Inverse for Complex Multiplication

Each element $z = x + i y$ of the set of non-zero complex numbers $\C_{\ne 0}$ has an inverse element $\dfrac 1 z$ under the operation of complex multiplication: :$\forall z \in \C_{\ne 0}: \exists \dfrac 1 z \in \C_{\ne 0}: z \times \dfrac 1 z = 1 + 0 i = \dfrac 1 z \times z$ This inverse can be expressed as: :$\dfrac 1 z = \dfrac {x - i y} {x^2 + y^2} = \dfrac {\overline z} {z \overline z}$ where $\overline z$ is the complex conjugate of $z$.


Additive Group of Integers is Normal Subgroup of Complex

Let $\struct {\Z, +}$ be the additive group of integers. Let $\struct {\C, +}$ be the additive group of complex numbers. Then $\struct {\Z, +}$ is a normal subgroup of $\struct {\C, +}$.


Rational Numbers form Subfield of Complex Numbers

Let $\struct {\Q, +, \times}$ denote the field of rational numbers. Let $\struct {\C, +, \times}$ denote the field of complex numbers. $\struct {\Q, +, \times}$ is a subfield of $\struct {\C, +, \times}$.


Subfield of Subfield is Subfield

Let $R$ be a ring with unity. Let $K_1, K_2$ be fields, such that: : $K_1$ is a subfield of $R$ : $K_2$ is a subfield of $K_1$ Then $K_2$ is a subfield of $R$.


Characterisation of Jacobson Radical

Let $A$ be a commutative ring with unity. Let $A^\times$ be the group of units of $A$. Let $\map {\operatorname {Jac} } A$ be the Jacobson radical of $A$. Then: :$\map {\operatorname {Jac} } A = \set {a \in A: 1_A - a x \in A^\times \text{ for all } x \in A}$ where $1_A$ is the unity of $A$.


Prime Ideal iff Quotient Ring is Integral Domain

Let $\struct {R, +, \circ}$ be a commutative ring with unity. Let $J$ be an ideal of $R$. Then $J$ is a prime ideal of $R$ {{iff}} the quotient ring $R / J$ is an integral domain.


Maximal Ideal of Commutative and Unitary Ring is Prime Ideal

Let $R$ be a commutative ring with unity. Let $M$ be a maximal ideal of $R$. Then $M$ is a prime ideal of $R$.


Radical Ideal iff Quotient Ring is Reduced

Let $\left({R, +, \circ}\right)$ be a commutative ring with unity. Let $J$ be an ideal of $R$. Then $J$ is a radical ideal {{iff}} the quotient ring $R / J$ is a reduced ring.


Properties of Degree

Let $\struct {R, +, \circ}$ be a ring with unity whose zero is $0_R$. Let $R \sqbrk X$ denote the ring of polynomial forms over $R$ in the indeterminate $X$. For $f \in R \sqbrk X$ let $\map \deg f$ denote the degree of $f$. Then the following hold:


Set of Monomials is Closed Under Multiplication

Let $M$ be the set of all monomials on the set $\set {X_j: j \in J}$, with multiplication $\circ$ defined by: :$\ds \paren {\prod_{j \mathop \in J} X_j^{k_j} } \circ \paren {\prod_{j \mathop \in J} X_j^{k_j'} } = \paren {\prod_{j \mathop \in J} X_j^{k_j + k_j'} }$ Then $M$ is closed under $\circ$.


Polynomial Addition is Commutative

Addition of polynomials is commutative.


Multiplication of Polynomials is Associative

Multiplication of polynomials is associative.


Polynomials Contain Multiplicative Identity

The set of polynomials has a multiplicative identity.


Multiplication of Polynomials is Commutative

Multiplication of polynomials is commutative.


Multiplication of Polynomials Distributes over Addition

Multiplication of polynomials is left- and right- distributive over addition.


Distributivity is Preserved in Induced Structure

Let $\struct {T, \oplus, \otimes}$ be an algebraic structure, and let $S$ be a set. Let $T^S$ denote the set of all mappings from $S$ to $T$. Let $\struct {T^S, \oplus}$ be the structure on $T^S$ induced by $\oplus$. Let $\struct {T^S, \otimes}$ be the structure on $T^S$ induced by $\otimes$. If $\otimes$ is distributive over $\oplus$, then the pointwise operation induced on $T^S$ by $\otimes$ is distributive over the operation induced by $\oplus$.


Dirichlet Series Absolute Convergence Lemma

Let $\displaystyle \map f s = \sum_{n \mathop = 1}^\infty \frac {a_n} {n^s}$ be a Dirichlet series. Suppose that $f$ converges absolutely at $s_0 = \sigma_0 + i t_0 \in \C$. Then $f$ converges absolutely at all points $s = \sigma + i t \in \C$ with $\sigma \ge \sigma_0$.


Existence of Abscissa of Absolute Convergence

Let $\displaystyle f \left({s}\right) = \sum_{n \mathop = 1}^\infty a_n n^{-s}$ be a Dirichlet series. Let the series $\displaystyle \sum_{n \mathop = 1}^\infty \left\vert { a_n n^{-s} } \right\vert$ not converge for all $s \in \C$, or diverge for all $s \in \C$. Then there exists a real number $\sigma_a$ such that $f \left({s}\right)$ converges absolutely for all $s = \sigma + it$ with $\sigma > \sigma_a$, and does not converge absolutely for all $s$ with $\sigma < \sigma_a$. We call $\sigma_a$ the '''abscissa of absolute convergence''' of the Dirichlet series.


Dirichlet Series Convergence Lemma

Let $\displaystyle f \left({s}\right) = \sum_{n \mathop = 1}^\infty \frac {a_n} {n^s}$ be a Dirichlet series. Let $f \left({s}\right)$ converge at $s_0 = \sigma_0 + i t_0$. Then $f \left({s}\right)$ converge for all $s = \sigma + i t$ where $\sigma > \sigma_0$.


Existence of Abscissa of Convergence/General

Let $s = \sigma + i t$ Let $\displaystyle \map f s = \sum_{n \mathop = 1}^\infty a_n e^{-\lambda_n s}$ be a general Dirichlet series. Then there exists a extended real number, $\sigma_0$, such that :$(1): \quad$ For $\sigma < \sigma_0$, $\map f s$ diverges :$(2): \quad$ For $\sigma > \sigma_0$, $\map f s$ converges.


Difference of Abscissae of Convergence

Let $\displaystyle \map f s = \sum_{n \mathop = 1}^\infty a_n n^{-s}$ be a Dirichlet series. Suppose that $\map f s$ has finite Abscissa of Convergence $\sigma_c$. Then the Abscissa of Absolute Convergence $\sigma_a$ is finite, and: :$0 \le \sigma_a - \sigma_c \le 1$


Completion Theorem (Metric Space)

Let $M = \struct {A, d}$ be a metric space. Then there exists a completion $\tilde M = \struct {\tilde A, \tilde d}$ of $\struct {A, d}$. Moreover, this completion is unique up to isometry. That is, if $\struct {\hat A, \hat d}$ is another completion of $\struct {A, d}$, then there is a bijection $\tau: \tilde A \leftrightarrow \hat A$ such that: :$(1): \quad \tau$ restricts to the identity on $x$: :::: $\forall x \in A: \map \tau x = x$ :$(2): \quad \tau$ preserves metrics: :::: $\forall x_1, x_2 \in A : \map {\hat d} {\map \tau {x_1}, \map \tau {x_2} } = \map {\tilde d} {x_1, x_2}$


Completeness Criterion (Metric Spaces)

Let $M = \struct {S, d}$ be a metric space. Let $A \subseteq S$ be a dense subset. Suppose that every Cauchy sequence in $A$ converges in $M$. Then $M$ is complete.


Natural Number Addition is Closed

The operation of addition on the set of natural numbers $\N$ is closed: :$\forall x, y \in \N: x + y \in \N$


Properties of Ordered Field

Let $\struct {F, +, \cdot}$ be an ordered field with unity $1$, zero $0$. Denote the strict order by $<$ and the weak order by $\le$. Let $\Char F$ denote the characteristic of $F$. Then the following hold for all $x, y, z \in k$: :$(1): \quad x < 0 \iff -x > 0$ :$(2): \quad x > y \iff x-y > 0$ :$(3): \quad x < y \iff -x > -y$ :$(4): \quad \paren {z < 0} \land \paren {x < y} \implies x z > y z$ :$(5): \quad x \ne 0 \implies x^2 > 0$ :$(6): \quad 1 > 0$ :$(7): \quad \Char k = 0$ :$(8): \quad x > y > 0 \iff y^{-1} > x^{-1} > 0$


Characterisation of Ordered Fields

Let $\struct {k, +, \cdot}$ be a field with unity $1$ and zero $0$. Then the following are equivalent: :$(1): \quad$ There exists a total ordering $\le$ on $k$ such that $\struct {k, \le}$ is an ordered field :$(2): \quad$ $-1$ cannot be written as a sum of squares of elements of $k$ :$(3): \quad$ $0$ cannot be written as a non-empty sum of squares of non-zero elements of $k$


Hahn-Banach Theorem

Let $E$ be a vector space over $\R$. Let $p: E \to \R$ be a Minkowski functional. Let $G \subseteq E$ be a linear subspace of $E$. Let $f: G \to \R$ be a linear functional such that: :$\forall x \in G: \map f x \le \map p x$ Then there exists a linear functional $\tilde f$ defined on the whole space $E$ which extends $f$. That is: :$\forall x \in G: \map {\tilde f} x = \map f x$ such that: :$\forall x \in E: \map {\tilde f} x \le \map p x$


Hardy-Littlewood Circle Method

Let $\AA$ be a subset of the non-negative integers. Let: :$\displaystyle \map T s = \sum_{a \mathop \in \AA} s^a$ be the generating function for $\AA$. For $N \in \N$, let $\map {r_{\AA, \ell} } N$ be the number of solutions $\tuple {x_1, \ldots, x_\ell} \in \AA^\ell$ to the equation: :$x_1 + \cdots + x_\ell = N$ Then: :$\displaystyle \forall \rho \in \openint 0 1: \map {r_{\AA, \ell} } N = \oint_{\size s \mathop = \rho} \frac {\map T s^\ell} {s^{N + 1} } \rd s$


Vinogradov Circle Method

$\newcommand{\A}{\mathcal A}$Let $\A$ be a subset of the non-negative integers. For $\alpha \in \R$, let: :$e \left({\alpha}\right) := \exp \left({2 \pi i \alpha} \right)$ Let: :$\displaystyle T_N \left({s}\right) = \sum_{\substack {a \mathop \in \A \\ a \mathop \le N}} s^a$ be the truncated generating function for $\A$. {{explain|"truncated" generating function}} Let: :$V_N \left({\alpha}\right) := T_N \left({e \left({\alpha}\right)}\right)$ Let $r_{\A, \ell} \left({N}\right)$ be the number of solutions $\left({x_1, \dotsc, x_\ell}\right) \in \A^\ell$ to the equation: :$(1): \quad \displaystyle x_1 + \cdots + x_l = N$ Then: :$\displaystyle r_{\A, \ell} \left({N}\right) = \int_0^1 V_N \left({\alpha}\right)^\ell e \left({-N \alpha}\right) \, \mathrm d \alpha$


Euler Product

Let $a_n : \N \to \C$ be an arithmetic function. Let $\displaystyle f \left({s}\right) = \sum_{n \mathop \in \N} a_n n^{-s}$ be its Dirichlet series. Let $\sigma_a$ be its abscissa of absolute convergence. Then for $\Re \left({s}\right) > \sigma_a$: :$\displaystyle \sum_{n \mathop = 1}^\infty a_n n^{-s} = \prod_p \frac 1 {1 - a_p p^{-s} }$ where $p$ ranges over the primes. This representation for $f$ is called an '''Euler product''' for the Dirichlet series. {{stub|Completely multiplicative hypothesis not mentioned. Needs also the statement: $\displaystyle f \left({s}\right) {{=}} \prod_p \left\{{\sum_{k \mathop \ge 1} a_{p^k} p^{-k s}}\right\}$ or however it goes for multiplicative functions which are not completely multiplicative}}


Correspondence between Linear Group Actions and Linear Representations

Let $\left({k, +, \cdot}\right)$ be a field. Let $V$ be a vector space over $k$ of finite dimension. Let $\left({G, *}\right)$ be a finite group. There is a one-to-one correspondence between linear group actions of $G$ on $V$ and linear representations of $G$ in $V$, as follows: Let $\phi : G \times V \to V$ be a group action. Let $\rho : G \to \operatorname{Sym}(V)$ be a permutation representation of $G$ on $V$. The following are equivalent: :$(1): \quad$ $\rho$ is the permutation representation associated to $\phi$ :$(2): \quad$ $\phi$ is the group action associated to $\rho$ If this is the case, the following are equivalent: :$(1): \quad$ $\rho$ is a linear representation :$(2): \quad$ $\phi$ is a linear group action


Set Difference with Disjoint Set

Let $S, T$ be sets. Then: :$S \cap T = \O \iff S \setminus T = S$ where: :$S \cap T$ denotes set intersection :$\O$ denotes the empty set :$S \setminus T$ denotes set difference.


Integer Subtraction is Closed

The set of integers is closed under subtraction: :$\forall a, b \in \Z: a - b \in \Z$


Rational Subtraction is Closed

The set of rational numbers is closed under subtraction: :$\forall a, b \in \Q: a - b \in \Q$


Real Subtraction is Closed

The set of real numbers is closed under subtraction: :$\forall a, b \in \R: a - b \in \R$


Complex Subtraction is Closed

The set of complex numbers is closed under subtraction: :$\forall a, b \in \C: a - b \in \C$


Positive Real Numbers Closed under Division

The set $\R_{>0}$ of strictly positive real numbers is closed under division: :$\forall a, b \in \R_{>0}: a \div b \in \R_{>0}$


Rational Numbers form Integral Domain

The set of rational numbers $\Q$ forms an integral domain under addition and multiplication: $\struct {\Q, +, \times}$.


