Definition 4.20.1 (Locally Compact Hausdorff Space). Let $X$ be a Hausdorff space, then the following are equivalent:
For any $x \in X$, there exists $K \in \cn(x)$ compact.
For any $x \in X$, $\cn(x)$ admits a fundamental system of neighbourhoods consisting of compact sets.
For any $x \in X$, $\cn(x)$ admits a fundamental system of neighbourhoods consisting of precompact sets.
If the above holds, then $X$ is a locally compact Hausdorff (LCH) space.
Proof. (1) $\Rightarrow$ (2): Let $K \in \cn(x)$ be compact and $U \in \cn(x)$, then $\overline{U \cap K}$ is closed. By Proposition 4.16.3, $K$ itself is closed, so $\overline{U \cap K}\subset K$ is a closed subset of a compact set, and compact by Proposition 4.16.2.
(2) $\Rightarrow$ (3): Let $U \in \cn(x)$, then there exists $K \in \cn(x)$ with $x \in K \subset U$. By Proposition 4.16.3, $K$ is closed, so $\overline{K^o}\subset K$ is compact by Proposition 4.16.2.$\square$