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Math Help - Compactification

  1. #1
    Junior Member
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    Compactification

    I want to prove that the one-point compactification of \mathbb{R}^n is homeomorphic to the n-sphere

    S^n=\{(y_1,\ldots,y_{n+1})\in\mathbb{R}^{n+1}:y_1^  2+\cdots +y_{n+1}^2=1\}.

    Thanks
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  2. #2
    Senior Member roninpro's Avatar
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    I'm not entirely familiar with compactification, but did you try using the stereographic projection as a homeomorphism?

    Stereographic projection - Wikipedia, the free encyclopedia
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by bram kierkels View Post
    I want to prove that the one-point compactification of \mathbb{R}^n is homeomorphic to the n-sphere

    S^n=\{(y_1,\ldots,y_{n+1})\in\mathbb{R}^{n+1}:y_1^  2+\cdots +y_{n+1}^2=1\}.

    Thanks
    Use the fact that \mathbb{R}^n_{\infty} is the unique, up to homeomorphism, space which is compact, Hausdorff, and contains a homeomorphic image of \mathbb{R}^n, and the complement of that image is a single point.

    Otherwise (if you don't know the above theorem), use roninpro's solution and map (1,0,\cdots,0)\mapsto\infty
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