Real Numbers form Integral Domain

The set of real numbers $\R$ forms an integral domain under addition and multiplication: $\struct {\R, +, \times}$.


Properties of Content

Let $f, g \in \Q \sqbrk X$ and $q \in \Q$. The content of a polynomial satisfies the following:


Complex Numbers form Integral Domain

The set of complex numbers $\C$ forms an integral domain under addition and multiplication: $\struct {\C, +, \times}$.


Root of Polynomial iff Divisible by Minimal Polynomial

Let $K$ be a field. Let $L / K$ be a field extension of $K$. Let $\alpha \in L$ be algebraic over $K$. Then there is a unique monic polynomial $\mu_\alpha \in K \left[{X}\right]$ of least degree such that $\mu_\alpha \left({\alpha}\right) = 0$. Moreover $f \in K \left[{X}\right]$ is such that $f \left({\alpha}\right) = 0$ {{iff}} $\mu_\alpha$ divides $f$. {{explain|Link to an appropriate page defining divisibility among polynomials.}}


Gauss's Lemma on Unique Factorization Domains

Let $R$ be a unique factorization domain. Then the ring of polynomials $R \sqbrk X$ is also a unique factorization domain.


Gröbner Basis

Let $F$ be a finite set of polynomials. {{explain|Explain what the underlying field of the polynomials is. Real numbers, abstract rings, general fields? Link to the appropriate definition for the case.}} Let $SP \left({f_1, f_2}\right)$ be the $S$-polynomial of $f_1$ and $f_2$. Let $g$ be a polynomial. Let $RF \left({g}\right)$ be the reduced form of $g$. Then $F$ is a Gröbner basis {{iff}}: :$\forall f_1, f_2 \in F \left({RF \left({F, SP \left({f_1, f_2}\right)}\right) = 0}\right)$ {{explain|The meaning of the above needs to be made clear. Presumably it should be interpreted something like: $\forall f_1, f_2 \in F: RF \left({F, SP \left({f_1, f_2}\right)}\right) {{=}} 0$. But further to this: $RF$ as defined takes a single argument, a polynomial (still to be defined upon what domain). As this statement stands, $RF$ seems to take two parameters: $F$ and $SP$. Clarification is needed.}}


Numbers of Type Integer a plus b root 2 form Subdomain of Reals

Let $\Z \sqbrk {\sqrt 2}$ denote the set: :$\Z \sqbrk {\sqrt 2} := \set {a + b \sqrt 2: a, b \in \Z}$ That is, all numbers of the form $a + b \sqrt 2$ where $a$ and $b$ are integers. Then the algebraic structure: :$\struct {\Z \sqbrk {\sqrt 2}, +, \times}$ where $+$ and $\times$ are conventional addition and multiplication on real numbers, form an integral subdomain of the real numbers $\R$.


Sufficient Conditions for Uncountability

Let $X$ be a set. The following are equivalent: :$(1): \quad X$ contains an uncountable subset :$(2): \quad X$ is uncountable :$(3): \quad $ Every sequence of distinct points $\sequence {x_n}_{n \mathop \in \N}$ in $X$ omits at least one $x \in X$ :$(4): \quad $ There is no surjection $\N \twoheadrightarrow X$ :$(5): \quad X$ is infinite and there is no bijection $X \leftrightarrow \N$ Assuming the Continuum Hypothesis holds, we also have the equivalent uncountability condition: :$(6): \quad $There exist extended real numbers $a < b$ and a surjection $X \to \closedint a b$


Square of Non-Zero Element of Ordered Integral Domain is Strictly Positive

Let $\struct {D, +, \times, \le}$ be an ordered integral domain whose zero is $0_D$. Then: :$\forall x \in D: x \ne 0_D \iff \map P {x \times x}$ where $\map P {x \times x}$ denotes that $x \times x$ has the (strict) positivity property. That is, the square of any element of an ordered integral domain is (strictly) positive {{iff}} that element is non-zero.


Simultaneous Equation With Two Unknowns

A pair of simultaneous linear equations of the form: {{begin-eqn}} {{eqn | l=a x + b y       | r=c }} {{eqn | l=d x + e y       | r=f }} {{end-eqn}} where $ ae \ne b d$, has as its only solution: {{begin-eqn}} {{eqn | l=x       | r= \frac {c e - b f} {a e - b d} }} {{eqn | l=y       | r=\frac {a f - c d} {a e - b d} }} {{end-eqn}}


Open Sets in Real Number Line

Every non-empty open set $I \subseteq \R$ can be expressed as a countable union of pairwise disjoint open intervals. If: :$\displaystyle I = \bigcup_{n \mathop \in \N} J_n$ :$\displaystyle I = \bigcup_{n \mathop \in \N} K_n$ are two such expressions, then there exists a permutation $\sigma$ of $\N$ such that: :$\forall n \in \N: J_n = K_{\map \sigma n}$


Unity of Ordered Integral Domain is Strictly Positive

Let $\struct {D, +, \times \le}$ be an ordered integral domain whose unity is $1_D$. Then: :$\map P {1_D}$ where $P$ is the (strict) positivity property.


Integers form Ordered Integral Domain

The integers $\Z$ form an ordered integral domain under addition and multiplication.


Rational Numbers form Ordered Integral Domain

The rational numbers $\Q$ form an ordered integral domain under addition and multiplication.


Real Numbers form Ordered Integral Domain

The set of real numbers $\R$ forms an ordered integral domain under addition and multiplication: $\struct {\R, +, \times, \le}$.


General Positivity Rule in Ordered Integral Domain

Let $\struct {D, +, \times}$ be an ordered integral domain, whose (strict) positivity property is denoted $P$. Let $S \subset D$ be a subset of $D$ such that: :$\forall s \in S: \map P x$ Then the following are true: :$\displaystyle \forall n \in \N_{>0}: \forall s_i \in S: \map P {\sum_{i \mathop = 1}^n s_i}$ :$\displaystyle \forall n \in \N_{>0}: \forall s_i \in S: \map P {\prod_{i \mathop = 1}^n s_i}$ where: :$\displaystyle \sum_{i \mathop = 1}^n s_i = s_1 + s_2 + \cdots + s_n$ :$\displaystyle \prod_{i \mathop = 1}^n s_i = s_1 \times s_2 \times \cdots \times s_n$ That is, the ring sum and ring product of any number of elements in $D$ which have the (strict) positivity property also have the (strict) positivity property.


Finite Integral Domain cannot be Ordered

Let $\struct {D, +, \times}$ be an integral domain whose unity is $1_D$ and whose order is finite. Then $D$ cannot be an ordered integral domain.


Ring of Integers Modulo m cannot be Ordered Integral Domain

Let $m \in \Z: m \ge 2$. Let $\struct {\Z_m, +, \times}$ be the ring of integers modulo $m$. Then $\struct {\Z_m, +, \times}$ cannot be an ordered integral domain.


Strict Positivity Property induces Total Ordering

Let $\struct {D, +, \times}$ be an integral domain whose zero is $0_D$. Let $D$ be endowed with a (strict) positivity property $P: D \to \set {\mathrm T, \mathrm F}$. Then there exists a total ordering $\le$ on $\struct {D, +, \times}$ induced by $P$ which is compatible with the ring structure of $\struct {D, +, \times}$.


Relation Induced by Strict Positivity Property is Transitive

Let $\struct {D, +, \times}$ be an ordered integral domain where $P$ is the (strict) positivity property. Let the relation $<$ be defined on $D$ as: :$\forall a, b \in D: a < b \iff \map P {-a + b}$ Then $<$ is a transitive relation.


Relation Induced by Strict Positivity Property is Compatible with Addition

Let $\struct {D, +, \times}$ be an ordered integral domain where $P$ is the (strict) positivity property. Let the relation $<$ be defined on $D$ as: :$\forall a, b \in D: a < b \iff \map P {-a + b}$ Then $<$ is compatible with $+$, that is: :$\forall x, y, z \in D: x < y \implies \paren {x + z} < \paren {y + z}$ :$\forall x, y, z \in D: x < y \implies \paren {z + x} < \paren {z + y}$


Relation Induced by Strict Positivity Property is Compatible with Multiplication

Let $\struct {D, +, \times}$ be an ordered integral domain where $P$ is the (strict) positivity property. Let the relation $<$ be defined on $D$ as: :$\forall a, b \in D: a < b \iff \map P {-a + b}$ Then $<$ is compatible with $\times$ in the following sense: :$\forall x, y, z \in D: x < y, \map P z \implies \paren {z \times x} < \paren {z \times y}$ :$\forall x, y, z \in D: x < y, \map P z \implies \paren {x \times z} < \paren {y \times z}$


Relation Induced by Strict Positivity Property is Trichotomy

Let $\struct {D, +, \times}$ be an ordered integral domain where $P$ is the (strict) positivity property. Let the relation $<$ be defined on $D$ as: :$\forall a, b \in D: a < b \iff \map P {-a + b}$ Then $\forall a, b \in D:$ exactly one of the following conditions applies: :$a < b$ :$a = b$ :$a > b$ That is, $<$ is a trichotomy.


Relation Induced by Strict Positivity Property is Asymmetric and Antireflexive

Let $\struct {D, +, \times}$ be an ordered integral domain where $P$ is the (strict) positivity property. Let the relation $<$ be defined on $D$ as: :$\forall a, b \in D: a < b \iff \map P {-a + b}$ Then $<$ is asymmetric and antireflexive.


Properties of Strict Negativity

Let $\struct {D, +, \times}$ be an ordered integral domain, whose (strict) positivity property is denoted $P$. Let $\le$ be the total ordering induced by $P$, and let $<$ be its strict total ordering counterpart. Let $N$ be the (strict) negativity property on $D$: :$\forall a \in D: \map N a \iff \map P {-a}$ Then the following properties apply for all $a, b \in D$:   === Strict Negativity is equivalent to Strictly Preceding Zero === {{:Strict Negativity is equivalent to Strictly Preceding Zero}}


Sum of Absolute Values on Ordered Integral Domain

Let $\struct {D, +, \times, \le}$ be an ordered integral domain. For all $a \in D$, let $\size a$ denote the absolute value of $a$. Then: :$\size {a + b} \le \size a + \size b$


One Succeeds Zero in Well-Ordered Integral Domain

Let $\struct {D, +, \times, \le}$ be a well-ordered integral domain. Let $0$ and $1$ be the zero and unity respectively of $D$. Then $0$ is the immediate predecessor of $1$: :$0 < 1$ :$\neg \exists a \in D: 0 < a < 1$


Square of Element Less than Unity in Ordered Integral Domain

Let $\struct {D, +, \times, \le}$ be an ordered integral domain. Let $x \in D$ such that $0 < x < 1$. Then: : $0 < x \times x < x$


Principle of Mathematical Induction/Well-Ordered Integral Domain

Let $\struct {D, +, \times, \le}$ be a well-ordered integral domain whose zero is $0_D$. Let the unity of $D$ be $1_D$. Let $S \subseteq D$ be such that: :$1_D \in S$ :$a \in S \implies a + 1_D \in S$ Then: :$D_{> 0_D} \subseteq S$ where $D_{> 0_D}$ denotes all the elements $d \in D$ such that $\map P d$. That is, $D_{> 0_D}$ is the set of all (strictly) positive elements of $D$.


Multiple of Divisor in Integral Domain Divides Multiple

Let $\struct {D, +, \times}$ be an integral domain. Let $a, b, c \in D$. Let $a \divides b$, where $\divides$ denotes divisibility. Then $a \times c$ is a divisor of $b \times c$.


Integer Divisor Results/One Divides all Integers

{{begin-eqn}} {{eqn | l = 1       | o = \divides       | r = n }} {{eqn | l = -1       | o = \divides       | r = n }} {{end-eqn}}


Integer Divisor Results/Integer Divides Itself

:$n \divides n$


Integer Divisor Results/Integer Divides its Negative

{{begin-eqn}} {{eqn | l = n       | o = \divides       | r = -n }} {{eqn | l = -n       | o = \divides       | r = n }} {{end-eqn}}


Integer Divisor Results/Integer Divides its Absolute Value

{{begin-eqn}} {{eqn | l = n       | o = \divides       | r = \size n }} {{eqn | l = \size n       | o = \divides       | r = n }} {{end-eqn}} where: :$\size n$ is the absolute value of $n$ :$\divides$ denotes divisibility.


Integer Divisor Results/Integer Divides Zero

:$n \divides 0$


Integer Divisor Results/Divisors of Negative Values

: $m \mathrel \backslash n \iff -m \divides n \iff m \divides -n \iff -m \divides -n$


Divisors of One

The only divisors of $1$ are $1$ and $-1$. That is: :$a \divides 1 \iff a = \pm 1$


Divisor Relation is Antisymmetric

''Divides'' is a antisymmetric relation on $\Z_{>0}$, the set of positive integers. That is: :$\forall a, b \in \Z_{>0}: a \divides b \land b \divides a \implies a = b$


Real Function is Continuous at Point iff Oscillation is Zero

Let $f: D \to \R$ be a real function where $D \subseteq \R$. Let $x$ be a point in $D$. Let $N_x$ be the set of open subset neighborhoods of $x$. Let $\map {\omega_f} x$ be the oscillation of $f$ at $x$: :$\map {\omega_f} x = \displaystyle \inf \set {\map {\omega_f} I: I \in N_x}$ where: :$\map {\omega_f} I = \displaystyle \sup \set {\size {\map f y - \map f z}: y, z \in I \cap D}$ Then $\map {\omega_f} x = 0$ {{iff}} $f$ is continuous at $x$.


Numbers of Type Integer a plus b root 2 are Not a Field

Let $\Z \sqbrk {\sqrt 2}$ denote the set: :$\Z \sqbrk {\sqrt 2} := \set {a + b \sqrt 2: a, b \in \Z}$ that is, all (real) numbers of the form $a + b \sqrt 2$ where $a$ and $b$ are integers. Then the algebraic structure: :$\struct {\Z \sqbrk {\sqrt 2}, +, \times}$ where $+$ and $\times$ are addition and multiplication on real numbers, is '''not''' a field.


