Has anyone studied the applications which map open sets to either open or closed sets? 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?
 A: Properties 1 and 2 seem difficult to work with, because the class of functions satisfying one of those properties isn't closed under composition. Similarly for properties 1' and 2'. However, the class of functions satisfying property 3 or 3' is closed under composition.
It's hard to say what would be useful about these functions -- the question is, are there any useful properties of a topological space that are preserved by these functions? It seems to me that any such properties will have to be similarly wishy-washy; that is, that they will also have to depend on something being open or closed, but not caring which.
Do you have a property in mind that makes these functions natural candidates?
A: Both 1' and 2' imply that f: X -> Y is a morphism of the underlying Borel spaces, c.f.
http://en.wikipedia.org/wiki/Borel_space
This sort of morphism is studied (I believe...) in measure theory, probability and descriptive set theory.
As pointed out above, the spaces and maps satisfying 3' form a category.  A natural (if somewhat vague) question is: does the category associated to 3' have more structure than the Borel category?  I suspect that the answer may be negative.
A: Well, since you mentioned "a generalization of open maps", I have studied a generalization of them in specific context (not exactly the way you defined them). I called the maps near-open maps. It is defined in the following manner: 
if X and Y are topological spaces and f : X --> Y is a function, we say that f is near open iff for any nonempty open subset U of X, f(U) has an interior point.
It is closely related to irreducible surjections when you consider surjections between Hausdorff spaces. I made some results using them that are algebraic (geometric) in nature when studying prime spectra between essential extensions of rings. Its in my dissertation you can take a peek of the latest version in my PhD Changelog.
A: 1' and 2' (and thus 3') are equivalent.
