Basis Problem for Turbulent Actions II: c 0 ‐Equalities

Let ( X n , d n ) be a sequence of finite metric spaces of uniformly bounded diameter. An equivalence relation D on the product ∏ n X n defined byx →Dy →if and only if lim sup n d n ( x n , y n ) = 0 is a c 0 ‐equality . A systematic study is made of c 0 ‐equalities and Borel reductions between them. Necessary and sufficient conditions, expressed in terms of combinatorial properties of metrics d n , are obtained for a c 0 ‐equality to be effectively reducible to the isomorphism relation of countable structures. It is proved that every Borel equivalence relation E reducible to a c 0 ‐equality D either reduces a c 0 ‐equality D ' additively reducible to D , or is Borel‐reducible to the equality relation on countable sets of reals. An appropriately defined sequence of metrics provides a c 0 ‐equality which is a turbulent orbit equivalence relation with no minimum turbulent equivalence relation reducible to it. This answers a question of Hjorth. It is also shown that, whenever E is an F σ ‐equivalence relation and D is a c 0 ‐equality, every Borel equivalence relation reducible to both D and to E has to be essentially countable. 2000 Mathematics Subject Classification : 03E15.