Consider two topological spaces X,Y and a function f from X to Y.

Are the following concepts already in use? How are they called?

1) f sends open subsets of X to either open or closed subsets of Y. 2) f sends closed subsets of X to either open or closed subsets of Y. 3) Both 1) and 2) simultaneously.

1') The preimage of every open subset of Y is either open or closed in X. 2') The preimage of every closed subset of Y is either open or closed in X. 3') Both 1') and 2') simultaneously.

(Obviously, those can be seen as weak generalizations for the definitions of open, closed and continuous maps).

Are there some useful results about them? Who has studied them and where?