Non-Trivial Particular Point Topology is not T3
Then $T$ is not a $T_3$ space.
We have that there are at least two distinct elements of $S$.
So, consider $x, p \in S: x \ne p$.
Then $X = \set x$ is closed in $T$ and $p \notin X$.
Suppose $U \in \tau_p$ is an open set in $T$ such that $X \subseteq U$.
We have that $\set p \in \tau_p$ such that $p \in \set p$.
But as $p \in U, p \in \set p$ we have that $U \cap \set p \ne \O$.
So $T$ is not a $T_3$ space.