Compactness

Definition

Consider a Topological Space $(X,\mathcal{T})$. The space is compact if every open covering (a collection of open subsets $\{U_i{\}}_{i\in I}$ of $X$ whose union is $X$) of $X$ has a finite subcovering. $\forall \{U_i{\}}_{i\in I}\subset \mathcal{T}\text{s.t.}{\bigcup }_{i\in I}{U}_i=X,\exists J\subset I\text{s.t.}|J|<\infty \text{and}{\bigcup }_{j\in J}{U}_j=X$

Examples

Consider a Standard Topology on real numbers $(\mathbb{R},\mathcal{U})$. Then, the subspace Topologies

• $\{\frac{1}{n}|n\in \mathbb{N}\}\subset \mathbb{R}$ is not compact

• $\{0\}\cup \{\frac{1}{n}|n\in \mathbb{N}\}\subset \mathbb{R}$ is compact

• $(0,1]\subset \mathbb{R}$ is not compact

• $\mathbb{R}$ is not compact

• $[0,1]\subset \mathbb{R}$ is compact (closed and bounded)

Facts

Consider a Subspace Topology $(Y,{\mathcal{T}}_Y)$ of a Topological Space $(X,\mathcal{T})$. Then, $Y$ is compact if and only if every covering of $Y$ by open sets in $X$ has a finite subcollection covering $Y$. $\forall \{U_i{\}}_{i\in I}\subset {\mathcal{T}}_Y\text{s.t.}{\bigcup }_{i\in I}{U}_i=Y,\exists J\subset I\text{s.t.}|J|<\infty \text{and}{\bigcup }_{j\in J}{U}_j=Y⟺\forall \{U_i{\}}_{i\in I}\subset \mathcal{T}\text{s.t.}{\bigcup }_{i\in I}{U}_i=Y,\exists J\subset I\text{s.t.}|J|<\infty \text{and}{\bigcup }_{j\in J}{U}_j=Y$

A closed subspace of a compact space is compact.

A compact subspace of a Hausdorff Space is closed.

The image of a compact space under a continuous map is compact.

Consider a Bijective continuous function $f:(X,{\mathcal{T}}_X)\to (Y,{\mathcal{T}}_Y)$ between two topological spaces. If $X$ is a compact space and $Y$ is a Hausdorff Space, then $f$ is a Homeomorphism

Tychonoff's Theorem

Definition

The Product Topology of any collection of compact topological spaces is compact.

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Consider a Topological Space $(X,\mathcal{T})$. $X$ is compact if and only if for every collection $\{C_i{\}}_{i\in I}\text{s.t.}(X\setminus C_i)\in \mathcal{T}$ of closed sets in $X$ that have the Finite Intersection Property, its entire intersection is a non-empty set $\bigcap _{i\in I}{C}_i\ne \varnothing$.

Every closed interval in Real Number $\mathbb{R}$ is compact.

Heine-Borel Theorem

Definition

Consider a Standard Topology on a Euclidean Space $({\mathbb{R}}^n,\mathcal{U})$ and a subset $A\subset {\mathbb{R}}^n$ Then, $A$ is closed and bounded if and only if $A$ is compact

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Consider a Topological Space $(X,\mathcal{T})$. If $X$ is compact then $X$ is Limit Point Compact

Consider a metrizable space $(X,\mathcal{T})$. Then, $X$ is compact if and only if $X$ is Limit Point Compact if and only if $X$ is Sequentially Compact

Every Compact metrizable space $(X,\mathcal{T})$ has a countable basis.

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A Compact Hausdorff Space is a normal space.

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Consider a Hausdorff Space $(X,\mathcal{T})$ and disjoint Compact subsets $A,B\subset X\text{s.t.}A\cap B=\varnothing$. Then, there exists disjoint open sets $U,V\in \mathcal{T}\text{s.t.}U\cap V=\varnothing$ satisfying $A\subset U$ and $B\subset V$.

Every compact space is Countably Compact.

Every Compact space has Bolzano–Weierstrass property.

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Every Compact space is locally compact.

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A Topological Space $(X,\mathcal{T})$ is Compact if $X$ has a basis $\mathcal{B}$ such that every open covering of $X$ by elements of $\mathcal{S}$ has a finite subcover.

Alexander Subbasis Theorem

Definition

A Topological Space $(X,\mathcal{T})$ is Compact if and only if $X$ has a Subbasis $\mathcal{S}$ such that every open covering of $X$ by elements of $\mathcal{S}$ has a finite subcover.

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