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Connected Metric Space Question

  1. Jun 30, 2009 #1
    Let X be a connected metric space, let a, b be distinct points of X and let r > 0. Is there a collection {B_i} of finitely many open balls of radius r such that their union is connected and contains a and b.

    I was trying to prove this by contradiction, but couldn't derive a contradiction. I can't think of any counterexamples either. Does anyone know whether this statements is true or false?
     
  2. jcsd
  3. Jun 30, 2009 #2

    Hurkyl

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    Hrm. Well, let's say that a and b are r-ball-connected if such a collection exists.

    A common argument regarding variants of connectedness is to consider something like

    A(a,r) = the set of all points that are r-ball connected to a.

    Now, A(a,r) is clearly an open set. Is it closed? (I'm assuming you know the ramifications of a set being both open and closed)
     
  4. Jun 30, 2009 #3
    Great tip. Thanks for clearing that up.
     
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