Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Effective refining of Borel coverings


Authors: Gabriel Debs and Jean Saint Raymond
Journal: Trans. Amer. Math. Soc. 361 (2009), 2831-2869
MSC (2000): Primary 03E15; Secondary 03E45, 54H05
DOI: https://doi.org/10.1090/S0002-9947-09-04930-7
Published electronically: January 22, 2009
MathSciNet review: 2485410
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Given a countable family $ (\mathbf{\Gamma}_i)_{i\in I}$ of additive or multiplicative Baire classes ( $ \mathbf{\Gamma}_i=\mathbf{\Sigma}^0_{\xi_i}$ or $ \mathbf{\Pi}^0_{\xi_i}$) we investigate the following complexity problem: Let $ (A_i)_{i\in I}$ be a Borel covering of $ \omega^\omega$ and assume that there exists some covering $ (B_i)_{i\in I}$ with $ B_i\subset A_i$ and $ B_i\in\mathbf{\Gamma}_i$ for all $ i$; can one find such a family $ (B_i)_{i\in I}$ in $ \varDelta^1_1(\alpha)$ where $ \alpha\in\omega^\omega$ is any reasonable code for the families $ (A_i)_{i\in I}$ and $ (\mathbf{\Gamma}_i)_{i\in I}$? The main result of the paper will give a full characterization of those families $ (\mathbf{\Gamma}_i)_{i\in I}$ for which the answer is positive. For example we will show that this is the case if $ I$ is finite or if all the Baire classes $ \mathbf{\Gamma}_i$ are additive, but in the general case the answer depends on the distribution of the multiplicative Baire classes inside the family $ (\mathbf{\Gamma}_i)_{i\in I}$.


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

  • 1. G. Debs and J. Saint Raymond, Borel liftings of Borel sets: Some decidable and undecidable statements, Memoirs of the Amer. Math. Soc. 187 (2007), no 876. MR 2308388 (2008b:03065)
  • 2. A. S. Kechris, Classical Descriptive Set Theory, Graduate Texts in Mathematics, Springer-Verlag, New York, 1995. MR 1321597 (96e:03057)
  • 3. A. Louveau, A separation theorem for $ \varSigma^1_1$ sets, Trans. Amer. Math. Soc. 260-2 (1980) 363-378. MR 0574785 (81j:04001)
  • 4. A. Louveau and J. Saint Raymond, Borel classes and closed games: Wadge-type and Hurewicz-type results, Trans. Amer. Math. Soc. 304-2 (1987) 431-467. MR 0911079 (89g:03068)
  • 5. T. Matrai, Hurewicz tests: Separating and reducing analytic sets on the conscious way, thesis, Central European University (2005).
  • 6. Y. N. Moschovakis, Descriptive Set Theory, North Holland, Amsterdam, 1980. MR 0561709 (82e:03002)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 03E15, 03E45, 54H05

Retrieve articles in all journals with MSC (2000): 03E15, 03E45, 54H05


Additional Information

Gabriel Debs
Affiliation: Analyse Fonctionnelle, Institut de Mathématique de Jussieu, Boîte 186, 4 place Jussieu, F-75252 Paris Cedex 05, France
Email: debs@math.jussieu.fr

Jean Saint Raymond
Affiliation: Analyse Fonctionnelle, Institut de Mathématique de Jussieu, Boîte 186, 4 place Jussieu, F-75252 Paris Cedex 05, France
Email: raymond@math.jussieu.fr

DOI: https://doi.org/10.1090/S0002-9947-09-04930-7
Keywords: Covering, separation, effectivity, Novikov Theorem, distinguished tree relations
Received by editor(s): January 30, 2006
Published electronically: January 22, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society