Normality and collectionwise normality of product spaces. II (Q1293160)

For Part I see [Topol. Proc. 22, 383-392 (1997; Zbl 0918.54027) ]. As for the product space \(X\times Y\), the author proves the equivalence of its normality and its collectionwise normality in the following cases which are generalizations of known results. (i) \(X\) is a collectionwise normal \(\Sigma\)-space and \(Y\) is a collectionwise normal first countable \(P\)-space. (ii) \(X\) is the closed, continuous image of a normal \(M\)-space and \(Y\) is a collectionwise normal first countable \(P\)-space. (iii) \(X\) is the closed, continuous image of a paracompact \(M\)-space and \(Y\) is a collectionwise normal \(P\)-space. It is also proved that in case (iii), if \(Y\) is a paracompact \(P\)-space, then \(X\times Y\) is paracompact if and only if \(X\times Y\) is normal. All spaces are assumed at least being Hausdorff.