# Quotient map

From Topospaces

*This article defines a property of continuous maps between topological spaces*

## Contents

## 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 be topological spaces and be continuous maps. is termed a quotient map if it is sujective and if is open iff is open in .

## Significance

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 .