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- W2067813723 abstract "It is well-known that Z is a perfectly normal space (normal P-space) if and only if X×Z is perfectly normal (normal) for every metric space X. Conversely, denote by Q (resp.N) the class of all spaces X whose products X×Z with all perfectly normal spaces (all normal P-spaces) Z are normal. It is natural to ask whether Q and N necessarily coincide with the class M of metrizable spaces. Clearly, M⊂N⊂Q. We prove that first countable members of Q are metrizable and that under V=L the classes M and N coincide, thus giving a consistency proof of Morita's conjecture. On the other hand, even though Q contains non-metrizable members, it is quite close to M: the class Q is countably productive and hereditary, and all members X of Q are stratifiable and satisfy c(X)=l(X)=w(X). In particular, locally Lindelöf or locally Souslin or locally p-spaces in Q are metrizable. The above results immediately lead to the consistency proof of another Morita's conjecture, stating that X is a metrizable σ-locally compact space if and only if X×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." @default.
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- W2067813723 date "1986-02-01" @default.
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- W2067813723 title "Normality of product spaces and Morita's conjectures" @default.
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- W2067813723 doi "https://doi.org/10.1016/0166-8641(86)90074-x" @default.
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