A remark on the Debs–Saint-Raymond theorem

Zelený, Miroslav

doi:10.1090/S0002-9939-01-05978-0

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