Abstract
We prove a selection theorem which characterizes C-spaces among paracompact spaces and give two applications: 1. (1) if 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; 2. (2) if 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.
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