I am looking at some topological dimension theory for product spaces, and in trying to construct a certain type of counterexample it's become relevant to consider the question in the title above. I am interested in finding a normal space $X$ whose products with itself is eventually non-normal, but not immediately.
It's not actually important for my application that it happens in three steps as opposed to more. An alternative question would be: Is there a normal space $X$ with $X \times X = Y$ normal, but $Y \times Y$ is not normal?
The original problem is here:
Thanks for any help!
As mentioned in a comment below, if we assume that $X$ is a compact Hausdorff space and that $X \times X \times X$ is completely normal, then $X$ is metrizable. Thus it stands to reason that a compact counterexample may be harder (if not impossible) to construct. The author in the linked paper wonders aloud if the complete normality of $X \times X$ is sufficient for the metrizability of $X$, so it may also be advisable to avoid cases where $X \times X$ is completely normal.