Homotopy extension property

Summary

Let $A$ be a subspace of a space $X$. The pair $(X, A)$ has the homotopy extension property when each homotopy on $A$ extends to all of $X$, once its start on $X$ is given. Pairs with this property let one move maps on a subspace without losing control of the whole space. It holds for CW pairs and is the defining property of cofibrations.