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 T n ⊂ X ( n = 1 , 2 , … ) such that every restriction f | T n is open or closed function onto Y n and Y n ( n = 0 , 1 , … ) cover Y. esult 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.