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Math Help - S^n\{x} isomorphic to R^n

  1. #1
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    S^n\{x} isomorphic to R^n

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

    Thank you so much.
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  2. #2
    Senior Member Shanks's Avatar
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    S^{n+1}-x is isomorphic to R^n, not S^n-x.
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  3. #3
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    Quote Originally Posted by Shanks View Post
    S^{n+1}-x is isomorphic to R^n, not S^n-x.
    Oh, thanks!
    And could you please tell me why is that?
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  4. #4
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    Quote Originally Posted by Shanks View Post
    S^{n+1}-x is isomorphic to R^n, not S^n-x.
    Well, that depends on your notation. The one I have seen the most is \mathbb{S} ^n = \{ x\in \mathbb{R} ^{n+1} : \Vert x \Vert =1 \} in which case it is indeed true that \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
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