Condition for Open Extension Space to be Separable

Theorem
Let $T = \left({S, \tau}\right)$ be a topological space.

Let $T^*_{\bar p} = \left({S^*_p, \tau^*_{\bar p}}\right)$ be the open extension space of $T$.

Then $T^*_{\bar p}$ is a separable space iff $T$ is.

Proof
Let $T = \left({S, \tau}\right)$ be a separable space.

Then there exists a countable subset $H \subseteq S$ which is everywhere dense in $T$.

That is, $H^- = S$ where $H^-$ is the closure of $H$ in $S$.

Hence $H$ is a countable subset $H \subseteq S^*_p$ which is everywhere dense in $T^*_{\bar p}$.

Now suppose $T = \left({S, \tau}\right)$ is not separable.