hcju/proofwikips huggingface dataset:
Excluded Point Topology is Topology
Let $T = \left({S, \tau_{\bar p}}\right)$ be an excluded point space. Then $\tau_{\bar p}$ is a topology on $S$, and $T$ is a topological space.
Excluded Point Topology is Open Extension Topology of Discrete Topology
Let $S$ be a set and let $p \in S$. Let $\tau_{\bar p}$ be the excluded point topology on $S$. Let $T = \struct {S \setminus \set p, \tau_D}$ be the discrete topological space on $S \setminus \set p$. Then $T^* = \struct {S, \tau_{\bar p} }$ is an open extension space of $T$.
Product of Countable Discrete Space with Sierpiński Space is Paracompact
Let $T_X = \struct {S, \tau}$ be a countable discrete space. Let $T_Y = \struct {\set {0, 1}, \tau_0}$ be a Sierpiński space. Let $T_X \times T_Y$ be the product space of $T_X$ and $T_Y$. Then $T_X \times T_Y$ is paracompact.
Reduction of Explicit ODE to First Order System
Let $\map {x^{\paren n} } t = \map F {t, x, x', \ldots, x^{\paren {n - 1} } }$, $\map x {t_0} = x_0$ be an explicit ODE with $x \in \R^m$. Let there exist $I \subseteq \R$ such that there exists a unique particular solution: :$x: I \to \R^m$ to this ODE. Then there exists a system of first order ODEs: :$y' = \map {\tilde F} {t, y}$ with $y = \tuple {y_1, \ldots, y_{m n} }^T \in \R^{m n}$ such that: :$\tuple {\map {y_1} t, \ldots, \map {y_m} t} = \map x t$ for all $t \in I$ and $\map y {t_0} = x_0$.
Open Continuous Image of Paracompact Space is not always Countably Metacompact
Let $T_A = \struct {X_A, \tau_A}$ be a topological space which is paracompact. Let $T_B = \struct {X_B, \tau_B}$ be another topological space. Let $\phi: T_A \to T_B$ be a mapping which is both continuous and open. Then it is not necessarily even the case that $T_B$ is countably metacompact, let alone paracompact.
Fixed Point Formulation of Explicit ODE
Let $x' = \map f {t, x}$ with $\map x {t_0} = x_0$ be an explicit ODE of dimension $n$. For $a, b \in \R$, let $\mathcal X = \map {\mathcal C} {\closedint a b; \R^n}$ be the space of continuous functions on the closed interval $\closedint a b$. Let $T: \mathcal X \to \mathcal X$ be the map defined by: :$\displaystyle \map {\paren {T x} } t = x_0 + \int_{t_0}^t \map f {s, \map x s} \rd s$ {{explain|Definition of $\paren {T x}$}} Then a fixed point of $T$ in $\mathcal X$ is a solution to the above ODE.
Cardinality of Infinite Union of Infinite Sets
Let $\kappa$ be an infinite cardinal. Let $X_i$ be sets of cardinality at most $\kappa$ indexed by a set $I$ of cardinality at most $\kappa$. Then their union $\displaystyle \bigcup_{i \mathop \in I} X_i$ has cardinality at most $\kappa$. Furthermore, if at least one of the $X_i$ is size $\kappa$, then the union has cardinality $\kappa$.
Well-Ordering Theorem
Every set is well-orderable.
Well-Ordering Theorem is Equivalent to Axiom of Choice
The Well-Ordering Theorem holds  the Axiom of Choice holds. That is, every set can be well-ordered  every collection of sets has a choice function.
Tychonoff Topology is Coarsest Topology such that Projections are Continuous
Let $\mathbb X = \family {\struct {X_i, \tau_i} }_{i \mathop \in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set. Let $X$ be the cartesian product of $\mathbb X$: :$\displaystyle X := \prod_{i \mathop \in I} X_i$ Let $\tau$ be the Tychonoff topology on $X$. For each $i \in I$, let $\pr_i : X \to X_i$ be the corresponding projection which maps each ordered tuple in $X$ to the corresponding element in $X_i$: :$\forall x \in X: \map {\pr_i} x = x_i$ Then $\tau$ is the coarsest topology on $X$ such that all the $\pr_i$ are continuous.
