Cofinal families of Borel equivalence relations and quasiorders

Abstract. Families of Borel equivalence relations and quasiorders that are coÞnal with respect to the Borel reducibility ordering, B, are constructed. There is an analytic ideal on ù generating a complete analytic equivalence relation and any Borel equivalence relation reduces to one generated by a Borel ideal. Several Borel equivalence relations, among them Lipschitz isomorphism of compact metric spaces, are shown to be Kó complete. x1. Introduction. For R and S binary relations on Polish spaces X and Y, re-spectively, one writes R B S and says thatR Borel reduces to S, if there is a Borel functionf: X! Y such that xRy () f(x)Sf(y) (i » (ff)1(S) = R). Usu-ally this ordering has been studied when R and S both are equivalence relations, but recently it has turned out that the study of quasiorders, i.e., reßexive, transitiv