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Math Help - Path in S2 empty interior

  1. #1
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    Path in S2 empty interior

    Hi,
    What is a good way to prove that the homeomorphic image I of the unit interval in S2 (unit sphere) has empty interior? I can see that any open ball does not lie in I, but do not know how to prove it.

    Any topology results can be assumed (it's for a measure theory course) so any suggestions are welcome!

    Thanks
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  2. #2
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    Quote Originally Posted by james123 View Post
    Hi,
    What is a good way to prove that the homeomorphic image I of the unit interval in S2 (unit sphere) has empty interior? I can see that any open ball does not lie in I, but do not know how to prove it.

    Any topology results can be assumed (it's for a measure theory course) so any suggestions are welcome!

    Thanks


    I don't know it the following suffices, but since the unit interval I has empty interior in \mathbb{R}^2 and we're talking here of a homeomorphism (which is topological equivalence), then ANY homeomorphic image of I in \mathbb{R}^2 will have the same topological characteristics as I itself...

    Tonio
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  3. #3
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    Quote Originally Posted by james123 View Post
    Hi,
    What is a good way to prove that the homeomorphic image I of the unit interval in S2 (unit sphere) has empty interior? I can see that any open ball does not lie in I, but do not know how to prove it.

    Any topology results can be assumed (it's for a measure theory course) so any suggestions are welcome!
    The unit interval had only two cut-points, namely the endpoints of the interval. (A cut-point in a connected space is one whose removal disconnects the space.)

    An interior point of a connected set in \mathbb{R}^2 is never a cut-point. So if the interior is nonempty then there are uncountably many cut points.

    To complete the proof, the number of cut-points is invariant under homeomorphism.
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