Compact Space is Strongly Locally Compact

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 Topological Space is Open and Closed in Itself, $S$ 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