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 \( \mathbf{Q} \) (resp. \( \mathbf{N} \)) 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 \( \mathbf{Q} \) and \( \mathbf{N} \) necessarily coincide with the class \( \mathbf{M} \) of metrizable spaces.
Clearly, \( \mathbf{M}\subset\mathbf{N}\subset\mathbf{Q} \). We prove that first countable members of \( \mathbf{Q} \) are metrizable and that under \( V=L \) the classes \( \mathbf{M} \) and \( \mathbf{N} \) coincide, thus giving a consistency proof of Morita’s conjecture. On the other hand, even though \( \mathbf{Q} \) contains non-metrizable members, it is quite close to \( \mathbf{M} \): the class \( \mathbf{Q} \) is countably productive and hereditary, and all members \( X \) of \( \mathbf{Q} \) are stratifiable and satisfy
\[ c(X)=l(X)=w(X) .\]
In particular, locally Lindelöf or locally Souslin or locally \( p \)-spaces in \( \mathbf{Q} \) 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.
In our investigation, 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.
