Infinite Particular Point Space is not Weakly Countably Compact

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

Then $T$ is not weakly countably compact.

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

$H$ is not open in $T$ by definition, so from Subsets of Particular Point Space Either Open or Closed it is closed

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 $T = \left({S, \vartheta_p}\right)$ is a finite particular point space, then Finite Space Satisfies All Compactness Properties applies.