Definition 3.13.1 (Locally Compact Hausdorff Space).label Let $X$ be a Hausdorff space, then the following are equivalent:
- (1)
For any $x\in X$, there exists $K\in\cali{N}(x)$ is compact.
- (2)
For any $x\in X$, $\cali{N}(x)$ admits a fundamental system of neighborhoods consisting of compact sets.
- (3)
For any $x\in X$, $\cali{N}(x)$ admits a fundamental system of neighborhoods consisting of precompact sets.
If the above holds, then $X$ is a locally compact Hausdorff space
Proof. (1)$\implies$(2) Let $K\in\cali{N}(x)$ be compact and $U\in\cali{N}^{o}(x)$ then there exists open sets $V\in\cali{N}(x)$ such that $V\suf K$ so $V\cap K\in\cali{N}^{o}(x)$ in the compact Hausdorff space $K$. By Proposition 3.10.6 there exists an open set $W\suf K$ satisfying
where $\cl{W}$ is unambigous by Proposition 3.13.2. Since $K$ is compact, so is $\cl{W}$.
(2)$\implies$(3) Let $U\in\cali{N}(x)$, then there exists $K\in\cali{N}(x)$ compact such that $x\in K\suf U$. Since $X$ is Hausdorff, $K$ is closed so $\cl{K^o}\suf K$ is compact.
For any $x\in X$, $\cali{N}(x)\neq\emp$ hence (2)/(3)$\implies$(1).$\square$
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