If $X times X$ is normal, then is $X times X times X$ normal?
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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 ...