On $\Game \mathbf {\Gamma }$-complete sets
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- by Gabriel Debs and Jean Saint Raymond
- Proc. Amer. Math. Soc. 148 (2020), 859-873
- DOI: https://doi.org/10.1090/proc/14731
- Published electronically: August 7, 2019
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Abstract:
Extending a result of A. Kechris we prove that under suitable assumptions on the class $\mathbf {\Gamma }$ of Borel sets, in particular when $\mathbf {\Gamma }$ is a Baire class, if any $\Game \mathbf {\Gamma }$ set is reducible to some $\Game \mathbf {\Gamma }$ set $A$ by a $\Game \mathbf {\Gamma }$-measurable function, then $A$ is $\Game \mathbf {\Gamma }$-complete.References
- N. Bourbaki, Éléments de mathématique. Topologie générale. Chapitres 1 à 4, Hermann, Paris, 1971. MR 0358652
- Gabriel Debs and Jean Saint Raymond, The descriptive complexity of the set of all closed zero-dimensional subsets of a Polish space, Topology Appl. 249 (2018), 43–66. MR 3864535, DOI 10.1016/j.topol.2018.07.010
- G. Debs and J. Saint Raymond, The descriptive complexity of connected Polish spaces, Fund. Math. (to appear).
- G. Debs and J. Saint Raymond, The game operator acting on Wadge classes of Borel sets, J. Symb. Log. (to appear).
- Leo A. Harrington and Alexander S. Kechris, On the determinacy of games on ordinals, Ann. Math. Logic 20 (1981), no. 2, 109–154. MR 622782, DOI 10.1016/0003-4843(81)90001-2
- Alexander S. Kechris, Forcing in analysis, Higher set theory (Proc. Conf., Math. Forschungsinst., Oberwolfach, 1977) Lecture Notes in Math., vol. 669, Springer, Berlin, 1978, pp. 277–302. MR 520191
- Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597, DOI 10.1007/978-1-4612-4190-4
- Alexander S. Kechris, On the concept of $\bfPi ^1_1$-completeness, Proc. Amer. Math. Soc. 125 (1997), no. 6, 1811–1814. MR 1372034, DOI 10.1090/S0002-9939-97-03770-2
- A. Louveau and J. Saint-Raymond, Les propriétés de réduction et de norme pour les classes de Boréliens, Fund. Math. 131 (1988), no. 3, 223–243 (French, with English summary). MR 978717, DOI 10.4064/fm-131-3-223-243
- Yiannis N. Moschovakis, Descriptive set theory, 2nd ed., Mathematical Surveys and Monographs, vol. 155, American Mathematical Society, Providence, RI, 2009. MR 2526093, DOI 10.1090/surv/155
Bibliographic Information
- Gabriel Debs
- Affiliation: Sorbonne Université, Université Paris Diderot, CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche, IMJ-PRG, F-75005, Paris, France – and – Université Le Havre Normandie, Institut Universitaire de Technologie, Rue Boris Vian, BP 4006 76610 Le Havre, France
- MR Author ID: 55795
- Email: gabriel.debs@imj-prg.fr
- Jean Saint Raymond
- Affiliation: Sorbonne Université, Université Paris Diderot, CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche, IMJ-PRG, F-75005, Paris, France
- MR Author ID: 153115
- Email: jean.saint-raymond@imj-prg.fr
- Received by editor(s): October 8, 2018
- Received by editor(s) in revised form: May 29, 2019
- Published electronically: August 7, 2019
- Communicated by: Heike Mildenberger
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 148 (2020), 859-873
- MSC (2010): Primary 03E15, 28A05; Secondary 54H05
- DOI: https://doi.org/10.1090/proc/14731
- MathSciNet review: 4052221