Existence of Normal Space which is not Completely Normal

Theorem

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

Proof

Let $T$ be a Tychonoff plank.

From Tychonoff Plank is Normal, $T$ is a normal space.

From Tychonoff Plank is not Completely Normal, $T$ is not a completely 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