Infinite Particular Point Space is not Compact

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

Then $T$ is not compact.

Proof
Consider the open cover of $T$:
 * $\CC = \set {\set {x, p}: x \in S, x \ne p}$

As $S$ is infinite, then so is $\CC$, as we can set up a bijection from $\phi: S \setminus \set p \leftrightarrow \CC$:
 * $\forall x \in S \setminus \set p: \map \phi x = \set {x, p}$

Hence $\CC$ has no finite subcover.

The result follows by definition of compactness.