Excluded Point Topology is T4/Proof 1

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Theorem

Let $T = \left({S, \tau_{\bar p}}\right)$ be an excluded point space.


Then $T$ is a $T_4$ space.


Proof

We have that an Excluded Point Space is Ultraconnected.

That means none of its closed sets are disjount.

Hence, vacuously, any two of its disjoint closed subsets of $S$ are separated by neighborhoods.

The result follows by definition of $T_4$ space.

$\blacksquare$


Sources