Weakly Locally Compact Hausdorff Space is Strongly Locally Compact

Theorem
Let $T = \left({X, \vartheta}\right)$ be a $T_2$ (Hausdorff) space.

Let $T$ be locally compact.

Then $T$ is a strongly locally compact space.

Proof
Let $T = \left({X, \vartheta}\right)$ be a $T_2$ (Hausdorff) space which is locally compact.

Let $x \in X$.

As $T$ is locally compact, $x$ is contained in a compact neighborhood $N_x$.

As $T$ is a $T_2$ (Hausdorff) space, we can use the result Compact Subspace of Hausdorff Space is Closed.

Thus the interior of $N_x$ has a closure which is compact.

Hence the result, from definition of strongly locally compact space.