbers are not Connected

The set of rational numbers $\Q$ is not a connected topological space.


Closure of Connected Set is Connected

Let $T$ be a topological space. Let $H$ be a connected set of $T$. Let $H^-$ denote the closure of $H$ in $T$.  Then $H^-$ is connected in $T$.


Closed Topologist's Sine Curve is Connected

Let $G$ be the graph of the function $y = \map \sin {\dfrac 1 x}$ for $x > 0$. Let $J$ be the line segment joining the points $\tuple {0, -1}$ and $\tuple {0, 1}$ in $\R^2$. Then $G \cup J$ is connected.


Separated Sets are Clopen in Union

Let $T = \left({S, \tau}\right)$ be a topological space. Let $A$ and $B$ be separated sets in $T$. Let $H = A \cup B$ be given the subspace topology. Then $A$ and $B$ are each both open and closed in $H$.


Compatibility of Atlases is Equivalence Relation

Let $M$ be a topological space. Let $d$ and $k$ be natural numbers. Let $\mathcal A$ denote the set of all $d$-dimensional atlases of class $\mathcal C^k$ on $M$. Define a relation $\sim$ on $\mathcal A$ by putting, for any two $\mathcal C^k$-atlases $\mathcal F$ and $\mathcal G$: :$\mathcal F \sim \mathcal G$ {{iff}} $\mathcal F$ and $\mathcal G$ are $C^k$-compatible. Then $\sim$ is an equivalence relation on $\mathcal A$.


Addition of Real and Imaginary Parts

Let $z_0, z_1 \in \C$ be two complex numbers. Then: :$\map \Re {z_0 + z_1} = \map \Re {z_0} + \map \Re {z_1}$ and: :$\map \Im {z_0 + z_1} = \map \Im {z_0} + \map \Im {z_1}$ Here, $\map \Re {z_0}$ denotes the real part of $z_0$, and $\map \Im {z_0}$ denotes the imaginary part of $z_0$.


Multiplication of Real and Imaginary Parts

Let $w, z \in \C$ be complex numbers. $(1)$ If $w$ is wholly real, then: :$\map \Re {w z} = w \, \map \Re z$ and: :$\map \Im {w z} = w \, \map \Im z$ $(2)$ If $w$ is wholly imaginary, then: :$\map \Re {w z} = -\map \Im w \, \map \Im z$ and: :$\map \Im {w z} = \map \Im w \, \map \Re z$ Here, $\map \Re z$ denotes the real part of $z$, and $\map \Im z$ denotes the imaginary part of $z$.


Rational Numbers are Totally Disconnected

Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$. Then $\struct {\Q, \tau_d}$ is a totally disconnected space.


Rational Numbers are not Discrete Space

The set of rational numbers $\Q$ does not form a discrete space.


Rational Numbers are not Extremally Disconnected

Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$. Then $\struct {\Q, \tau_d}$ does not form an extremally disconnected space.


Real and Imaginary Part Projections are Continuous

Define the real-valued functions $x, y: \C \to \R$ by: :$\forall z \in \C: \map x z = \map \Re z$ :$\forall z \in \C: \map y z = \map \Im z$ Equip $\R$ with the usual Euclidean metric. Equip $\C$ with the usual Euclidean metric. {{refactor|Reconsider the above link so that it goes to an actual definition page.}} Then both $x$ and $y$ are continuous functions.


Continuous Complex Function is Complex Riemann Integrable

Let $\closedint a b$ be a closed real interval. Let $f: \closedint a b \to \C$ be a continuous complex function. Then $f$ is complex Riemann integrable over $\closedint a b$.


Compact Subspace of Metric Space is Sequentially Compact in Itself

Let $M = \struct {A, d}$ be a metric space. Let $C \subseteq M$ be a subspace of $M$ such that $C$ is compact. Then $C$ is sequentially compact in itself.


Sum of Complex Integrals on Adjacent Intervals

Let $\left[{a \,.\,.\, b}\right]$ be a closed real interval. Let $f: \left[{a \,.\,.\, b}\right] \to \C$ be a continuous complex function. Let $c \in \left[{a \,.\,.\, b}\right]$. Then: :$\displaystyle \int_a^c f \left({t}\right) \ \mathrm dt + \int_c^b f \left({t}\right) \ \mathrm dt = \int_a^b f \left({t}\right) \ \mathrm dt$


Equivalence of Definitions of Metric Space Continuity at Point

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 $A_1$. {{TFAE|def = Continuous at Point of Metric Space|continuity at a point}}


Ordering Principle

Let $S$ be a set. Then there exists a total ordering on $S$.


Number Smaller than Lebesgue Number is also Lebesgue Number

Let $M = \left({A, d}\right)$ be a metric space. Let $\epsilon \in \R_{>0}$ be a Lebesgue number for $M$. Let $\epsilon' \in \R_{>0}: \epsilon' < \epsilon$. Then $\epsilon'$ is also a Lebesgue number for $M$.


Open Cover may not have Lebesgue Number

Let $M = \left({A, d}\right)$ be a metric space. Let $\mathcal C$ be an open cover of $M$. Then it may not necessarily be the case that $\mathcal C$ has a Lebesgue number.


Burnside's Theorem

Let $G$ be a finite group. Let the order of $G$ be $p^m q^n$ where: : $p, q$ are prime : $m, n \in \N$ Then $G$ is solvable.


Constant Function is Uniformly Continuous/Metric Space

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 uniformly continuous throughout $A_1$ with respect to $d_1$ and $d_2$.


Constant Function is Uniformly 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 uniformly continuous on $\R$.


Linear Combination of Complex Integrals

Let $\closedint a b$ be a closed real interval. Let $f, g: \closedint a b \to \C$ be complex Riemann integrable functions over $\closedint a b$. Let $\lambda, \mu \in \C$ be complex constants. Then: :$\displaystyle \int_a^b \paren {\lambda \map f t + \mu \map g t} \rd t = \lambda \int_a^b \map f t \rd t + \mu \int_a^b \map g t \rd t$


Modulus Larger than Real Part and Imaginary Part

Let $z \in \C$ be a complex number. Let $\operatorname{Re} \left({z}\right)$ denote the real part of $z$, and $\operatorname{Im} \left({z}\right) $ the imaginary part of $z$. Then:


Pointwise Difference of Simple Functions is Simple Function

Let $\left({X, \Sigma}\right)$ be a measurable space. Let $f,g : X \to \R$ be simple functions. Then $f - g: X \to \R, \left({f - g}\right) \left({x}\right) := f \left({x}\right) - g \left({x}\right)$ is also a simple function.


