Closed Extension Topology is not T1

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Theorem

Let $T = \left({S, \tau}\right)$ be a topological space.

Let $T^*_p = \left({S^*_p, \tau^*_p}\right)$ be the closed extension space of $T$.


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


Proof

By definition:

$\tau^*_p = \left\{{U \cup \left\{{p}\right\}: U \in \tau}\right\} \cup \left\{{\varnothing}\right\}$


Let $x \in S^*_p, x \ne p$.

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

Then $U \in \tau^*_p$ such that $p \in U, x \notin U$.

But there is no $V \in \tau^*_p$ such that $x \in V, p \notin V$, by definition of the closed extension topology.

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

$\blacksquare$


Sources