Compact Space is Strongly Locally Compact

Theorem

Let $T = \struct {S, \tau}$ be a compact space.

Then $T$ is a strongly locally compact space.

Proof

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 $T = \struct {S, \tau}$ itself is compact).

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

$\blacksquare$

Also see

• Compact Space is Weakly Locally Compact

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $3$: Compactness: Localized Compactness Properties