Modulus and Argument of Complex Exponential

Let $z \in \C$ be a complex number. Let $\hointr a {a + 2 \pi}$ be a half open interval of length $2 \pi$. Let $r \in \hointr 0 {+\infty}$ and $\theta \in \hointr a {a + 2 \pi}$. Then: :$r = \cmod z$ and $\theta = \map \arg z$ {{iff}}: :$z = r e^{i \theta}$ where: :$\cmod z$ denotes the modulus of $z$ :$\map \arg z$ denotes the argument of $z$ :$x \mapsto e^x$ is the complex exponential function. If $z = 0$ or $r = 0$, then $\theta$ may be any number in $\hointr a {a + 2 \pi}$.


Bases of Vector Space have Equal Cardinality

Let $k$ be a division ring. Let $V$ be a vector space over $k$. Let $X$ and $Y$ be bases of $V$. Then $X$ and $Y$ are equinumerous.


Equivalence of Definitions of Symmetric Difference/(1) iff (2)

Let $S$ and $T$ be sets. {{TFAENocat|def = Symmetric Difference|view = symmetric difference $S * T$ between $S$ and $T$}}


Equivalence of Definitions of Symmetric Difference/(1) iff (3)


Equivalence of Definitions of Symmetric Difference/(2) iff (4)


Subset of Countable Set is Countable

A subset of a countable set is countable.


Sine Exponential Formulation

:$\sin z = \dfrac {\map \exp {i z} - \map \exp {-i z} } {2 i}$


Sine and Cosine Exponential Formulation

For any complex number $x$:


Sine of Sum

:$\map \sin {a + b} = \sin a \cos b + \cos a \sin b$


Hyperbolic Sine Function is Odd

:$\map \sinh {-x} = -\sinh x$


Exponential of Sum/Complex Numbers

Let $z_1, z_2 \in \C$ be complex numbers. Let $\exp z$ be the exponential of $z$. Then: :$\map \exp {z_1 + z_2} = \paren {\exp z_1} \paren {\exp z_2}$


Hyperbolic Cosine Function is Even

:$\map \cosh {-x} = \cosh x$


Hyperbolic Tangent Function is Odd

Let $\tanh: \C \to \C$ be the hyperbolic tangent function on the set of complex numbers. Then $\tanh$ is odd: :$\map \tanh {-x} = -\tanh x$


Hyperbolic Sine in terms of Sine

Let $z \in \C$ be a complex number. Then: :$i \sinh z = \map \sin {i z}$ where: :$\sin$ denotes the complex sine :$\sinh$ denotes the hyperbolic sine :$i$ is the imaginary unit: $i^2 = -1$.


Hyperbolic Cosine in terms of Cosine

Let $z \in \C$ be a complex number. Then: :$\cosh z = \map \cos {i z}$ where: :$\cos$ denotes the complex cosine :$\cosh$ denotes the hyperbolic cosine :$i$ is the imaginary unit: $i^2 = -1$.


Hyperbolic Tangent in terms of Tangent

Let $z \in \C$ be a complex number. Then: :$i \tanh z = \map \tan {i z}$ where: :$\tan$ denotes the tangent function :$\tanh$ denotes the hyperbolic tangent :$i$ is the imaginary unit: $i^2 = -1$.


Order-Extension Principle

Let $S$ be a set. Let $\preceq$ be an ordering on $S$. Then there exists a total ordering $\le$ on $S$ such that: :$\forall a, b \in S: \left({a \preceq b \implies a \le b}\right)$


Union of Nest of Orderings is Ordering

Let $S$ be a set. Let $C$ be a nonempty nest of orderings on $S$. Then $\bigcup C$ is an ordering on $S$.


Ordering can be Expanded to compare Additional Pair

Let $\left({S, \preceq}\right)$ be an ordered set. Let $a$ and $b$ be non-comparable elements of $S$. That is, let: :$a \not\preceq b$ and: :$b \not\preceq a$ Let ${\preceq'} = {\preceq} \cup \left\{ {\left({a, b}\right)} \right\}$. Let $\preceq'^+$ be the transitive closure of $\preceq'$. Then: :$\preceq'^+$ is an ordering. $\preceq'^+$ can be defined by letting $p \preceq'^+ q$ {{iff}}: :$p \preceq q$ or :$p \preceq a$ and $b \preceq q$.


Contour Integral is Well-Defined

Let $C$ be a contour defined by a finite sequence $C_1, \ldots, C_n$ of directed smooth curves in the complex plane$\C$. Let $C_i$ be parameterized by the smooth path $\gamma_i: \left[{a_i \,.\,.\, b_i}\right] \to \C$ for all $i \in \left\{ {1, \ldots, n}\right\}$. Let $f: \operatorname{Im} \left({C}\right) \to \C$ be a continuous complex function, where $\operatorname{Im} \left({C}\right)$ denotes the image of $C$. Suppose that $\sigma_i: \left[{c_i \,.\,.\, d_i}\right] \to \C$ is a reparameterization of $C_i$ for all $i \in \left\{ {1, \ldots, n}\right\}$. Then: :$\displaystyle \sum_{i \mathop = 1}^n \int_{a_i}^{b_i} f \left({\gamma_i \left({t}\right) }\right) \gamma_i' \left({t}\right) \rd t = \sum_{i \mathop = 1}^n \int_{c_i}^{d_i} f \left({\sigma_i \left({t}\right) }\right) \sigma_i' \left({t}\right) \rd t$ and all complex Riemann integrals in the equation are defined.


Boolean Prime Ideal Theorem

Let $\left({S, \le}\right)$ be a Boolean algebra. Let $I$ be an ideal in $S$. Let $F$ be a filter on $S$. Let $I \cap F = \varnothing$. Then there exists a prime ideal $P$ in $S$ such that: : $I \subseteq P$ and: : $P \cap F = \varnothing$


Tangent Exponential Formulation

Let $z$ be a complex number. Let $\tan z$ denote the tangent function and $i$ denote the imaginary unit: $i^2 = -1$. Then:


Arcsine Logarithmic Formulation

For any real number $x: -1 \le x \le 1$: :$\arcsin x = \dfrac 1 i \map \ln {i x + \sqrt {1 - x^2} }$ where $\arcsin x$ is the arcsine and $i^2 = -1$.


Arccosine Logarithmic Formulation

For any real number $x: -1 \le x \le 1$: :$\arccos x = \dfrac 1 i \map \ln {x + \sqrt {x^2 - 1} }$ where $\arccos x$ is the arccosine and $i^2 = -1$.


Arctangent Logarithmic Formulation

For any real number $x$: :$\arctan x = \dfrac 1 2 i \, \map \ln {\dfrac {1 - i x} {1 + i x} }$ where $\arctan x$ is the arctangent and $i^2 = -1$.


Two Ring is Boolean Ring

Let $2$ be the two ring. Then $2$ is a Boolean ring.


Idempotent Ring is Commutative

Let $\struct {R, +, \circ}$ be an idempotent ring. Denote with $0_R$ the zero of $R$. Then $\struct {R, +, \circ}$ is a commutative ring.


