# Existence of Completely Normal Space whose Product Space is Not Normal

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## Theorem

There exists at least one example of a completely normal topological space $T$ such that the product space $T \times T$ is not a normal topological space.

## Proof

Let $T$ be the real number line with the right half-open interval topology.

Let $T' = T \times T$ be Sorgenfrey's half-open square topology.

From Right Half-Open Interval Topology is Completely Normal, $T$ is a completely normal space.

From Sorgenfrey's Half-Open Square Topology is Not Normal, $T'$ is not a normal space.

Hence the result.

$\blacksquare$

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{I}: \ \S 2$: Functions, Products, and Subspaces