Locally Compact Space is Weakly Locally Compact

Theorem

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

Then $T$ is weakly locally compact.

We are given $T$ is locally compact, so every point of $S$ has a neighborhood basis consisting of compact sets.

Let $x \in S$ be arbitrary.

From Space is Neighborhood of all its Points, $S$ is a neighborhood of $x$ in $S$.

So, every point contained in $S$ has a compact neighborhood.

Hence, $T$ is weakly locally compact.

$\blacksquare$