hcju/proofwikips huggingface dataset:
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 $T$  $A$ 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.
Closed Interval in Reals is Uncountable
Let $a, b$ be extended real numbers such that $a < b$. Then the closed interval $\set {x \in \R : a \le x \le b} \subseteq \R$ is uncountable.
Metric Space is T5
Let $M = \struct {A, d}$ be a metric space. Then $M$ is a $T_5$ space.
Metric Space is Perfectly T4
Let $M = \struct {A, d}$ be a metric space. Then $M$ is a perfectly $T_4$ space.
Metric Space fulfils all Separation Axioms
Let $M = \struct {A, d}$ be a metric space. Then $M$, considered as a topological space, fulfils all separation axioms: :$M$ is a $T_0$ (Kolmogorov) space :$M$ is a $T_1$ (Fréchet) space :$M$ is a $T_2$ (Hausdorff) space :$M$ is a semiregular space :$M$ is a $T_{2 \frac 1 2}$ (completely Hausdorff) space  :$M$ is a $T_3$ space  :$M$ is a regular space :$M$ is an Urysohn space :$M$ is a $T_{3 \frac 1 2}$ space :$M$ is a Tychonoff (completely regular) space :$M$ is a $T_4$ space :$M$ is a normal space :$M$ is a $T_5$ space  :$M$ is a completely normal space :$M$ is a perfectly $T_4$ space :$M$ is a perfectly normal space
Metric Space is Fully T4
Let $M = \struct {A, d}$ be a metric space. Then $M$ is a fully $T_4$ space.
Metric Space is Fully Normal
Let $M = \struct {A, d}$ be a metric space. Then $M$ is a fully normal space.
Metric Space is Paracompact
Let $M = \struct {A, d}$ be a metric space. Then $M$ is a paracompact space.
Metric Space is First-Countable
Let $M = \struct {A, d}$ be a metric space. Then $M$ is first-countable.
Metric Space is Separable iff Second-Countable
A metric space is separable  it is second-countable.
Metric Space is Lindelöf iff Second-Countable
Let $M = \struct {A, d}$ be a metric space. Then $M$ is Lindelöf  $M$ is second-countable.
Metric Space is Countably Compact iff Sequentially Compact
Let $M$ be a metric space. Then $M$ is countably compact  $M$ is sequentially compact.
Metric Space is Weakly Countably Compact iff Countably Compact
Let $M = \struct {A, d}$ be a metric space. Then $M$ is weakly countably compact  $M$ is countably compact.
Countably Compact Metric Space is Compact
Let $M = \struct {A, d}$ be a metric space. Let $M$ be countably compact. Then $M$ is compact.
Metric Space is Weakly Locally Compact iff Strongly Locally Compact
Let $M = \struct {A, d}$ be a metric space. Then $M$ is weakly locally compact  $M$ is strongly locally compact.
Sequence of Implications of Metric Space Compactness Properties
Let $P_1$ and $P_2$ be compactness properties and let: :$P_1 \implies P_2$ mean: :If a metric space $M$ satsifies property $P_1$, then $M$ also satisfies property $P_2$. Then the following sequence of implications holds: {| |- | align="center" | Sequentially Compact || | align="center" | $\implies$ || | align="center" | Weakly $\sigma$-Locally Compact || | align="center" | $\implies$ || | align="center" | Weakly Locally Compact || |- | align="center" | $\Big\Updownarrow$ || | align="center" | || | align="center" | $\Big\Downarrow$ || | align="center" | || | align="center" | $\Big\Updownarrow$ || |- | align="center" | Countably Compact || | align="center" | || | align="center" | $\sigma$-Compact || | align="center" | || | align="center" | Strongly Locally Compact || |- | align="center" | $\Big\Updownarrow$ || | align="center" | || | align="center" | $\Big\Downarrow$ || |- | align="center" | Compact || | align="center" | || | align="center" | Separable || |- | align="center" | $\Big\Updownarrow$ || | align="center" | || | align="center" | $\Big\Updownarrow$ || |- | align="center" | Weakly Countably Compact || | align="center" | || | align="center" | Lindelöf || |- | align="center" | || | align="center" | || | align="center" | $\Big\Updownarrow$ || |- | align="center" | || | align="center" | || | align="center" | Second-Countable || |}
Extremally Disconnected Metric Space is Discrete
Let $M = \struct {A, d}$ be a metric space which is extremally disconnected. Then $M$ is the discrete topology.
Indicator is Well-Defined
Let $G$ be a finite group, and $a \in G$. Let $H$ be a subgroup of $G$. Then the indicator of $a$ in $H$ is well defined.
