Compact perfect sets in weak analytic spaces (Q1183634)

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 \textit{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 \textit{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.