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

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- by Miroslav Zelený
- Proc. Amer. Math. Soc.
**125**(1997), 3027-3032 - DOI: https://doi.org/10.1090/S0002-9939-97-04041-0
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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.## 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** - Alexander S. Kechris and Alain Louveau,
*Descriptive set theory and the structure of sets of uniqueness*, London Mathematical Society Lecture Note Series, vol. 128, Cambridge University Press, Cambridge, 1987. MR**953784**, DOI 10.1017/CBO9780511758850 - 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 - C. J. Everett Jr.,
*Annihilator ideals and representation iteration for abstract rings*, Duke Math. J.**5**(1939), 623–627. MR**13** - Carlos E. Uzcátegui A.,
*The covering property for $\sigma$-ideals of compact sets*, Fund. Math.**141**(1992), no. 2, 119–146. MR**1183328**, DOI 10.4064/fm-141-2-119-146

## Bibliographic Information

**Miroslav Zelený**- Affiliation: Faculty of Mathematics and Physics, Charles University, Sokolovská 83, Prague 186 00, Czech Republic
- Email: zeleny@karlin.mff.cuni.cz
- 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
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**125**(1997), 3027-3032 - MSC (1991): Primary 03E15, 28A05; Secondary 42A63
- DOI: https://doi.org/10.1090/S0002-9939-97-04041-0
- MathSciNet review: 1415378