Infinite Particular Point Space is not Strongly Locally Compact

From ProofWiki
Jump to navigation Jump to search

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