Condition for Closed Extension Space to be T0 Space
Let $T = \struct {S, \tau}$ be a topological space. Let $T^*_p = \struct {S^*_p, \tau^*_p}$ be the closed extension space of $T$. Then $T^*_p$ is a $T_0$ (Kolmogorov) space  $T$ is.
Excluded Point Space is T0
Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space. Then $T$ is a $T_0$ (Kolmogorov) space.
Open Extension Topology is not T1
Let $T = \struct {S, \tau}$ be a topological space. Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the open extension space of $T$. Then $T^*_{\bar p}$ is not a $T_1$ (Fréchet) space.
Extreme Value Theorem
Let $X$ be a compact metric space and $Y$ a normed vector space. Let $f: X \to Y$ be a continuous mapping. Then $f$ is bounded, and there exist $x, y \in X$ such that: :$\forall z \in X: \norm {\map f x} \le \norm {\map f z} \le \norm {\map f y}$ where $\norm {\map f x}$ denotes the norm of $\map f x$. Moreover, $\norm f$ attains its minimum and maximum.
Continuous Functions on Compact Space form Banach Space
Let $X$ be a compact Hausdorff space. Let $Y$ be a Banach space. Let $\CC = \CC \struct {X; Y}$ be the set of all continuous mappings $X \to Y$. {{explain|Work out what convention ProofWiki has already evolved for $\CC \struct {X; Y}$.}} Let $\norm {\,\cdot\,}_\infty$ be the supremum norm on $\CC$. Then $\struct {\CC, \norm {\,\cdot\,} }$ is a Banach space.
Supremum Norm is Norm
Let $S$ be a set. Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space over $K \in \set {\R, \C}$. Let $\BB$ be the set of bounded mappings $S \to X$. Let $\norm {\, \cdot \,}_\infty$ be the supremum norm on $\BB$. Then $\norm {\, \cdot \,}_\infty$ is a norm on $\BB$. {{MissingLinks|Add a link that establishes that $\BB$ is a vector space}}
One-Step Vector Subspace Test
Let $V$ be a vector space over a division ring $K$. Let $U \subseteq V$ be a non-empty subset of $V$ such that: :$\forall u, v \in U: \forall \lambda \in K: u + \lambda v \in U$ Then $U$ is a subspace of $V$.
Ax-Grothendieck Theorem
Let $f: \C^n \to \C^n$ be a polynomial map. Let $f$ be injective. Then $f$ is surjective.
Condition for Open Extension Space to be T0 Space
Let $T = \struct {S, \tau}$ be a topological space. Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the open extension space of $T$. Then $T^*_{\bar p}$ is a $T_0$ (Kolmogorov) space  $T$ is.
Non-Trivial Excluded Point Topology is not T1
Let $T = \struct {S, \tau_{\bar p} }$ be a excluded point space such that $S$ is not a singleton. Then $T$ is not a $T_1$ (Fréchet) space.
Limit Points in Open Extension Space
Let $T = \struct {S, \tau}$ be a topological space. Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the open extension space of $T$. Let $x \in S$. Then $p$ is a limit point of $x$.
Limit Points in Excluded Point Space
Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space. Let $x \in S$ such that $x \ne p$. Then $p$ is the only limit point of $x$. Similarly, let $U \subseteq S$. Then $p$ is the only limit point of $U$.
Excluded Point Space is T5
Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space. Then $T$ is a $T_5$ space.
Excluded Point Space is Compact
Let $T = \left({S, \tau_{\bar p}}\right)$ be an excluded point space. Then $T$ is a compact space.