Idempotent Ring has Characteristic Two

Let $\struct {R, +, \circ}$ be an idempotent non-null ring. Denote with $0_R$ the zero of $R$. Then $\struct {R, +, \circ}$ has characteristic $2$.


Two-Valued Functions form Boolean Ring

Let $S$ be a set, and let $2$ be the two ring. Let $2^S$ be the set of all $2$-valued functions on $S$. Denote with $+$ and $\cdot$ the pointwise operations induced on $2^S$ by $+_2$ and $\times_2$, respectively. Then $\struct {2^S, +, \cdot}$ is a Boolean ring.


Complex Integration by Substitution

Let $\left[{a \,.\,.\, b}\right]$ be a closed real interval. Let $\phi: \left[{a \,.\,.\, b}\right] \to \R$ be a real function which has a derivative on $\left[{a \,.\,.\, b}\right]$. Let $f: A \to \C$ be a continuous complex function, where $A$ is a subset of the image of $\phi$. If $\phi \left({a}\right) \le \phi \left({b}\right)$, then: :$\displaystyle \int_{\phi \left({a}\right)}^{\phi \left({b}\right)} f \left({t}\right) \, \mathrm d t = \int_a^b f \left({\phi \left({u}\right)}\right) \phi' \left({u}\right) \, \mathrm d u$ If $\phi \left({a}\right) > \phi \left({b}\right)$, then: :$\displaystyle \int_{\phi \left({b}\right)}^{\phi \left({a}\right)} f \left({t}\right) \, \mathrm  dt = -\int_a^b f \left({\phi \left({u}\right)}\right) \phi' \left({u}\right) ,\ \mathrm d u$


Continuous Injection of Interval is Strictly Monotone

Let $I$ be a real interval. Let $f: I \to \R$ be an injective continuous real function. Then $f$ is strictly monotone.


Maximal Ideal WRT Filter Complement is Prime in Distributive Lattice

Let $L$ be a distributive lattice. {{explain|Full notational specification needed}} Let $F$ be a filter in $L$. Let $M$ be an ideal in $L$ which is disjoint from $F$ such that no ideal in $L$ larger than $M$ is disjoint from $F$. Then $M$ is a prime ideal.


Prime Ideal in Lattice

Let $\left({L, \le}\right)$ be a lattice. Let $I$ be an ideal in $L$. Then $I$ is a prime ideal {{iff}}: :$\forall a, b \in L: a \wedge b \in I \implies a \in I \text{ or } b \in I$ where $a \wedge b$ denotes $\min \left\{{a, b}\right\}$, the meet of $a$ and $b$.


Inverse of Increasing Bijection need not be Increasing

Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be ordered sets. Let $\phi: S \to T$ be a bijection which is increasing. Then $\phi^{-1}: T \to S$ is not necessarily also increasing.


Order Isomorphic Sets are Equivalent

Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be ordered sets. Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be order isomorphic. Then $S$ and $T$ are equivalent.


Linear Combination of Contour Integrals

Let $C$ be a contour in $\C$. Let $f, g: \Img C \to \C$ be continuous complex functions, where $\Img C$ denotes the image of $C$. Let $\lambda, \mu \in \C$ be complex constants. Then: :$\displaystyle \int_C \paren {\lambda \map f z + \mu \map g z} \rd z = \lambda \int_C \map f z \rd z + \mu \int_C \map g z \rd z$


P-Product Metric Induces Product Topology

Let $M_A = \struct{A, d_A}$ and $M_B = \struct{B, d_B}$ be metric spaces. Let $\tau_A$ and $\tau_B$ be the topologies on $A$ and $B$ induced by $d_A$ and $d_B$, respectively. Let $p \ge 1$ be an extended real number. Let $M = \struct{A \times B, d}$ be the $p$-product of $M_A$ and $M_B$. We have that $M$ is a metric space. Let $\tau$ be the topology on $A \times B$ induced by $d$. Then $\struct{A \times B, \tau}$ is the product space of $\struct{A, \tau_A}$ and $\struct{B, \tau_B}$.


P-Product Metrics are Lipschitz Equivalent

Let $M_A = \left({A, d_A}\right)$ and $M_B = \left({B, d_B}\right)$ be metric spaces. Let $\tau_A$ and $\tau_B$ be the topologies on $A$ and $B$ induced by $d_A$ and $d_B$, respectively. For all extended real numbers $p \ge 1$, let $M_p = \left({A \times B, d_p}\right)$ be the $p$-product of $M_A$ and $M_B$. Then all the metrics $\left\{{d_p: p \ge 1}\right\}$ are Lipschitz equivalent.


Ordered Sum of Tosets is Totally Ordered Set/General Result

Let $S_1, S_2, \ldots, S_n$ all be tosets. Let $T_n$ be the ordered sum of $S_1, S_2, \ldots, S_n$: :$\forall n \in \N_{>0}: T_n = \begin{cases} S_1 & : n = 1 \\ T_{n-1} + S_n & : n > 1 \end{cases}$ Then $T_n$ is a toset.


Supremum of Suprema

Let $\struct {S, \preceq}$ be an ordered set. Let $\mathbb T \subseteq \powerset S$, where $\powerset S$ is the power set of $S$. Suppose all $T \in \mathbb T$ admit a supremum $\sup T$ in $S$. Then: :$\sup \bigcup \mathbb T = \sup {\set {\sup T: T \in \mathbb T} }$ if one of these two quantities exists (in $S$).


Ordered Product of Tosets is Totally Ordered Set/General Result

Let $S_1, S_2, \ldots, S_n$ all be tosets. Let $T_n$ be the ordered product of $S_1, S_2, \ldots, S_n$: :$\forall n \in \N_{>0}: T_n = \begin{cases} S_1 & : n = 1 \\ T_{n-1} \cdot S_n & : n > 1 \end{cases}$ Then $T_n$ is a toset.


Derivative of Complex Composite Function

Let $f: D \to \C$ be a complex-differentiable function, where $D \subseteq \C$ is an open set. Let $g: \operatorname{Im} \left({f}\right) \to \C$ be a complex-differentiable function, where $\operatorname{Im} \left({f}\right)$ denotes the image of $f$. Define $h = f \circ g: D \to C$ as the composite of $f$ and $g$. Then $h$ is complex-differentiable, and its derivative is defined as: :$\forall z \in D: h' \left({z}\right) =  f' \left({g \left({z}\right)}\right) g' \left({z}\right)$


Ring of Idempotents is Idempotent Ring

Let $\struct {R, +, \circ}$ be a commutative ring. Let $\struct {A, \oplus, \circ}$ be its ring of idempotents. Then $\struct {A, \oplus, \circ}$ is an idempotent ring.


Kelley's Theorem

Let $\struct {D, \preceq}$ be a directed set, Let $S$ be a non-empty set. Let $n: D \to S$ be a net in $S$. Then $n$ has a universal subnet.


