Weakly normal topological spaces and products, (Q676974)

\textit{A. V. Arkhangel'skij} defined weakly normal spaces over a class \(\mathcal P\) of spaces in [Math. Res. 67, 13-26 (1992; Zbl 0792.54003)] and posed several problems concerning productivity of weak normality. Among others, the present paper shows that: (1) Weak normality is not closed under sums; (2) A Tikhonov space \(X\) is paracompact iff \(X\times cX\) is weakly normal over metric spaces for some compactification \(cX\); (3) The product of the Tikhonov plank with any first countable Hausdorff space is weakly normal; (4) If \(X\times cX\) is weakly normal for a pseudocompact Tikhonov space \(X\) and its compactification \(cX\), then \(X\) is realcompact (i.e., compact). Some results deal with hereditary weak normality.