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Borel cardinalities below $ c_0$

Author(s): Michael Ray Oliver
Journal: Proc. Amer. Math. Soc. 134 (2006), 2419-2425.
MSC (2000): Primary 03E15; Secondary 37A20
Posted: March 14, 2006
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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.

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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)

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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
Email: oliver@cs.ucla.edu

DOI: 10.1090/S0002-9939-06-08207-4
PII: S 0002-9939(06)08207-4
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
Posted: March 14, 2006
Communicated by: Carl G. Jockusch, Jr.
Copyright of article: Copyright 2006, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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