Quotient map

Symbol-free definition
A continuous map between topological spaces is termed a quotient map if it is surjective, and if a set in the range space is open iff its inverse image is open in the domain space.

Definition with symbols
Let $$X,Y$$ be topological spaces and $$f:X \to Y$$ be continuous maps. $$f$$ is termed a quotient map if it is sujective and if $$U \subset Y$$ is open iff $$f^{-1}(U)$$ is open in $$X$$.

Significance
Given a topological space $$X$$, a set $$Y$$ and a surjective map $$f:X \to Y$$, we can prescribe a unique topology on $$Y$$, the so-called quotient topology, such that $$f$$ is a quotient map. Moreover, this is the coarsest topology for which $$f$$ becomes continuous.

Also, the study of a quotient map is equivalent to the study of the equivalence relation on $$X$$ given by $$x \simeq y \iff f(x) = f(y)$$.

Stronger properties

 * Open map
 * Closed map
 * Perfect map
 * Proper map