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Math Help - Connectedness of a subset of R^2

  1. #1
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    Connectedness of a subset of R^2

    Hi I need help with this problem. This is not for a grade. Please give a detailed proof, this is just to help me study.

    1) Let C be a countable subset of the plane  \mathbb{R}^2 . Prove the complement of C in  \mathbb{R}^2 is connected.
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  2. #2
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    Quote Originally Posted by ElieWiesel View Post
    Hi I need help with this problem. This is not for a grade. Please give a detailed proof, this is just to help me study.

    1) Let C be a countable subset of the plane  \mathbb{R}^2 . Prove the complement of C in  \mathbb{R}^2 is connected.
    Every path-connected space is connected. So I think it is easier to show that it is path-connected. The proof for path-connectedness of the above problem can be found here.
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  3. #3
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    Quote Originally Posted by aliceinwonderland View Post
    Every path-connected space is connected. So I think it is easier to show that it is path-connected. The proof for path-connectedness of the above problem can be found here.
    There's a slightly simpler argument than the one presented. Given any two points x and y in R^2 - C, there are uncountably many pairwise disjoint (except at the endpoints) paths from x to y. (E.g., take any point z on a given line perpendicular to that between x and y, and then draw the path from x to z to y.) Since there are only countably many points in C, at least one of these paths (actually, uncountably many) must be free of any of these points. Hence R^2 - C is path-connected.
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  4. #4
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    Awesome, thanks! I actually came up with something similar. Its nice to know I was on the right "path"...get it...get it. "Path connected" oh man i crack myself up.

    Thanks again.
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