Existence of Compact Hausdorff Space which is not T5

Theorem

There exists at least one example of a compact $T_2$ (Hausdorff) space which is not a $T_5$ space.

Proof

Let $T$ be the Tychonoff plank.

From Tychonoff Plank is Compact, $T$ is a compact topological space.

From Tychonoff Plank is Hausdorff, $T$ is a $T_2$ (Hausdorff) space.

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

From $T_2$ Space is $T_1$ Space, $T$ is also a $T_1$ (Fréchet) space.

As $T$ is a $T_1$ space, it follows by definition of completely normal space that $T$ is not a $T_5$ space.

$\blacksquare$

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