Existence of Maximal Compact Topological Space which is not Hausdorff

Theorem
Let $S$ be a set.

Let $\tau$ be the minimal subset of the power set $\mathcal P \left({S}\right)$ such that $\left({S, \tau}\right)$ is a compact topological space.

Then it is not necessarily the case that $\left({S, \tau}\right)$ is a Hausdorff space.

Proof
Let $T = \left({S, \tau}\right)$ be the canonical maximal compact non-Hausdorff space.

This space has been so named on in order to allow reference to it without needing to describe it whenever it is mentioned.

By Canonical Maximal Compact Non-Hausdorff Space is Maximal Compact, $\tau$ is the maximal subset of the power set $\mathcal P \left({S}\right)$ such that $T$ is compact.

By Canonical Maximal Compact Non-Hausdorff Space is not Hausdorff, $T$ is not a Hausdorff space.

Hence the result.