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Homework Help: Metric space topology problem

  1. Jul 25, 2007 #1

    quasar987

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    1. The problem statement, all variables and given/known data
    Show that if {K_n} is a decreasing family of compact connected sets in a metric space, then their intersection is connected as well. Illustrate with an example why 'compact' is necessary instead of just 'closed'.


    3. The attempt at a solution

    Well, I have a example for the second part of the question. Consider F_n = R²\{(x,y): -n<y<n, -1<x<1}. Then each F_n is closed and (path-)connected, but their intersection is the plane separated in half along the y axis by this open band of width 2, which is not connected.

    For the first part though, I can visualize why it's true for simple examples, but I don't know how to approach a general proof.
     
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  3. Jul 25, 2007 #2

    Dick

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    Ok. Suppose the limit set K is disconnected. That means there are two open sets A and B that disconnect K, right? So A intersect B is empty but K is contained in AUB. Since K is compact we can define A and B so that A, B and K are all contained in the interior of a closed ball R. Consider the compact set R-(AUB). K_n is connected so it must intersect R-(AUB). So for each K_n there is a point in K_n, say x_n, contained in the compact set R-(AUB). I'll let you take over now...
     
  4. Jul 26, 2007 #3

    quasar987

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    Nice, thx.
     
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