When we study topology, we typically study topological spaces and continuous maps between them. From a categorical perspective, this is "wrong," because continuous maps are not the structure preserving maps. If "the structure of a topological space" is it's topology (i.e., it's open sets) then it seems like the natural choice for the definition of a structure preserving map would be an open map. So why do we like continuous maps instead of open maps?
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http://math.stackexchange.com/questions/932942/why-are-continuous-functions-the-right-morphisms-between-topological-spaces – Najib Idrissi Sep 17 '14 at 09:38
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Oops, forgot that I could close unilaterally. If someone strongly disagrees, feel free to reopen. – Zhen Lin Sep 17 '14 at 09:40