Closed Extension Topology is not Hausdorff

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

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

Then $T^*_p$ is not a $T_2$ (Hausdorf) space.

Proof
$T^*_p$ is not a $T_2$ (Hausdorf) space.

From $T_2$ Space is $T_1$ Space, $T^*_p$ is a $T_1$ (Fréchet) space.

But this contradicts Closed Extension Topology is not $T_1$

Hence by Proof by Contradiction $T^*_p$ can not be a $T_2$ (Hausdorf) space.