Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-28T15:40:40.916Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
Get access Rights & Permissions [Opens in a new window]

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.

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[DS]Debs, G. and Saint Raymond, J., Ensembles d’unicité et d’unicité au sens large, Annales de l’Institut Fourier (Grenoble), Vol. 37 (1987), fasc. 3, pp. 217–239.Google Scholar
[EKM]Van Engelen, F., Kunen, K., and Miller, A., Two Remarks About Analytic Sets, Lecture Notes in Mathematics, vol. 1401, Springer-Verlag. Berlin and New York, 1989, pp. 68–72.Google Scholar
[F]Feng, Q., Homogeneity of open partitions of pairs of the reals, Transactions of the American Mathematical Society, vol. 339 (1993), pp. 659–684.CrossRefGoogle Scholar
[J]Jech, T., Set theory, Academic Press, New York, San Francisco, and London, 1978.Google Scholar
[K]Kechris, A. S., On a notion of smallness for subsets of the Baire space, Transactions of the American Mathematical Society, vol. 229 (1977), pp. 191–207.CrossRefGoogle Scholar
[KL]Kechris, A. S. and Louveau, A., Descriptive set theory and harmonic analysis this Journal, vol. 57 (1992), pp. 413–441.Google Scholar
[KLW]Kechris, A. S., Louveau, A., and Woodin, W. H., The structure of σ-ideals of compact sets, Transactions of the American Mathematical Society, vol. 301 (1987), pp. 263–288.Google Scholar
[L]Louveau, A., σ-idéaux engendrés par des ensembles fermés et théorémes d’approximation, Transactions of the American Mathematical Society, vol. 257 (1980), pp. 143–169.Google Scholar
[L1]Louveau, A., circulated manuscript.Google Scholar
[L2]Louveau, A., Recursivity and capacity theory, Proceedings of Symposia in Pure Mathematics, vol. 42, American Mathematical Society, Providence, Rhode Island, 1985, pp. 285–301.Google Scholar
[MS]Martin, D. A., and Steel, J. R., A proof of projective determinacy, Journal of the American Mathematical Society, Journal of the American Mathematical Society, vol. 2 (1989), pp. 71–125.Google Scholar
[M]Mycielski, J., Independent sets in topological algebras, Fundamenta Mathematicae, vol. 55 (1964), pp. 139–147.CrossRefGoogle Scholar
[O]Oxtoby, J., Measure and category, Springer-Verlag, New York and Heidelberg, 1971.CrossRefGoogle Scholar
[P]Petruska, G. Y., On Borel sets with small covers, Real Analysis Exchange, vol. 18 (1992–1993), pp. 330–338.CrossRefGoogle Scholar
[S]Solovay, R. M., On the cardinality of sets of reals, Foundations of mathematics, Springer-Verlag. Berlin and New York. 1969, pp. 58–73.Google Scholar
[SR]Saint Raymond, J., Approximation des sous-ensembles analytiques par I’interieur, Comptes Rendus des Séances de l’Académie des Sciences, Série I. Mathématique, vol. 281 (1975), pp. 85–87.Google Scholar