Why are inverse images of functions more central to mathematics than the image?


I have a sequence of related questions:


1. Why the fixation on continuous maps as opposed to open maps? (Is there an epsilon-delta definition of open maps in metric spaces?)
2. Is there an inverse-image characterization of homomorphisms in algebraic categories? (What kind of map do you get if you look at a map from a group to another group, where inverse images of subgroups are subgroups?)
3. Inverse images have better set-theoretic properties than the image (for instance, commuting with unions, intersections, etc..) This clearly is a direct consequence of definition of a function. There is an asymmetry in the definition of a function (the domain and codomain behave differently with respect to the function). I think this also has consequences for differences between existence and uniqueness of left and right inverses for one-to-one or onto functions. Why this asymmetry? What are the historical reasons for the asymmetry? Whats sort of mathematics do we have if the definition of a function was purely symmetric? (For instance, f(a) may give multiple values, just like f^-1(a) may have multiple values).
4. Is it accurate to say the definitions for monomorphisms and epimorphisms in category theory correct for the asymmetry? (And hence, the notion of epimorphisms and onto-morphisms in concrete categories don't coincide)



Why are inverse images more important than images in mathematics?
Open sets can be identified with maps from a space to the Sierpinski space, and maps out of a space pull back under morphisms. (In other words, if you believe that the essence of what it means to be a topological space has to do with functions out of the space, you are privileging inverse images over images. A related question was discussed [here](https://mathoverflow.net/questions/19152/why-is-a-topology-made-up-of-open-sets/).) I think essentially this kind of reasoning underlies the basic appearances of inverse images in mathematics. For example, in the category of sets, subsets can be identified with maps from a set to the two-point set, and again these maps pull back under morphisms. This should be responsible for the nice properties of inverse image with respect to Boolean operations.


Your third question was asked, closed, and deleted once; I started a blog discussion about it [here](http://qchu.wordpress.com/2010/03/25/the-definition-of-a-function/).


