Borel Cardinal Invariant properties of countable Borel equivalence relations

Details Borel Cardinal Invariant properties of countable Borel equivalence relations Borel Cardinal Invariant properties of countable Borel equivalence relations

2011-03-11

arxiv.org/abs/1103.2312v1

Mathematics

Logic

30 pages

Scientific paper

Boykin and Jackson recently introduced a property of countable Borel equivalence relations called Borel boundedness, which they showed is closely related to the unions problem for hyperfinite relations. In this paper, we introduce a family of properties of countable Borel equivalence relations which correspond to combinatorial cardinal invariants 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 invariants.

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