Homeomorphism proof

1. Nov 18, 2009

1. The problem statement, all variables and given/known data
Prove that $$D^n/S^{n-1}$$ is homeomorphic to $$S^n$$. (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 $$Y \cap Z$$ non-empty is such that p(Z) is open in X/Y if and only if $$Y \cup X$$ is open in X.

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

I need a function mapping open sets to open sets. For n = 1 this was easy since I knew the basis for $$S^1$$ 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