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Transactions of the American Mathematical Society
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
Published electronically: January 22, 2009
MathSciNet review: 2485410
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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}$.


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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: http://dx.doi.org/10.1090/S0002-9947-09-04930-7
PII: S 0002-9947(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.