Countable Fort Space is Perfectly Normal

Theorem
Let $T = \left({S, \tau_p}\right)$ be a Fort space on a countably infinite set $S$.

Then $T$ is a perfectly normal space.

Proof
Let's proof first that every closed set is a $G_\delta$ set.


 * Let $H \subseteq S$ be closed in $T$.


 * Then by definition of Fort space, $H$ is finite or contains $p$.


 * Consider the family of sets defined by $\mathcal D=\{S\setminus\{z\}\ :\ z\notin H\}$. This collection is countable (because $S$ is countable) and is formed by open sets by definition.


 * $H\subseteq {\displaystyle\bigcap_{V\in\mathcal D}V}\ $ because if $r\in H$, then $r\in S\setminus\{z\}$ for all $z\notin H$.


 * Even more, $H= {\displaystyle\bigcap_{V\in\mathcal D}V}$, because if $z\notin H$, then $z\notin S\setminus\{z\}\in\mathcal D$.


 * Thus $H$ is a $G_\delta$

From Fort Space is T5 and T5 Space is T4 Space, we get that Fort space is $T_4$.

Finally from Fort Space is T1 the result by definition