Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A remark on the Debs-Saint-Raymond theorem


Author: Miroslav Zelený
Journal: Proc. Amer. Math. Soc. 129 (2001), 3711-3714
MSC (2000): Primary 03E15, 28A05, 54H05
DOI: https://doi.org/10.1090/S0002-9939-01-05978-0
Published electronically: April 24, 2001
MathSciNet review: 1860506
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract:

A theorem of Debs and Saint-Raymond gives sufficient conditions for a $\sigma $-ideal of compact sets to have the covering property. We discuss the necessity of these conditions. Namely, we show that there exists a $\boldsymbol \Pi _{\mathbf{1}}^{\mathbf{1}}$ $\sigma $-ideal that is locally non-Borel, has no Borel basis and has the covering property. This partially answers a question posed by Kechris.


References [Enhancements On Off] (What's this?)

  • [DSR] G. Debs, J. Saint-Raymond, Ensembles boréliens d'unicité et d'unicité au sens large, Ann. Inst. Fourier (Grenoble) 37 (1987), 217-239. MR 89d:04007
  • [H] W. Hurewicz, Relative perfekte Teile von Punktmengen und Mengen(A), Fund. Math. 12 (1928), 78-109.
  • [K1] A. S. Kechris, The descriptive set theory of $\sigma $-ideals of compact sets, Logic Colloquium '88 (1989), 117-138. MR 90h:03032
  • [K2] A. S. Kechris, Classical Descriptive Set Theory, Springer-Verlag, 1994. MR 96e:03057
  • [KLW] A. S. Kechris, A. Louveau, W. H. Woodin, The structure of $\sigma $-ideals of compact sets, Trans. Amer. Math. Soc. 301 (1987), 263-288. MR 88f:03042
  • [L] A. Louveau, Ensembles analytiques et boréliens dans les espaces produits, Astérisque 78 (1980). MR 82j:03062

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 03E15, 28A05, 54H05

Retrieve articles in all journals with MSC (2000): 03E15, 28A05, 54H05


Additional Information

Miroslav Zelený
Affiliation: Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, Prague 186 00, Czech Republic
Email: zeleny@karlin.mff.cuni.cz

DOI: https://doi.org/10.1090/S0002-9939-01-05978-0
Received by editor(s): January 7, 2000
Received by editor(s) in revised form: April 9, 2000
Published electronically: April 24, 2001
Additional Notes: The author’s research was supported by GAUK 190/1996, GAČR 201/97/1161, and CEZ J13/98113200007
Communicated by: Alan Dow
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society