Particular Point Space is Weakly Locally Compact

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

Then $T$ is weakly locally compact.

Proof
By Compact Space in Particular Point Space, $\set p$ is compact in $T$.

But by definition $\set p$ is open in $T$.

So $p$ is contained in a compact neighborhood, that is, $\set p$.

Now let $x \in S: x \ne p$.

Then $\set {x, p}$ is open in $T$.

We have that $\set {x, p}$ trivially has an open cover, that is, $\set {\set {x, p} }$ itself.

Hence any open cover of $\set {x, p}$ has a finite subcover: any one set that contains $x$ and $p$ is a cover for $\set {x, p}$.

So $\set {x, p}$ is a neighborhood of $x$ which is compact in $T$.

So $x$ is contained in a compact neighborhood.

Hence the result, by definition of weakly locally compact.

Also see

 * Particular Point Space is Locally Compact