Infinite Particular Point Space is not Strongly Locally Compact

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Let $T = \struct {S, \tau_p}$ be an infinite particular point space.

Then $T$ is not strongly locally compact.


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.