Non-Trivial Particular Point Topology is not T1

Theorem
Let $T = \left({S, \vartheta_p}\right)$ be a particular point space such that $S$ is not a singleton.

Then $T$ is not a $T_1$ (Fréchet) space.

Proof
Let $x, p \in S, x \ne p$.

Let $U = \left\{{p}\right\}$.

Then $U \in \vartheta_p$ such that $p \in U, x \notin U$.

But there is no $V \in \vartheta_p$ such that $x \in v, p \notin V$, by definition of the particular point topology.

Hence $T$ can not be a $T_1$ (Fréchet) space.