Contour Integral is Independent of Parameterization

Let $C$ be a contour defined by a finite sequence $C_1, \ldots, C_n$ of directed smooth curves. Let $C_i$ be parameterized by the smooth path $\gamma_i: \left[{a_i \,.\,.\, b_i}\right] \to \C$ for all $i \in \left\{ {1, \ldots, n}\right\}$. Let $f: \operatorname{Im} \left({C}\right) \to \C$ be a continuous complex function, where $\operatorname{Im} \left({C}\right)$ denotes the image of $C$. Suppose that $\sigma_i: \left[{c_i \,.\,.\, d_i}\right] \to \C$ is a reparameterization of $C_i$ for all $i \in \left\{ {1, \ldots, n}\right\}$. Then: :$\displaystyle \int_C f \left({z}\right) \rd z = \sum_{i \mathop = 1}^n \int_{a_i}^{b_i} f \left({\gamma_i \left({t}\right) }\right) \gamma_i' \left({t}\right) \rd t = \sum_{i \mathop = 1}^n \int_{c_i}^{d_i} f \left({\sigma_i \left({t}\right) }\right) \sigma_i' \left({t}\right) \rd t$


Unity is Unity in Ring of Idempotents

Let $\left({R, +, \circ}\right)$ be a commutative and unitary ring whose unity is $1_R$. Let $\left({A, \oplus, \circ}\right)$ be the ring of idempotents of $R$. Then $1_R$ is also a unity for $\left({A, \oplus, \circ}\right)$.


Ring of Idempotents of Commutative and Unitary Ring is Boolean Ring

Let $\struct {R, +, \circ}$ be a commutative and unitary ring. Let $\struct {A, \oplus, \circ}$ be its ring of idempotents. Then $\struct {A, \oplus, \circ}$ is a Boolean ring.


Definition:Boolean Group

Let $\struct {G, \circ}$ be a group. Then $\struct {G, \circ}$ is a '''Boolean group''' {{iff}} all its elements, other than the identity, have order $2$.


Boolean Ring has Proper Zero Divisor

Let $\left({R, +, \circ}\right)$ be a Boolean ring whose zero is $0_R$. Suppose that $R$ has more than two elements. Then $R$ has a proper zero divisor.


Continuous Real Function on Closed Interval is Bijective iff Strictly Monotone

Let $\closedint a b$ and $\closedint c d$ be closed real intervals. Let $f: \closedint c d \to \closedint a b$ be a continuous real function. Let $\map f c, \map f d \in \set {a, b}$. Then $f$ is bijective {{iff}} $f$ is strictly monotone.


Concatenation of Contours is Contour

Let $C$ and $D$ be contours. That is, $C$ is a finite sequence of directed smooth curves $C_1, \ldots, C_n$. Let $C_i$ be parameterized by the smooth path $\gamma_i: \left[{a_i\,.\,.\,b_i}\right] \to \C$ for all $i \in \left\{ {1, \ldots, n}\right\}$. Similarly, $D$ is a finite sequence of directed smooth curves $D_1, \ldots, D_m$. Let $D_i$ be parameterized by the smooth path $\sigma_i: \left[{c_i\,.\,.\,d_i}\right] \to \C$ for all $i \in \left\{ {1, \ldots, m}\right\}$. Suppose $\gamma_n \left({b_n}\right) = \sigma_1 \left({c_1}\right)$. Then the finite sequence: :$C_1, \ldots, C_n, D_1, \ldots, D_m$ defines a contour.


Contour Integral of Concatenation of Contours

Let $C$ and $D$ be contours in $\C$. Suppose that the end point of $C$ is equal to the start point of $D$, so the concatenation $C \cup D$ is defined. Let $f: \operatorname{Im} \left({C \cup D}\right) \to \C$ be a continuous complex function, where $\operatorname{Im} \left({C \cup D}\right)$ denotes the image of $C \cup D$. Then: :$\displaystyle \int_{C \cup D} f \left({z}\right) \ \mathrm dz = \int_C f \left({z}\right) \rd z + \int_D f \left({z}\right) \rd z$


Regular Expression is Accepted by Finite State Machine

Let $R$ be a regular expression. Then there exists a finite state machine $F$ such that its accepted language $L \left({F}\right)$ is exactly $L \left({R}\right)$, the language defined by $R$.


Directed Smooth Curve Relation is Equivalence

Let $\sim$ denote a relation on the set of all smooth paths: $\left\{ {\gamma: I \to \C : \text{$I$ is a closed real interval, $\gamma$ is a smooth path} }\right\}$. Let $\gamma : \left[{ a \,.\,.\, b }\right] \to \C$ and $\sigma : \left[{c \,.\,.\, d}\right] \to \C$ be smooth paths. Define $\sim$ as follows:  :$\gamma \sim \sigma$ {{iff}} there exists a bijective differentiable strictly increasing real function $\phi: \left[{c \,.\,.\, d}\right] \to \left[{a \,.\,.\, b}\right]$ such that $\sigma = \gamma \circ \phi$. Then $\sim$ is an equivalence relation on the set of all smooth paths.


Max Operation Representation on Real Numbers

Let $x, y \in \R$. Then: :$\max \left\{{x, y}\right\} = \frac 1 2 \left({x + y + \left\vert{x - y}\right\vert}\right)$ where $\operatorname{max}$ denotes the max operation.


Reparameterization of Directed Smooth Curve Preserves Image

Let $\left[{a \,.\,.\, b}\right]$ and $\left[{c\,.\,.\, d}\right]$ be closed real intervals. Let $\gamma : \left[{a \,.\,.\, b}\right] \to \C$ be a smooth path. Let $C$ be a directed smooth curve with parameterization $\gamma$. Suppose that $\sigma : \left[{c \,.\,.\, d}\right] \to \C$ is a reparameterization of $C$. Then $\operatorname{Im} \left({\gamma}\right) = \operatorname{Im} \left({\sigma}\right)$, where $\operatorname{Im} \left({\gamma}\right)$ denotes the image of $\gamma$.


Reparameterization of Directed Smooth Curve Maps Endpoints To Endpoints

Let $\R^n$ be a real cartesian space of $n$ dimensions. Let $\left[{a \,.\,.\, b}\right]$ and $\left[{c \,.\,.\, d}\right]$ be closed real intervals. Let $\gamma: \left[{a \,.\,.\, b}\right] \to \R^n$ be a smooth path in $\R^n$. Let $C$ be a directed smooth curve with parameterization $\gamma$. Suppose that $\sigma: \left[{c \,.\,.\, d}\right] \to \C$ is a reparameterization of $C$. Then the start points and end points of $\gamma$ and $\sigma$ are identical: :$\gamma \left({a}\right) = \sigma \left({c}\right)$ :$\gamma \left({b}\right) = \sigma \left({d}\right)$


Product of Ring Negatives

Let $\struct {R, +, \circ}$ be a ring. Then: :$\forall x, y \in \struct {R, +, \circ}: \paren {-x} \circ \paren {-y} = x \circ y$ where $\paren {-x}$ denotes the negative of $x$.


