Open Extension Topology is not T3

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Let $T = \struct {S, \tau}$ be a topological space.

Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the open extension space of $T$.

Then $T^*_{\bar p}$ is not a $T_3$ space.


As $S$ is (trivially) open in $T$, it is also open in $T^*_{\bar p}$.

As $S^*_p = S \cup \set p$, it follows that $\set p$ is closed in $T^*_{\bar p}$.

But by definition, the only open set of $T^*_{\bar p}$ that contains $\set p$ is $S^*_p$ itself.

So there can be no $x \in S^*_p: x \notin \set p$ contained in an open set of $T^*_{\bar p}$ which is disjoint from $S^*_p$.

Hence the result.