Homotopy equivalence of topological spaces: Difference between revisions
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Suppose <math>A</math> and <math>B</math> are [[topological space]]s. A homotopy equivalence between <math>A</math> and <math>B</math> is a map <math>f:A \to B</math> such that there exists a map <math>g:B \to A</math> for which <math>f \circ g</math> is homotopic to the identity on <math>B</math> and <math>g \circ f</math> is homotopic to the identity on <math>B</math>. | Suppose <math>A</math> and <math>B</math> are [[topological space]]s. A homotopy equivalence between <math>A</math> and <math>B</math> is a map <math>f:A \to B</math> such that there exists a map <math>g:B \to A</math> for which <math>f \circ g</math> is homotopic to the identity on <math>B</math> and <math>g \circ f</math> is homotopic to the identity on <math>B</math>. | ||
==Relation with other properties== | |||
===Stronger properties=== | |||
* [[Homeomorphism]] | |||
* [[Strong deformation retraction]] | |||
* [[Homotopy retraction]] | |||
===Weaker properties=== | |||
* [[Weak homotopy equivalence of topological spaces]] | |||
==Related notions== | ==Related notions== | ||
* [[Homotopy equivalence of chain complexes]] | * [[Homotopy equivalence of chain complexes]] | ||
Revision as of 23:32, 24 October 2007
Definition
Suppose and are topological spaces. A homotopy equivalence between and is a map such that there exists a map for which is homotopic to the identity on and is homotopic to the identity on .