Open Extension Space is Compact
Let $T = \struct {S, \tau}$ be a topological space. Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the open extension space of $T$. Then $T^*_{\bar p}$ is a compact space.
Open Extension Space is Connected
Let $T = \struct {S, \tau}$ be a topological space. Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the open extension space of $T$. Then $T^*_{\bar p}$ is a connected space.
Excluded Point Space is Connected
Let $T = \left({S, \tau_{\bar p}}\right)$ be an excluded point space. Then $T^*_{\bar p}$ is a connected space.
Open Extension Space is Ultraconnected
Let $T = \struct {S, \tau}$ be a topological space. Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the open extension space of $T$. Then $T^*_{\bar p}$ is ultraconnected.
Excluded Point Space is Ultraconnected
Let $T = \left({S, \tau_{\bar p}}\right)$ be an excluded point space. Then $T$ is ultraconnected.
Excluded Point Space is not Irreducible
Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space with at least three points. Then $T^*_{\bar p}$ is not irreducible.
Excluded Point Space is Path-Connected
Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space. Then $T^*_{\bar p}$ is path-connected.
Open Extension Space is Path-Connected
Let $T = \struct {S, \tau}$ be a topological space. Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the open extension space of $T$. Then $T^*_{\bar p}$ is path-connected.
Excluded Point Space is not Arc-Connected
Let $S$ be a set with at least two distinct elements. Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space. Then $T$ is not arc-connected.
Basis for Excluded Point Space
Let $T = \left({S, \tau_{\bar p}}\right)$ be an excluded point space. Consider the set $\mathcal B$ defined as: :$\mathcal B = \left\{{\left\{{x}\right\}: x \in S \setminus \left\{{p}\right\}}\right\} \cup \left\{{S}\right\}$ Then $B$ is a basis for $S$.
Partition Topology is Zero Dimensional
Let $T = \struct {S, \tau}$ be a partition space. Then $T$ is zero dimensional.
Discrete Space is Zero Dimensional
Let $T = \struct {S, \tau}$ be a discrete space. Then $T$ is zero dimensional.
Discrete Space is Scattered
Let $T = \struct {S, \tau}$ be a topological space where $\tau$ is the discrete topology on $S$. Then $T$ is a scattered space.
Discrete Space is Extremally Disconnected
Let $T = \struct {S, \tau}$ be a topological space where $\tau$ is the discrete topology on $S$. Then $T$ is extremally disconnected.
Totally Separated Space is Totally Disconnected
Let $T = \struct {S, \tau}$ be a topological space which is totally separated. Then $T$ is totally disconnected.
Open Extension Topology is not Perfectly T4
Let $T = \struct {S, \tau}$ be a topological space. Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the open extension space of $T$. Then $T^*_{\bar p}$ is not a perfectly $T_4$ space.
Excluded Point Space is not Perfectly T4
Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space. Then $T$ is not a perfectly $T_4$ space.
Type is Realized in some Elementary Extension
Let $\mathcal M$ be an $\mathcal L$-structure. Let $A$ be a subset of the universe of $\mathcal M$. Let $p$ be an $n$-type over $A$. There exists an elementary extension of $\mathcal M$ which realizes $p$.
Subset of Excluded Point Space is not Dense-in-itself
Let $T = \struct {S, \tau_{\bar p} }$ be a excluded point space such that $S$ is not a singleton. Let $H \subseteq S$. Then $H$ is not dense-in-itself.
Excluded Point Space is Scattered
Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space. Then $T$ is a scattered space.
Dispersion Point of Excluded Point Space
Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space. Then $p$ is a dispersion point of $T$.
Excluded Point Space is First-Countable
Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space. Then $T$ is first-countable.
Countable Excluded Point Space is Second-Countable
Let $T = \struct {S, \tau_{\bar p} }$ be a countable excluded point space. Then $T$ is a second-countable space.
Uncountable Excluded Point Space is not Second-Countable
Let $T = \struct {S, \tau_{\bar p} }$ be an uncountable excluded point space. Then $T$ is not second-countable.
