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
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
- Closed subset in a Hausdorff space: For full proof, refer: Cofibration implies closed subset in Hausdorff space
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