Compact Hausdorff Space is Locally Compact/Proof 1
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Theorem
Let $T = \left({S, \tau}\right)$ be a Hausdorff space which is compact.
Then $T$ is locally compact.
Proof
By Compact Space is Weakly Locally Compact, $T$ is weakly locally compact.
Thus $T$ is a locally compact Hausdorff space.
$\blacksquare$