Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Borel classes and closed games: Wadge-type and Hurewicz-type results


Authors: A. Louveau and J. Saint-Raymond
Journal: Trans. Amer. Math. Soc. 304 (1987), 431-467
MSC: Primary 03E15; Secondary 04A15, 28A05, 54H05
MathSciNet review: 911079
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For each countable ordinal $ \xi $ and pair $ ({A_0},\,{A_1})$ of disjoint analytic subsets of $ {2^\omega }$, we define a closed game $ {J_\xi }({A_0},\,{A_1})$ and a complete $ \Pi _\xi ^0$ subset $ {H_\xi }$ of $ {2^\omega }$ such that (i) a winning strategy for player I constructs a $ \sum _\xi ^0$ set separating $ {A_0}$ from $ {A_1}$; and (ii) a winning strategy for player II constructs a continuous map $ \varphi :{2^\omega } \to {A_0} \cup {A_1}$ with $ {\varphi ^{ - 1}}({A_0}) = {H_\xi }$. Applications of this construction include: A proof in second order arithmetics of the statement "every $ \Pi _\xi ^0$ non $ \sum _\xi ^0$ set is $ \Pi _\xi ^0$-complete"; an extension to all levels of a theorem of Hurewicz about $ \sum _2^0$ sets; a new proof of results of Kunugui, Novikov, Bourgain and the authors on Borel sets with sections of given class; extensions of results of Stern and Kechris. Our results are valid in arbitrary Polish spaces, and for the classes in Lavrentieff's and Wadge's hierarchies.