Numbers of Type Rational a plus b root 2 form Field

Let $\Q \sqbrk {\sqrt 2}$ denote the set: :$\Q \sqbrk {\sqrt 2} := \set {a + b \sqrt 2: a, b \in \Q}$ that is, all numbers of the form $a + b \sqrt 2$ where $a$ and $b$ are rational numbers. Then the algebraic structure: :$\struct {\Q \sqbrk {\sqrt 2}, +, \times}$ where $+$ and $\times$ are conventional addition and multiplication on real numbers, is a field.


Characteristic of Ring of Integers Modulo Prime

Let $\struct {\Z_p, +, \times}$ be the ring of integers modulo $p$, where $p$ is a prime number. The characteristic of $\struct {\Z_p, +, \times}$ is $p$.


Gamma Function Extends Factorial

:$\forall n \in \N: \map \Gamma {n + 1} = n!$


Uniform Limit of Analytic Functions is Analytic

Let $U$ be an open subset of $\C$. Let $\left\{ {f_n}\right\}_{n \mathop \in \N}$ be a sequence of analytic functions $f_n : U \to \C$. Let $\left\{ {f_n}\right\}$ converge locally uniformly to $f$ on $U$. Then $f$ is analytic.


Modulus of Complex Integral

Let $\left[{a \,.\,.\, b}\right]$ be a closed real interval. Let $f : \left[{a \,.\,.\, b}\right] \to \C$ be a continuous complex function. Then: :$\displaystyle \left|{\int_a^b f \left({t}\right) \rd t}\right| \le \int_a^b \left|{f \left({t}\right)}\right| \rd t$ where the first integral is a complex Riemann integral, and the second integral is a definite real integral.


Triangle Inequality for Contour Integrals

Let $C$ be a contour. Let $f: \Img C \to \C$ be a continuous complex function, where $\Img C$ denotes the image of $C$. Then: :$\displaystyle \size {\int_C \map f z \rd z} \le \max_{z \mathop \in \Img C} \size {\map f z} \map L C$ where $\map L C$ denotes the length of $C$.


Jensen's Formula

Let $S$ be an open subset of the complex plane containing the closed disk: :$D_r = \set {z \in \C : \cmod z \le r}$ of radius $r$ about $0$. Let $f: S \to \C$ be holomorphic on $S$.  Let $f$ have no zeroes on the circle $\cmod z = r$. Let $\map f 0 \ne 0$. Let $\rho_1, \ldots, \rho_n$ be the zeroes of $f$ in $D_r$, counted with multiplicity. Then: :$(1): \quad \displaystyle \frac 1 {2 \pi} \int_0^{2 \pi} \ln \cmod {\map f {r e^{i \theta} } } \rd \theta = \ln \cmod {\map f 0} + \sum_{k \mathop = 1}^n \paren {\ln r - \ln \size {\rho_k} }$


Residue Theorem

Let $U$ be a simply connected open subset of the complex plane $\C$. Let $a_1, a_2, \dots, a_n$ be finitely many points of $U$. Let $f: U \to \C$ be analytic in $U \setminus \set {a_1, a_2, \dots, a_n}$.  Let $L = \partial U$ be oriented counterclockwise. Then: :$\displaystyle \oint_L \map f z \rd z = 2 \pi i \sum_{k \mathop = 1}^n \Res f {a_k}$


Cauchy's Integral Formula

Let $D = \set {z \in \C: \cmod z \le r}$ be the closed disk of radius $r$ in $\C$. Let $f: U \to \C$ be holomorphic on some open set containing $D$. Then for each $a$ in the interior of $D$: :$\displaystyle \map f a = \frac 1 {2 \pi i} \int_{\partial D} \frac {\map f z} {\paren {z - a} } \rd z$ where $\partial D$ is the boundary of $D$, and is traversed anticlockwise.


Cauchy's Integral Formula/General Result

Let $n \in \N$ be a natural number. Then for each $a$ in the interior of $D$: :$\displaystyle f^{\paren n} \paren a = \dfrac {n!} {2 \pi i} \int_{\partial D} \frac {\map f z} {\paren {z - a}^{n + 1} } \rd z$ where $\partial D$ is the boundary of $D$, and is traversed anticlockwise.


Characteristic of Subfield of Complex Numbers is Zero

The characteristic of any subfield of the field of complex numbers is $0$.


Characteristic times Ring Element is Ring Zero

Let $\struct {R, +, \circ}$ be a ring with unity. Let the zero of $R$ be $0_R$ and the unity of $R$ be $1_R$. Let the characteristic of $R$ be $n$. Then: :$\forall a \in R: n \cdot a = 0_R$


Field of Rational Functions is Field

Let $K$ be a field. Let $K \sqbrk x$ be the integral domain of polynomial forms on $K$. Let $\map K x$ be the field of rational functions on $K$. Then $\map K x$ forms a field. If the characteristic of $K$ is $p$, then the characteristic of $\map K x$ is non-zero.


Hadamard Factorization Theorem

Let $f: \C \to \C$ be an entire function of finite order $\omega$. Let $0$ be a zero of $f$ of multiplicity $m\geq0$. Let $\left\langle{a_n}\right\rangle$ be the sequence of nonzero zeroes of $f$, repeated according to multiplicity. Then $f$ has finite rank $p\leq\omega$ and there exists a polynomial $g$ of degree at most $\omega$ such that: :$\displaystyle f \left({z}\right) = z^m e^{g(z)} \prod_{n \mathop = 1}^\infty E_p\left( \frac z{a_n} \right)$ where $E_p$ denotes the $p$th Weierstrass elementary factor.


Properties of Gamma Function

The gamma function $\Gamma \left({z}\right)$ has the following properties:


Poisson Summation Formula

Let $f: \R \to \C$ be a Schwarz function. Let $\hat f$ be its Fourier transform. Then: :$\displaystyle \sum_{n \mathop \in \Z} \map f n = \sum_{m \mathop \in \Z} \map {\hat f} m$


Harmonic Properties of Schwarz Functions

Let $f, g : \R \to \C$ be Schwarz functions. Let $\hat f$, $\hat g$ be the Fourier transforms of $f$ and $g$ respectively. Then: :$(1): \quad \hat f$, $\hat g$ are Schwarz functions. :$(2): \quad \map {\widehat {\paren {\hat f} } } x = \map f {-x}$ for all $x \in \R$. :$(3): \quad$ If $f * g$ is the convolution of $f$ and $g$, then: {{disambiguate|Definition:Convolution}} ::::$\widehat {f * g} = \hat f \hat g$


Euler's Reflection Formula

:$\forall z \notin \Z: \map \Gamma z \, \map \Gamma {1 - z} = \dfrac \pi {\map \sin {\pi z} }$


Legendre's Duplication Formula

:$\forall z \notin \set {-\dfrac n 2: n \in \N}: \map \Gamma z \, \map \Gamma {z + \dfrac 1 2} = 2^{1 - 2 z} \sqrt \pi \, \map \Gamma {2 z}$ where $\N$ denotes the natural numbers.


Approximation to Reciprocal times Derivative of Gamma Function

For all $z \in \C$ such that $\cmod {\map \arg z} < \pi - \epsilon, \cmod z > 1$: :$\dfrac {\map {\Gamma'} z} {\map \Gamma z} = \ln z + \map {\mathcal O_\epsilon} {z^{-1} }$ where: :$\map {\mathcal O} {z^{-1} }$ denotes big-O notation :the implied constant depends on $\epsilon$.


Complex Conjugate of Gamma Function

:$\forall z \in \C \setminus \set {0, -1, -2, \ldots}: \map \Gamma {\overline z} = \overline {\map \Gamma z}$


Intersection of Integer Ideals is Lowest Common Multiple

Let $\ideal m$ and $\ideal n$ be ideals of the integers $\Z$. Let $\ideal k$ be the intersection of $\ideal m$ and $\ideal n$. Then $k = \lcm \set {m, n}$.


LCM iff Divides All Common Multiples

Let $a, b \in \Z$ such that $a b \ne 0$. Let $m \in \Z: d > 0$. Then $m = \lcm \set {a, b}$ {{iff}}: : $(1): \quad a \divides m \land b \divides m$ : $(2): \quad a \divides n \land b \divides n \implies m \divides n$ That is, in the set of positive integers, $m$ is the LCM of $a$ and $b$ {{iff}} $m$ is a common multiple of $a$ and $b$, and $m$ also divides any other common multiple of $a$ and $b$.


LCM Divides Common Multiple

Let $a, b \in \Z$ such that $a b \ne 0$. Let $n$ be any common multiple of $a$ and $b$. That is, let $n \in \Z: a \divides n, b \divides n$. Then: :$\lcm \set {a, b} \divides n$ where $\lcm \set {a, b}$ is the lowest common multiple of $a$ and $b$. {{:Euclid:Proposition/VII/35}}


Sum of Integer Ideals is Greatest Common Divisor

Let $\ideal m$ and $\ideal n$ be ideals of the integers $\Z$. Let $\ideal d = \ideal m + \ideal n$. Then $d = \gcd \set {m, n}$.


Quotient Ring of Integers with Principal Ideal

Let $\struct {\Z, +, \times}$ be the integral domain of integers. Let $n \in \Z$. Let $\ideal n$ be the principal ideal of $\struct {\Z, +, \times}$ generated by $n$. The quotient ring $\struct {\Z, +, \times} / \ideal n$ is isomorphic to $\struct {\Z_n, +_n, \times_n}$, the ring of integers modulo $n$. Note the special cases where $n = 0$ or $1$: :Quotient Ring of Integers and Zero :Quotient Ring of Integers and Principal Ideal from Unity


Quotient Ring by Null Ideal

Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$. Let $\struct {\set {0_R}, +, \circ}$ be the null ideal of $\struct {R, +, \circ}$. Let $\struct {R / \set {0_R}, +, \circ}$ be the quotient ring of $R$ defined by $\set {0_R}$. Then $\struct {R / \set {0_R}, +, \circ}$ is isomorphic to $\struct {R, +, \circ}$.


Ring Homomorphism Preserves Negatives

Let $\phi: \struct {R_1, +_1, \circ_1} \to \struct {R_2, +_2, \circ_2}$ be a ring homomorphism. Then: :$\forall x \in R_1: \map \phi {-x} = -\paren {\map \phi x}$


Integral Representation of Riemann Zeta Function in terms of Jacobi Theta Function

Let $\zeta$ be the Riemann zeta function. Let $s \in \C$ be a complex number with real part $\sigma>1$. Then: :$\displaystyle \pi^{-s / 2} \Gamma \left({\frac s 2}\right) \zeta \left({s}\right) = - \frac 1 {s \left({1 - s}\right)} + \int_1^\infty \left({x^{s / 2 - 1} + x^{- \left({s + 1}\right) / 2} }\right) \omega \left({x}\right) \rd x$ where: : $\Gamma$ is the gamma function : $\displaystyle \omega \left({x}\right) = \sum_{n \mathop = 1}^\infty e^{- \pi n^2 x}$


Mapping on Integers a plus b root 2 to Conjugate is Automorphism

Let $\Z \sqbrk {\sqrt 2}$ denote the set: :$\Z \sqbrk {\sqrt 2} := \set {a + b \sqrt 2: a, b \in \Z}$ that is, all numbers of the form $a + b \sqrt 2$ where $a$ and $b$ are integers. Then the mapping $\phi: \Z \sqbrk {\sqrt 2} \to \Z \sqbrk {\sqrt 2}$ defined as: :$\forall x = a + b \sqrt 2 \in \Z \sqbrk {\sqrt 2}: \map \phi {a + b \sqrt 2} = a - b \sqrt 2$ is a ring automorphism.


Ring Epimorphism from Integers to Integers Modulo m

Let $\struct {\Z, +, \times}$ be the ring of integers. Let $\struct {\Z_m, +_m, \times_m}$ be the ring of integers modulo $m$. Let $\phi: \struct {\Z, +, \times} \to \struct {\Z_m, +_m, \times_m}$ be the mapping defined as: :$\forall x \in \Z: \map \phi x = \eqclass x m$ where $\eqclass x m$ is the residue class modulo $m$. Then $\phi$ is a ring epimorphism, but specifically not a ring monomorphism. The image of $\phi$ is $\struct {\Z_m, +_m, \times_m}$. The kernel of $\phi$ is $m \Z$, the set of integer multiples of $m$.


Monotone Additive Function is Linear

Let $f: \R \to \R$ be a monotone real function which is additive, that is: :$\forall x, y \in \R: \map f {x + y} = \map f x + \map f y$ Then: :$\exists a \in \R: \forall x \in \R: \map f x = a x$


Units of Ring of Polynomial Forms over Field

Let $\struct {F, +, \circ}$ be a field whose zero is $0_F$ and whose unity is $1_F$. Let $F \sqbrk X$ be the ring of polynomial forms in an indeterminate $X$ over $F$. Then the units of $F \sqbrk X$ are all the elements of $F \sqbrk X$ whose degree is $0$.


Units of Numbers of Type Integer a plus b root 2

Let $\Z \sqbrk {\sqrt 2}$ denote the set: :$\Z \sqbrk {\sqrt 2} := \set {a + b \sqrt 2: a, b \in \Z}$ that is, all numbers of the form $a + b \sqrt 2$ where $a$ and $b$ are integers. Let $\struct {\Z \sqbrk {\sqrt 2}, +, \times}$ be the integral domain where $+$ and $\times$ are conventional addition and multiplication on real numbers. Then numbers of the form $a + b \sqrt 2$ such that $a^2 - 2 b^2 = \pm 1$ are all units of $\struct {\Z \sqbrk {\sqrt 2}, +, \times}$.


Units of Gaussian Integers

Let $\struct {\Z \sqbrk i, +, \times}$ be the ring of Gaussian integers. The set of units of $\struct {\Z \sqbrk i, +, \times}$ is $\set {1, i, -1, -i}$.


