Existence of Semiregular Topological Space which is not Completely Hausdorff

Theorem

There exists at least one example of a semiregular topological space which is not a completely Hausdorff space.

Proof

Let $T$ be a simplified Arens square.

From Simplified Arens Square is Semiregular, $T$ is a semiregular space.

From Simplified Arens Square is not Completely Hausdorff, $T$ is not a $T_3$ 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: Additional Separation Properties