Open Extension Topology is T4

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

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

Then $T^*_{\bar p}$ is a $T_4$ space.

Proof
We have that an Open Extension Space is Ultraconnected.

That means none of its closed sets are disjoint.

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

The result follows by definition of $T_4$ space.