Existence of T4 Space which is not T3 1/2

Theorem
There exists at least one example of a topological space which is a $T_4$ space, but is not also a $T_{3 \frac 1 2}$ space.

Proof
Let $T$ be a Hjalmar Ekdal space.

From Hjalmar Ekdal Space is $T_4$, $T$ is a $T_4$ space.

From Hjalmar Ekdal Space is not $T_{3 \frac 1 2}$ Space, $T$ is not a $T_{3 \frac 1 2}$ space.

Hence the result.