## 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)

### Citations:

Zbl 0593.54028; Zbl 0574.54041
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### References:

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