Integers are Euclidean Domain

The integers $\Z$ with the mapping $\nu: \Z \to \Z$ defined as: :$\forall x \in \Z: \map \nu x = \size x$ form a Euclidean domain.


Gaussian Integers form Euclidean Domain

Let $\struct {\Z \sqbrk i, +, \times}$ be the integral domain of Gaussian Integers. Let $\nu: \Z \sqbrk i \to \R$ be the real-valued function defined as:  :$\forall a \in \Z \sqbrk i: \map \nu a = \cmod a^2$  where $\cmod a$ is the (complex) modulus of $a$. Then $\nu$ is a Euclidean valuation on $\Z \sqbrk i$. Hence $\struct {\Z \sqbrk i, +, \times}$ with $\nu: \Z \sqbrk i \to \Z$ forms a Euclidean domain.


Degree of Product of Polynomials over Ring

:$\forall f, g \in R \left[{X}\right]: \deg \left({f g}\right) \le \deg \left({f}\right) + \deg \left({g}\right)$


Degree of Sum of Polynomials

:$\forall f, g \in R \sqbrk X: \map \deg {f + g} \le \max \set {\map \deg f, \map \deg g}$


Degree of Product of Polynomials over Integral Domain not Less than Degree of Factors

:$\forall f, g \in R \sqbrk X: \map \deg {f g} \ge \map \deg f$


Polynomial Forms over Field is Euclidean Domain

Let $\struct {F, +, \circ}$ be a field whose zero is $0_F$ and whose unity is $1_F$. Let $X$ be transcendental in $F$. Let $F \sqbrk X$ be the ring of polynomial forms in $X$ over $F$. Then $F \sqbrk X$ is a Euclidean domain.


Euclidean Domain is Principal Ideal Domain

A Euclidean domain is a principal ideal domain.


Elements of Euclidean Domain have Greatest Common Divisor

Let $\struct {D, +, \times}$ be a Euclidean domain. Then any two elements $a, b \in D$ have a greatest common divisor $d$ such that: :$d \divides a \land d \divides b$ :$x \divides a \land x \divides b \implies x \divides d$ and $d$ is written $\gcd \set {a, b}$. For any $a, b \in D$: :$\exists s, t \in D: s a + t b = d$ Any two greatest common divisors of any $a, b$ are associates.


Associates in Ring of Polynomial Forms over Field

Let $F \sqbrk X$ be the ring of polynomial forms over the field $F$. Let $\map d X$ and $\map {d'} X$ be polynomial forms in $F \sqbrk X$. Then $\map d X$ is an associate of $\map {d'} X$ {{iff}} $\map d X = c \cdot \map {d'} X$ for some $c \in F, c \ne 0_F$. Hence any two polynomials in $F \sqbrk X$ have a unique monic GCD.


Zero Element Generates Null Ideal

Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$. For $r \in R$, let $\ideal r$ denote the ideal generated by $r$. Then $\ideal {0_R}$ is the null ideal.


Irreducible Elements of Ring of Integers

Let $\struct {\Z, +, \times}$ be the ring of integers. The irreducible elements of $\struct {\Z, +, \times}$ are the prime numbers and their negatives.


Euclid's Lemma for Euclidean Domains

Let $\struct {D, +, \times}$ be a Euclidean domain whose unity is $1$. Let $a, b, c \in D$. Let $a \divides b \times c$, where $\divides$ denotes divisibility. Let $a \perp b$, where $\perp$ denotes relative primeness. Then $a \divides c$.


Rational Polynomial is Content Times Primitive Polynomial

Let $\Q \sqbrk X$ be the ring of polynomial forms over the field of rational numbers in the indeterminate $X$. Let $\map f X \in \Q \sqbrk X$. Then: :$\map f X = \cont f \, \map {f^*} X$ where: :$\cont f$ is the content of $\map f X$ :$\map {f^*} X$ is a primitive polynomial. For a given polynomial $\map f X$, both $\cont f$ and $\map {f^*} X$ are unique.


Product of Rational Polynomials

Let $\Q \sqbrk X$ be the ring of polynomial forms over the field of rational numbers in the indeterminate $X$. Let $\map f X, \map g X \in \Q \sqbrk X$. Using Rational Polynomial is Content Times Primitive Polynomial, let these be expressed as: :$\map f X = \cont f \cdot \map {f^*} X$ :$\map g X = \cont g \cdot \map {g^*} X$ where: :$\cont f, \cont g$ are the content of $f$ and $g$ respectively :$f^*, g^*$ are primitive. Let $\map h X = \map f X \, \map g X$ be the product of $f$ and $g$. Then: :$\map {h^*} X = \map {f^*} X \, \map {g^*} X$


Parallelogram Law

Let $\mathbf a$ and $\mathbf b$ be vector quantities. Consider a parallelogram, two of whose adjacent sides represent $\mathbf a$ and $\mathbf b$ (in magnitude and direction). :300px Then the diagonal of the parallelogram through that common point represents the magnitude and direction of $\mathbf a + \mathbf b$, the sum of $\mathbf a$ and $\mathbf b$.


Equivalence of Definitions of Perfect Set

{{TFAE|def = Perfect Set}} Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$ be a subset of $S$.


Complement of Interior equals Closure of Complement

Let $T$ be a topological space. Let $H \subseteq T$. Let $H^-$ denote the closure of $H$ and $H^\circ$ denote the interior of $H$. Let $\map \complement H$ be the complement of $H$ in $T$: :$\map \complement H = T \setminus H$ Then: :$\map \complement {H^\circ} = \paren {\map \complement H}^-$ and similarly: :$\paren {\map \complement H}^\circ = \map \complement {H^-}$ These can alternatively be written: :$T \setminus H^\circ = \paren {T \setminus H}^-$ :$\paren {T \setminus H}^\circ = T \setminus H^-$ which, it can be argued, is easier to follow.


Interior of Finite Intersection equals Intersection of Interiors

Let $T$ be a topological space. Let $n \in \N$. Let: :$\forall i \in \set {1, 2, \dotsc, n}: H_i \subseteq T$ Then: :$\displaystyle \left({\bigcap_{i \mathop = 1}^n H_i}\right)^\circ = \bigcap_{i \mathop = 1}^n H_i^\circ$ where $H_i^\circ$ denotes the interior of $H_i$.


Finite Union of Regular Closed Sets is Regular Closed

Let $T$ be a topological space. Let $n \in \N$. Suppose that: :$\forall i \in \set {1, 2, \dotsc, n}: H_i \subseteq T$ where all the $H_i$ are regular closed in $T$. That is: :$\forall i \in \set {1, 2, \dotsc, n}: H_i = H_i^{\circ -}$ where $H_i^{\circ -}$ denotes the closure of the interior of $H_i$ Then $\displaystyle \bigcup_{i \mathop = 1}^n H_i$ is regular closed in $T$. That is: :$\displaystyle \bigcup_{i \mathop = 1}^n H_i = \paren {\bigcup_{i \mathop = 1}^n H_i}^{\circ -}$


Boundary is Intersection of Closure with Closure of Complement

Let $T$ be a topological space. Let $X \subseteq T$. Let $\partial X$ denote the boundary of $X$, defined as: :$\partial X = X^- \setminus X^\circ$ Let $\overline X = T \setminus X$ denote the complement of $X$ in $T$. Let $X^-$ denote the closure of $X$. Then: :$\partial X = X^- \cap \paren {\overline X}^-$


Set is Closed iff it Contains its Boundary

Let $T$ be a topological space, and let $H \subseteq T$. Then $H$ is closed in $T$ {{iff}}: :$\partial H \subseteq H$ where $\partial H$ is the boundary of $H$.


Set is Open iff Disjoint from Boundary

Let $T$ be a topological space, and let $H \subseteq T$. Then $H$ is open in $T$ {{iff}}: :$\partial H \cap H = \O$ where $\partial H$ is the boundary of $H$.


Set is Clopen iff Boundary is Empty

Let $T$ be a topological space, and let $H \subseteq T$. Then $H$ is both closed and open in $T$ {{iff}}: :$\partial H = \O$ where $\partial H$ is the boundary of $H$.


Boundary of Set is Closed

Let $T$ be a topological space, and let $H \subseteq T$. Let $\partial H$ denote the boundary of $H$. Then $\partial H$ is closed in $T$.


Boundary of Boundary is Contained in Boundary

Let $T$ be a topological space. Let $H \subseteq T$. Then: :$\map \partial {\partial H} \subseteq \partial H$ where $\partial H$ is the boundary of $H$. That is, the boundary of the boundary of $H$ is contained in the boundary of $H$.


Equivalence of Definitions of Exterior

{{TFAE|def = Exterior (Topology)|view = Exterior|context = Topology (Mathematical Branch)|contextview = Topology}} Let $T$ be a topological space. Let $H \subseteq T$.


Interior is Subset of Exterior of Exterior

Let $T$ be a topological space. Let $H \subseteq T$. Let $H^e$ denote the exterior of $H$, and let $H^\circ$ denote the interior of $H$. Then: :$H^\circ \subseteq \paren {H^e}^e$


Interior is Subset of Interior of Closure

Let $T$ be a topological space. Let $H \subseteq T$. Let $H^\circ$ denote the interior of $H$. Let $H^-$ denote the closure of $H$. Then: :$H^\circ \subseteq \left({H^-}\right)^\circ$


Closure of Union contains Union of Closures

Let $T = \struct {S, \tau}$ be a topological space. Let $\mathbb H$ be a set of subsets of $S$. That is, let $\mathbb H \subseteq \powerset S$ where $\powerset S$ denotes the power set of $S$. Then the union of the closures of the elements of $\mathbb H$ is a subset of the closure of the union of $\mathbb H$: :$\displaystyle \bigcup_{H \mathop \in \mathbb H} \map \cl H \subseteq \map \cl {\bigcup_{H \mathop \in \mathbb H} H}$


Set is Subset of its Topological Closure

Let $T$ be a topological space. Let $H \subseteq T$. Let $H^-$ be the closure of $H$ in $T$. Then: : $H \subseteq H^-$


Set Closure as Intersection of Closed Sets

Let $T$ be a topological space. Let $H \subseteq T$. Let the closure of $H$ (in $T$) be defined as: :$H^- := H \cup H'$ where $H'$ is the derived set of $H$. Let $\mathbb K$ be defined as: :$\mathbb K := \left\{{K \supseteq H: K}\right.$ is closed in $\left.{T}\right\}$ That is, let $\mathbb K$ be the set of all closed sets of $T$ which contain $H$. Then the closure of $H$ (in $T$) can be defined as: :$\displaystyle H^- := \bigcap \mathbb K$ that is, as the intersection of all the closed sets of $T$ which contain $H$.


Set Closure is Smallest Closed Set/Topology

Let $T$ be a topological space. Let $H \subseteq T$. Let $H^-$ denote the closure of $H$ in $T$. Then $H^-$ is the smallest superset of $H$ that is closed in $T$.


Equivalence of Definitions of Interior (Topology)

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


Intersection of Interiors contains Interior of Intersection

Let $T$ be a topological space. Let $\mathbb H$ be a set of subsets of $T$. That is, let $\mathbb H \subseteq \powerset T$ where $\powerset T$ is the power set of $T$. Then the interior of the intersection of $\mathbb H$ is a subset of the intersection of the interiors of the elements of $\mathbb H$. :$\displaystyle \paren {\bigcap_{H \mathop \in \mathbb H} H}^\circ \subseteq \bigcap_{H \mathop \in \mathbb H} H^\circ$


Exterior of Intersection contains Union of Exteriors

Let $T$ be a topological space. Let $\mathbb H$ be a set of subsets of $T$. That is, let $\mathbb H \subseteq \powerset T$ where $\powerset T$ is the power set of $T$. Then: :$\displaystyle \bigcup_{H \mathop \in \mathbb H} H^e \subseteq \paren {\bigcap_{H \mathop \in \mathbb H} H}^e$ where $H^e$ denotes the exterior of $H$.


Separated Sets are Disjoint

Let $T = \struct {S, \tau}$ be a topological space. Let $A, B \subseteq S$ such that $A$ and $B$ are separated in $T$. Then $A$ and $B$ are disjoint: :$A \cap B  = \O$


Second-Countable Space is First-Countable

Let $T = \struct {S, \tau}$ be a topological space which is second-countable. Then $T$ is also first-countable.


Second-Countability is Hereditary

Let $T = \struct {S, \tau}$ be a topological space which is second-countable. Let $T_H = \struct {H, \tau_H}$, where $\O \subset H \subseteq S$, be a subspace of $T$. Then $T_H$ is second-countable.


Basis for Topological Subspace

Let $T = \struct {A, \tau}$ be a topological space. Let $\O \subseteq H \subseteq A$ and so let $T_H = \struct {H, \tau_H}$ be a subspace of $T$. Let $\BB$ be a (synthetic) basis for $T$. Let $\BB_H$ be defined as: :$B_H = \set {U \cap H: U \in \BB}$ Then $\BB_H$ is a (synthetic) basis for $H$.


Continuity Defined by Closure

Let $T_1 = \struct {X_1, \tau_1}$ and $T_2 = \struct {X_2, \tau_2}$ be topological spaces. Let $f: T_1 \to T_2$ be a mapping. Then $f$ is continuous {{iff}}: :$\forall H \subseteq X_1: f \sqbrk {H^-} \subseteq \paren {f \sqbrk H}^-$ where $H^-$ denotes the closure of $H$ in $T_1$. That is, {{iff}} the image of the closure is a subset of the closure of the image.


Valuation Ring is Local

Let $R$ be a valuation ring. Then $R$ is a local ring.


Bijection is Open iff Inverse is Continuous

Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces. Let $f: T_1 \to T_2$ be a bijection. Then $f$ is open {{iff}} $f^{-1}$ is continuous.


