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.