# Infinite Particular Point Space is not Strongly Locally Compact

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## Theorem

Let $T = \left({S, \tau_p}\right)$ be an infinite particular point space.

Then $T$ is not strongly locally compact.

## Proof

By definition, $T$ is strongly locally compact if and only if every point of $S$ is contained in an open set whose closure is compact.

Let $x \in S: x \ne p$.

Let $x \in U$ where $U$ is open in $T$.

From Closure of Open Set of Particular Point Space we have:

- $U^- = S$

where $U^-$ is the closure of $U$.

But from Infinite Particular Point Space is not Compact we have that $S$ is not compact.

Hence the result, by definition of strongly locally compact.

$\blacksquare$

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{II}: \ 9 - 10: \ 5, 15$