On exact Milyutin mappings

We introduce the notion of an exact Milyutin mapping as a Milyutin mapping for which supports of values of its associated map coincide with point-preimages. We prove that every open continuous surjection between Polish spaces is an exact Milyutin mapping. For regular mappings we prove that the measure of singletons in the preimages equals zero for some exact Milyutin mapping. As a corollary, we obtain a proof of the local triviality of regular mappings with one-dimensional (not necessary compact) polyhedral fibers and a new proof of the same result for compact fibers.