Countable open and closed functions
  We define two natural classes of functions, called 2-open and 2-closed, that are closest to open and closed functions.   We show that they have the following property: there are $X_i \subset X$ $ (i=1,2,...$) such that $f|X_i$ are open or closed functions onto $f(X_i)$ and $f(X_i)$ cover $Y.$ 
Alexey Ostrovsky
math.GN