Excluded Point Topology is T4

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

Then $T$ is a $T_4$ space.

Proof 1
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 $X$ are separated by neighborhoods.

The result follows by definition of $T_4$ space.

Proof 2

 * Excluded Point Topology is Open Extension Topology of Discrete Topology
 * Open Extension Topology is $T_4$

Proof 3

 * Excluded Point Topology is $T_5$
 * $T_5$ Space is $T_4$