Abstract
Let X be the set of all metrics on ω, and let Xcpt be the set of all metrics r on ω such that the completion of (ω,r) is compact. We define the Cauchy sequence equivalence relation Ecs on X as: rEcss iff the set of Cauchy sequences in (ω,r) is same as in (ω,s). We also denote Ecsc=Ecs↾Xcpt.We show that Ecs is a Π11-complete equivalence relation, while Ecsc is a Π30 equivalence relation. We also show that Ecsc is Borel bireducible to an orbit equivalence relation. Furthermore, we investigate the Borel reducibility between Ecsc and some benchmark equivalence relations. For instance, we show that =+ and Rω/c0 are Borel reducible to Ecsc, and E1 is not. Restrictions of Ecsc on some special invariant subsets of Xcpt are also considered.
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