Double Pointed Countable Complement Topology is Weakly Countably Compact

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

Let $T \times D$ be the double pointed topology on $T$.

Then $T \times D$ is weakly countably compact.

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

Let $D = \left\{{0, 1}\right\}$.

Let $\left({p, 0}\right)$ belong to some infinite $A \subseteq S$.

Then its twin $\left({p, 1}\right)$ is a limit point of $A$.

Hence the result by definition of weakly countably compact.