## 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?

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Seems like a strange question... Also, why write "applications" in the question when you mean "functions"? – Scott Morrison Oct 28 2009 at 5:29
"Application" is another term for "function" or "mapping." I believe it is of French origin. – Dave Penneys Oct 28 2009 at 6:03
Oui, une "application" est grosso modo le meme chose qu'une "fonction" - roughly synonymous with "mapping" as far as I can tell. – Yemon Choi Oct 29 2009 at 8:28

1' and 2' (and thus 3') are equivalent.

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 Interesting! Is this a direct consequence of the arithmetic of preimages and complements or is there something more deep into it? – Jose Brox Nov 7 2009 at 10:12 The former. The preimage of a closed set C is the complement of the preimage of the complement of C. The complement of C is open, so if we assume 1' then the preimage of C is the complement of an open or closed set, and hence is closed or open. So 1' -> 2' and the converse is similar. – Jonas Meyer Nov 7 2009 at 18:38 Thank you, Jonas! – Jose Brox Nov 10 2009 at 7:52

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.

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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.

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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?

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In a similar vein, the set of subsets which are either open or closed isn't closed under infinite union or infinite intersection, so it's not a very nice set to work with. And if you allow infinite unions and infinite intersections, then in a Hausdorff space you just get every subset. – Qiaochu Yuan Oct 28 2009 at 14:23
If you allow only countable unions and intersections you get a very nice set of subsets to work with, of course, namely the Borel sets, which is certainly not every subset (in "reasonable" topological spaces.) – Apollo Nov 5 2009 at 2:00