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Proceedings of the American Mathematical Society
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
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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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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: http://dx.doi.org/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
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.