Cofibration: Difference between revisions
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A [[subspace]] <math>A</math> of a [[topological space]] <math>X</math> is said to be a '''cofibration''', or to have the '''homotopy extension property''' if the following holds: given any map <math>f_0: X \to Y</math> and a homotopy <math>F:A \times I \to Y</math> such that <math>F(a,0) = f(a) \ \forall \ a \in A</math>, we have a homotopy <math>\tilde{F}:X \times I \to Y</math> whose restriction to <math>A</math> is <math>F</math>, and such that <math>\tilde{F}(x,0) = f(x) \ \forall \ x \in X</math>. | A [[subspace]] <math>A</math> of a [[topological space]] <math>X</math> is said to be a '''cofibration''', or to have the '''homotopy extension property''' if the following holds: given any map <math>f_0: X \to Y</math> and a homotopy <math>F:A \times I \to Y</math> such that <math>F(a,0) = f(a) \ \forall \ a \in A</math>, we have a homotopy <math>\tilde{F}:X \times I \to Y</math> whose restriction to <math>A</math> is <math>F</math>, and such that <math>\tilde{F}(x,0) = f(x) \ \forall \ x \in X</math>. | ||
==Relation with other properties== | |||
===Stronger properties=== | |||
* [[CW pair]] | |||
* [[Simplicial pair]] | |||
===Weaker properties=== | |||
* [[Closed subset]] in a Hausdorff space: {{proofat|[[Cofibration implies closed subset in Hausdorff space]]}} | |||
==Metaproperties== | |||
{{transitive subspace property}} | |||
Latest revision as of 19:40, 11 May 2008
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
A subspace of a topological space is said to be a cofibration, or to have the homotopy extension property if the following holds: given any map and a homotopy such that , we have a homotopy whose restriction to is , and such that .
Relation with other properties
Stronger properties
Weaker properties
- Closed subset in a Hausdorff space: For full proof, refer: Cofibration implies closed subset in Hausdorff space
Metaproperties
Transitivity
This property of subspaces of topological spaces is transitive. In other words, if satisfies the property as a subspace of and satisfies the property as a subspace of then satisfies the property as a subspace of
