Definition with symbols
Let be topological spaces and be continuous maps. is termed a quotient map if it is sujective and if is open iff is open in .
Given a topological space , a set and a surjective map , we can prescribe a unique topology on , the so-called quotient topology, such that is a quotient map. Moreover, this is the coarsest topology for which becomes continuous.
Also, the study of a quotient map is equivalent to the study of the equivalence relation on given by .