Hatcher, Algebraic Topology, Chapter 0
9. Show that a retract of a contractible space is contractible.
Proof. Let be a topological space which is contractible, a subspace, and suppose that retracts onto via the map . To be more precise, suppose is a point and that is a map with homotopy inverse for which and . Let (resp. ) denote the homotopy connecting and (resp., connecting and ). It suffices to find homotopies connecting to and vice versa.
Note that being a retract implies immediately that the inclusion satisfies . Solution of this problem, then, boils down to finding maps and so that the diagram in Figure 1 below commutes and for which , .
The obvious candidates here are and , which by definition ensure that the diagram in Figure 1 commutes. Moreover,
whereby it follows that and satisfy the desired properties with regards to the identity maps. Hence, the claim is proven.
Note: The choices for and can be rectified with the “usual homotopy notation” by modifying the definitions of and given above simply by working relative to throughout. To be specific, let and be the homotopies mentioned above sending and , respectively. In particular, , , , and for all . Next, define homotopies and so that and . It follows immediately, then, that , , , and for all . Thus, has the same homotopy type as by way of the explicit homotopies and .