Compact Space in Particular Point Space

Theorem
Let $T = \left({S, \vartheta_p}\right)$ be a particular point space.

Then $\left\{{p}\right\}$ is compact in $T$.

Proof
$\left\{{p}\right\}$ has an open cover, trivially, i.e. $\left\{{\left\{{p}\right\}}\right\}$ itself.

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

So $\left\{{p}\right\}$ is compact in $T$.