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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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##### References:
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