Strongly Locally Compact Space is Weakly Locally Compact

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

Then $T$ is weakly locally compact.

Proof
Let $T = \left({S, \tau}\right)$ be strongly locally compact.

Let $x \in S$.

By definition, there exists an open set $U_x$ of $T$ such that:
 * $x \in U_x$
 * ${U_x}^-$ (the closure of $U_x$) is compact.

From Set is Subset of its Topological Closure, $U_x \subseteq U_x^-$ and so $x \in U_x^-$.

Thus $x$ is contained in a compact neighborhood.

As this holds for all $x$, $T$ is a weakly locally compact.

Also see

 * Locally Compact Space is Weakly Locally Compact
 * Sequence of Implications of Local Compactness Properties