Fractional Sobolev Embedding Theorem
Let $S'$ denote the space of tempered distributions. Let $\mathcal F : S' \to S'$ denote the Fourier transform. For each $s \in \R$ and $p \in [1, \infty]$ let $W^{s,p} \left({\R^n}\right) = \left\{{u \in S': \langle \xi \rangle^s \hat{u} \in L^p \left({\R^n}\right)}\right\}$ where $\displaystyle \left \langle{\xi}\right \rangle = \left({1 + \left|{\xi}\right|^2}\right)^\frac 1 2$. Then : 1. If $s > t$ then $W^{s,p} \left({\R^n}\right)$ embeds continuously into $W^{t,q}\left({\R^n}\right)$ where $q$ is given by $\displaystyle \frac 1 q = \frac 1 p - \frac {s-t} n$.  {{namedfor|Sergei Lvovich Sobolev}}
Countable Excluded Point Space is Separable
Let $T = \struct {S, \tau_{\bar p} }$ be a countable excluded point space. Then $T$ is a separable space.
Uncountable Excluded Point Space is not Separable
Let $T = \struct {S, \tau_{\bar p} }$ be an uncountable excluded point space. Then $T$ is not separable.
Excluded Set Topology is Topology
Let $T = \struct {S, \tau_{\bar H} }$ be an excluded set space. Then $\tau_{\bar H}$ is a topology on $S$, and $T$ is a topological space.
Either-Or Topology is Topology
Let $T = \struct {S, \tau}$ be the either-or space. Then $\tau$ is a topology on $T$.
Closed Sets of Either-Or Topology
Let $T = \struct {S, \tau}$ be the either-or space. Then the closed sets of $T$ are: :$\O$ :$S$ :$\set {-1}$ :$\set 1$ :$\set {-1, 1}$ :Any subset of $\closedint {-1} 1$ containing $\set 0$ as a subset.
Type Space is Compact
Let $\mathcal M$ be an $\mathcal L$-structure, and let $A$ be a subset of the universe of  $\mathcal M$. The type space $S_n^{\mathcal M}(A)$ of $n$-types over $A$ is compact.
Omitting Types Theorem
Let $\mathcal L$ be the language of predicate logic with a countable signature. Let $T$ be an $\mathcal L$-theory. Let $\left\{{p_i: i \in \N}\right\}$ be a countable set of non-isolated $n$-types of $T$. There is a countable $\mathcal L$-structure $\mathcal M$ such that $\mathcal M \models T$ and $\mathcal M$ omits each $p_i$.
Vinogradov's Theorem
Let $\Lambda$ be the von Mangoldt function. For $N \in \Z$, let: :$\displaystyle \map R N = \sum_{n_1 + n_2 + n_3 \mathop = N} \map \Lambda {n_1} \, \map \Lambda {n_2} \, \map \Lambda {n_3}$ be a weighted count of the number of representations of $N$ as a sum of three prime powers. Let $\SS$ be the arithmetic function: :$\displaystyle \map \SS N = \prod_{p \mathop \nmid N} \paren {1 + \frac 1 {\paren {p - 1}^3} } \prod_{p \mathop \divides N} \paren {1 - \frac 1 {\paren {p - 1}^2} }$ where: :$p$ ranges over the primes :$p \nmid N$ denotes that $p$ is not a divisor of $N$ :$p \divides N$ denotes that $p$ is a divisor of $N$. Then for any $A > 0$ and sufficiently large odd integers $N$: :$\map R N = \dfrac 1 2 \map \SS N N^2 + \map \OO {\dfrac {N^2} {\paren {\log N}^A} }$ where $\OO$ denotes big-O notation.
