Quotient map

This article defines a property of continuous maps between topological spaces

Definition

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 $x \simeq y \iff f(x) = f(y)$.

Relation with other properties

Stronger properties

• Open map
• Closed map
• Perfect map
• Proper map