Compact Metric Space is Totally Bounded
Let $M = \struct {A, d}$ be a metric space which is compact. Then $M$ is totally bounded.
Subgroup Generated by Subgroup and Element
Let $G$ be a finite abelian group. Let $H$ be a proper subgroup of $G$. Let $a \in G \setminus H$. Let $n$ be the indicator of $a$ in $H$. Then: :$K = \left\{{x a^k: x \in H, \ 0 \le k < n}\right\}$ is a subgroup of $G$ such that $H \subseteq K$, and each element of $K$ has a unique representation in this form. Moreover: :$K = \left\langle{H, a}\right\rangle$ where $\left\langle{H, a}\right\rangle$ denotes the subgroup generated by $\left\{{H, a}\right\}$, and: :$\left\vert{K}\right\vert = n \left\vert{H}\right\vert$ where $\left\vert{K}\right\vert$ denotes the order of $K$.
Number of Characters on Finite Abelian Group
Let $G$ be a finite abelian group. Then the number of characters $G \to \C^\times$ is $\left\vert{G}\right\vert$.
Sequentially Compact Metric Space is Complete
Let $M = \struct {A, d}$ be a metric space which is sequentially compact. Then $M$ is complete.
Topological Completeness is Topological Property
Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces which are homeomorphic. Let $T_1$ be topologically complete. Then $T_2$ is also topologically complete.
Topological Completeness is Weakly Hereditary
Let $T = \struct {S, \tau}$ be a topological space which is topologically complete. Let $V \subseteq S$ be a closed subspace of $T$. Then $V$ is also topologically complete. That is, topological completeness is weakly hereditary.
Metric Space Compact iff Complete in All Equivalent Metrics
Let $M_1 = \struct {A, d_1}$ be a metric space. Then $M_1$ is compact  $M_2 = \struct {A, d_2}$ is a complete metric space whenever $d_2$ is equivalent to $d_1$.
Urysohn's Metrization Theorem
Let $T = \struct {S, \tau}$ be a topological space which is regular and second-countable. Then $T$ is metrizable.
Metrizable iff Regular and has Sigma-Locally Finite Basis
Let $T = \struct {S, \tau}$ be a topological space. Then $T$ is metrizable  $T$ is regular and has a $\sigma$-locally finite basis.
Uniformity iff Quasiuniformity has Symmetric Basis
Let $S$ be a set. Let $\UU$ be a quasiuniformity on $S$. Then $\UU$ is a uniformity  $\UU$ has a symmetric filter basis.
Quasiuniformity Induces Topology
Let $\UU$ be a quasiuniformity on a set $S$. Let $\tau \subseteq \powerset S$ be a subset of the power set of $S$, created from $\UU$ by: :$\tau := \set {\map u x: u \in \UU, x \in S}$ where: :$\forall x \in S: \map u x = \set {y: \tuple {x, y} \in u}$ That is, where $\map u x$ is the image of $x$ under $u$, where $u$ is considered as a relation on $S$. Then $\tau$ is a topology on $S$. That is, the quasiuniform space $\struct {\struct {S, \UU}, \tau}$ is also the topological space $\struct {S, \tau}$.
Topological Space is Quasiuniformizable
Every topological space is quasiuniformizable.
T3 1/2 Space is Uniformizable
Let $T$ be a topological space. Then $T$ is uniformizable  $T$ is a $T_{3 \frac 1 2}$ space.
Pseudometric Space generates Uniformity
Let $P = \struct {A, d}$ be a pseudometric space. Let $\UU$ be the set of sets defined as: :$\UU := \set {u \in A \times A: \exists \epsilon \in \R_{>0}: u_\epsilon \subseteq u}$ where: :$\R_{>0}$ is the set of strictly positive real numbers :$u_\epsilon$ is defined as: ::$u_\epsilon := \set {\tuple {x, y}: \map d {x, y} < \epsilon}$ Then $\UU$ is a uniformity on $A$ which generates a uniform space with the same topology as the topology induced by $d$.
Definition:Pseudometrizable Uniformity
Let $P = \struct {A, d}$ be a pseudometric space. Let $\UU$ be the uniformity on $A$ defined as: :$\UU := \set {u_\epsilon: \epsilon \in \R_{>0} }$ where: :$\R_{>0}$ is the set of strictly positive real numbers :$u_\epsilon$ is defined as: ::$u_\epsilon := \set {\tuple {x, y}: \map d {x, y} < \epsilon}$ Then $\UU$ is defined as '''pseudometrizable'''.
