Analytic spaces and paracompactness of X 2\(\backslash \Delta\) (Q1098135)

The first author [ibid. 17, 287-304 (1984; Zbl 0547.54016)] proved that every compact Hausdorff space X with X \(2\setminus \Delta\) paracompact, where \(\Delta\) is the diagonal in \(X\times X\), is metrizable. The main theorem of the authors is that every regular \(\Sigma\)-space X such that X \(2\setminus \Delta\) is paracompact has a \(G_{\delta}\)-diagonal (hence X is a \(\sigma\)-space). Here \(\Sigma\)-space and \(\sigma\)-space are used in the sense of \textit{K. Nagami} [Fundam. Math. 65, 169-192 (1969; Zbl 0181.507)]. A corollary to this is that every completely regular p-space X (in the sense of Arkhangel'skij) such that X \(2\setminus \Delta\) is paracompact is metrizable. In Czech. Math. J. 35(110), 43-51 (1985; Zbl 0583.54023), \textit{Z. Frolík} asked whether every analytic space X with X \(2\setminus \Delta\) paracompact is point-analytic. By showing that Souslin-F subsets of paracompact p-spaces are strong \(\Sigma\)-spaces the authors answer Frolík's question in the affirmative.