Metric Space is Weakly Locally Compact iff Strongly Locally Compact

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

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

Proof
We have from Strongly Locally Compact Space is Weakly Locally Compact that strong local compactness implies weak 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 weakly locally compact then it is strongly locally compact.

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

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

Also see

 * Definition:Locally Compact Hausdorff Space