Fortissimo Space is T5
Then $T$ is a $T_5$ space.
Let $A, B \in \tau_p$ such that $A$ and $B$ are separated.
If $p \notin A$ and $p \notin B$ then $A$ and $B$ are both open.
Otherwise $p$ must be in exactly one of them, because if $p$ were in both they could not be separated.
Without loss of generality, suppose $p \in A$.
Then $p \notin B$ so $B$ is open.
Now suppose $B$ were not closed.
So $p$ would be in $B^-$, where $B^-$ is the closure of $B$.
But then $p \in A$, by hypothesis.
So $A \cap B^- \ne \O$ and so $A$ and $B$ are not separated.
So $B$ must be closed.