Distance Function for Distinct Elements in Metric Space is Strictly Positive

Let $A$ be a set. Let $d: A \times A \to \R$ be a real-valued function on $A$ with the following properties: {{begin-axiom}} {{axiom|n = M1'        |q = \forall x, y \in A        |m = d \left({x, y}\right) = 0 \iff x = y }} {{axiom|n = M2        |q = \forall x, y, z \in A        |m = d \left({x, y}\right) + d \left({y, z}\right) \ge d \left({x, z}\right) }} {{axiom|n = M3        |q = \forall x, y \in A        |m = d \left({x, y}\right) = d \left({y, x}\right) }} {{end-axiom}} which can be considered as being an alternative formulation of the metric space axioms. Then: :$\forall x, y \in A: x \ne y \implies d \left({x, y}\right) > 0$ which is metric space axiom $(M4)$. Thus $d$ is a distance function, so making $M := \left({A, d}\right)$ a metric space.


Transpose of Upper Triangular Matrix is Lower Triangular

The transpose of an upper triangular matrix is a lower triangular matrix.


Derivative iff Right and Left Derivative

Let $f$ be a real function. Then $f$ is differentiable {{iff}} it has both right- and left-hand derivatives which agree.


Composition of Ring Homomorphisms is Ring Homomorphism

Let: :$\struct {R_1, +_1, \circ_1}$ :$\struct {R_2, +_2, \circ_2}$ :$\struct {R_3, +_3, \circ_3}$ be rings. Let: :$\phi: \struct {R_1, +_1, \circ_1} \to \struct {R_2, +_2, \circ_2}$ :$\psi: \struct {R_2, +_2, \circ_2} \to \struct {R_3, +_3, \circ_3}$ be homomorphisms. Then the composite of $\phi$ and $\psi$ is also a homomorphism.


Composition of Ring Epimorphisms is Ring Epimorphism

Let: :$\struct {R_1, +_1, \circ_1}$ :$\struct {R_2, +_2, \circ_2}$ :$\struct {R_3, +_3, \circ_3}$ be rings. Let: :$\phi: \struct {R_1, +_1, \circ_1} \to \struct {R_2, +_2, \circ_2}$ :$\psi: \struct {R_2, +_2, \circ_2} \to \struct {R_3, +_3, \circ_3}$ be (ring) epimorphisms. Then the composite of $\phi$ and $\psi$ is also a (ring) epimorphism.


Composition of Ring Isomorphisms is Ring Isomorphism

Let: * $\left({R_1, +_1, \circ_1}\right)$ * $\left({R_2, +_2, \circ_2}\right)$ * $\left({R_3, +_3, \circ_3}\right)$ be rings. Let: * $\phi: \left({R_1, +_1, \circ_1}\right) \to \left({R_2, +_2, \circ_2}\right)$ * $\psi: \left({R_2, +_2, \circ_2}\right) \to \left({R_3, +_3, \circ_3}\right)$ be (ring) isomorphisms. Then the composite of $\phi$ and $\psi$ is also a (ring) isomorphism.


Composition of Ring Endomorphisms is Ring Endomorphism

Let $R$ be a set. Let: : $\left({R, +_1, \circ_1}\right)$ : $\left({R, +_2, \circ_2}\right)$ : $\left({R, +_3, \circ_3}\right)$ be rings. Let: : $\phi: \left({R, +_1, \circ_1}\right) \to \left({R, +_2, \circ_2}\right)$ : $\psi: \left({R, +_2, \circ_2}\right) \to \left({R, +_3, \circ_3}\right)$ be (ring) endomorphisms. Then the composite of $\phi$ and $\psi$ is also a (ring) endomorphism.


Composition of Ring Automorphisms is Ring Automorphism

Let $R$ be a set. Let: * $\left({R, +_1, \circ_1}\right)$ * $\left({R, +_2, \circ_2}\right)$ * $\left({R, +_3, \circ_3}\right)$ be rings. Let: * $\phi: \left({R, +_1, \circ_1}\right) \to \left({R, +_2, \circ_2}\right)$ * $\psi: \left({R, +_2, \circ_2}\right) \to \left({R, +_3, \circ_3}\right)$ be (ring) automorphisms. Then the composite of $\phi$ and $\psi$ is also a (ring) automorphism.


Set has Rank

Let $S$ be a set. Then $S$ has a rank.


Quotient Epimorphism is Epimorphism/Group

Let $G$ be a group. Let $N$ be a normal subgroup of $G$. Let $G / N$ be the quotient group of $G$ by $N$. Let $q_N: G \to G / N$ be the quotient epimorphism from $G$ to $G / N$: :$\forall x \in G: \map {q_N} x = x N$ Then $q_N$ is a group epimorphism whose kernel is $N$.


Strictly Increasing Mapping on Well-Ordered Class

Let $ \left({S, \prec}\right)$ be a strictly well-ordered class. Let $\left({T, <}\right)$ be a strictly ordered class. Let $f$ be a mapping from $S$ to $T$. For each $i \in S$ such that $i$ is not maximal in $S$, let: : $f \left({i}\right) < f \left({\operatorname{succ} \left({i}\right)}\right)$ where $\operatorname{succ} \left({i}\right)$ is the immediate successor element of $i$. Let: :$\forall i, j \in S: i \preceq j \implies f \left({i}\right) \le f \left({j}\right)$ Then for each $i, j \in S$ such that $i \prec j$: :$f \left({i}\right) < f \left({j}\right)$ {{MissingLinks|In the above, link to definitions of increasing mapping and strictly increasing mapping, in order to correlate with page title.}}


Non-Maximal Element of Well-Ordered Class has Immediate Successor

Let $\left({C, \le}\right)$ be a well-ordered class. Let $x \in C$. Suppose that $x$ is maximal in $C$. {{mistake|maximal?}} Then $x$ has an immediate successor in $C$.


Set Contained in Smallest Transitive Set

Let $S$ be a set. Then there exists a transitive set $G$ such that: :$S \subseteq G$ and: :if $Q$ is any transitive set such that $S \subseteq Q$, then $G \subseteq Q$.


Transitive Set Contained in Von Neumann Hierarchy Level

Let $G$ be a transitive set. Then for some ordinal $i$, $G \subseteq V_i$.


External Direct Product Closure/General Result

Let $\displaystyle \struct {S, \circ} = \prod_{k \mathop = 1}^n S_k$ be the external direct product of the algebraic structures $\struct {S_1, \circ_1}, \struct {S_2, \circ_2}, \ldots, \struct {S_n, \circ_n}$. Let $\struct {S_1, \circ_1}, \struct {S_2, \circ_2}, \ldots, \struct {S_n, \circ_n}$ all be closed algebraic structures. Then $\struct {S, \circ}$ is also a closed algebraic structure.