Metric Space generates Uniformity
Let $M = \struct {A, d}$ be a metric space. Let $\UU$ be the set of sets defined as: :$\UU := \set {u \in A \times A: \exists \epsilon \in \R_{>0}: u_\epsilon \subseteq u}$ where: :$\R_{>0}$ is the set of strictly positive real numbers :$u_\epsilon$ is defined as: ::$u_\epsilon := \set {\tuple {x, y}: \map d {x, y} < \epsilon}$ Then $\UU$ is a uniformity on $A$ which generates a uniform space with the same topology as the topology induced by $d$.
Discrete Topology is Topology
:$\tau$ is a topology on $S$.
Discrete Topology is Finest Topology
:$\tau$ is the finest topology on $S$.
Set in Discrete Topology is Clopen
:$\forall U \subseteq S: U$ is both closed and open in $\struct {S, \tau}$.
Topological Space is Discrete iff All Points are Isolated
:$\tau$ is the discrete topology on $S$  all points in $S$ are isolated points of $T$.
Point in Discrete Space is Adherent Point
Let $T = \struct {S, \tau}$ be a discrete topological space. Let $U \subseteq S$. Then $x$ is an adherent point of $U$  $x \in U$.
Interior Equals Closure of Subset of Discrete Space
Let $A \subseteq S$. Then: :$A = A^\circ = A^-$ where: :$A^\circ$ is the interior of $A$ :$A^-$ is the closure of $A$.
Mapping from Discrete Space is Continuous
Let $T_1 = \struct {S_1, \tau_1}$ be the discrete topological space on $S_1$. Let $T_2 = \struct {S_2, \tau_2}$ be any other topological space. Let $\phi: S_1 \to S_2$ be a mapping. Then $\phi$ is continuous.
Exponential Function is Superfunction
The function $f : \C \to \C$, defined as :$f \left({ z }\right) = c^z$ is a superfunction for any complex number $c$.
Standard Discrete Metric induces Discrete Topology
Let $M = \struct {A, d}$ be the (standard) discrete metric space on $A$. Then $d$ induces the discrete topology on $A$. Thus the discrete topology is metrizable.
Discrete Space satisfies all Separation Properties
Let $T = \struct {S, \powerset S}$ be the discrete topological space on $S$. Then $T$ fulfils all separation axioms: :$T$ is a $T_0$ (Kolmogorov) space :$T$ is a $T_1$ (Fréchet) space :$T$ is a $T_2$ (Hausdorff) space :$T$ is a semiregular space :$T$ is a $T_{2 \frac 1 2}$ (completely Hausdorff) space  :$T$ is a $T_3$ space  :$T$ is a regular space :$T$ is an Urysohn space :$T$ is a $T_{3 \frac 1 2}$ space :$T$ is a Tychonoff (completely regular) space :$T$ is a $T_4$ space :$T$ is a normal space :$T$ is a $T_5$ space  :$T$ is a completely normal space :$T$ is a perfectly $T_4$ space :$T$ is a perfectly normal space
Point in Discrete Space is Neighborhood
Let $T = \struct {S, \tau}$ be a discrete topological space. Let $x \in S$. Then $\set x$ is a neighborhood of $x$ in $T$.
Discrete Space is Strongly Locally Compact
Let $T = \struct {S, \tau}$ be a discrete topological space. Then $T$ is strongly locally compact.
Discrete Space is First-Countable
Let $T = \struct {S, \tau}$ be a discrete topological space. Then $T$ is first-countable.
Discrete Space has Open Locally Finite Cover
Let $T = \struct {S, \tau}$ be a discrete topological space. Consider the set $\CC$ of all singleton subsets of $S$: :$\CC := \set {\set x: x \in S}$ Then $\CC$ is an open cover of $T$ which is locally finite. This cover is the finest cover on $S$. That is, if $\VV$ is a cover of $T$, then $\CC$ is a refinement of $\VV$.
Discrete Space is Paracompact
Let $T = \struct {S, \tau}$ be a discrete topological space. Then $T$ is paracompact.
Singleton Set in Discrete Space is Compact
Let $T = \struct {S, \tau}$ be a topological space where $\tau$ is the discrete topology on $S$. Let $x \in S$. Then $\set x$ is compact.
Countable Discrete Space is Sigma-Compact
Let $T = \struct {S, \tau}$ be a countable discrete topological space. Then $T$ is $\sigma$-compact.
Countable Discrete Space is Lindelöf
Let $T = \struct {S, \tau}$ be a countable discrete topological space. Then $T$ is a Lindelöf space.
Basis for Discrete Topology
Let $S$ be a set. Let $\tau$ be the discrete topology on $S$. Let $\BB$ be the set of all singleton subsets of $S$: :$\BB := \set {\set x: x \in S}$. Then $\BB$ is a basis for $T$.
