# Compact Space is Strongly Locally Compact

From ProofWiki

## Theorem

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

Then $T$ is a strongly locally compact space.

## Proof

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

From Underlying Set of Topological Space is Clopen, $S$ is clopen in $T$.

From Closed Set Equals its Closure, $S = S^-$.

So every point of $S$ is contained in an open set (that is, $S$) whose closure (that is, $S$ again) is compact (as $T = \left({S, \tau}\right)$ itself is compact).

That is precisely the definition of a strongly locally compact space.

$\blacksquare$

## Sources

- Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(1970)... (previous)... (next): $\text{I}: \ \S 3$: Localized Compactness Properties