Existence of Completely Normal Space whose Product Space is Not Normal

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.