# Metric Space is Weakly Locally Compact iff Strongly Locally Compact

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## Contents

## Theorem

Let $M = \struct {A, d}$ be a metric space.

Then $M$ is weakly locally compact if and only if $M$ 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.

$\blacksquare$

## Also see

## Sources

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $5$: Metric Spaces