Locally Compact Metric Space is Strongly Locally Compact

Theorem

Let $M = \left({A, d}\right)$ be a metric space.

Then $M$ is locally compact iff $M$ is strongly locally compact.

Proof

We have from Strongly Locally Compact Space is Locally Compact that strong local compactness implies local compactness in all topological spaces regardless of whether they are metric spaces or not.

So all we need to do is demonstrate that if $M$ is locally compact then it is strongly locally compact.

We have that a metric space is a $T_2$ (Hausdorff) space.

The result follows from Locally Compact Hausdorff Space is Strongly Locally Compact.

$\blacksquare$

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 5$