Vinogradov's Theorem/Major Arcs
Let $B \in \R_{>0}$. Then: :$\displaystyle \int_\mathcal M \map F \alpha^3 \map e {-N \alpha} \rd \alpha = \frac {N^2} 2 \map {\mathcal S} N + \map {\mathcal O} {\frac {N^2} {\paren {\ln N}^{B/2} } }$ where the implied constant depends only on $B$.
Exponential Dominates Polynomial
Let $\exp$ denote the real exponential function. For any fixed $k \in \N$ and $\alpha > 0$ there exists $N \in \N$ such that $x^k < \exp \left({\alpha x}\right)$ for all real $x > N$. {{expand|it seems that the proof would apply to $x^k, x \in \R$ without any changes to the proof. Changes made as suggested Nov. 2016.}}
Power Dominates Logarithm
Let $\epsilon \in \R_{>0}$. Let $B \in \N$ be arbitrary. Then there exists $N \in \N$ such that: :$\forall n > N: \left({\ln n}\right)^B < n^\epsilon$
Condition for Open Extension Space to be T5 Space
Let $T = \struct {S, \tau}$ be a topological space. Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the open extension space of $T$. Then $T^*_{\bar p}$ is a $T_5$ space  $T$ is a $T_5$ space.
Open Extension Topology is T4
Let $T = \struct {S, \tau}$ be a topological space. Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the open extension space of $T$. Then $T^*_{\bar p}$ is a $T_4$ space.
Open Extension Topology is not T3
Let $T = \struct {S, \tau}$ be a topological space. Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the open extension space of $T$. Then $T^*_{\bar p}$ is not a $T_3$ space.
Excluded Point Topology is not T3
Let $T = \struct {S, \tau_{\bar p} }$ be a excluded point space. Then $T$ is not a $T_3$ space.
Condition for Open Extension Space to be Separable
Let $T = \struct {S, \tau}$ be a topological space. Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the open extension space of $T$. Then $T^*_{\bar p}$ is a separable space  $T$ is.
Finitely Satisfiable Theory has Maximal Finitely Satisfiable Extension
Let $T$ be a finitely satisfiable $\mathcal L$-theory. Then there exists a finitely satisfiable $\mathcal L$-theory $T'$ which contains $T$ as a subset such that: :for all $\mathcal L$-sentences $\phi$, either $\phi \in T'$ or $\neg \phi \in T'$.
Condition for Open Extension Space to be Second-Countable
Let $T = \struct {S, \tau}$ be a topological space. Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the open extension space of $T$. Then $T^*_{\bar p}$ is a second-countable space  $T$ is.
Limit Points of Either-Or Topology
Let $T = \struct {S, \tau}$ be the either-or space. Let $H \subseteq S$ be any subset of $S$. Then no element of $S$ can be a limit point of $H$ except $0$.
Either-Or Topology is T5
Let $T = \struct {S, \tau}$ be the either-or space. Then $T$ is a $T_5$ space.
Either-Or Topology is Lindelöf
Let $T = \struct {S, \tau}$ be the either-or space. Then $T$ is a Lindelöf space.
Subspace of Either-Or Space less Zero is not Lindelöf
Let $T = \struct {S, \tau}$ be the either-or space. Let $H = S \setminus \set 0$ be the set $S$ without zero. Then the topological subspace $T_H = \struct {H, \tau_H}$ is not a Lindelöf space.
Basis for Either-Or Topology
Let $T = \left({S, \tau}\right)$ be the either-or space. Let $\mathcal B$ be the set: :$\mathcal B := \left\{{\left\{{x}\right\}: x \in S, x \ne 0}\right\} \cup \left\{{ \left({-1 \,.\,.\, 1}\right) }\right\}$ ... that is, the set of all singleton subsets of $S$ less $\left\{{0}\right\}$ and including the open real interval $\left({-1 \,.\,.\, 1}\right)$. Then $\mathcal B$ is a basis for $T$.
Either-Or Topology is Locally Path-Connected
Let $T = \struct {S, \tau}$ be the either-or space. Then $T$ is a locally path-connected space.
