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Math Help - Topology: Connectedness

  1. #1
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    Topology: Connectedness

    Hello!

    Can anybody help me with the following exercise?

    Prove that the one-point compactification of \mathbb{R}^{n} is homeomorphic to the n-sphere
    S^n = {( y_1, ... , y_{n+1} ) \in \mathbb{R}^{n+1} : y^2_1 + ... + y^2_{n+1} = 1}.

    Hint: It suffices to show that S^n\(0,0,...,0,1) is homeomorphic to \mathbb{R}^{n}.


    From a different exercise I know that the circle in R^2 is homeomorphic with R. Can I use it somehow?

    Thanks for your help!
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by sssitex View Post
    Hello!

    Can anybody help me with the following exercise?

    Prove that the one-point compactification of \mathbb{R}^{n} is homeomorphic to the n-sphere
    S^n = {( y_1, ... , y_{n+1} ) \in \mathbb{R}^{n+1} : y^2_1 + ... + y^2_{n+1} = 1}.

    Hint: It suffices to show that S^n\(0,0,...,0,1) is homeomorphic to \mathbb{R}^{n}.


    From a different exercise I know that the circle in R^2 is homeomorphic with R. Can I use it somehow?

    Thanks for your help!
    Personally, I don't like that hint. Note though that you need only prove that f:\mathbb{S}^n\to\left(\mathbb{R}^n\right)_{\infty  } is bijective and continuous since \mathbb{S}^n is compact and \left(\mathbb{R}_n\right)_{\infty} is Hausdorff (since \mathbb{R}^n is locally compact).

    P.S. Define the mapping in the obvious way
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  3. #3
    CSM
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    Quote Originally Posted by Drexel28 View Post
    Personally, I don't like that hint. Note though that you need only prove that f:\mathbb{S}^n\to\left(\mathbb{R}^n\right)_{\infty  } is bijective and continuous since \mathbb{S}^n is compact and \left(\mathbb{R}_n\right)_{\infty} is Hausdorff (since \mathbb{R}^n is locally compact).

    P.S. Define the mapping in the obvious way
    Correct. I understand the topology in your answer.

    But how about the mapping (calculus) of a n-Sphere to \mathbb{R}^n? I know it is defined by the intersections of the line that goes through (0,0,....1) and the plane. But what is the parametrization?
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by CSM View Post
    Correct. I understand the topology in your answer.

    But how about the mapping (calculus) of a n-Sphere to \mathbb{R}^n? I know it is defined by the intersections of the line that goes through (0,0,....1) and the plane. But what is the parametrization?
    Oh god, you're testing my memory here. You mean stereographic projection obviously, right? Look here and here starting on page 13.
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  5. #5
    CSM
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    Quote Originally Posted by Drexel28 View Post
    Oh god, you're testing my memory here. You mean stereographic projection obviously, right? Look here and here starting on page 13.
    Hahaha, yes I get the topology in this topic, but the calculus is gone for years now . Thanks for the second link, i'll take a look at it. I see it's an exercise there too.
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