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Analytic spaces and paracompactness of X 2\(\backslash \Delta\). (English) Zbl 0636.54025
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 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), 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.
Reviewer: H.H.Wicke

MSC:
54E18 \(p\)-spaces, \(M\)-spaces, \(\sigma\)-spaces, etc.
54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
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