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Compact perfect sets in weak analytic spaces. (English) Zbl 0759.54016

This paper introduces the concept of a cover-analytic space, which is a generalization of that of a Čech-analytic space. In particular, a Hausdorff space \(X\) is cover-analytic if there is a cover-complete [i.e., has a complete sequence of exhaustive covers — a concept studied by E. Michael in Proc. Am. Math. Soc. 96, 513-522 (1986; Zbl 0593.54028)] subspace of \(X\times \mathbb{N}^ \mathbb{N}\) which projects onto \(X\). The main theorem in this paper has three corollaries which each generalize a known result about \(\sigma\)-scattered sets or compact perfect sets. For example, one corollary (which generalizes a result of G. Koumoullis in Topology Appl. 21, 59-71 (1985; Zbl 0574.54041)) is that a cover-analytic regular Hausdorff space \(X\) is either \(\sigma\)- scattered or else contains a nonempty compact perfect set, and these conditions are mutually exclusive.

MSC:

54E52 Baire category, Baire spaces
54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
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References:

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