# If $X \times X$ is normal, then is $X \times X \times X$ normal?

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:

https://mathoverflow.net/questions/315657/if-textdimx-times-x-2-textdimx-does-textdimxn-n-textdim

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.

Theorem 1. For every $$k$$ and $$m$$ such that $$1 \leq k \leq m \leq \omega$$ there exists a separable and first coutnable space $$X = X(k,m)$$ such that
1. $$X^n$$ is paracompact (Lindelöf, subparacompact) if and only if $$n < k$$,
2. $$X^n$$ is normal (collectionwise normal) if and only if $$n < m$$.