Countably open and closed functions

It is obvious that every open function f : X → Y has for every y ∈ Y a cover ε y of the subspace f − 1 ( y ) by singletons such that(⁎) open neighborhood of every x ∈ ε y also contains x ′ ∈ ε y ′ for every point y ′ from some neighborhood of y. and x ′ ) in (⁎) can be two-point set with the same image, we obtain a simple generalization of the notion of open function. In this case we prove that there exist X i ⊂ X ( i = 1 , 2 , … ) such that each restriction f | X i is an open function onto f ( X i ) and the sets f ( X i ) cover Y.