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Homeomorphism proof

  1. Nov 18, 2009 #1
    1. The problem statement, all variables and given/known data
    Prove that [tex] D^n/S^{n-1} [/tex] is homeomorphic to [tex] S^n [/tex]. (Hint - try the cases n = 1,2,3 first).


    2. Relevant equations
    X/Y is defined as the union of the complement X\Y with one point. I showed in a previous section that the equivalence relation x~x' if x and x' are in Y gives X/~ = X/Y. The identification space topology was then used, and I showed that a subset Z of X with [tex] Y \cap Z [/tex] non-empty is such that p(Z) is open in X/Y if and only if [tex] Y \cup X [/tex] is open in X.

    3. The attempt at a solution
    First I define the open sets. A set in [tex] D^n [/tex] is open if [tex] U = D^n \cap V [/tex] for some V open in [tex] R^n [/tex]. A subset U in [tex] D^n /S^{n-1} [/tex] is open if [tex] U \cap S^{n-1} [/tex] is empty and U is open in [tex] D^n [/tex] or if the intersection is non-empty as in the relevant equations section.
    A set U in [tex] S^n [/tex] is open if [tex] U = V \cap S^n [/tex] for V open in [tex] R^{n+1} [/tex].

    I need a function mapping open sets to open sets. For n = 1 this was easy since I knew the basis for [tex] S^1 [/tex] was open arcs, so f = (cos pi(x+1), sin pi(x+1)) (I think this is a homeomorphism) worked. The only was I could explicitly do this for n = 2 would be to use the spherical polar coordinates that I know from physics and other maths courses, which I wouldn't be able to generalise anyway. I'm not really sure how to proceed here.
     
    Last edited: Nov 18, 2009
  2. jcsd
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