Condition for Closed Extension Space to be T5 Space

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 a $T_5$ space $T$ is a $T_5$ space vacuously, and $T^*_p$ in this case is also a $T_5$ space vacuously.