Homotopy retract: Difference between revisions
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A [[subspace]] of a [[topological space]] is termed a '''homotopy retract''' if the identity map from the whole space to itself is homotopic to the [[retraction]] onto that subspace. | A [[subspace]] of a [[topological space]] is termed a '''homotopy retract''' if the identity map from the whole space to itself is homotopic to the [[retraction]] onto that subspace. | ||
===Definition with symbols=== | |||
A subspace <math>A</math> of a [[topological space]] <math>z</math> is termed a '''homotopy retract''' of <math>X</math> if there exists a map <math>F: X \times I \to X</math> such that: | |||
* <math>F(x,0) = x \ \forall \ x \in X</math> | |||
* <math>F(x,1) \in A \ \forall \ x \in X</math> | |||
Note that unlike in the stronger notion of deformation retract, we do not require that at intermediate times, <math>F</math> should restrict to the identity on <math>A</math>. | |||
==Relation with other properties== | ==Relation with other properties== | ||
Revision as of 22:07, 26 September 2007
This article defines a property over pairs of a topological space and a subspace, or equivalently, properties over subspace embeddings (viz, subsets) in topological spaces
Definition
Symbol-free definition
A subspace of a topological space is termed a homotopy retract if the identity map from the whole space to itself is homotopic to the retraction onto that subspace.
Definition with symbols
A subspace of a topological space is termed a homotopy retract of if there exists a map such that:
Note that unlike in the stronger notion of deformation retract, we do not require that at intermediate times, should restrict to the identity on .