Limit Points of Infinite Subset of Finite Complement Space
Let $T = \struct {S, \tau}$ be a finite complement space. Let $H \subseteq S$ be an infinite subset of $S$. Then every point of $S$ is a limit point of $H$.
Finite Complement Topology is Separable
Let $T = \struct {S, \tau}$ be a finite complement topology on an infinite set $S$. Then $T$ is a separable space.
Closure of Infinite Subset of Finite Complement Space
Let $T = \struct {S, \tau}$ be a finite complement space. Let $H \subseteq S$ be an infinite subset of $S$. Then $H^- = S$ where $H^-$ is the closure of $S$.
Subspace of Finite Complement Topology is Compact
Let $T = \struct {S, \tau}$ be a finite complement topology on an infinite set $S$. Then every topological subspace of $T$, including $T$ itself, is a compact space.
Kluyver's Formula for Ramanujan's Sum
Let $q \in \N_{>0}$. Let $n \in \N$. Let $\map {c_q} n$ be Ramanujan's sum. Let $\mu$ denote the Möbius function. Then: :$\displaystyle \map {c_q} n = \sum_{d \mathop \divides \gcd \set {q, n} } d \map \mu {\frac q d}$ where $\divides$ denotes divisibility.
Condition for Complex Root of Unity to be Primitive
Let $n, k \in \N$. Then $\alpha_k = \map \exp {\dfrac {2 \pi i k} n}$ is a primitive $n$th root of unity  $\gcd \set {n, k} = 1$.
Relation of Boubaker Polynomials to Chebyshev Polynomials
The Boubaker polynomials are related to Chebyshev polynomials through the equations: : $B_n \left({2x}\right) = \dfrac {4x}n \dfrac{\mathrm d}{\mathrm d x} T_n \left({x}\right) - 2 T_n \left({x}\right)$ : $B_n \left({2x}\right) = -2 T_n \left({x}\right) + 4 x U_{n-1} \left({x}\right)$ where: : $T_n$ denotes the Chebyshev polynomials of the first kind : $U_n$ denotes the Chebyshev polynomials of the second kind.
Relation of Boubaker Polynomials to Fermat Polynomials
The Boubaker polynomials are related to Fermat polynomials by: : $B_n \left({x}\right) = \dfrac 1 {\left({\sqrt 2}\right)^n} F_n \left({\dfrac {2 \sqrt 2 x} 3}\right) + \dfrac 3 {\left({\sqrt 2}\right)^{n-2}} F_{n-2} \left({\dfrac {2 \sqrt 2 x} 3}\right): \quad n = 0, 1, 2, \ldots$
F-Sigma and G-Delta Subsets of Uncountable Finite Complement Space
Let $T = \struct {S, \tau}$ be a finite complement topology on an uncountable set $S$. Then countably infinite subsets of $S$ are $F_\sigma$ sets and are neither open nor closed sets. Their relative complements in $S$ are $G_\delta$ sets, and are also neither open nor closed sets.
Skolem's Paradox
Let $\mathcal L$ be a countable first-order language. Let $T$ be an $\mathcal L$-theory which axiomatizes some version of set theory (for example, ZFC). There is a countable model of $T$.
Uncountable Finite Complement Space is not First-Countable
Let $T = \struct {S, \tau}$ be a finite complement topology on an uncountable set $S$. Then $T$ is not first-countable.
Separable Space need not be First-Countable
Let $T = \struct {S, \tau}$ be a topological space which is separable. Then $T$ does not necessarily have to be first-countable.
Asymptotic Growth of Euler Phi Function
Let $\phi$ be the Euler $\phi$ function. For any $\epsilon > 0$ and sufficiently large $n$: :$n^{1 - \epsilon} < \phi \left({n}\right) < n$
Finite Complement Space is T1
Let $T = \struct {S, \tau}$ be a finite complement topology on any set $S$ which contains at least two points. Then $T$ is a $T_1$ (Fréchet) space.
