Consider the category whose objects are topological spaces and whose morphisms are the open maps (or closed maps, open continuous maps, closed continuous maps … that is, one whose isomorphisms are precisely the homeomorphisms). How does such a category compare with the usual one whose objects are topological spaces and whose morphisms are continuous maps? For example, what limits and colimits exist?

I'm probably missing something obvious, but why don't products typically exist in the category with open maps? The projections from the usual product (in the category with continuous maps) are open, yielding a canonical open map from the usual product to the putative unusual product.

Todd's observation is true enough: the product in the usual topology (continuous maps) typically fails to realize the corresponding universal property in the unusual topology (open maps). Nevertheless, some other object might realize that universal property. Is it even clear that if such a space exists its underlying set should be naturally identifiable with the underlying set of the factors? After all, while one point spaces are still terminal, maps out of such objects tend not to be open: it seems one would thereby only extract the subset of isolated points. In any event, http://christianmarks.wordpress.com/category/bagatelle
treats the special case of squares. The appropriate space is $X\times X$ with the weakest topology (stronger than the usual) which makes the diagonal embedding open. This construction is clearly not available for products of distinct spaces. My question concerns whether there isn't (as that post suggests there isn't) some devious workaround.