Infinite Particular Point Space is not Weakly Countably Compact

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

Then $T$ is not weakly countably compact.

Proof
By definition, $T$ is weakly countably compact every infinite subset of $S$ has a limit point in $S$.

Let $H \subseteq S$ be an infinite subset of $S$ where $p \notin H$.

$H$ is not open in $T$ by definition.

So from Subset of Particular Point Space is either Open or Closed, $H$ is closed in $T$.

Then we have that a Closed Set in Particular Point Space has no Limit Points.

The result follows from definition of weakly countably compact.

If $T = \struct {S, \tau_p}$ is a finite particular point space, then Finite Space Satisfies All Compactness Properties applies.

Mistakes in Sources
See : $\text{II}: \ 8 - 10: \ 12$ where it is stated that:

Particular Point Space is not Weakly Countably Compact