Fortissimo Space is not Separable

Theorem
Let $T = \struct {S, \tau_p}$ be the fortissimo space 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 the fortissimo space, $U$ is closed.

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

Thus, by definition, $U$ is not everywhere dense in $T$.

Thus, there exists no countable subset of $S$ which is everywhere dense in $T$.

So, by definition, $T$ is not a separable space.