Infinite Particular Point Space is not Countably Metacompact

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

Then $T$ is not countably metacompact.

Proof
Suppose $T$ is a countable particular point space.

Let $\CC$ be the open cover of $T$ defined as:
 * $\CC = \set {\set {x, p}: x \in S}$

$\CC$ is countable and has no open refinement except $\CC$ itself.

But $\CC$ is not point finite because $\forall U \in \CC: p \in U$, and $\CC$ is (countably) infinite.

Now suppose $T$ is an uncountable particular point space.

Let $S' \subseteq S$ be a countable subset of $S$.

Let $\CC$ be the open cover of $T$ defined as:
 * $\CC = \set {\set {x, p}: x \in S'} \cup \set {S \setminus S' \cup \set p}$

$\CC$ is also countable and has no open refinement except $\CC$ itself.

And similarly, $\CC$ is not point finite because $\forall U \in \CC: p \in U$, and $\CC$ is (countably) infinite.

Hence the result by definition of countably metacompact.