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 $\powerset S$ such that $\struct {S, \tau}$ is a compact topological space.

Then it is not necessarily the case that $\struct {S, \tau}$ is a Hausdorff space.

Proof

Let $T = \struct {S, \tau}$ be the canonical maximal compact non-Hausdorff space.

This space has been so named on $\mathsf{Pr} \infty \mathsf{fWiki}$ 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 $\powerset S$ such that $T$ is compact.

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

Hence the result.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $3$: Compactness: Compactness Properties and the $T_i$ Axioms