hcju/proofwikips huggingface dataset:
Set is Open iff Disjoint from Boundary
Let $T$ be a topological space, and let $H \subseteq T$. Then $H$ is open in $T$ : :$\partial H \cap H = \O$ 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 $T$ : :$\partial H = \O$ 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 : :$\forall H \subseteq X_1: f \sqbrk {H^-} \subseteq \paren {f \sqbrk H}^-$ where $H^-$ denotes the closure of $H$ in $T_1$. That is,  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  $f^{-1}$ 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  $f$ 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  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  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  $T$ 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.  :$(1): \quad T$ 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  $T$ 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 : :$\map f 1 \ne 0$
Möbius Inversion Formula
Let $f$ and $g$ be arithmetic functions. Then: :$(1): \quad \displaystyle \map f n = \sum_{d \mathop \divides n} \map g d$ : :$(2): \quad \displaystyle \map g n = \sum_{d \mathop \divides n} \map f d \, \map \mu {\frac n d}$ 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: :$\displaystyle \langle \eta, \chi \rangle_G = \sum_{x \mathop \in G} \eta \left({x}\right) \overline{\chi} \left({x}\right)$