Strongly Locally Compact Space is Weakly Locally Compact

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

Then $T$ is a locally compact space.

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

Let $x \in X$.

By definition, $x \in U_x$ where:
 * $U_x$ is open in $T$
 * $\overline{U_x}$ (the closure of $U_x$) is compact.

From Set is Subset of its Topological Closure, $U_x \subseteq \overline{U_x}$ and so $x \in \overline{U_x}$.

Thus $x$ is contained in a compact neighborhood.

As this holds for all $x$, $T$ is a locally compact space (watch out here: this only holds if $T$ is a Hausdorff-space).