A generic factorization theorem

Abstract

Let F:Z→X be a minimal usco map from the Baire space Z into the compact space X. Then a complete metric space P and a minimal usco G:P→X can be constructed so that for every dense Gδ-subset P1 of P there exist a dense Gδ Z1 of Z and a (single-valued) continuous map f: Z1→P1 such that F(Z)⊂G(f(z)) for every z∈Z1. In particular, if G is single valued on a dense Gδ-subset of P, then F is also single-valued on a dense Gδ-subset of its domain. The above theorem remains valid if Z is Čech complete space and X is an arbitrary completely regular space. These factorization theorems show that some generalizations of a theorem of Namioka concerning generic single-valuedness and generic continuity of mappings defined in more general spaces can be derived from similar results for mappings with complete metric domains. The theorems can be used also as a tool to establish that certain topological spaces contain dense completely metrizable subspaces.