Quotient map: Difference between revisions

Revision as of 21:10, 10 November 2007

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

@@ Line 14: / Line 14: @@
 Given a topological space <math>X</math>, a ''set'' <math>Y</math> and a surjective map <math>f:X \to Y</math>, we can prescribe a unique topology on <math>Y</math>, the so-called [[quotient topology]], such that <math>f</math> is a quotient map. Moreover, this is the [[coarser topology|coarsest]] topology for which <math>f</math> becomes continuous.
+Also, the study of a quotient map is equivalent to the study of the equivalence relation on <math>X</math> given by <math>x \simeq y \iff f(x) = f(y)</math>.
 ==Relation with other properties==