Countable Discrete Space is Second-Countable
Let $T = \struct {S, \tau}$ be a countable discrete topological space. Then $T$ is second-countable.
Countable Discrete Space is Separable
Let $T = \struct {S, \tau}$ be a countable discrete topological space. Then $T$ is separable.
Uncountable Discrete Space is not Separable
Let $T = \struct {S, \tau}$ be an uncountable discrete topological space. Then $T$ is not separable.
Uncountable Discrete Space is not Second-Countable
Let $T = \struct {S, \tau}$ be an uncountable discrete topological space. Then $T$ is not second-countable.
Uncountable Discrete Space is not Lindelöf
Let $T = \struct {S, \tau}$ be an uncountable discrete topological space. Then $T$ is not a Lindelöf space.
Uncountable Discrete Space is not Sigma-Compact
Let $T = \struct {S, \tau}$ be an uncountable discrete topological space. Then $T$ is not $\sigma$-compact.
Discrete Space is Compact iff Finite
Let $T = \left({S, \tau}\right)$ be a topological space where $\tau$ is the discrete topology on $S$. Then $T$ is compact  $S$ is a finite set, thereby making $\tau$ the finite discrete topology on $S$.
Finite Space is Sequentially Compact
Let $T = \struct {S, \tau}$ be a topological space where $S$ is a finite set. Then $T$ is sequentially compact.
Discrete Space is Fully Normal
Let $T = \struct {S, \tau}$ be a discrete topological space. Then $T$ is fully normal.
Discrete Space is Fully T4
Let $T = \struct {S, \tau}$ be a discrete topological space. Then $T$ is fully $T_4$.
Finite Topological Space is Compact
Let $T = \struct {S, \tau}$ be a topological space where $S$ is a finite set. Then $T$ is compact.
Finite Space is Second-Countable
Let $T = \struct {S, \tau}$ be a topological space where $S$ is a finite set. Then $T$ is a second-countable space.
Finite Discrete Space satisfies all Compactness Properties
Let $T = \struct {S, \tau}$ be a finite discrete topological space. Then $T$ satisfies the following compactness properties: :$T$ is compact. :$T$ is Sequentially Compact. :$T$ is Countably Compact. :$T$ is Weakly Countably Compact. :$T$ is a Lindelöf Space :$T$ is Pseudocompact. :$T$ is $\sigma$-Compact. :$T$ is Locally Compact. :$T$ is Weakly Locally Compact. :$T$ is Strongly Locally Compact. :$T$ is $\sigma$-Locally Compact. :$T$ is Weakly $\sigma$-Locally Compact. :$T$ is Fully Normal. :$T$ is Fully $T_4$. :$T$ is Paracompact. :$T$ is Countably Paracompact. :$T$ is Metacompact. :$T$ is Countably Metacompact.
Finite Space Satisfies All Compactness Properties
Let $T = \struct {S, \tau}$ be a topological space where $S$ is a finite set. Then $T$ satisfies the following compactness properties: : $T$ is compact. : $T$ is sequentially compact. : $T$ is countably compact. : $T$ is weakly countably compact. : $T$ is a Lindelöf space. : $T$ is pseudocompact. : $T$ is $\sigma$-compact. : $T$ is strongly locally compact. : $T$ is $\sigma$-locally compact. : $T$ is weakly $\sigma$-locally compact. : $T$ is locally compact. : $T$ is weakly locally compact. : $T$ is paracompact. : $T$ is countably paracompact. : $T$ is metacompact. : $T$ is countably metacompact.
Discrete Space is not Dense-In-Itself
Let $T = \struct {S, \tau}$ be a discrete topological space. Then $T$ is not dense-in-itself.
Non-Trivial Discrete Space is not Connected
Let $T = \struct {S, \tau}$ be a non-trivial discrete topological space. Then $T$ is not connected. Thus also:
Discrete Space is Locally Path-Connected
Let $T = \struct {S, \tau}$ be a discrete topological space. Then $T$ is locally path-connected.
Point is Path-Connected to Itself
Let $T = \left({S, \tau}\right)$ be a topological space. Let $a \in S$. Then $a$ is path-connected to itself.
Discrete Uniformity is Uniformity
Let $S$ be a set. Let $\UU$ be the discrete uniformity on $S$ Then $\UU$ is indeed a uniformity.
Indiscrete Topology is Topology
:$\tau$ is a topology on $S$.
Indiscrete Topology is Coarsest Topology
:$\tau$ is the coarsest topology on $S$.
Open and Closed Sets in Indiscrete Topology
Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space. Let $H \subseteq S$.
Subset of Indiscrete Space is Compact and Sequentially Compact
Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space. Let $H \subseteq S$.