Quotient map - Revision history

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2008-05-11T19:57:45Z

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2007-11-10T21:10:15Z

Significance

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	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.		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==		==Relation with other properties==

Vipul at 21:09, 10 November 2007

2007-11-10T21:09:25Z

New page

{{continuous map property}}

==Definition==

===Symbol-free definition===

A [[continuous map]] between [[topological space]]s 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 <math>X,Y</math> be topological spaces and <math>f:X \to Y</math> be continuous maps. <math>f</math> is termed a quotient map if it is sujective and if <math>U \subset Y</math> is open iff <math>f^{-1}(U)</math> is open in <math>X</math>.

==Significance==

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.

==Relation with other properties==

===Stronger properties===

* [[Open map]]
* [[Closed map]]
* [[Perfect map]]
* [[Proper map]]