Existence of Minimal Hausdorff Space which is not Compact

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 Hausdorff space.

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

Proof
Let $T = \left({S, \tau}\right)$ be the canonical minimal Hausdorff non-compact 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 Minimal Hausdorff Non-Compact Space is Minimal Hausdorff, $\tau$ is the minimal subset of the power set $\mathcal P \left({S}\right)$ such that $T$ is a Hausdorff space.

By Canonical Minimal Hausdorff Non-Compact Space is not Compact, $T$ is not a compact topological space.

Hence the result.