Finite Complement Space is Irreducible
Let $T = \struct {S, \tau}$ be a finite complement topology on an infinite set $S$. Then $T$ is irreducible.
Infinite Subset of Finite Complement Space Intersects Open Sets
Let $T = \struct {S, \tau}$ be a finite complement topology on an infinite set $S$. Let $H \subseteq S$ be an infinite subset of $S$. Then the intersection of $H$ with any non-empty open set of $T$ is infinite.
Irreducible Hausdorff Space is Singleton
Let $T = \left({S, \tau}\right)$ be a topological space which is irreducible and Hausdorff. Then $T$ contains only one point.
Finite Complement Space is not T2
Let $T = \struct {S, \tau}$ be a finite complement topology on an infinite set $S$. Then $T$ is not a $T_2$ (Hausdorff) space.
Finite Complement Space is not T3, T4 or T5
Let $T = \struct {S, \tau}$ be a finite complement topology on an infinite set $S$. Then $T$ is not a $T_3$ space, $T_4$ space or $T_5$ space.
Double Pointed Finite Complement Topology is Compact
Let $T = \struct {S, \tau}$ be a finite complement topology on an infinite set $S$. Let $T \times D$ be the double pointed topology on $T$. Then $T \times D$ is compact.
Double Pointed Finite Complement Topology fulfils no Separation Axioms
Let $T = \struct {S, \tau}$ be a finite complement topology on an infinite set $S$. Let $T \times D$ be the double pointed topology on $T$. Then $T \times D$ is not a $T_0$ space, $T_1$ space, $T_2$ space, $T_3$ space, $T_4$ space or $T_5$ space.
Vaughan's Identity
Let $\Lambda$ be von Mangoldt's function. Let $\mu$ be the Möbius function. Then for $y, z \ge 1$ and $n > z$: :$\displaystyle \map \Lambda n = \sum_{\substack {d \mathop \divides n \\ d \mathop \le y}} \map \mu d \, \map \ln {\frac n d} - \mathop {\sum \sum}_{\substack {d c \mathop \divides n \\ d \mathop \le y, \, c \mathop \le z}} \map \mu d \, \map \Lambda c + \mathop {\sum \sum}_{\substack {d c \mathop \divides n \\ d \mathop > y, \, c \mathop > z}} \map \mu d \, \map \Lambda c$ where $\divides$ denotes divisibility.
Sum Over Divisors of von Mangoldt is Logarithm
Let $\Lambda$ be von Mangoldt's function. Then for $n \geq 1$ :$\displaystyle \sum_{d | n} \Lambda(d) = \log n$
Multiplicative Function that Converges to Zero on Prime Powers
Let $f$ be a multiplicative function such that: :$\displaystyle \lim_{p^k \mathop \to \infty} \map f {p^k} = 0$ where $p^k$ runs though ''all'' prime powers. Then: :$\displaystyle \lim_{n \mathop \to \infty} \map f n = 0$ where $n$ runs through the integers.
Finite T1 Space is Discrete
Let $S$ be a finite set. Let $T = \left({S, \tau}\right)$ be a $T_1$ (Fréchet) space. Then $\tau$ is the discrete topology on $S$.
Finite Complement Topology is Minimal T1 Topology
Let $T = \struct {S, \tau}$ be a finite complement space. Let $\tau'$ be a topology on $S$ such that $T' = \struct {S, \tau'}$ is a $T_1$ (Fréchet) space. Then $\tau$ is comparable with $\tau'$ such that $\tau$ is coarser than $\tau'$. That is, of all the topologies on $S$ fulfilling the $T_1$ separation axiom, the finite complement space is the smallest. Thus the finite complement topology is known as the '''minimal $T_1$ topology''' on any given set.
Finite Complement Space is Connected
Let $T = \struct {S, \tau}$ be a finite complement topology on an infinite set $S$. Then $T$ is a connected space.