External Direct Product of Abelian Groups is Abelian Group/General Result

The external direct product of a finite sequence of abelian groups is itself an abelian group.


Homomorphism of External Direct Products/General Result

Let $n \in \N_{>0}$. Let: : $\displaystyle \left({\mathcal S_n, \circledcirc_n}\right) := \prod_{k \mathop = 1}^n S_k = \left({S_1, \circ_1}\right) \times \left({S_2, \circ_2}\right) \times \cdots \times \left({S_n, \circ_n}\right)$ : $\displaystyle \left({\mathcal T_n, \circledast_n}\right) := \prod_{k \mathop = 1}^n T_k = \left({T_1, \ast_1}\right) \times \left({T_2, \ast_2}\right) \times \cdots \times \left({T_n, \ast_n}\right)$ be external direct products of algebraic structures. Let $\Phi_n: \left({\mathcal S_n, \circledcirc_n}\right) \to \left({\mathcal T_n, \circledast_n}\right)$ be the mapping defined as: :$\Phi_n: \left({s_1, \ldots, s_n}\right) := \begin{cases} \phi_1 \left({s_1}\right) & : n = 1 \\ \left({\phi_1 \left({s_1}\right), \phi_2 \left({s_2}\right)}\right) & : n = 2 \\ \left({\Phi_n \left({s_1, \ldots, s_{n - 1} }\right), \phi_n \left({s_n}\right)}\right) & : n > 2 \\ \end{cases}$ That is: :$\Phi_n: \left({s_1, \ldots, s_n}\right) := \left({\phi_1 \left({s_1}\right), \phi_2 \left({s_2}\right), \ldots, \phi_n \left({s_n}\right)}\right)$ Let $\phi_k: \left({S_k, \circ_k}\right) \to \left({T_k, \ast_k}\right)$ be a homomorphism for each $k \in \left\{{1, 2, \ldots, n}\right\}$. Then $\Phi_n$ is a homomorphism from $\left({\mathcal S_n, \circledcirc_n}\right)$ to $\left({\mathcal T_n, \circledast_n}\right)$.


Cesàro Mean

Let $\sequence {a_n}$ be a sequence of complex numbers. Suppose that $\sequence {a_n}$ converges to $\ell$ in $\C$: :$\displaystyle \lim_{n \mathop \to \infty} a_n = \ell$ Then also: :$\displaystyle \lim_{n \mathop \to \infty} \frac {a_1 + \dotsb + a_n} n = \ell$


Characterization of Minimal Element

Let $C$ be a class. Let $\prec$ be a relation on $C$. Let $B$ be a subclass of $C$. Let $x \in B$. Let $S_x = \left\{ {y \in C: y \prec x \text{ and } y \ne x}\right\}$ be the initial segment of $x$ in $C$. Then $x$ is a minimal element of $B$ {{iff}} $B \cap S_x = \varnothing$.


Non-Empty Class has Element of Least Rank

Let $C$ be a class. Let $C \ne \varnothing$. Then $C$ has an element of least rank. That is: :$\exists x \in C: \forall y \in C: \operatorname{rank} \left({x}\right) \le \operatorname {rank}\left({y}\right)$ where $\operatorname{rank}\left({x}\right)$ is the rank of $x$.


Reversed Directed Smooth Curve is Directed Smooth Curve

Let $C$ be a directed smooth curve in $\C$. Let $C$ be parameterized by the smooth path $\gamma: \left[{a \,.\,.\, b}\right] \to \C$. Define $\psi: \left[{a \,.\,.\, b}\right] \to \left[{a \,.\,.\, b}\right]$ by $\psi \left({t}\right) = a + b - t$. Define $\rho: \left[{a \,.\,.\, b}\right] \to \C$ by $\rho = \gamma \circ \psi$. Then $\rho$ is a smooth path which parameterizes a directed smooth curve $-C$. The directed smooth curve $-C$ is independent of the parameterization $\gamma$.


Reversed Contour is Contour

Let $\R^n$ be a real cartesian space of $n$ dimensions. Let $C$ be a contour in $\R^n$ that is defined as a concatenation of a finite sequence $C_1, \ldots, C_n$ of directed smooth curves in $\R^n$. Then the finite sequence of reversed directed smooth curves: :$-C_n, -C_{n - 1}, \ldots, -C_1$ defines a contour that is independent of the parameterization of $C_1, \ldots, C_n$.


Contour Integral along Reversed Contour

Let $C$ be a contour. Let $f: \Img C \to \C$ be a continuous complex functions, where $\Img C$ denotes the image of $C$. Then the contour integral of $f$ along the reversed contour $-C$ is: :$\displaystyle \int_{-C} \map f z \rd z = -\int_C \map f z \rd z$


Well-Founded Relation is Strongly Well-Founded/Lemma

Let $A$ be a non-empty class. Let $\prec$ be a foundational relation on $A$. Then $A$ has a $\prec$-minimal element.


Alternative Differentiability Condition

Let $\mathbb K$ be either $\R$ or $\C$. Let $f: D \to \mathbb K$ be a continuous mapping, where $D \subseteq \mathbb K$ is an open set. Let $z \in \mathbb K$. Then $f$ is differentiable at $z$ {{iff}} there exist $\alpha \in \mathbb K$ and $r \in \R_{>0}$ such that for all $h \in \map {B_r} 0 \setminus \set 0$: :$\map f {z + h} = \map f z + h \paren {\alpha + \map \epsilon h}$ where $\map {B_r} 0$ denotes an open ball of $0$, and $\epsilon: \map {B_r} 0 \setminus \set 0 \to \mathbb K$ is a continuous mapping with $\displaystyle \lim_{h \mathop \to 0} \map \epsilon h = 0$. If the conditions are true, then $\alpha = \map {f'} z$.


Von Neumann Hierarchy is Cumulative

For any two ordinals $x$ and $y$, If $x < y$ then $V(x) \subsetneqq V(y)$.


Odd Number Theorem/Corollary

A recurrence relation for the square numbers is: :$S_n = S_{n - 1} + 2 n - 1$


Effect of Sequence of Elementary Row Operations on Determinant

Let $\hat o_1, \ldots, \hat o_m$ be a finite sequence of elementary row operations.  Here, $\hat o_i$ denotes an elementary row operation on a square matrix of order $n$ over a commutative ring with unity $\struct {R, +, \circ}$. Here, $i \in \set {1, \ldots, m}$. Then there exists $c \in R$ such that for all square matrices of order $n$ $\mathbf A$ over $R$: :$\map \det {\mathbf A} = c \map \det {\mathbf A'}$ where $\mathbf A'$ is the square matrix of order $n$ that results from applying the elementary row operations $\hat o_1, \ldots, \hat o_m$ on $\mathbf A$.


