## Normality of product spaces and Morita’s conjectures.(English)Zbl 0575.54007

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.

### MSC:

 54B10 Product spaces in general topology 54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.) 54A35 Consistency and independence results in general topology
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### References:

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