Existence of Regular Space which is not Tychonoff

Theorem

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

Proof

Let $T$ be a Tychonoff corkscrew.

From Tychonoff Corkscrew is Regular, $T$ is a regular space.

From Tychonoff Corkscrew is not Completely Regular, $T$ is not a Tychonoff 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