Hatcher, Algebraic Topology, Chapter 0
27. Given a pair and a homotopy equivalence , show that the natural map is a homotopy equivalence if satisfies the homotopy extension property [Hint: Consider and use the preceding problem]. An interesting case is when is a quotient map, hence the map is the quotient map identifying each set to a point. When is a point, this gives another proof of Proposition 0.17.