homotopy extension property

In mathematics, in the area of algebraic topology, the homotopy extension property indicates which homotopies defined on a subspace can be extended to a homotopy defined on a larger space. The homotopy extension property of cofibrations is dual to the homotopy lifting property that is used to define fibrations. Definition Let be a topological space, and let . We say that the pair has the homotopy extension property if, given a homotopy and a map such that then there exists an extension of to a homotopy such that . That is, the pair has the homotopy extension property if any map can be extended to a map (i.e. and agree on their common domain). If the pair has this property only for a certain codomain , we say that has the homotopy extension property with respect to .