Local Compactness

Definition

Consider a Topological Space $(X,\mathcal{T})$. The set $X$ is locally compact if $\forall x\in X,\exists \text{compact}C\subset X\text{s.t.}\exists {\mathcal{N}}_x,{\mathcal{N}}_x\subset C$ where ${\mathcal{N}}_x$ is a Neighborhood of $x$.

Examples

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

• $\mathbb{R}$ is not Compact but locally compact ($\because \forall x\in \mathbb{R},\exists \text{compact}[x-1,x+1]\supset (x-1,x+1)\ni x$).
• $\mathbb{Q}$ is not locally compact.

Every Simply Ordered Set $X$ having the Least Upper Bound Property is locally compact $\because$ All closed interval $[a,b]\subset X$ is Compact by the theorem

Facts

Consider a Hausdorff Space $(X,\mathcal{T})$. $X$ is locally compact if and only if given $x\in X$ and Neighborhood $U$ of $x$, there exists another Neighborhood $V$ of $x$ such that its Closure $\bar{V}$ is Compact and is included in $U$ $\forall x\in X,\forall U:={\mathcal{N}}_x,\exists V:={\mathcal{N}}_x\text{s.t.}\bar{V}\text{is compact}\text{and}\bar{V}\subset U$

Consider a locally compact Hausdorff Space $(X,\mathcal{T})$ and a subset $A\subset X$. $A\in \mathcal{T}\text{or}(X\setminus A)\in \mathcal{T}⟹A\text{is locally compact}$

A space $(X,\mathcal{T})$ locally compact Hausdorff Space if and only if $X$ is homeomorphic to an open subspace of a Compact Hausdorff Space.

Every Compact space is locally compact.