Bijection is Open iff Closed

Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces. Let $f: T_1 \to T_2$ be a bijection. Then $f$ is open {{iff}} $f$ is closed.


T2 Space is T1 Space

Let $\struct {S, \tau}$ be a $T_2$ (Hausdorff) space. Then $\struct {S, \tau}$ is also a $T_1$ (Fréchet) space.


T1 Space is T0 Space

Let $\struct {S, \tau}$ be a Fréchet ($T_1$) space. Then $\struct {S, \tau}$ is also a Kolmogorov ($T_0$) space.


T5 Space is T4 Space

Let $\struct {S, \tau}$ be a $T_5$ space. Then $\struct {S, \tau}$ is also a $T_4$ space.


Completely Normal Space is Normal Space

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


Regular Space is T2 Space

Let $\struct {S, \tau}$ be a regular space. Then $\struct {S, \tau}$ is also a $T_2$ (Hausdorff) space.


Normal Space is T3 Space

Let $\struct {S, \tau}$ be a normal space. Then $\struct {S, \tau}$ is also a $T_3$ space.


Completely Hausdorff Space is Hausdorff Space

Let $\struct {S, \tau}$ be a $T_{2 \frac 1 2}$ (completely Hausdorff) space. Then $\struct {S, \tau}$ is also a $T_2$ (Hausdorff) space.


Regular Space is Completely Hausdorff Space

Let $\struct {S, \tau}$ be a regular space. Then $\struct {S, \tau}$ is also a $T_{2 \frac 1 2}$ (completely Hausdorff) space.


Equivalence of Definitions of T0 Space

{{TFAE|def = T0 Space|view = $T_0$ (Kolmogorov) space}} Let $T = \struct {S, \tau}$ be a topological space.


Equivalence of Definitions of T1 Space

{{TFAE|def = T1 Space|view = $T_1$ (Fréchet) space}} Let $T = \struct {S, \tau}$ be a topological space.


T3 1/2 Space is T3 Space

Let $T$ be a $T_{3 \frac 1 2}$ space. Then $T$ is also a $T_3$ space.


Tychonoff Space is Regular, T2 and T1

Let $\struct {S, \tau}$ be a Tychonoff space. Then $\struct {S, \tau}$ is also: :a regular space :a $T_2$ (Hausdorff) space :a $T_1$ (Fréchet) space.


Normal Space is Tychonoff Space

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


T4 and T3 Space is T 3 1/2

Let $T = \struct {S, \tau}$ be: :a $T_4$ space and also: :a $T_3$ space. Then $T$ is also a $T_{3 \frac 1 2}$ space.


Separation Axioms Preserved under Homeomorphism

The separation axioms are preserved under homeomorphism. Let $T_A = \left({S_A, \tau_A}\right), T_B = \left({S_B, \tau_B}\right)$ be topological spaces. Let $\phi: T_A \to T_B$ be a homeomorphism.


T0 Space is Preserved under Homeomorphism

If $T_A$ is a $T_0$ (Kolmogorov) space, then so is $T_B$.


T1 Space is Preserved under Homeomorphism

If $T_A$ is a $T_1$ (Fréchet) space, then so is $T_B$.


T2 Space is Preserved under Homeomorphism

If $T_A$ is a $T_2$ (Hausdorff) space, then so is $T_B$.


Completely Hausdorff Space is Preserved under Homeomorphism

If $T_A$ is a $T_{2 \frac 1 2}$ (completely Hausdorff) space, then so is $T_B$.


Regular Space is Preserved under Homeomorphism

If $T_A$ is a regular space, then so is $T_B$.


T3 Space is Preserved under Homeomorphism

If $T_A$ is a $T_3$ space, then so is $T_B$.


T3 1/2 Space is Preserved under Homeomorphism

If $T_A$ is a $T_{3 \frac 1 2}$ space, then so is $T_B$.


T4 Space is Preserved under Homeomorphism

If $T_A$ is a $T_4$ space, then so is $T_B$.


T5 Space is Preserved under Homeomorphism

If $T_A$ is a $T_5$ space, then so is $T_B$.


Inverse of Homeomorphism is Homeomorphism

Let $T, T'$ be topological spaces. Let $f: T \to T'$ be a homeomorphism. Then $f^{-1}: T' \to T$ is also a homeomorphism.


T0 Space is Preserved under Closed Bijection

Let $T_A = \struct {S_A, \tau_A}$ and $T_B = \struct {S_B, \tau_B}$ be topological spaces. Let $\phi: T_A \to T_B$ be a closed bijection. If $T_A$ is a $T_0$ (Kolmogorov) space, then so is $T_B$.


Completely Hausdorff Space is Preserved under Closed Bijection

Let $T_A = \struct {S_A, \tau_A}$ and $T_B = \struct {S_B, \tau_B}$ be topological spaces. Let $\phi: T_A \to T_B$ be a closed bijection. If $T_A$ is a $T_{2 \frac 1 2}$ (completely Hausdorff) space, then so is $T_B$.


T1 Space is Preserved under Closed Bijection

Let $T_A = \struct {S_A, \tau_A}$ and $T_B = \struct {S_B, \tau_B}$ be topological spaces. Let $\phi: T_A \to T_B$ be a closed bijection. If $T_A$ is a $T_1$ (Fréchet) space, then so is $T_B$.


T2 Space is Preserved under Closed Bijection

Let $T_A = \struct {S_A, \tau_A}$ and $T_B = \struct {S_B, \tau_B}$ be topological spaces. Let $\phi: T_A \to T_B$ be a closed bijection. If $T_A$ is a $T_2$ (Hausdorff) space, then so is $T_B$.


Separation Properties Preserved by Expansion

These separation properties are preserved under expansion: :$T_0$ (Kolmogorov) Space :$T_1$ (Fréchet) Space :$T_2$ (Hausdorff) Space :$T_{2 \frac 1 2}$ (Completely Hausdorff) Space


Identity Mapping to Expansion is Closed

Let $S$ be a set on which $\tau_1$ and $\tau_2$ are topologies such that: :$\tau_1 \subseteq \tau_2$ that is, such that $\tau_2$ is an expansion of $\tau_1$. Let $I_X: \struct {S, \tau_1} \to \struct {S, \tau_2}$ be the identity mapping from $\struct {S, \tau_1}$ to $\struct {S, \tau_2}$. Then $I_S$ is closed.


Separation Properties Preserved under Topological Product

Let $\mathbb S = \family {\struct {S_i, \tau_i} }_{i \mathop \in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set. Let $\displaystyle T = \struct {S, \tau} = \prod_{i \mathop \in I} \struct{S_i, \tau_i}$ be the product space of $\mathbb S$. Then $T$ has one of the following properties {{iff}} each of $\struct {S_i, \tau_i}$ has the same property: :$T_0$ (Kolmogorov) Property :$T_1$ (Fréchet) Property :$T_2$ (Hausdorff) Property :$T_{2 \frac 1 2}$ (Completely Hausdorff) Property :$T_3$ Property :$T_{3 \frac 1 2}$ Property If $T = \struct {S, \tau}$ has one of the following properties then each of $\struct {S_i, \tau_i}$ has the same property: :$T_4$ Property :$T_5$ Property but the converse does not necessarily hold.


Separation Properties Preserved in Subspace

Let $T = \struct {S, \tau}$ be a topological space. Let $T_H$ be a subspace of $T$. If $T$ has one of the following properties then $T_H$ has the same property: :$T_0$ (Kolmogorov) Property :$T_1$ (Fréchet) Property :$T_2$ (Hausdorff) Property :$T_{2 \frac 1 2}$ (Completely Hausdorff) Property :$T_3$ Property :$T_{3 \frac 1 2}$ Property :$T_5$ Property That is, the above properties are all hereditary.


T4 Property Preserved in Closed Subspace

Let $T = \struct {S, \tau}$ be a topological space. Let $T_K$ be a subspace of $T$ such that $K$ is closed in $T$. If $T$ is a $T_4$ space then $T_K$ is also a $T_4$ space. That is, the property of being a $T_4$ space is weakly hereditary.


Completely Normal iff Every Subspace is Normal

Let $T = \struct {S, \tau}$ be a topological space. Then $T$ is a completely normal space {{iff}} every subspace of $T$ is normal.


Tychonoff Space is Urysohn Space

Let $\struct {S, \tau}$ be a Tychonoff space. Then $\struct {S, \tau}$ is also an Urysohn space.


Urysohn Space is Completely Hausdorff Space

Let $\struct {S, \tau}$ be an Urysohn space. Then $\struct {S, \tau}$ is also a $T_{2 \frac 1 2}$ (completely Hausdorff) space.


Perfectly Normal Space is Completely Normal Space

Let $T = \struct {S, \tau}$ be a perfectly normal space. Then $T$ is also a completely normal space.


T3 Space is Semiregular

Let $T = \struct {S, \tau}$ be a $T_3$ space. Then $T$ is a semiregular space.


Regular Space is Semiregular Space

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


Compact Space is Sigma-Compact

Every compact space is $\sigma$-compact.


Sigma-Compact Space is Lindelöf

Every $\sigma$-compact space is a Lindelöf space.


Sequentially Compact Space is Countably Compact

A sequentially compact topological space is also countably compact.


Countably Compact Space is Weakly Countably Compact

Every countably compact space is a weakly countably compact space.


T1 Space is Weakly Countably Compact iff Countably Compact

Let $T = \struct {S, \tau}$ be a $T_1$ (Fréchet) topological space. Then $T$ is weakly countably compact {{iff}} $T$ is countably compact.


Countably Compact Space is Pseudocompact

Let $T = \struct {S, \tau}$ be a countably compact space. Then $T$ is a pseudocompact space.


Countably Compact Lindelöf Space is Compact

Let $T = \struct {S, \tau}$ be a Lindelöf space which is also countably compact. Then $T$ is compact.


Compact Space is Weakly Locally Compact

Let $T = \left({S, \tau}\right)$ be a compact space. Then $T$ is a weakly locally compact.


Completed Riemann Zeta Function has Order One

The completed Riemann zeta function $\xi$ has order at most $1$.


Product Equation for Riemann Zeta Function

There exists a constant $B$ such that :$\displaystyle \frac{\zeta'(s)}{\zeta(s)} = B - \frac 1{s-1} +\frac12 \log \pi  - \frac12 \frac{\Gamma'(s/2+1)}{\Gamma(s/2+1)} + \sum_\rho \left( \frac 1{s-\rho} + \frac1\rho \right)$ where $\zeta$ is the Riemann zeta function, $\rho$ runs over the non-trivial zeros of $\zeta$, and $\Gamma$ is the gamma function.


Zeros of Functions of Finite Order

Let $\map f z$ be an entire function which satisfies: :$\map f 0 \ne 0$ :$\cmod {\map f z} \ll \map \exp {\map \alpha {\cmod z} }$ for all $z \in \C$ and some function $\alpha$, where $\ll$ is the order notation. For $T \ge 1$, let: :$\map N T = \# \set {\rho \in \C: \map f r = 0, \ \cmod \rho < T}$ where $\#$ denotes the cardinality of a set. Then: :$\map N T \ll \map \alpha {2 T}$


Strongly Locally Compact Space is Weakly Locally Compact

Let $T = \struct {S, \tau}$ be a strongly locally compact space. Then $T$ is weakly locally compact.


Weakly Locally Compact Hausdorff Space is Strongly Locally Compact

Let $T = \struct {S, \tau}$ be a $T_2$ (Hausdorff) space. Let $T$ be weakly locally compact. Then $T$ is strongly locally compact.


Weakly Sigma-Locally Compact iff Weakly Locally Compact and Lindelöf

Let $T = \struct {S, \tau}$ be a topological space. {{TFAE}} :$(1): \quad T$ is weakly $\sigma$-locally compact :$(2): \quad T$ is weakly locally compact and Lindelöf


Second-Countable Space is Lindelöf

Let $T = \struct {S, \tau}$ be a topological space which is second-countable. Then $T$ is also a Lindelöf space.


Separable Space satisfies Countable Chain Condition

Let $T = \struct {S, \tau}$ be a separable space. Then $T$ satisfies the countable chain condition.


Logarithmic Derivative of Riemann Zeta Function

Let $\zeta$ be the Riemann zeta function: :$\displaystyle \forall s \in \C: \map \Re s > 1: \map \zeta s = \sum_{n \mathop \ge 1} n^{-s}$ Then for all $s$ with $\map \Re s > 1$: :$\displaystyle -\frac {\map {\zeta'} s} {\map \zeta s} = \sum_{n \mathop \ge 1} \map \Lambda n n^{-s}$ where $\Lambda$ is von Mangoldt's function.


Riemann Zeta Has No Zeros With Real Part One

Let $\zeta$ be the Riemann zeta function. Then for all $t \in \R$, $\zeta(1 + it) \neq 0$


Compact Space is Countably Compact

Let $T = \struct {S, \tau}$ be a compact space. Then $T$ is countably compact.


Second-Countable Space is Compact iff Countably Compact

Let $T = \struct {S, \tau}$ be a second-countable space. Then $T$ is compact {{iff}} $T$ is countably compact.


Unsymmetric Functional Equation for Riemann Zeta Function

Let $\zeta$ be the Riemann zeta function. Let $\Gamma$ be the gamma function. Then for all $s \in \C$, :$\displaystyle \map \zeta {1 - s} = 2^{1 - s} \pi^{-s} \, \map \cos {\frac {\pi s} 2} \map \Gamma s \, \map \zeta s$


Countably Compact First-Countable Space is Sequentially Compact

A countably compact first-countable topological space is also sequentially compact.


Residue at Simple Pole

Let $f: \C \to \C$ be a function meromorphic on some region, $D$, containing $a$.  Let $f$ have a simple pole at $a$.  Then the residue of $f$ at $a$ is given by: :$\displaystyle \operatorname{Res} \left({f, a}\right) = \lim_{z \mathop \to a} \left({z - a}\right) f \left({z}\right)$


Poles of Riemann Zeta Function

Let $\zeta$ be the Riemann zeta function. Then $\zeta$ has a simple pole at $s = 1$ with residue $1$, and no other poles.


