Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Borel cardinalities below $ c_0$

Author: Michael Ray Oliver
Journal: Proc. Amer. Math. Soc. 134 (2006), 2419-2425
MSC (2000): Primary 03E15; Secondary 37A20
Published electronically: March 14, 2006
MathSciNet review: 2213716
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The Borel cardinality of the quotient of the power set of the natural numbers by the ideal $ \mathcal{Z}_0$ of asymptotically zero-density sets is shown to be the same as that of the equivalence relation induced by the classical Banach space $ c_0$. We also show that a large collection of ideals introduced by Louveau and Velickovic, with pairwise incomparable Borel cardinality, are all Borel reducible to $ c_0$. This refutes a conjecture of Hjorth and has facilitated further work by Farah.

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

  • [AK00] Scot Adams and Alexander S. Kechris, Linear algebraic groups and countable Borel equivalence relations, J. Amer. Math. Soc. 13 (2000), no. 4, 909-943. MR 1775739 (2001g:03086)
  • [Far01] Ilijas Farah, Basis problem for turbulent actions. II. $ c\sb 0$-equalities, Proc. London Math. Soc. (3) 82 (2001), 1-30. MR 1794255 (2002c:03075)
  • [Hjo00] Greg Hjorth, Actions by the classical Banach spaces, J. Symb. Logic 65 (2000), no. 1, 392-420. MR 1782128 (2001h:03088)
  • [HK97] Greg Hjorth and Alexander S. Kechris, New dichotomies for Borel equivalence relations, Bull. Symb. Logic 3 (1997), no. 3, 329-346. MR 1476761 (98m:03101)
  • [JK84] Winfried Just and Adam Krawczyk, On certain Boolean algebras $ \mathscr{P}{\omega}/{I}$, Trans. Amer. Math. Soc. 285 (1984), no. 1, 411-429. MR 0748847 (86f:04003)
  • [KL97] Alexander S. Kechris and Alain Louveau, The classification of hypersmooth Borel equivalence relations, J. Amer. Math. Soc. 10 (1997), no. 1, 215-242. MR 1396895 (97e:03067)
  • [LV94] Alain Louveau and Boban Velickovic, A note on Borel equivalence relations, Proc. Amer. Math. Soc. 120 (1994), no. 1, 255-259. MR 1169042 (94f:54076)
  • [Maz00] Krzysztof Mazur, A modification of Louveau and Velickovic's construction for $ {F}\sb \sigma$-ideals, Proc. Amer. Math. Soc. 128 (2000), no. 5, 1475-1479. MR 1626442 (2000j:03067)
  • [Sol96] S\lawomir Solecki, Analytic ideals, Bull. Symb. Logic 2 (1996), no. 3, 339-348. MR 1416872 (97i:04002)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 03E15, 37A20

Retrieve articles in all journals with MSC (2000): 03E15, 37A20

Additional Information

Michael Ray Oliver
Affiliation: Department of Mathematics, University of California, Box 951555, Los Angeles, California 90095–1555
Address at time of publication: Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, Ontario, Canada M3J 1P3

Keywords: Borel equivalence relations
Received by editor(s): April 5, 2004
Received by editor(s) in revised form: October 25, 2004, and February 18, 2005
Published electronically: March 14, 2006
Communicated by: Carl G. Jockusch, Jr.
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society