How to establish a homeomorphism between $\mathbb{R}^{n+1} \setminus \{0\}$ and $S^n\times\mathbb{R}$?
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Thomas and Git both give ways of doing this for $n=1$. This is probably the most important step for solving the problem. However the second part of solving it is generalizing what you know about $n=1$ to $n > 1$. For $n > 1$, you can still define a radius and this is what is sent to the $\mathbb{R}$ component (with a homeomorphism $\mathbb{R}^+ \rightarrow \mathbb{R}$). It should then also be clear what you need to send to $S^n$. What might not be clear is that this is a homeomorphism. For this, write down everything in coordinates and you will be able to see that the maps are continuous. – RghtHndSd Jul 21 '13 at 15:15
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possible duplicate of Finding a homeomorphism $\mathbb{R} \times S^1 \to \mathbb{R}^2 \setminus {(0,0)}$ – Seirios Jul 22 '13 at 08:24
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this question is broader, however – mau Jul 22 '13 at 09:16
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Hint: Consider $n=1$ and polar coordinates.
Thomas Andrews
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Consider the map $f:\mathbb{R}^{n+1}\setminus\{0\} \to S^n \times \mathbb{R}_{++}$ given by $$f(x):=(x/||x||,||x||).$$ It's a homeomorphism, you just need to compose it with a homeomorphism $\mathbb{R}_{++}\to \mathbb{R}$.
Edoardo Lanari
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