Hurewicz-like tests for Borel subsets of the plane
Author:
Dominique Lecomte
Journal:
Electron. Res. Announc. Amer. Math. Soc. 11 (2005), 95-102
MSC (2000):
Primary 03E15; Secondary 54H05
DOI:
https://doi.org/10.1090/S1079-6762-05-00152-6
Published electronically:
December 15, 2005
MathSciNet review:
2191690
Full-text PDF Free Access
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Abstract: Let $\xi \geq 1$ be a countable ordinal. We study the Borel subsets of the plane that can be made $\boldsymbol \Pi ^{0}_{\xi }$ by refining the Polish topology on the real line. These sets are called potentially $\boldsymbol \Pi ^{0}_{\xi }$. We give a Hurewicz-like test to recognize potentially $\boldsymbol \Pi ^{0}_{\xi }$ sets.
[D-SR]D-SR G. Debs and J. Saint Raymond, Borel liftings of Borel sets: some decidable and undecidable statements, to appear in Memoirs of the Amer. Math. Soc.
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[D-SR]D-SR G. Debs and J. Saint Raymond, Borel liftings of Borel sets: some decidable and undecidable statements, to appear in Memoirs of the Amer. Math. Soc.
[H-K-Lo]H-K-Lo L. A. Harrington, A. S. Kechris, and A. Louveau, A Glimm-Effros dichotomy for Borel equivalence relations, J. Amer. Math. Soc. 3 (1990), 903–928.
[K]K A. S. Kechris, Classical Descriptive Set Theory, Springer-Verlag, 1995.
[K-S-T]K-S-T A. S. Kechris, S. Solecki, and S. Todorčević, Borel chromatic numbers, Adv. Math. 141 (1999), 1–44.
[L1]L1 D. Lecomte, Classes de Wadge potentielles et théorèmes d’uniformisation partielle, Fund. Math. 143 (1993), 231–258.
[L2]L2 D. Lecomte, Complexité des boréliens à coupes dénombrables, Fund. Math. 165 (2000), 139–174.
[Lo]Lo A. Louveau, Ensembles analytiques et boréliens dans les espaces produit, Astérisque (S. M. F.) 78 (1980).
[M]M Y. N. Moschovakis, Descriptive Set Theory, North-Holland, 1980.
[Lo-SR]Lo-SR A. Louveau and J. Saint Raymond, Borel classes and closed games: Wadge-type and Hurewicz-like results, Trans. Amer. Math. Soc. 304 (1987), 431–467.
[SR]SR J. Saint Raymond, La structure borélienne d’Effros est-elle standard?, Fund. Math. 100 (1978), 201–210.
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Additional Information
Dominique Lecomte
Affiliation:
Université Paris 6, Equipe d’Analyse Fonctionnelle, tour 46-0, boîte 186, 4, place Jussieu, 75 252 Paris Cedex 05, France, and Université de Picardie, I.U.T. de l’Oise, site de Creil, 13, allée de la faïencerie, 60 107 Creil, France
MR Author ID:
336400
Email:
lecomte@moka.ccr.jussieu.fr
Keywords:
Potentially,
Baire classes,
reduction,
Hurewicz’s Theorem
Received by editor(s):
July 29, 2005
Published electronically:
December 15, 2005
Communicated by:
Alexander Kechris
Article copyright:
© Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
