  We study groups that can be defined as Polish, pro-countable groups, as
non-archimedean groups with an invariant metric or as quasi-countable groups,
i.e., closed subdirect products of countable, discrete groups, endowed with the
product topology. We show, among other results, that for every non-locally
compact, abelian quasi-countable group G there exists a closed subgroup L of G,
and a closed, non-locally compact subgroup K of G/L which is a direct product
of discrete, countable groups. As an application we prove that for every
abelian Polish group G of the form H/L, where H,L are closed subgroups of
Iso(X) and X is a locally compact separable metric space (e.g., G is abelian,
quasi-countable), G is locally compact iff every continuous action of G on a
Polish space Y induces an orbit equivalence relation that is reducible to an
equivalence relation with countable classes.
