# Weakly Locally Compact Hausdorff Space is Strongly Locally Compact

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## Contents

## Theorem

Let $T = \struct {S, \tau}$ 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

## Sources

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $3$: Compactness: Localized Compactness Properties