Existence of Completely Hausdorff Space which is not Regular

Theorem

There exists at least one example of a topological space which is a completely Hausdorff space, but is not also a regular space.

Proof

Let $T$ be a half-disc space.

From Half-Disc Space is Completely Hausdorff, $T$ is a completely Hausdorff space.

From Half-Disc Space is not Regular, $T$ is not a regular 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 $2$: Separation Axioms: Completely Regular Spaces