  In this paper we introduce a new class of metric actions on separable (not
necessarily connected) metric spaces called "Cauchy-indivisible" actions. This
new class coincides with that of proper actions on locally compact metric
spaces and, as examples show, it may be different in general. The concept of
"Cauchy-indivisibility" follows a more general research direction proposal in
which we investigate the impact of basic notions in substantial results, like
the impact of local compactness and connectivity in the theory of proper
transformation groups. In order to provide some basic theory for this new class
of actions we embed a "Cauchy-indivisible" action of a group of isometries of a
separable metric space in a proper action of a semigroup in the completion of
the underlying space. We show that, in case this subgroup is a group, the
original group has a "Weil completion" and vice versa. Finally, in order to
establish further connections between "Cauchy-indivisible" actions on separable
metric spaces and proper actions on locally compact metric spaces we
investigate the relation between "Borel sections" for "Cauchy-indivisible"
actions and "fundamental sets" for proper actions. Some open questions are
added.
