Weakly Locally Compact Hausdorff Space is Strongly Locally Compact

Theorem

Let $T = \left({S, \tau}\right)$ be a $T_2$ (Hausdorff) space.

Let $T$ be weakly locally compact.

Then $T$ is strongly locally compact.

Proof

Let $x \in S$.

As $T$ is weakly 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.

$\blacksquare$

Also see

• Existence of Weakly Locally Compact Space which is not Strongly Locally Compact

Sources

• 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology ... (previous) ... (next): $\text{I}: \ \S 3$: Localized Compactness Properties