A GENERIC FACTORIZATION THEOREM

Let F:Z-->X be a minimal usco map from the Baire space Z into the comp act space X. Then a complete metric space P and a minimal usco G:P-->X can be constructed so that for every dense G(delta)-subset P-1 of P t here exist a dense G(delta)-subset Z(1) of Z and a (single-valued) con tinuous map f:Z(1)-->P-1 such that F(z)subset of G(f(z)) for every z i s an element of Z(1). In particular, if G is single-valued on st dense Gs-subset of P, then F is also single-valued on a dense G(delta)-subs et of its domain. The above theorem remains valid if Z is Cech complet e space and X is an arbitrary completely regular space. These factoriz ation theorems show that some generalizations of a theorem of Namioka concerning generic single-valuedness and generic continuity of mapping s defined in more general spaces can be derived from similar results f or mappings with complete metric domains. The theorems can be used als o as a tool to establish that certain topological spaces contain dense completely metrizable subspaces.