# Category of topological spaces with open or closed maps

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 (contiuous 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 the 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.

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The isomorphisms in the category of topological spaces and continuous maps are precisely the homeomorphisms. Did you mean to say "bijection" instead of "isomorphism"? –  Zhen Lin Dec 25 '12 at 0:17
I think he means what he wrote, namely, to what extent these three categories differ, taking into account that they have the same objects ands the same isomorphisms. –  Pietro Majer Dec 25 '12 at 11:35
Regarding the category whose morphisms are open maps, there are a few remarks here: christianmarks.wordpress.com/category/bagatelle For example, it is asserted that all coproducts exist, that not all products exist, and that "something analogous to powers of a single space" exists. –  Adam Epstein Dec 26 '12 at 20:36
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. After this I am stuck. –  Adam Epstein Jan 3 '13 at 20:10

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