Cofibration: Difference between revisions
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==Definition== | ==Definition== | ||
A [[subspace]] <math>A</math> of a [[topological space]] <math>X</math> is said 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>. | ||
Revision as of 22:00, 24 October 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
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 .
