Could someone please tell me why is $\displaystyle S^n-{x}$ isomorphic to $\displaystyle \mathbb{R}^n$?

Thank you so much.

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- Nov 21st 2009, 04:32 AMgeorgelS^n\{x} isomorphic to R^n
Could someone please tell me why is $\displaystyle S^n-{x}$ isomorphic to $\displaystyle \mathbb{R}^n$?

Thank you so much. - Nov 21st 2009, 05:14 AMShanks
$\displaystyle S^{n+1}-x$ is isomorphic to $\displaystyle R^n$, not $\displaystyle S^n-x$.

- Nov 21st 2009, 05:18 AMgeorgel
- Nov 21st 2009, 08:32 AMJose27
Well, that depends on your notation. The one I have seen the most is $\displaystyle \mathbb{S} ^n = \{ x\in \mathbb{R} ^{n+1} : \Vert x \Vert =1 \}$ in which case it is indeed true that $\displaystyle \mathbb{S} ^n \setminus {e_n} \cong \mathbb{R} ^n$

To prove it, try to generalize the proof of the stereographic projection used to build the Riemann sphere which can be found in any (good) book on basic complex analysis