Non-Trivial Particular Point Topology is not T1

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Theorem

Let $T = \struct {S, \tau_p}$ 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 $T_1$.

$\blacksquare$


Sources