Properties of Dirichlet Convolution

Let $f, g, h$ be arithmetic functions. Let $*$ denote Dirichlet convolution. Let $\iota$ be the identity arithmetic function. Then the following properties hold:


Ring of Arithmetic Functions is Ring with Unity

Let $\AA$ be the set of all arithmetic functions. Let $*$ denote Dirichlet convolution, and $+$ the pointwise sum of functions. The ring of arithmetic functions $\struct {\AA, +, *}$ is a commutative ring with unity.


Units of Ring of Arithmetic Functions

Let $f$ be an arithmetic function. Then $f$ is a unit in the ring of arithmetic functions {{iff}}: :$\map f 1 \ne 0$


Möbius Inversion Formula

Let $f$ and $g$ be arithmetic functions. Then: :$(1): \quad \displaystyle \map f n = \sum_{d \mathop \divides n} \map g d$ {{iff}}: :$(2): \quad \displaystyle \map g n = \sum_{d \mathop \divides n} \map f d \, \map \mu {\frac n d}$ where: : $d \divides n$ denotes that $d$ is a divisor of $n$ : $\mu$ is the Möbius function.


Fully Normal Space is Normal Space

Let $T = \left({S, \tau}\right)$ be a fully normal space. Then $T$ is a normal space.


Paracompact Space is Countably Paracompact

Let $T = \struct {S, \tau}$ be a paracompact space. Then $T$ is a countably paracompact space.


Metacompact Space is Countably Metacompact

Let $T = \struct {S, \tau}$ be a metacompact space. Then $T$ is a countably metacompact space.


Fully T4 Space is T4 Space

Let $T = \struct {S, \tau}$ be a fully $T_4$ space. Then $T$ is a $T_4$ space.


Fully Normal Space is Paracompact

Let $T = \struct {S, \tau}$ be a fully normal space. Then $T$ is paracompact.


Subcover is Refinement of Cover

Let $S$ be a set. Let $\mathcal C$ be a cover for $S$. Let $\mathcal V$ be a subcover of $\mathcal C$. Then $\mathcal V$ is a refinement of $\mathcal C$.


Compact Space is Paracompact

Let $T = \struct {S, \tau}$ be a compact space. Then $T$ is paracompact.


Paracompact Space is Metacompact

Let $T = \struct {S, \tau}$ be a paracompact space. Then $T$ is also metacompact.


Countably Paracompact Space is Countably Metacompact

Let $T = \struct {S, \tau}$ be a countably paracompact space. Then $T$ is countably metacompact.


Countably Compact Space is Countably Paracompact

Let $T = \struct {S, \tau}$ be a countably compact space. Then $T$ is countably paracompact.


Metacompact Countably Compact Space is Compact

Let $T = \struct {S, \tau}$ be a countably compact space which is also metacompact. Then $T$ is compact.


Countably Paracompact Lindelöf Space is Paracompact

Let $T = \struct {S, \tau}$ be a Lindelöf space which is also countably paracompact. Then $T$ is paracompact.


Convergence of Dirichlet Series with Bounded Partial Sums

Let $\left\langle{a_n}\right\rangle_{n \mathop \in \N}$ be a sequence in $\C$. Suppose that there exists $B > 0$ such that for all $n, m \in \N$: :$\displaystyle \left\vert{\sum_{k \mathop = m}^n a_n}\right\vert \le B$ Then the Dirichlet series: : $\displaystyle f \left({s}\right) = \sum_{n \mathop \ge 1} a_n n^{-s}$ converges locally uniformly to an analytic function on $\Re \left({s}\right) > 0$.


Convergence of Dirichlet Series with Bounded Coefficients

Let $\left\langle{a_n}\right\rangle_{n \mathop \in \N}$ be a bounded sequence in $\C$. Then the Dirichlet series: :$\displaystyle f \left({s}\right) = \sum_{n \mathop \ge 1} a_n n^{-s}$ converges absolutely and locally uniformly to an analytic function on $\Re \left({s}\right) > 1$.


Dirichlet L-Function from Trivial Character

Let $\chi_0$ be the trivial Dirichlet character modulo $q$. {{explain|Trivial character we got, Dirichlet character we got, we still need a page for trivial Dirichlet character. There exists on another page a link to Definition:Dirichlet Character/Trivial Character which ought to be straightforward to construct.}} Let $\zeta$ be the Riemann zeta function. Then: :$\displaystyle \map L {s, \chi_0} = \map \zeta s \cdot \prod_{p \mathop \divides q} \paren {1 - p^{-s} }$ where $\divides$ denotes divisibility.


Separable Metacompact Space is Lindelöf

Let $T = \left({S, \tau}\right)$ be a separable topological space which is also metacompact. Then $T$ is a Lindelöf space.


Functional Equation for Dirichlet L-Functions

Let $\chi$ be a primitive Dirichlet character to the modulus $q \geq 1$. Let $\map \Lambda {s, \chi}$ be the completed $L$-function for $\chi$. Let $\map \tau \chi$ denote the Gaussian sum. Then for all $s \in \C$: :$\map \Lambda {s, \chi} = i^{-\kappa} \dfrac {\map \tau \chi} {\sqrt q} \map \Lambda {1 - s, \overline \chi}$ where $\kappa = \dfrac 1 2 \paren {1 - \map \chi {-1} }$.


Orthogonality Relations for Characters

Let $G$ be a finite abelian group with identity $e$. Let $G^*$ be the dual group of characters $\chi : G \to \C_{\ne 0}$. Let $\chi_0$ be the trivial character on $G$. Let $\psi: G \to \C_{\ne 0}$ be any character. Let $y \in G$ be arbitrary. Then: :$\displaystyle \sum_{x \mathop \in G} \psi \left({x}\right) = \begin{cases} \left\lvert{G}\right\rvert & : \psi = \chi_0 \\ 0 & : \psi \ne \chi_0 \end{cases}$ and: :$\displaystyle \sum_{\chi \mathop \in G^*} \chi \left({y}\right) = \begin{cases} \left\lvert{G^*}\right\rvert & : y = e \\ 0 & : y \ne e \end{cases}$


Analytic Continuation of Dirichlet L-Function

Let $\chi : G := \left({\Z / q \Z}\right)^\times \to \C^\times$ be a Dirichlet character modulo $q$. {{explain|$\C^\times$}} Let $L \left({s, \chi}\right)$ be the Dirichlet $L$-function for $\chi$. Let $\chi$ be the trivial character. Then $L \left({s, \chi}\right)$ has an analytic continuation to $\C$ except for a simple pole at $s = 1$. Let $\chi$ be non-trivial. Then $L \left({s, \chi}\right)$ is analytic on $\Re \left({s}\right) > 0$.


L-Function does not Vanish at One

Let $\psi$ be a non-trivial Dirichlet charater modulo $q$. Let $\map L {s, \chi}$ be the Dirichlet $L$-function associated to $\chi$. Then $\map L {1, \chi} \ne 0$.


Logarithm of Dirichlet L-Functions

Let $\chi$ be a Dirichlet character modulo $q$. The Dirichlet series :$\map f s = \displaystyle \sum_{n \mathop \ge 1} \sum_p \frac {\map \chi p^n} {n p^{n s} }$ converges absolutely to an analytic function, where $p$ ranges over the primes. Moreover, $\map f s$ defines a branch of $\ln \map L {s, \chi}$


Dirichlet's Theorem on Arithmetic Sequences

Let $a, q$ be coprime integers. Let $\PP_{a, q}$ be the set of primes $p$ such that $p \equiv a \pmod q$. Then $\PP_{a, q}$ has Dirichlet density: :$\map \phi q^{-1}$ where $\phi$ is Euler's phi function. In particular, $\PP_{a, q}$ is infinite.


Discrete Fourier Transform on Abelian Group

Let $G$ be a finite abelian group. Let $G^*$ be the dual group of characters $G \to \C^\times$. Let $\eta: G \to \C$ be a mapping from $G$ to the set of complex numbers. Then for all $x \in G$: :$\displaystyle \eta \left({x}\right) = \frac 1 {\phi \left({q}\right)} \sum_{\chi \mathop \in G^*} \langle \eta, \chi \rangle_G \chi(x)$ where: :$\displaystyle \langle \eta, \chi \rangle_G = \sum_{x \mathop \in G} \eta \left({x}\right) \overline{\chi} \left({x}\right)$


Disjoint Compact Sets in Hausdorff Space have Disjoint Neighborhoods

Let $T = \struct {S, \tau}$ be a Hausdorff space. Let $V_1$ and $V_2$ be compact sets in $T$. Then $V_1$ and $V_2$ have disjoint neighborhoods.


Compact Subsets of T3 Spaces

Let $T = \struct {S, \tau}$ be a $T_3$ space. Let $A \subseteq S$ be compact in $T$. Then for each $U \in \tau$ such that $A \subseteq U$: :$\exists V \in \tau: A \subseteq V \subseteq V^- \subseteq U$ where $V^-$ denotes the closure of $V$.


Compact Hausdorff Space is T4

Let $T = \struct {S, \tau}$ be a compact Hausdorff space. Then $T$ is a $T_4$ space.


Maximum Cardinality of Separable Hausdorff Space

Let $T = \struct {S, \tau}$ be a Hausdorff space which is separable. Then $S$ can have a cardinality no greater than $2^{2^{\aleph_0} }$.


Cluster Point of Ultrafilter is Unique

Let $S$ be a set. Let $\FF$ be an ultrafilter on $S$. Let $x \in S$ be a cluster point of $\FF$. Then there is no point $y \in S: y \ne x$ such that $y$ is also a cluster point of $\FF$.


Sequence of Implications of Separation Axioms

Let $P_1$ and $P_2$ be separation axioms and let: :$P_1 \implies P_2$ mean: :If a topological space $T$ satsifies separation axiom $P_1$, then $T$ also satisfies separation axiom $P_2$. Then the following sequence of separation axioms holds: {| |- | align="center" | || | align="center" | || | align="center" | Perfectly Normal || | align="center" | $\implies$ || | align="center" | Perfectly $T_4$ || | align="center" | $\implies$ || | align="center" | $T_4$ || |- | align="center" | || | align="center" | || | align="center" | $\Big\Downarrow$ || | align="center" | || | align="center" | || |- | align="center" | || | align="center" | || | align="center" | Completely Normal || | align="center" | $\implies$ || | align="center" | $T_5$ || |- | align="center" | || | align="center" | || | align="center" | $\Big\Downarrow$ || | align="center" | || | align="center" | $\Big\Downarrow$ || |- | align="center" | || | align="center" | || | align="center" | Normal || | align="center" | $\implies$ || | align="center" | $T_4$ || |- | align="center" | || | align="center" | || | align="center" | $\Big\Downarrow$ || | align="center" | || | align="center" | || |- | align="center" | $T_{3 \frac 1 2}$ || | align="center" | $\impliedby$ || | align="center" | Completely Regular (Tychonoff) || | align="center" | $\implies$ || | align="center" | Urysohn || |- | align="center" | $\Big\Downarrow$ || | align="center" | || | align="center" | $\Big\Downarrow$ || | align="center" | || | align="center" | $\Big\Downarrow$ || |- | align="center" | $T_3$ || | align="center" | $\impliedby$ || | align="center" | Regular || | align="center" | $\implies$ || | align="center" | $T_{2 \frac 1 2}$ (Completely Hausdorff) || |- | align="center" | || | align="center" | || | align="center" | $\Big\Downarrow$ || | align="center" | || | align="center" | $\Big\Downarrow$ || |- | align="center" | || | align="center" | || | align="center" | Semiregular || | align="center" | $\implies$ || | align="center" | $T_2$ (Hausdorff) || |- | align="center" | || | align="center" | || | align="center" | || | align="center" | || | align="center" | $\Big\Downarrow$ || |- | align="center" | || | align="center" | || | align="center" | || | align="center" | || | align="center" | $T_1$ (Fréchet) || |- | align="center" | || | align="center" | || | align="center" | || | align="center" | || | align="center" | $\Big\Downarrow$ || |- | align="center" | || | align="center" | || | align="center" | || | align="center" | || | align="center" | $T_0$ (Kolmogorov) || |}


Sequence of Implications of Global Compactness Properties

Let $P_1$ and $P_2$ be compactness properties and let: :$P_1 \implies P_2$ mean: :If a topological space $T$ satsifies property $P_1$, then $T$ also satisfies property $P_2$. Then the following sequence of implications holds: {| |- | align="center" | || | align="center" | || | align="center" | Sequentially Compact || | align="center" | || | align="center" | || |- | align="center" | || | align="center" | || | align="center" | $\Big\Downarrow$ || | align="center" | || | align="center" | || |- | align="center" | Compact || | align="center" | $\implies$ || | align="center" | Countably Compact || | align="center" | $\implies$ || | align="center" | Pseudocompact || |- | align="center" | $\Big\Downarrow$ || | align="center" | || | align="center" | $\Big\Downarrow$ || | align="center" | || | align="center" | || |- | align="center" | $\sigma$-Compact || | align="center" | || | align="center" | Weakly Countably Compact || | align="center" | || | align="center" | || |- | align="center" | $\Big\Downarrow$ || | align="center" | || | align="center" | || | align="center" | || | align="center" | || |- | align="center" | Lindelöf Space || | align="center" | || | align="center" | || | align="center" | || | align="center" | || |}


Sequence of Implications of Local Compactness Properties

Let $P_1$ and $P_2$ be compactness properties and let: :$P_1 \implies P_2$ mean: :If a topological space $T$ satsifies property $P_1$, then $T$ also satisfies property $P_2$. Then the following sequence of implications holds: {| |- | align="center" | Compact || | align="center" | $\implies$ || | align="center" | Strongly Locally Compact || |- | align="center" | $\Big\Downarrow$ || | align="center" | || | align="center" | $\Big\Downarrow$ || |- | align="center" | Weakly $\sigma$-Locally Compact || | align="center" | $\implies$ || | align="center" | Weakly Locally Compact || | align="center" | $\Longleftarrow$ || | align="center" | Locally Compact || |- | align="center" | $\Big\Downarrow$ || | align="center" | || | align="center" | || |- | align="center" | $\sigma$-Compact || | align="center" | || | align="center" | || |- | align="center" | $\Big\Downarrow$ || | align="center" | || | align="center" | || |- | align="center" | Lindelöf Space || | align="center" | || | align="center" | || |}


Equivalence of Definitions of Integral Dependence

Let $A$ be an extension of a commutative ring with unity $R$. For $x \in A$, the following are equivalent: {{begin-eqn}} {{eqn | n = 1       | o =       | r = \) $x$ is integral over $R$ \( }} {{eqn | n = 2       | o =       | r = \) The $R$-module $R \left[{x}\right]$ is finitely generated \( }} {{eqn | n = 3       | o =       | r = \) There exists a subring $B$ of $A$ such that $x \in B$, $R \subseteq B$ and $B$ is a finitely generated $R$-module \( }} {{eqn | n = 4       | o =       | r = \) There exists a subring $B$ of $A$ such that $x B \subseteq B$ and $B$ is finitely generated over $R$ \( }} {{eqn | n = 5       | o =       | r = \) There exists a faithful $R \left[{x}\right]$-module $B$ that is finitely generated as an $R$-module \( }} {{end-eqn}}


Transitivity of Finite Generation

Let $A \subseteq B \subseteq C$ be rings. Suppose $B$ is a finitely generated $A$-module, and $C$ is a finitely generated $B$-module. Then $C$ is a finitely generated $A$-module.


