Let $(X_n,d_n)$ be a sequence of finite metric spaces of uniformly bounded diameter. An equivalence relation $D$ on the product $\prod_n X_n$ defined by $\vec x\, D\,\vec y$ if and only if $\limsup_n d_n(x_n,y_n)=0$ is a {\em $c_0$-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 $F_\sigma$-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.