  Let $G$ be a finitely generated group, $\mathrm{Sub}(G)$ the (compact,
metric) space of all subgroups of $G$ with the Chaubuty topology and $X!$ the
(Polish) group of all permutations of a countable set $X$. We show that the
following properties are equivalent: (i) Every finitely generated subgroup is
closed in the profinite topology, (ii) the finite index subgroups are dense in
$\mathrm{Sub}(G)$, (iii) A Baire generic homomorphism $\phi: G \rightarrow X!$
admits only finite orbits. Property (i) is known as the LERF property. We
introduce a new family of groups which we call {\it{A-separable}} groups. These
are defined by replacing, in (ii) above, the word "finite index" by the word
"co-amenalbe". The class of A-separable groups contains all LERF groups, all
amenable groups and more. We investigate some properties of these groups.
