Cofibration

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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 A of a topological space X is said to be a cofibration, or to have the homotopy extension property if the following holds: given any map f_0: X \to Y and a homotopy F:A \times I \to Y such that F(a,0) = f(a) \ \forall \ a \in A, we have a homotopy \tilde{F}:X \times I \to Y whose restriction to A is F, and such that \tilde{F}(x,0) = f(x) \ \forall \ x \in X.

Relation with other properties

Stronger properties

Weaker properties

Metaproperties

Transitivity

This property of subspaces of topological spaces is transitive. In other words, if A satisfies the property as a subspace of B and B satisfies the property as a subspace of C then A satisfies the property as a subspace of C