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

  • [B 1] J. Bourgain, $ {F_{\sigma \delta }}$ sections of Borel sets, Fund. Math. 107 (1980), 129-133. MR 584665 (81k:54068a)
  • [B 2] -, Borel sets with $ {F_{\sigma \delta }}$ sections, Fund. Math. 107 (1980), 149-159. MR 584668 (81k:54068b)
  • [Da] M. Davis, Infinite games of perfect information, Ann. of Math. Studies, no. 52, Princeton Univ. Press, Princeton, N.J., 1964, pp. 445-448. MR 0170727 (30:965)
  • [D] G. Debs, Un résultat d'uniformisation borélienne, Proc. Amer. Math. Soc. 92 (1984), 445-448. MR 759671 (85k:54043)
  • [De] C. Dellacherie, Ensembles analytiques: Théorèmes de séparation et applications, Sém. Probabilités Strasbourg IX, (1973-74), Lecture Notes in Math., vol. 465, Springer-Verlag, Berlin and New York, 1975. MR 0428306 (55:1331)
  • [F] H. Friedman, Higher set theory and mathematical practice. Ann. of Math. Logic 2 (1971), 326-357. MR 0284327 (44:1556)
  • [Ha] L. Harrington, Analytic determinacy and $ {0^\char93 }$, J. Symbolic Logic 43 (1978), 685-693. MR 518675 (80b:03065)
  • [Hu] W. Hurewicz, Relativ perfekte Teile von Punktmengen und Mengen $ (A)$, Fund. Math. 12 (1928), 78-109.
  • [J] T. John, Thesis, Berkeley, 1983.
  • [Ke] A. S. Kechris, A basis theorem for $ \Delta _3^1$ Borel sets, circulated notes.
  • [K-M] K. Kunen and A. W. Miller, Borel and projective sets from the point of view of compact sets, Math. Proc. Cambridge Philos. Soc. 94 (1983), 399-409. MR 720790 (85b:03087)
  • [Kun] K. Kunugui, Contributions à la théorie des ensembles boréliens et analytiques III, J. Fac. Sci. Hokkaido Imp. Univ. 8 (1939-40), 79-108. MR 0001818 (1:301f)
  • [Kur] K. Kuratowski, Topologie, vol. 1, PWN, Warszawa, 1958.
  • [L-SR] A. Louveau et J. Saint Raymond, Caractérisation par des jeux fermés de la classe de Baire des Boréliens, C. R. Acad. Sci. Paris 300 (1985). MR 802643 (86h:03084)
  • [Lo 1] A. Louveau, Recursivity and compactness, Higher Set Theory, Proc. Oberwolfach 1977, (G. H. Müller and D. S. Scott, eds.), Lecture Notes in Math., vol. 669, Springer-Verlag, Berlin and New York, 1978, pp. 303-338. MR 520192 (80j:03071)
  • [Lo 2] -, A separation theorem for $ \sum _1^1$ sets, Trans. Amer. Math. Soc. 260 (1980), 363-378. MR 574785 (81j:04001)
  • [Lo 3] -, Borel sets and the analytical hierarchy, Proc. Herbrand Sympos., Logic Coll. 81 (J. Stern, ed.), North-Holland, 1982, pp. 209-215. MR 757030 (85j:03081)
  • [Lo 4] -, Some results in the Wadge Hierarchy of Borel sets, Cabal Sem. 79-81, (A. S. Kechris, D. A. Martin and Y. N. Moschovakis, eds.), Lecture Notes in Math., vol. 1019, Springer-Verlag, 1983, pp. 28-55. MR 730585
  • [Lu] N. Luzin, Leçons sur les ensembles analytiques et leurs applications, 2nd ed., Chelsea, New York, 1972. MR 0392465 (52:13282)
  • [Ma] D. A. Martin, Borel determinacy, Ann. of Math. (2) 102 (1975), 363-371. MR 0403976 (53:7785)
  • [M-K] D. A. Martin and A. S. Kechris, Infinite games and effective descriptive set theory, Analytic Sets (C. A. Rogers et al.), Academic Press, New York, 1980. MR 562614 (81i:04003)
  • [Mo] Y. N. Moschovakis, Descriptive set theory, North-Holland, 1980. MR 561709 (82e:03002)
  • [N] P. S. Novikov, Sur une propriété des ensembles analytiques, Dokl. Akad. Nauk SSSR 3(5), (1934), 273-276.
  • [SR 1] J. Saint Raymond, Boréliens à coupes $ {K_\sigma }$, Bull. Soc. Math. France 104 (1976), 389-406. MR 0433418 (55:6394)
  • [SR 2] -, La structure borélienne d'Effros est-elle standard?, Fund. Math. 100 (1978), 201-210. MR 509546 (80g:54044)
  • [SR 3] -, Fonctions boréliennes sur un quotient, Bull. Sci. Math. 100 (1976), 141-147. MR 0460578 (57:571)
  • [S] J. R. Steel, Analytic sets and Borel isomorphisms, Fund. Math. 108 (1980), 83-88. MR 594307 (82b:03091)
  • [St] J. Stern, On Luzin's restricted continuum problem Ann. of Math. (2) 120 (1984), 7-37. MR 750715 (85h:03051)
  • [vE 1] F. van Engelen, Homogeneous Borel sets, Preprint. MR 826501 (87i:54080)
  • [vE 2] -, Characterizations of the countable infinite product of rationals and related problems, Preprint.
  • [vE-vM] F. van Engelen and J. van Mill, Borel sets in compact spaces: Some Hurewicz-type theorems, Fund. Math. 124 (1984), 271-286. MR 774518 (86j:54067)
  • [vW] R. van Wesep, Wadge degrees and descriptive set theory, Cabal Sem. 76-77, (A. S. Kechris and Y. N. Moschovakis, eds.), Lecture Notes in Math., vol. 689, Springer-Verlag, Berlin and New York, 1978, pp. 151-170. MR 526917 (80i:03058)
  • [W 1] W. W. Wadge, Degrees of complexity of subsets of the Baire space, Notices Amer. Math. Soc. (1972), A-714.
  • [W 2] -, Thesis, Berkeley, 1984.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 03E15, 04A15, 28A05, 54H05

Retrieve articles in all journals with MSC: 03E15, 04A15, 28A05, 54H05


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1987-0911079-0
PII: S 0002-9947(1987)0911079-0
Keywords: Borel classes, closed games, Wadge games, determinacy, Hurewicz' theorem
Article copyright: © Copyright 1987 American Mathematical Society