Normality of product spaces and Morita's conjectures (Q1064560)

It is well known that Z is a perfectly normal space (normal P-space) if and only if \(X\times Z\) is perfectly normal (normal) for every metric space X. Conversely, denote by \({\mathbb{Q}}\) (resp. \({\mathbb{N}})\) the class of all spaces X whose products \(X\times Z\) with all perfectly normal spaces (all normal P-spaces) Z are normal. It is natural to ask whether \({\mathbb{Q}}\) and \({\mathbb{N}}\) necessarily coincide with the class \({\mathbb{M}}\) of metrizable spaces. Clearly, \({\mathbb{M}}\subset {\mathbb{N}}\subset {\mathbb{Q}}\). We prove that first countable members of \({\mathbb{Q}}\) are metrizable and that under V\(=L\) the classes \({\mathbb{M}}\) and \({\mathbb{N}}\) coincide, thus giving a consistency proof of Morita's conjecture. On the other hand, even though \({\mathbb{Q}}\) contains non-metrizable members, it is quite close to \({\mathbb{M}}:\) the class \({\mathbb{Q}}\) is countably productive and hereditary, and all members X of \({\mathbb{Q}}\) are stratifiable and satisfy \(c(X)=l(X)=w(X)\). In particular, locally Lindelöf or locally Souslin or locally p-spaces in \({\mathbb{Q}}\) are metrizable. The above results immediately lead to the consistency proof of another Morita's conjecture, stating that X is a metrizable \(\sigma\)-locally compact space if and only if \(X\times Y\) is normal for every normal countably paracompact space Y. No additional set-theoretic assumptions are necessary if X is first countable. An important role is played by the famous Bing examples of normal, non-collectionwise normal spaces. Answering Dennis Burke's question, we prove that products of two Bing-type examples are always non-normal.