Infinite iterations of a functor of probability measures and open mappings (Q753185)

The following result is proved: Let \(f:X\to Y\) be a continuous open map between compact metric spaces with no singleness points. Denote \(P^{\omega}(f):=\lim P^ n(f)\) the limit of an inverse sequence of the n-th iterations \(P^ n(f):P^ n(X)\to P^ n(Y)\) of the functor of probability measures P, then \(P^{\omega}(f)\) is a Q-fibration, i.e., a trivial fibration with a fiber homeomorphic to the Hilbert cube Q.