Hatcher, Algebraic Topology, Chapter 0
2. Construct an explicit deformation retraction of onto .
Proof. This problem is essentially a problem from Calculus III. Note that for a vector in , the normalized vector lies on . It suffices, then, to do the normalization process in a way that’s continuous for a time parameter , and one way to accomplish this is to define a family so that, for each in the domain,
As noted in problem 1, the function is continuous for each . Moreover, , , and due to the fact that for all .