Strongly Locally Compact Space is Weakly Locally Compact

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Theorem

Let $T = \left({S, \tau}\right)$ be a strongly locally compact space.


Then $T$ is weakly locally compact.


Proof

Let $T = \left({S, \tau}\right)$ be strongly locally compact.

Let $x \in S$.

By definition, there exists an open set $U_x$ of $T$ such that:

$x \in U_x$
${U_x}^-$ (the closure of $U_x$) is compact.

From Set is Subset of its Topological Closure, $U_x \subseteq {U_x}^-$ and so $x \in {U_x}^-$.

Thus $x$ is contained in a compact neighborhood.

As this holds for all $x$, $T$ is a weakly locally compact.

$\blacksquare$


Also see


Sources