Compact Hausdorff Space is Locally Compact

Theorem

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

Then $T$ is locally compact.

Proof 1

By Compact Space is Weakly Locally Compact, $T$ is weakly locally compact.

Thus $T$ is a locally compact Hausdorff space.

$\blacksquare$

Proof 2

By Neighborhood in Compact Hausdorff Space Contains Compact Neighborhood, every point has a neighborhood basis of compact sets.

$\blacksquare$