Countably open and closed functions

It is obvious that every open function f:X→Y has for every y∈Y a cover εy of the subspace f−1(y) by singletons such that(⁎)every open neighborhood of every x∈εy also contains x′∈εy′ for every point y′ from some neighborhood of y.If x (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 Xi⊂X (i=1,2,…) such that each restriction f|Xi is an open function onto f(Xi) and the sets f(Xi) cover Y.