×

Covering analytic sets by families of closed sets. (English) Zbl 0808.03031

Summary: We prove that for every family \(I\) of closed subsets of a Polish space each \(\Sigma^ 1_ 1\) set can be covered by countably many members of \(I\) or else contains a nonempty \(\Pi^ 0_ 2\) set which cannot be covered by countably many members of \(I\). We prove an analogous result for \(\kappa\)-Souslin sets and show that if \(A^ \#\) exists for any \(A\subset \omega^ \omega\), then the above result is true for \(\Sigma^ 1_ 2\) 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 “big” \(\Sigma^ 1_ 1\) and \(\Sigma^ 1_ 2\) sets are consequences of our results.

MSC:

03E15 Descriptive set theory
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Set theory (1978)
[2] Transactions of the American Mathematical Society 229 pp 191–207– (1977)
[3] Two Remarks About Analytic Sets 1401 pp 68–72– (1989)
[4] Annales de l’Institut Fourier (Grenoble) 37 pp 217–239– (1987)
[5] Comptes Rendus des Séances de l’Académie des Sciences, Série I. Mathématique 281 pp 85–87– (1975)
[6] On the cardinality of sets of reals, Foundations of mathematics pp 58–73– (1969)
[7] Real Analysis Exchange 18 pp 330–338– (1992)
[8] Measure and category (1971)
[9] Fundamenta Mathematicae 55 pp 139–147– (1964)
[10] Journal of the American Mathematical Society, Journal of the American Mathematical Society 2 pp 71–125– (1989)
[11] Proceedings of Symposia in Pure Mathematics 42 pp 285–301– (1985)
[12] Transactions of the American Mathematical Society 257 pp 143–169– (1980)
[13] Transactions of the American Mathematical Society 301 pp 263–288– (1987)
[14] Descriptive set theory and harmonic analysis 57 pp 413–441– (1992)
[15] Transactions of the American Mathematical Society 339 pp 659–684– (1993)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.