Integral Closure is Subring

Let $A$ be an extension of a commutative ring with unity $\struct {R, +, \circ}$. Let $C$ be the integral closure of $R$ in $A$. Then $C$ is a subring of $A$.


Localization of Ring is Unique

Let $A$ be a commutative ring with unity. Let $S \subseteq A$ be a multiplicatively closed subset. Let $\left({A_S, \iota}\right)$ and $\left({\tilde A_S, \tilde \iota}\right)$ both satisfy the definition of the localization of $A$ at $S$. Then there is a canonical isomorphism $\phi: A_S \to \tilde A_S$.


Spectrum of Ring is Nonempty

Let $A$ be a non-trivial commutative ring with unity. Then its prime spectrum is non-empty: :$\Spec A \ne \O$


Compact Space is Strongly Locally Compact

Let $T = \struct {S, \tau}$ be a compact space. Then $T$ is a strongly locally compact space.


Compact Space is Weakly Sigma-Locally Compact

Let $T = \struct {S, \tau}$ be a compact space. Then $T$ is a weakly $\sigma$-locally compact space.


Sequence of Implications of Paracompactness Properties

Let $P_1$ and $P_2$ be paracompactness properties and let: :$P_1 \implies P_2$ mean: :If a topological space $T$ satsifies property $P_1$, then $T$ also satisfies property $P_2$. Then the following sequence of implications holds: {| |- | align="center" | || | align="center" | || | align="center" | Compact || | align="center" | $\implies$ || | align="center" | Countably Compact || |- | align="center" | || | align="center" | || | align="center" | $\Big\Downarrow$ || | align="center" | || | align="center" | $\Big\Downarrow$ || |- | align="center" | Fully Normal || | align="center" | $\implies$ || | align="center" | Paracompact || | align="center" | $\implies$ || | align="center" | Countably Paracompact || |- | align="center" | $\Big\Downarrow$ || | align="center" | || | align="center" | $\Big\Downarrow$ || | align="center" | || | align="center" | $\Big\Downarrow$ || |- | align="center" | Fully $T_4$ || | align="center" | || | align="center" | Metacompact || | align="center" | $\implies$ || | align="center" | Countably Metacompact || |- | align="center" | $\Big\Downarrow$ || | align="center" | || | align="center" | || | align="center" | || | align="center" | || |- | align="center" | $T_4$ || | align="center" | || | align="center" | || | align="center" | || | align="center" | || |}


Compactness Properties in Hausdorff Spaces

Let $P_1$ and $P_2$ be compactness properties and let: :$P_1 \implies P_2$ mean: :If a $T_2$ (Hausdorff) space $T$ satsifies property $P_1$, then $T$ also satisfies property $P_2$. Then the following sequence of implications holds: {| |- | align="center" | Fully $T_4$ || | align="center" | $\iff$ || | align="center" | Paracompact || |- | align="center" | || | align="center" | || | align="center" | $\Big\Downarrow$ || |- | align="center" | Weakly $\sigma$-Locally Compact || | align="center" | $\implies$ || | align="center" | $T_4$ || |- | align="center" | $\Big\Downarrow$ || | align="center" | || | align="center" | $\Big\Downarrow$ || |- | align="center" | Weakly Locally Compact || | align="center" | $\implies$ || | align="center" | $T_{3 \frac 1 2}$ || |- | align="center" | || | align="center" | || | align="center" | $\Big\Downarrow$ || |- | align="center" | First-Countable and Countably Compact|| | align="center" | $\implies$ || | align="center" | $T_3$ || |}


Zero Locus of Set is Zero Locus of Generated Ideal

Let $k$ be a field. Let $n\geq1$ be a natural number. Let $A = k \left[{X_1, \ldots, X_n}\right]$ be the ring of polynomial functions in $n$ variables over $k$. Let $T \subseteq A$ be a subset, and $V \left({T}\right)$ the zero locus of $T$. Let $J = \left({T}\right)$ be the ideal generated by $T$. Then: : $V \left({T}\right) = V \left({J}\right)$


Compactness Properties in T3 Spaces

Let $P_1$ and $P_2$ be compactness properties and let: :$P_1 \implies P_2$ mean: :If a $T_3$ space $T$ satsifies property $P_1$, then $T$ also satisfies property $P_2$. Then the following sequence of implications holds: {| |- | align="center" | Second-Countable || | align="center" | $\implies$ || | align="center" | Lindelöf || |- | align="center" | $\Big\Downarrow$ || | align="center" | || | align="center" | $\Big\Downarrow$ || |- | align="center" | $\Downarrow$|| | align="center" | || | align="center" | Fully $T_4$ $\iff$ Paracompact || |- | align="center" | $\Big\Downarrow$ || | align="center" | || | align="center" | $\Big\Downarrow$ || |- | align="center" | $T_5$ || | align="center" | $\implies$ || | align="center" | $T_4$ || |- |}


Finite Product of Sigma-Compact Spaces is Sigma-Compact

Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $\set {\struct {S_i, \tau_i}: 1 \le i \le n}$ be a finite set of topological spaces. Let $\displaystyle \struct {S, \tau} = \prod_{i \mathop = 1}^n \struct {S_i, \tau_i}$ be the product space of $\set {\struct {S_i, \tau_i}: 1 \le i \le n}$. Let each of $\struct {S_i, \tau_i}$ be $\sigma$-compact. Then $\struct {S, \tau}$ is also $\sigma$-compact.


Countable Product of Sequentially Compact Spaces is Sequentially Compact

Let $I$ be an indexing set with countable cardinality. Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a family of topological spaces indexed by $I$. Let $\displaystyle \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$. Let each of $\struct {S_\alpha, \tau_\alpha}$ be sequentially compact. Then $\struct {S, \tau}$ is also sequentially compact.


Finite Product of Weakly Locally Compact Spaces is Weakly Locally Compact

Let $n \in \Z_{\ge 0}$ be a (strictly) positive integer. Let $\set {\struct {S_i, \tau_i}: 1 \le i \le n}$ be a finite set of topological spaces. Let $\displaystyle \struct {S, \tau} = \prod_{i \mathop = 1}^n \struct {S_i, \tau_i}$ be the product space of $\set {\struct {S_i, \tau_i}: 1 \le i \le n}$. Let each of $\struct {S_i, \tau_i}$ be weakly locally compact. Then $\struct {S, \tau}$ is also weakly locally compact.


Countable Product of Second-Countable Spaces is Second-Countable

Let $I$ be an indexing set with countable cardinality. Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a family of topological spaces indexed by $I$. Let $\displaystyle \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$. Let each of $\struct {S_\alpha, \tau_\alpha}$ be second-countable. Then $\struct {S, \tau}$ is also second-countable.


Countable Product of Separable Spaces is Separable

Let $I$ be an indexing set with countable cardinality. Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a family of topological spaces indexed by $I$. Let $\displaystyle \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$. Let each of $\struct {S_\alpha, \tau_\alpha}$ be separable. Then $\struct {S, \tau}$ is also separable.


Compactness Properties Preserved under Continuous Surjection

Let $T_A = \strict {S_A, \tau_A}$ and $T_B = \strict {S_B, \tau_B}$ be topological spaces. Let $\phi: T_A \to T_B$ be a continuous surjection. If $T_A$ has one of the following properties, then $T_B$ has the same property: :Compact Space :$\sigma$-Compact Space :Countable Compact Space :Sequential Compact Space :Lindelöf Space


Projection from Product Topology is Open

Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces. Let $T = \struct {T_1 \times T_2, \tau}$ be the product space of $T_1$ and $T_2$, where $\tau$ is the Tychonoff topology on $S$. Let $\pr_1: T \to T_1$ and $\pr_2: T \to T_2$ be the first and second projections from $T$ onto its factors. Then both $\pr_1$ and $\pr_2$ are open.


Weak Local Compactness is Preserved under Open Continuous Surjection

Let $T_A = \struct {S_A, \tau_A}$ and $T_B = \struct {S_B, \tau_B}$ be topological spaces. Let $\phi: T_A \to T_B$ be a continuous mapping which is also an open mapping and a surjection. If $T_A$ is weakly locally compact, then $T_B$ is also weakly locally compact.


Compactness Properties Preserved under Projection Mapping

Let $I$ be an indexing set with countable cardinality. Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a family of topological spaces indexed by $I$. Let $\displaystyle \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$. Let $\pr_\alpha: \struct {S, \tau} \to \struct {S_\alpha, \tau_\alpha}$ be the projection on the $\alpha$ coordinate. Then $\pr_\alpha$ preserves the following compactness properties. That is, if $\struct {S, \tau}$ has one of the following properties, then each of $\struct {S_\alpha, \tau_\alpha}$ has the same property: :Compact Space :$\sigma$-Compact Space :Countably Compact Space :Sequentially Compact Space :Lindelöf Space :Locally Compact Space :Weakly Locally Compact Space :Paracompact Space


Countability Properties Preserved under Projection Mapping

Let $\sequence{\struct{S_\alpha, \tau_\alpha}}$ be a sequence of topological spaces. Let $\displaystyle \struct{S, \tau} = \prod \struct{S_\alpha, \tau_\alpha}$ be the product space of $\sequence {\struct{S_\alpha, \tau_\alpha}}$. Let $\pr_\alpha: \struct{S, \tau} \to \struct{S_\alpha, \tau_\alpha}$ be the projection on the $\alpha$ coordinate. Then $\pr_\alpha$ preserves the following countability properties. That is, if $\struct{S, \tau}$ has one of the following properties, then each of $\struct{S_\alpha, \tau_\alpha}$ has the same property. :Separability :First-Countability :Second-Countability


Countability Axioms Preserved under Open Continuous Surjection

Let $T_A = \left({S_A, \tau_A}\right)$ and $T_B = \left({S_B, \tau_B}\right)$ be topological spaces. Let $\phi: T_A \to T_B$ be a surjective open mapping which is also continuous. If $T_A$ has one of the following properties, then $T_B$ has the same property: :First-Countability :Second-Countability


Connectedness of Points is Equivalence Relation

Let $T = \struct {S, \tau}$ be a topological space. Let $a \sim b $ denote the relation: :$a \sim b \iff a$ is connected to $b$ where $a, b \in S$. Then $\sim$ is an equivalence relation.


Surgery for Rings

Let $R$ and $S$ be commutative rings with unity, and $\phi: R \to S$ a ring monomorphism. Then there is a ring $T$ isomorphic to $S$ that contains $R$ as a subring.


Clopen Set contains Components of All its Points

Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$ be both closed and open in $T$. Then $H$ contains the components of all of its points.


Connectedness Between Two Points is an Equivalence Relation

Let $T = \struct {S, \tau}$ be a topological space. Let $a \sim b $ denote the relation: :$a \sim b \iff \exists S \subseteq T: S$ is connected between the two points  $a$ and $b$ where $a, b \in X$. Then $\sim$ is an equivalence relation.


Quasicomponent is Intersection of Clopen Sets

Let $T = \struct {S, \tau}$ be a topological space. Let $p \in S$. Then the quasicomponent containing $p$ equals the intersection of all sets which are both open and closed which contain $p$.


Localization of Ring Exists

Let $A$ be a commutative ring with unity. Let $S \subseteq A$ be a multiplicatively closed subset with $0 \notin S$. Then there exists a localization $\struct {A_S, \iota}$ of $A$ at $S$.


Transitivity of Integrality

Let $A \subseteq B \subseteq C$ be extensions of commutative rings with unity. Suppose that $C$ is integral over $B$, and $B$ is integral over $A$. Then $C$ is integral over $A$.


Universal Property for Field of Quotients

Let $\struct {D, +, \circ}$ be an integral domain. Let $\struct {F, \oplus, \cdot}$ be a field of quotients of $D$. Then $F$ satisfies the following universal property: There exists a (ring) homomorphism $\iota : D \to F$ such that: ::for every field $\tilde F$ and :and: ::for every (ring) homomorphism $\phi: D \to \tilde F$ :there exists a unique field homomorphism $\psi: F \to \tilde F$ satisfying: :::$\psi \iota = \phi$ That is, the following diagram commutes: :Universal Property Namely we may take: :$\psi: a / b \mapsto \map \phi a \map \phi b^{-1}$


Constant Mapping is Continuous

Let $T_A = \left({A, \tau_A}\right)$ and $T_B = \left({B, \tau_B}\right)$ be topological spaces. Let $b \in B$ be any point in $B$. Let $f_b: A \to B$ be the constant mapping defined by: :$\forall x \in A: f_b \left({x}\right) = b$ Then $f_b$ is continuous.


