Strongly Locally Compact Space is Weakly Locally Compact

Theorem

Let $T = \struct {S, \tau}$ be a strongly locally compact space.

Then $T$ is weakly locally compact.

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

$\blacksquare$

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