Alternative Differentiability Condition/Proof 1

Let $\mathbb K$ be either $\R$ or $\C$. Let $f: D \to \mathbb K$ be a continuous mapping, where $D \subseteq \mathbb K$ is an open set. Let $z \in \mathbb K$. Then: : $f$ is differentiable at $z$ {{iff}}: : there exist $\alpha \in \mathbb K$ and $r \in \R_{>0}$ such that for all $h \in B_r \left({0}\right) \setminus \left\{ {0}\right\}$: ::$f \left({z + h}\right) = f \left({z}\right) + h \left({\alpha + \epsilon \left({h}\right) }\right)$ where: : $B_r \left({0}\right)$ denotes an open ball of $0$ : $\epsilon: B_r \left({0}\right) \setminus \left\{ {0}\right\} \to \mathbb K$ is a continuous mapping with $\displaystyle \lim_{h \to 0} \ \epsilon \left({h}\right) = 0$. If the conditions are true, then $\alpha = f' \left({z}\right)$.


Alternative Differentiability Condition/Proof 2

Let $\mathbb K$ be either $\R$ or $\C$. Let $f: D \to \mathbb K$ be a continuous mapping, where $D \subseteq \mathbb K$ is an open set. Let $z \in \mathbb K$. Then $f$ is differentiable at $z$ {{iff}} there exist $\alpha \in \mathbb K$ and $r \in \R_{>0}$ such that for all $h \in B_r \left({0}\right) \setminus \left\{ {0}\right\}$: :$f \left({z + h}\right) = f \left({z}\right) + h \left({\alpha + \epsilon \left({h}\right) }\right)$ where $B_r \left({0}\right)$ denotes an open ball of $0$, and $\epsilon: B_r \left({0}\right) \setminus \left\{ {0}\right\} \to \mathbb K$ is a continuous mapping with $\displaystyle \lim_{h \to 0} \ \epsilon \left({h}\right) = 0$. If the conditions are true, then $\alpha = f' \left({z}\right)$.


Every Filter has Limit Point implies Every Ultrafilter Converges

Let $T = \left({S, \tau}\right)$ be a topological space. Let each filter on $S$ have a limit point in $S$. Then each ultrafilter on $S$ converges to a point in $S$.


Every Ultrafilter Converges implies Every Filter has Limit Point

Let $T = \left({S, \tau}\right)$ be a topological space. Let every ultrafilter on $S$ be convergent. Then every filter on $S$ has a limit point.


Left Distributive and Commutative implies Distributive

Let $\struct {S, \circ, *}$ be an algebraic structure. Let the operation $\circ$ be left distributive over the operation $*$. Let $\circ$ be commutative. Then $\circ$ is distributive over $*$.


Filter on Product Space Converges iff Projections Converge

Let $\family{X_i}_{i \mathop \in I}$ be an indexed family of non-empty topological spaces where $I$ is an arbitrary index set. Let $\displaystyle X := \prod_{i \mathop \in I} X_i$ be the corresponding product space. Let $\pr_i: X \to X_i$ denote the projection from $X$ onto $X_i$. Let $\FF \subset \powerset X$ be a filter on $X$. Then $\FF$ converges {{iff}} for all $i \in I$ the image filter $\map {\pr_i} \FF$ converges.


Tychonoff's Theorem for Hausdorff Spaces

Let $\family {X_i}_{i \mathop \in I}$ be an indexed family of non-empty Hausdorff spaces, where $I$ is an arbitrary index set. Let $\displaystyle X = \prod_{i \mathop \in I} X_i$ be the corresponding product space. Then $X$ is compact {{iff}} each $X_i$ is compact. == Proof == First assume that $X$ is compact.  From Projection from Product Topology is Continuous, the projections $\pr_i : X \to X_i$ are continuous. From Continuous Image of Compact Space is Compact, it follows that the $X_i$ are compact. Assume now that each $X_i$ is compact. By Equivalent Definitions of Compactness it is enough to show that every ultrafilter on $X$ converges. Thus let $\FF$ be an ultrafilter on $X$. From Image of Ultrafilter is Ultrafilter, for each $i \in I$, the image filter $\map {\pr_i} \FF$ is an ultrafilter on $X_i$. Each $X_i$ is compact by assumption. So by Equivalent Definitions of Compactness, each $\map {\pr_i} \FF$ converges. From Filter on Product of Hausdorff Spaces Converges iff Projections Converge, $\FF$ converges. {{explain|Find the appropriate link for "converges" and explain why this proves the result.}} {{qed}}


Filter on Product of Hausdorff Spaces Converges iff Projections Converge

Let $\family{X_i}_{i \mathop \in I}$ be an indexed family of non-empty Hausdorff spaces where $I$ is an arbitrary index set. Let $\displaystyle X := \prod_{i \mathop \in I} X_i$ be the corresponding product space. Let $\pr_i: X \to X_i$ denote the projection from $X$ onto $X_i$. Let $\FF \subset \powerset X$ be a filter on $X$. Then $\FF$ converges {{iff}} for each $i \in I$ the image filter $\map {\pr_i} \FF$ converges.


Tychonoff's Theorem/General Case

Let $\family {X_i}_{i \mathop \in I}$ be an indexed family of non-empty topological spaces, where $I$ is an arbitrary index set. Let $\displaystyle X = \prod_{i \mathop \in I} X_i$ be the corresponding product space. Then $X$ is compact {{iff}} each $X_i$ is compact.


Positive Real has Real Square Root

Let $x \in \R_{>0}$ be a (strictly) positive real number. Then: :$\exists y \in \R: x = y^2$


Join is Idempotent

Let $\struct {S, \vee, \preceq}$ be a join semilattice. Then $\vee$ is idempotent.


Supremum of Singleton

Let $\struct {S, \preceq}$ be an ordered set. Then for all $a \in S$: :$\sup \set a = a$ where $\sup$ denotes supremum.


Infimum of Singleton

Let $\struct {S, \preceq}$ be an ordered set. Then for all $a \in S$: :$\inf \set a = a$ where $\inf$ denotes infimum.


Meet is Idempotent

Let $\struct {S, \wedge, \preceq}$ be a meet semilattice. Then $\wedge$ is idempotent.


GCD from Prime Decomposition

Let $a, b \in \Z$. From Expression for Integers as Powers of Same Primes, let: {{begin-eqn}} {{eqn | l = a       | r = p_1^{k_1} p_2^{k_2} \ldots p_r^{k_r} }} {{eqn | l = b       | r = p_1^{l_1} p_2^{l_2} \ldots p_r^{l_r} }} {{eqn | ll= \forall i \in \set {1, 2, \dotsc, r}:       | l = p_i       | o = \divides       | r = a       | c =  }} {{eqn | lo= \lor       | l = p_i       | o = \divides       | r = b }} {{end-eqn}} That is, the primes given in these prime decompositions may be divisors of ''either'' of the numbers $a$ or $b$. Then: :$\gcd \set {a, b} = p_1^{\min \set {k_1, l_1} } p_2^{\min \set {k_2, l_2} } \ldots p_r^{\min \set {k_r, l_r} }$ where $\gcd \set {a, b}$ denotes the greatest common divisor of $a$ and $b$.


