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Covering analytic sets by families of closed set

Published online by Cambridge University Press:  12 March 2014

Sławomir Solecki
Sławomir Solecki*
Affiliation:
Department of Mathematics 253-37, California Institute of Technology, Pasadena, California 91125, E-mail: solecki@cco.caltech.edu
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Abstract

We prove that for every family I of closed subsets of a Polish space each set can be covered by countably many members of I or else contains a nonempty set which cannot be covered by countably many members of I. We prove an analogous result for κ-Souslin sets and show that if A# exists for any A ⊂ ωω, then the above result is true for sets. A theorem of Martin is included stating that this result is also true for weakly homogeneously Souslin sets. As an application of our results we derive from them a general form of Hurewicz's theorem due to Kechris. Louveau, and Woodin and a theorem of Feng on the open covering axiom. Also some well-known theorems on finding “big” closed sets inside of “big” and are consequences of our results.

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 59 , Issue 3 , September 1994 , pp. 1022 - 1031
Copyright
Copyright © Association for Symbolic Logic 1994

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References

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