# Non-Trivial Particular Point Topology is not T1

## Theorem

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

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

## Proof

Follows directly from:

Particular Point Topology is Closed Extension Topology of Discrete Topology
Closed Extension Topology is not $T_1$

$\blacksquare$