On properties of normality type in products (Q1406413)

According to the paper [\textit{J. Mack}, Trans. Am. Math. Soc. 148, 265--272 (1970; Zbl 0209.26904)] the author investigates the definition of a \(\delta\)-normal space. Theorems are proved on the behavior of \(\delta\)-normality in products and relations to compactness, e.g.: Theorem 1. If the product \(X\times Y\) is \(F_\sigma\)-\(\delta\)-normal, then either \(X\) is normal or all the countable subsets of \(Y\) are closed. Theorem 2. If the product \(X\times Y\) is hereditarily \(\delta\)-normal, then either \(X\) is perfectly normal or all the countable subsets in \(Y\) are closed. Theorem 3. Any bicompact cube which is hereditarily \(\delta\)-normal is metrizable. Theorem 4. The \(F_\sigma\)-\(\delta\)-normality of \(\exp(X)\) implies the bicompactness of \(X\).