Clopen Points in Fort Space

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

Let $q \in S: q \ne p$.

Then $\left\{{q}\right\}$ is both open and closed in $T$.

$\left\{{p}\right\}$ itself is closed, but not open.

Proof
We have that $\left\{{q}\right\}$ is finite so $\complement_S \left({\left\{{q}\right\}}\right)$ is cofinite.

So $\complement_S \left({\left\{{q}\right\}}\right)$ is open and so $\left\{{q}\right\}$ is closed.

Then we have that $p \notin \left\{{q}\right\}$ so $\left\{{q}\right\}$ is open.

However, $p \notin \complement_S \left({\left\{{p}\right\}}\right)$ and $\complement_S \left({\left\{{p}\right\}}\right)$ is infinite.

So $\left\{{p}\right\}$ is not open in $S$.

But as $\left\{{p}\right\}$ is finite so $\complement_S \left({\left\{{q}\right\}}\right)$ is cofinite.

Hence, as for $\left\{{q}\right\}$, we have that $\left\{{p}\right\}$ is closed.