Particular Point Space is T0/Proof 2

Theorem
Let $T = \left({S, \tau_p}\right)$ be a particular point space.

Then $T$ is a $T_0$ (Kolmogorov) space.

Proof
We have:
 * Particular Point Topology is Closed Extension Topology of Discrete Topology


 * Discrete Space satisfies all Separation Properties (including being a $T_0$ space)

Then by Condition for Closed Extension Space to be $T_0$ Space, as a discrete space is $T_0$ then so is its closed extension.