Countable Complement Space is not Separable

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

Then $T$ is not a separable space.

Proof
Let $U$ be a countable subset of $S$.

By the definition of $T$, $U$ is closed.

From Closed Set Equals its Closure, $U^-=U\ne S$.

Thus, there is no countable set dense in $T$.