Partition Topology is not Hausdorff

Theorem
Let $S$ be a set and let $\PP$ be a partition on $S$ which is not the (trivial) partition of singletons.

Let $T = \struct {S, \tau}$ be the partition space whose basis is $\PP$.

Then $T$ is not a $T_2$ (Hausdorff) space.

Proof
$T$ is a $T_2$ (Hausdorff) space.

Then from $T_2$ Space is $T_1$ Space, $T$ is a $T_1$ (Fréchet) space.

This contradicts the result Partition Topology is not $T_1$.

Hence the result, by Proof by Contradiction.