derived smooth geometry
Quotient objects in the category of vector spaces also traditionally use the term ‘quotient space’, but they are really just a special case of quotient modules, very different from the other kinds of quotient space. Quotient TVSes, however, combine both aspects.
Let be a topological space and an equivalence relation on (the underlying set of) . (Since monomorphisms in Top are just injective continuous maps, to give an equivalence relation on the underlying set of a topological space is the same as to give a congruence on that space in .) Let be the quotient set and the quotient map.
Obviously, up to homeomorphism, all that matters is the surjective function . For the above definition, we don’t even need it to be surjective, and we could generalize to a sink instead of a single map; in such a case one generally says final topology or strong topology. See topological concrete category.
Recall that a map is open if is open in whenever is open in . It is not the case that a quotient map is necessarily open. Indeed, the identification map , where the endpoints of are identified with , takes the open point of the domain to a non-open point in .
Nor is it the case that a quotient map is necessarily a closed map; the classic example is the projection map , which projects the closed locus onto a non-closed subset of . (This is a quotient map, by the next remark.)
It is easy to prove that a continuous open surjection is a quotient map. For instance, projection maps are quotient maps, provided that is inhabited. Likewise, a continuous closed surjection is a quotient map: is open is closed is closed is open. For example, a continuous surjection from a compact space to a Hausdorff space is a quotient map.
(More details needed.)