Quotient Map

A quotient map associated with an equivalence relation on a set is the surjection that sends to its equivalence class in the quotient set. It is also called the canonical projection. A function that is constant on each equivalence class factors through . The unique factor is given by .

For a group with a normal subgroup , the quotient map to the quotient group sends to the coset . It is a surjective group homomorphism with kernel . A group homomorphism from whose kernel contains factors uniquely through this map as a group homomorphism (Milne 2011, pp. 40-42).

In topology, a quotient map is a surjection between topological spaces such that a subset of is an open set iff its inverse image is an open set in (Munkres 2000, p. 137). This is the condition that carry the quotient topology induced by . For a function to a topological space , is continuous iff is continuous (Munkres 2000, p. 142). The canonical projection onto a quotient space is therefore a quotient map.

The identity map on a set with at least two elements is a continuous surjection when the domain has the discrete topology and the codomain has the trivial topology. It fails the quotient condition because every inverse image is an open set, but some nonempty proper subset of the codomain is not an open set.