  The purpose of this paper is to contribute to the theory of profinite
semigroups by considering the special class consisting of those all of whose
finitely generated closed subsemigroups are countable, which are said to be
locally countable. We also call locally countable a pseudovariety V (of finite
semigroups) for which all pro-V semigroups are locally countable. We
investigate operations preserving local countability of pseudovarieties and
show that, in contrast with local finiteness, several natural operations do not
preserve it. We also investigate the relationship of a finitely generated
profinite semigroup being countable with every element being expressable in
terms of the generators using multiplication and the idempotent (omega) power.
The two properties turn out to be equivalent if there are only countably many
group elements, gathered in finitely many regular J-classes. We also show that
the pseudovariety generated by all finite ordered monoids satisfying the
inequality $1\le x^n$ is locally countable if and only if $n=1$.
