A selection theorem for C-spaces

We prove a selection theorem which characterizes C-spaces among paracompact spaces and give two applications: 1. f: X → Y is a map between compact spaces with k-dimensional fibres and Y has property C, then f can be approximated by maps with fibres contained in k-dimensional polyhedra; f: X → Y is an open map with infinite fibres between compact metric spaces, then the map P(f): P(X) → P(Y) between the spaces of probability measures is a trivial Q-bundle, provided that Y has property C.