Why is the infinite sphere contractible? Why is the infinite sphere contractible?
I know a proof from Hatcher p. 88, but I don't understand how this is possible. I really understand the statement and the proof, but in my imagination this is weird.
Thanks for your help.

I like to think of this in term of classifying spaces (I don't know which proof Hatcher presents). The category with two objects and an isomorphism between them (aka: the free-living isomorphism) is a contractible category, i.e., it is equivalent to a terminal category. Now, the classifying space of the free-living iso is easily seen to be the infinite dimensional sphere. So now the categorical equivalence between the free-living iso and a terminal category means the infinite dimension sphere is homotopic to the classifying space of a terminal category, namely a point.  
