An alternative approach to the decomposition of functions

Let a function f:X→Y with compact fibers takes any open set into a union of an open and a closed set. Then there exist subsets Tn⊂X (n=1,2,…) such that every restriction f|Tn is open or closed function onto Yn and Yn (n=0,1,…) cover Y.This result is related to the problem of preservation of Borel classes by open-Borel functions and to Luzinʼs question about decomposition of Borel-measurable functions into countably many continuous ones.