Existence of Tychonoff Space which is not Normal

Theorem

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

Proof

Let $T$ be a Niemytzki's tangent disc space.

From Niemytzki's Tangent Disc Space is Tychonoff, $T$ is a Tychonoff space.

From Niemytzki's Tangent Disc Space is not Normal, $T$ is not a normal 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