Properties of countable separation and implicit function theorem

We consider some properties of countable separation for familiesA andD of subsets of a setX by means of elements of a fixed familyM⊂P(X). We give necessary and sufficient conditions, in terms of measurability of some sets constructed by means of multifunctions, in order that the familiesA andD satisfy such a property. As an application we derive an implicit function theorem for functions ofCarathéodory-type. Supported by 60% and 40% research fund of M.P.I. of Italy.