LCM from Prime Decomposition

Let $a, b \in \Z$. From Expression for Integers as Powers of Same Primes, let: {{begin-eqn}} {{eqn | l = a       | r = p_1^{k_1} p_2^{k_2} \ldots p_r^{k_r} }} {{eqn | l = b       | r = p_1^{l_1} p_2^{l_2} \ldots p_r^{l_r} }} {{eqn | ll= \forall i \in \set {1, 2, \dotsc, r}:       | l = p_i       | o = \divides       | r = a       | c =  }} {{eqn | lo= \lor       | l = p_i       | o = \divides       | r = b }} {{end-eqn}} That is, the primes given in these prime decompositions may be divisors of ''either'' of the numbers $a$ or $b$. Then: :$\lcm \set {a, b} = p_1^{\max \set {k_1, l_1} } p_2^{\max \set {k_2, l_2} } \ldots p_r^{\max \set {k_r, l_r} }$ where $\lcm \set {a, b}$ denotes the lowest common multiple of $a$ and $b$.


Product of Positive Element and Element Greater than One

Let $\left({R, +, \circ, \le}\right)$ be an ordered ring with unity $1_R$ and zero $0_R$. Let $x,y \in R$. Suppose that $x > 0_R$ and $y > 1_R$. Then $x \circ y > x$ and $y \circ x > x$.


Strictly Positive Power of Strictly Positive Element Greater than One Succeeds Element

Let $\left({R, +, \circ, \le}\right)$ be an ordered ring with unity. Let $x \in R$ with $x > 1$ and $x > 0$. Let $n \in \N_{>0}$. Then: : $\circ^n x \ge x$


Directed Set has Strict Successors iff Unbounded Above

Let $\left({S, \le}\right)$ be a directed set. Then every element of $S$ has a strict successor in $S$ iff $S$ has no upper bound in $S$.


Complex Numbers are Uncountable

The set of complex numbers $\C$ is uncountably infinite.


Infimum of Infima

Let $\left({S, \preceq}\right)$ be an ordered set. Let $\mathbb T$ be a collection of subsets of $S$. Suppose all $T \in \mathbb T$ admit an infimum $\inf T$ in $S$. Then: :$\inf \bigcup \mathbb T = \inf \left\{{\inf T: T \in \mathbb T}\right\}$ as soon as one of these two quantities exists.


Join Semilattice is Ordered Structure

Let $\left({S, \vee, \preceq}\right)$ be a join semilattice. Then $\left({S, \vee, \preceq}\right)$ is an ordered structure. That is, $\preceq$ is compatible with $\vee$.


Meet Semilattice is Ordered Structure

Let $\left({S, \wedge, \preceq}\right)$ be a meet semilattice. Then $\left({S, \wedge, \preceq}\right)$ is an ordered structure.


Join Semilattice is Semilattice

Let $\struct {S, \vee, \preceq}$ be a join semilattice. Then $\struct {S, \vee}$ is a semilattice.


Meet Semilattice is Semilattice

Let $\struct {S, \wedge, \preceq}$ be a meet semilattice. Then $\struct {S, \wedge}$ is a semilattice.


Semilattice Induces Ordering

Let $\struct {S, \circ}$ be a semilattice. Let $\preceq$ be the relation on $S$ defined by, for all $a, b \in S$: :$a \preceq b$ {{iff}} $a \circ b = b$ Then $\preceq$ is an ordering.


Elementary Row Operations Commute with Matrix Multiplication

Let $\struct {R, +, \circ}$ be a commutative ring. Let $\mathbf A = \sqbrk a_{m n}$ be an $m \times n$ matrix over $R$. Let $\mathbf B = \sqbrk b_{n p}$ be an $n \times p$ matrix over $R$. Let $\hat o_1, \ldots, \hat o_{\hat n}$ be a finite sequence of elementary row operations that can be performed on a matrix over $R$ with $m$ rows. Let $\mathbf A'$ denote the $m \times n$-matrix that results from using $\hat o_1, \ldots, \hat o_{\hat n}$ on $\mathbf A$. Let $\mathbf C = \mathbf A \mathbf B$ be the matrix product of $\mathbf A$ and $\mathbf B$. Let $\mathbf C'$ denote the $m \times p$-matrix that results from using $\hat o_1, \ldots, \hat o_{\hat n}$ on $\mathbf C$. Then: :$\mathbf C' = \mathbf A' \mathbf B$


Hypothetical Syllogism/Formulation 1

:$p \implies q, q \implies r \vdash p \implies r$


Hypothetical Syllogism/Formulation 2/Proof 1

: $p \implies q, q \implies r, p \vdash r$


Hypothetical Syllogism/Formulation 2/Proof 3


Union of Relations Compatible with Operation is Compatible

Let $\left({S, \circ}\right)$ be a closed algebraic structure. Let $\mathcal F$ be a family of relations on $S$. Let each element of $\mathcal F$ be compatible with $\circ$. Let $\mathcal Q = \bigcup \mathcal F$. Then $Q$ is a relation compatible with $\circ$.


Complement of Relation Compatible with Group is Compatible

Let $\left({G, \circ}\right)$ be a group. Let $\mathcal R$ be a relation on $G$. Let $\mathcal R$ be compatible with $\circ$. Let $\mathcal Q = \complement_{G \times G} \mathcal R$, so that: :$\forall a, b \in G: a \mathop {\mathcal Q} b \iff \neg \left({a \mathop {\mathcal R} b}\right)$ Then $\mathcal Q$ is a relation compatible with $\circ$.


Relation Compatible with Group Operation is Strongly Compatible

Let $\left({S,\circ}\right)$ be a group. Let $\mathcal R$ be an endorelation on $S$ compatible with $\circ$. Let $x, y, z \in S$. Then $\mathcal R$ is strongly compatible with $\circ$. That is, the following equivalences hold: :$x \mathrel{\mathcal R} y \iff x \circ z \mathrel{\mathcal R} y \circ z$ :$x \mathrel{\mathcal R} y \iff z \circ x \mathrel{\mathcal R} z \circ y$


Intersection of Relations Compatible with Operation is Compatible

Let $\struct {S, \circ}$ be a closed algebraic structure. Let $\mathscr F$ be a indexed family of relations on $S$. Suppose that each element of $\mathscr F$ is compatible with $\circ$. Let $\QQ = \bigcap \mathscr F$ be the intersection of $\mathscr F$. Then $\QQ$ is a relation compatible with $\circ$.


Inverse of Relation Compatible with Operation is Compatible

Let $\left({S, \circ}\right)$ be a closed algebraic structure. Let $\mathcal R$ be a relation on $S$ which is compatible with $\circ$. Let $\mathcal Q$ be the inverse relation of $\mathcal R$. Then $\mathcal Q$ is compatible with $\circ$.