On \(F_\sigma\text{-}\delta\)-normality and hereditary \(\delta\)-normality (Q1292756)

By \textit{J. Mack} [Trans. Am. Math. Soc. 148, 265-272 (1970; Zbl 0209.26904)] a topological space \(X\) is called \(\delta\)-normal if one can separate (by open sets) each closed set and each disjoint regularly closed \(G_\delta\)-set. It is shown that if \(X\times Y\) is hereditarily (or \(F_\sigma\)-hereditarily) \(\delta\)-normal, then either \(X\) is perfectly normal (or normal and countably paracompact, resp.) or all countable subsets of \(Y\) are closed. The results are applied to \(C_p(X)\): every \(F_\sigma\)-hereditarily \(\delta\)-normal \(C_p(X)\) is normal, and every hereditarily \(\delta\)-normal \(C_p(X)\) is perfectly normal.