Proceedings of the American Mathematical Society

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



Calibrated thin $\boldsymbol \Pi _{\mathbf {1}}^{\mathbf {1}}$ $\sigma $-ideals are $\boldsymbol G_{\delta }$

Author: Miroslav Zelený
Journal: Proc. Amer. Math. Soc. 125 (1997), 3027-3032
MSC (1991): Primary 03E15, 28A05; Secondary 42A63
MathSciNet review: 1415378
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Abstract: Let $E$ be a compact metric space, and let $I \subset \mathcal {K} (E) $ be a calibrated thin $\boldsymbol \Pi _{\mathbf {1}}^{\mathbf {1}}$ $\sigma $-ideal. Then $I$ is $\boldsymbol G_{\delta }$. This solves an open problem, which was posed by Kechris, Louveau and Woodin. Using our result we obtain a new proof of Kaufman's theorem concerning $U$-sets and $U_{0}$-sets.

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Additional Information

Miroslav Zelený
Affiliation: Faculty of Mathematics and Physics, Charles University, Sokolovská 83, Prague 186 00, Czech Republic

Received by editor(s): May 5, 1996
Additional Notes: Research supported by Research Grants GAUK 362, GAUK 363 and GAČR 201/94/0474.
Communicated by: Franklin D. Tall
Article copyright: © Copyright 1997 American Mathematical Society