# Definition:Locally Compact Hausdorff Space

## Definition

Let $T = \left({S, \tau}\right)$ be a Hausdorff topological space.

### Definition 1

$T$ is a locally compact Hausdorff space if and only if each point of $S$ has a compact neighborhood.

That is, if and only if $T$ is weakly locally compact.

### Definition 2

$T$ is a locally compact Hausdorff space if and only if each point has a neighborhood basis consisting of compact sets.

That is, if and only if $T$ is locally compact (in the general sense).