Compact Space in Particular Point Space

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

Then $\set p$ is compact in $T$.

Proof
Any open cover of $\set p$ has a finite subcover: any single set that contains $p$ is a cover for $\set p$.

So $\set p$ is compact in $T$.

Also see

 * Finite Topological Space is Compact