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 Sorgenfrey line.
Let $T' = T \times T$ be Sorgenfrey's half-open square topology.
From Sorgenfrey Line satisfies all Separation Axioms, $T$ 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
- 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: Functions, Products, and Subspaces