A remark on the Debs–Saint-Raymond theorem
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- by Miroslav Zelený PDF
- Proc. Amer. Math. Soc. 129 (2001), 3711-3714 Request permission
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
- G. Debs and J. Saint-Raymond, Ensembles boréliens d’unicité et d’unicité au sens large, Ann. Inst. Fourier (Grenoble) 37 (1987), no. 3, 217–239 (French, with English summary). MR 916281
- W. Hurewicz, Relative perfekte Teile von Punktmengen und Mengen(A), Fund. Math. 12 (1928), 78-109.
- Alexander S. Kechris, The descriptive set theory of $\sigma$-ideals of compact sets, Logic Colloquium ’88 (Padova, 1988) Stud. Logic Found. Math., vol. 127, North-Holland, Amsterdam, 1989, pp. 117–138. MR 1015324, DOI 10.1016/S0049-237X(08)70267-2
- Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597, DOI 10.1007/978-1-4612-4190-4
- A. S. Kechris, A. Louveau, and W. H. Woodin, The structure of $\sigma$-ideals of compact sets, Trans. Amer. Math. Soc. 301 (1987), no. 1, 263–288. MR 879573, DOI 10.1090/S0002-9947-1987-0879573-9
- Alain Louveau, Ensembles analytiques et boréliens dans les espaces produits, Astérisque, vol. 78, Société Mathématique de France, Paris, 1980 (French). With an English summary. MR 606933
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
- 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
- © Copyright 2001 American Mathematical Society
- 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
- MathSciNet review: 1860506