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In this paper we introduce a new class of metric actions on separable (notnecessarily connected) metric spaces called "Cauchy-indivisible" actions. Thisnew class coincides with that of proper actions on locally compact metricspaces and, as examples show, it may be different in general. The concept of"Cauchy-indivisibility" follows a more general research direction proposal inwhich we investigate the impact of basic notions in substantial results, likethe impact of local compactness and connectivity in the theory of propertransformation groups. In order to provide some basic theory for this new classof actions we embed a "Cauchy-indivisible" action of a group of isometries of aseparable metric space in a proper action of a semigroup in the completion ofthe underlying space. We show that, in case this subgroup is a group, theoriginal group has a "Weil completion" and vice versa. Finally, in order toestablish further connections between "Cauchy-indivisible" actions on separablemetric spaces and proper actions on locally compact metric spaces weinvestigate the relation between "Borel sections" for "Cauchy-indivisible"actions and "fundamental sets" for proper actions. Some open questions areadded.

Properness, Cauchy-indivisibility and the Weil completion of a group of isometries