Countable Complement Space is not Weakly Countably Compact

Theorem
Let $T = \left({S, \tau}\right)$ be a countable complement topology on an uncountable set $S$.

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 $S$.

From Limit Points of Countable Complement Space, if $H \subseteq S$ is countable, it contains all its limit points.