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.