Compact Space is Strongly Locally Compact

Theorem
Let $T = \left({X, \vartheta}\right)$ be a compact space.

Then $T$ is a strongly locally compact space.

Proof
Let $T = \left({X, \vartheta}\right)$ be a compact space.

From Open and Closed Sets in a Topological Space, $X$ is clopen in $T$.

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

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

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