TITLE: Is there a conceptual reason why topological spaces have quotient structures while metric spaces don't?
QUESTION [16 upvotes]: Of the mathematical objects that I am familiar with, it is normally the case that the product of 2 objects is an object of the same type and that an equivalence relation on an object induces a quotient object of the same type.
I think I have some understanding as to why the product of 2 fields is not a field, because a field is not an algebra in the universal algebra sense. But I don't see a reason as to why an equivalence relation on a metric space fails to induce a quotient structure, apart from the fact that it just doesn't work.

REPLY [14 votes]: One way to get around the lack of quotients is to work in the category of Uniform Spaces. This is more general than a metric space, but still gives you a way to talk about how "close" two points are. Uniform spaces are also cool because they simultaneously generalize metric spaces and topological groups. I believe their study was popular back in the 60s and 70s for instance because this category of uniform spaces has much nicer categorical properties than either of the categories it generalizes. For instance, with uniform maps it's complete and cocomplete, as can be seen here.
Even if your maps are just continuous maps, you still get quotients as can be seen here. Isbell's book is a great reference for this subject.
Side note: I like Eric Wofsey's answer a lot and I wonder if it's secretly related to this answer. The wikipedia page points out the connection between uniform spaces and pseudo-metrics but for categorical things it's probably easy to work with the former than the latter.