Forcing, genericity and CBERS

Abstract

In this paper we continue the study of equivalence of generics filters started by Smythe in [Smy22]. We fully characterize those forcing posets for which the corresponding equivalence of generics is smooth using the purely topological property of condensation. Next we leverage our characterization to show that there are non-homogeneous forcing for which equivalence of generics is not smooth. Then we prove hyperfiniteness in the case of Prikry forcing and some additional results addressing the problem whether generic equivalence for Cohen forcing is hyperfinite.