Path-Connectedness is Equivalence Relation

Let $T = \struct {S, \tau}$ be a topological space. Let $a \sim b $ denote the relation: :$a \sim b \iff a$ is path-connected to $b$ where $a, b \in S$. Then $\sim$ is an equivalence relation.


Arc-Connectedness is Equivalence Relation

Let $T = \struct {S, \tau}$ be a topological space. Let $a \sim b $ denote the relation: :$a \sim b \iff a$ is arc-connected to $b$ where $a, b \in S$. Then $\sim$ is an equivalence relation.


Joining Arcs makes Another Arc

Let $T$ be a topological space. Let $\mathbb I \subseteq \R$ be the closed unit interval $\left[{0 \,.\,.\, 1}\right]$. Let $a,b,c$ be three distinct points of $T$. Let $f, g: \mathbb I \to T$ be arcs in $T$ from $a$ to $b$ and from $b$ to $c$ respectively. Let $h: \mathbb I \to T$ be the mapping given by: :$h \left({x}\right) = \begin{cases} f \left({2 x}\right) & : x \in \left[{0 \,.\,.\, \dfrac 1 2}\right] \\ g \left({2 x - 1}\right) & : x \in \left[{\dfrac 1 2 \,.\,.\, 1}\right] \end{cases}$ Then either: :$h$ is an arc in $T$ or :There exists some restriction of $h$ which, possibly after reparametrisation, is an arc in $T$.


Arc in Topological Space is Path

Let $T = \struct {X, \tau}$ be a topological space. Let $f$ be an arc in $T$. Then $f$ is a path in $T$.


Vector Times Magnitude Same Length As Magnitude Times Vector

Let $\mathbf u$ and $\mathbf v$ be two vectors in the vector space $\struct {G, +_G, \circ}_K$ over a division ring $\struct {K, +_K, \times}$ with subfield $\R$ such that $\R \subseteq \map Z K$ with $\map Z K$ the center of $K$ Let $\norm {\mathbf u}$ and $\norm {\mathbf v}$ be the lengths of $\mathbf u$ and $\mathbf v$ respectively. Then: :$\norm {\paren {\norm {\mathbf v} \circ \mathbf u} } = \norm {\paren {\norm {\mathbf u} \circ \mathbf v} }$


Increasing Sequence in Ordered Set Terminates iff Maximal Element

Let $\left({P, \le}\right)$ be an ordered set. {{TFAE}} : $(1): \quad$ Every increasing sequence $x_1 \leq x_2 \leq x_3 \leq \cdots$ with $x_i \in P$ eventually terminates: there is $n \in \N$ such that $x_n = x_{n+1} = \cdots$. : $(2): \quad$ Every non-empty subset of $P$ has a maximal element.


Characterisation of UFDs

Let $A$ be an integral domain. {{TFAE}} : $(1): \quad A$ is a unique factorisation domain  : $(2): \quad A$ is a GCD domain satisfying the ascending chain condition on principal ideals. : $(3): \quad A$ satisfies the ascending chain condition on principal ideals and every irreducible element of $A$ is a prime element of $A$.


Ultraconnected Space is Path-Connected

Let $T = \struct {S, \tau}$ be a topological space which is ultraconnected. Then $T$ is path-connected.


Continuous Real-Valued Function on Irreducible Space is Constant

Let $T = \struct {S, \tau}$ be a topological space which is irreducible. Let $f: S \to \R$ be a continuous real-valued function. Then $f$ is constant, that is: :$\exists a \in \R: \forall x \in S: \map f x = a$


Irreducible Space is Pseudocompact

Let $T = \struct {S, \tau}$ be a topological space which is irreducible. Then $T$ is pseudocompact.


Arc-Connected Space is Path-Connected

Let $T = \struct {S, \tau}$ be a topological space which is arc-connected. Then $T$ is path-connected.


Irreducible Space is Connected

Let $T = \struct {S, \tau}$ be a topological space which is irreducible. Then $T$ is connected.


Non-Trivial Ultraconnected Space is not T1

Let $T = \struct {S, \tau}$ be a topological space which is ultraconnected. If $S$ has more than one element, then $T$ is not a $T_1$ (Fréchet) space. That is, if $T$ is a $T_1$ (Fréchet) space with more than one element, it is not ultraconnected.


Dot Product of Vector with Itself

:$\mathbf u \cdot \mathbf u = \norm {\mathbf u}^2$


Cosine Formula for Dot Product

Let $\mathbf v,\mathbf w$ be two non-zero vectors in $\R^n$. The dot product of $\mathbf v$ and $\mathbf w$ can be calculated by: :$\mathbf v \cdot \mathbf w = \norm {\mathbf v} \norm {\mathbf w} \cos \theta$ where: :$\norm {\, \cdot \,}$ denotes vector length and :$\theta$ is the angle between $\mathbf v$ and $\mathbf w$.


Quasicomponents and Components are Equal in Locally Connected Space

Let $T = \struct {S, \tau}$ be a topological space which is locally connected. Then $A \subseteq S$ is a component of $T$ {{iff}} $A \subseteq S$ is a quasicomponent of $T$.


Irreducible Space is Locally Connected

Let $T = \struct {S, \tau}$ be a topological space which is irreducible. Then $T$ is locally connected.


Quasicomponents and Path Components are Equal in Locally Path-Connected Space

Let $T = \struct {S, \tau}$ be a topological space which is locally path-connected. Then $A \subseteq S$ is a path component of $T$ {{iff}} $A \subseteq S$ is a quasicomponent of $T$.


Quasicomponents and Arc Components are Equal in Locally Arc-Connected Space

Let $T = \struct {S, \tau}$ be a topological space which is locally arc-connected. Then $A \subseteq S$ is an arc component of $T$ {{iff}} $A$ is a quasicomponent of $T$.


Locally Path-Connected Space is Locally Connected

Let $T = \struct {S, \tau}$ be a topological space which is locally path-connected. Then $T$ is also locally connected.


UFD is GCD Domain

Let $A$ be a unique factorisation domain. Then $A$ is a GCD domain.


Totally Disconnected Space is T1

Let $T = \struct {S, \tau}$ be a topological space which is totally disconnected. Then $T$ is a $T_1$ (Fréchet) space.


Determinant of Linear Operator is Well Defined

Let $V$ be a nontrivial finite dimensional vector space over a field $K$. Let $A: V \to V$ be a linear operator of $V$. Then the determinant $\det A$ of $A$ is well defined, that is, does not depend on the choice of a basis of $V$.


Localization Preserves Integral Closure

Let $A \subseteq B$ be an extension of commutative rings with unity. Let $C$ be the integral closure of $A$ in $B$. Let $S \subseteq A$ be a multiplicatively closed subset. Then $C_S$ is the integral closure of $A_S$ in $B_S$, where subscript $S$ indicates the localization at $S$.


Integrally Closed is Local Property

Let $A$ be an integral domain. For a prime ideal $\mathfrak p$ of $A$, let $A_{\mathfrak p}$ denote the localization at $S = A \divides \mathfrak p$. Then the following are equivalent: :$(1): \quad A$ is integrally closed :$(2): \quad A_{\mathfrak p}$ is integrally closed for all prime ideals $\mathfrak p$. :$(3): \quad A_{\mathfrak m}$ is integrally closed for all maximal ideals $\mathfrak m$.


Totally Disconnected and Locally Connected Space is Discrete

Let $T = \struct {S, \tau}$ be a topological space which is both totally disconnected and locally connected. Then $T$ is the discrete space on $S$.


Totally Separated Space is Completely Hausdorff and Urysohn

Let $T = \struct {S, \tau}$ be a topological space which is totally separated. Then $T$ is completely Hausdorff and Urysohn.


Identity Morphism is Unique

Let $\mathbf C$ be a category, and $X$ and object of $\mathbf C$. Then the identity morphism $\operatorname{id}_X : X \to X$ is unique.


Category of Sets is Category

Let $\mathbf{Set}$ be the category of sets. Then $\mathbf{Set}$ is a metacategory.


Projection from Product Category

Let $\mathcal C$ and $\mathcal D$ be categories. Let $\mathcal C \times \mathcal D$ be the product category. The ''projections'': :$\pi_{\mathcal C} : \mathcal C \times \mathcal D \to \mathcal C : (f,g) \mapsto f$ :$\pi_{\mathcal D} : \mathcal C \times \mathcal D \to \mathcal D : (f,g) \mapsto g$ where $(f,g) \in \operatorname{ob}(\mathcal C \times \mathcal D)$ or $(f,g)\in \operatorname{mor}(\mathcal C \times \mathcal D)$ are functors. Moreover, $\pi_{\mathcal C}$ and $\pi_{\mathcal D}$ satisfy the following universal property: :''For any category $\mathcal E$ and any functors $F : \mathcal E \to \mathcal C$, $G : \mathcal E \to \mathcal D$, there exists a unique functor $H : \mathcal E \to \mathcal C \times \mathcal D$ such that $F = \pi_{\mathcal C}H$ and $G = \pi_{\mathcal D}H$'' That is, the following diagram commutes: :380px {{improve|XyJax?}}


Extremally Disconnected Space is Totally Separated

Let $T = \struct {S, \tau}$ be an extremally disconnected topological space. Then $T$ is totally separated.


Zero Dimensional Space is T3

Let $T = \struct {S, \tau}$ be a zero dimensional topological space. Then $T$ is a $T_3$ space.


Zero Dimensional T0 Space is Totally Separated

Let $T = \struct {S, \tau}$ be a zero dimensional topological space which is also a $T_0$ (Kolmogorov) space. Then $T$ is totally separated.


Scattered T1 Space is Totally Disconnected

Let $T = \struct {S, \tau}$ be a scattered topological space which is also a $T_1$ (Fréchet) space. Then $T$ is totally disconnected.


Non-Trivial Connected Set in T1 Space is Dense-in-itself

Let $T = \struct {S, \tau}$ be a $T_1$ (Fréchet) topological space. Let $H \subseteq S$ be connected in $T$. If $H$ has more than one element, then $H$ is dense-in-itself.


Sequence of Implications of Disconnectedness Properties

Let $P_1$ and $P_2$ be disconnectedness properties and let: :$P_1 \implies P_2$ mean: :If a topological space $T$ satsifies property $P_1$, then $T$ also satisfies property $P_2$. Then the following sequence of implications holds: {| |- | align="center" | Regular|| | align="center" | $\impliedby$ || | align="center" | Zero Dimensional and $T_0$|| | align="center" | $\impliedby$ || | align="center" | Discrete Space || | align="center" | $\implies$ || | align="center" | $\implies$ || | align="center" | $\implies$ || | align="center" | Scattered and $T_1$ || |- | align="center" | || | align="center" | || | align="center" | $\Big\Downarrow$ || | align="center" | || | align="center" | $\Big\Downarrow$ || | align="center" | || | align="center" | || | align="center" | || | align="center" | $\Big\Downarrow$ || |- | align="center" | || | align="center" | || | align="center" | $\Big\Downarrow$ || | align="center" | || | align="center" | Extremally Disconnected || | align="center" | || | align="center" | || | align="center" | || | align="center" | $\Big\Downarrow$ || |- | align="center" | || | align="center" | || | align="center" | $\Big\Downarrow$ || | align="center" | || | align="center" | $\Big\Downarrow$ || | align="center" | || | align="center" | || | align="center" | || | align="center" | $\Big\Downarrow$ || |- | align="center" | || | align="center" | || | align="center" | $\implies$ || | align="center" | $\implies$ || | align="center" | Totally Separated || | align="center" | $\implies$ || | align="center" | Urysohn || | align="center" | || | align="center" | $\Big\Downarrow$ || |- | align="center" | || | align="center" | || | align="center" | || | align="center" | || | align="center" | $\Big\Downarrow$ || | align="center" | || | align="center" | || | align="center" | || | align="center" | $\Big\Downarrow$ || |- | align="center" | || | align="center" | || | align="center" | $T_1$ || | align="center" | $\impliedby$ || | align="center" | Totally Disconnected || | align="center" | $\impliedby$ || | align="center" | $\impliedby$ || | align="center" | $\impliedby$ || | align="center" | $\impliedby$ || |- | align="center" | || | align="center" | || | align="center" | || | align="center" | || | align="center" | $\Big\Downarrow$ || |- | align="center" | || | align="center" | || | align="center" | || | align="center" | || | align="center" | Totally Pathwise Disconnected || |}


Set with Dispersion Point is Biconnected

Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$ be a connected set in $T$. Let $p \in H$ be a dispersion point of $H$. Then $H$ is biconnected.


Totally Disconnected Space is Punctiform

Let $T = \struct {S, \tau}$ be a topological space which is totally disconnected. Then $T$ is punctiform.


Distance in Pseudometric is Non-Negative

Let $X$ be a set on which a pseudometric $d: X \times X \to \R$ has been imposed. Then: :$\forall x, y \in X: \map d {x, y} \ge 0$


Metric Defines Norm iff it Preserves Linear Structure

Let $\struct {k, \norm{\,\cdot\,}_k}$ be a valued field. Let $V$ be a vector space over the valued field $\struct {k, \norm{\,\cdot\,}_k}$. Let $d : V \times V \to k$ be a metric on $V$. Then the function $\norm{ v } := d(v,0)$ is a norm on $V$ if and only if for all $x,y,z \in V$, $\lambda \in k$: :$(1): \quad d(x+z,y+z) = d(x,y)$ (homogeneity or translation invariance) :$(2): \quad d(\lambda x, \lambda y) = \norm{\lambda}_k d(x,y)$ (the enlargement property)


Set of all Sets

Forming the set $\SS$ of all sets leads to a contradiction.


Real Numbers form Perfect Set

Consider the set of real numbers $\R$ as a (complete) metric space with the usual (Euclidean) metric. Then $\R$ forms a perfect set.