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arXiv · 1103.2312

Cardinal characteristics and countable Borel equivalence relations

Samuel Coskey·Scott Schneider
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Abstract

Boykin and Jackson recently introduced a property of countable Borel equivalence relations called Borel boundedness, which they showed is closely related to the union problem for hyperfinite equivalence relations. In this paper, we introduce a family of properties of countable Borel equivalence relations which correspond to combinatorial cardinal characteristics of the continuum in the same way that Borel boundedness corresponds to the bounding number $\mathfrak b$. We analyze some of the basic behavior of these properties, showing for instance that the property corresponding to the splitting number $\mathfrak s$ coincides with smoothness. We then settle many of the implication relationships between the properties; these relationships turn out to be closely related to (but not the same as) the Borel Tukey ordering on cardinal characteristics.

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BibTeXRIS

Samuel Coskey, Scott Schneider. 2016-06-22. Cardinal characteristics and countable Borel equivalence relations. https://doi.org/10.1002/malq.201400111

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