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.

Relation with other properties

Stronger properties