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 .

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