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Math Help - descending sequence proof

  1. #1
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    descending sequence proof

    Hi, I need this problem to study off of. It is not for a grade. Please provide a complete prove if you can. Thanks,

    Suppose  K_1 \supset K_2 \supset K_3 \supset ... is a descending sequence of nonempty sonnected subsets of a metric space M. Prove: if M is compact, the intersection  \bigcap K_n is a connected subset of M.

    Also give a counterexample to show this is false without compactness.


    Thanks.
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  2. #2
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    Quote Originally Posted by ElieWiesel View Post
    Hi, I need this problem to study off of. It is not for a grade. Please provide a complete prove if you can. Thanks,

    Suppose  K_1 \supset K_2 \supset K_3 \supset ... is a descending sequence of nonempty sonnected subsets of a metric space M. Prove: if M is compact, the intersection  \bigcap K_n is a connected subset of M.
    See this thread for some hints on how to do a similar problem.

    Quote Originally Posted by ElieWiesel View Post
    Also give a counterexample to show this is false without compactness.
    In the space \mathbb{R}^2 let A_n = \{(x,y):|x|<1,\ y<n\}, and let K_n be the complement of A_n, K_n = \mathbb{R}^2\setminus A_n. Then K_n is the whole of \mathbb{R}^2 except for a strip of width 2 around the y-axis going from -\infty up to the height y=n. This gives a descending sequence of nonempty connected subsets K_n whose intersection is the whole of \mathbb{R}^2 except for the entire strip of width 2 around the y-axis. That is clearly not a connected set.
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  3. #3
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    Edit: Never mind, I gave an incorrect proof.
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