Definition 8

Let X be a topological space and let A be a subset of X. We say that the pair $(X,A)$ possesses the homotopy extension property if, for every topological space Y, every continuous mapping $f: X\rightarrow Y$ and every homotopy $\sigma : A\times \mathbf{I}\rightarrow Y$ whose term is the mapping $f|A$, there exists a homotopy $\tau : X\times \mathbf{I}\rightarrow Y$ extending $\sigma$ and whose term is the mapping $f$.