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Proving an open ball is connected in a metric space X

  1. Oct 9, 2013 #1
    The problem statement, all variables and given/known data.

    Let ##B(a,ε) (ε>0)## in a metric space ##(X,d)##. Decide whether this subset of ##(X,d)## is connected or not.

    The attempt at a solution.
    Well, I know open intervals in the real line are connected. I suppose that an open ball in a given metric space can be imagined as an open interval of a more general metric space instead of the real line; at least, that's the way I see it. So, by this analogy, I think that any open ball in a given metric space is always connected. My problem is I don't know how to prove it. I've tried it by the absurd:

    So, suppose we can disconnect ##B(a,ε)##. Then there exist ##U## and ##V##, nonempty open subsets of ##(X,d)## such that
    i)##B(a,ε)= U \cup V##
    ii)##U \cap V=\emptyset##

    How can I come to an absurd? By all the things above, I know that there exists ##x \in B(a,ε)## : ##x \in U## and ##x \not\in V##. On the other hand, ##U## is an open set, so for some ##δ>0##, ##B(x,δ) \subset U##. That's all I got up to now. How can I continue to arrive to an absurd?
     
  2. jcsd
  3. Oct 9, 2013 #2
    Is it possible that the statement that you are trying to prove, that every open ball in every metric space is connected, is not true?
     
  4. Oct 9, 2013 #3

    Dick

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    The real line is too simple an example to make judgements based on it. Can't you think of a metric space that has disconnected open balls?
     
  5. Oct 9, 2013 #4
    I got really mixed up trying to generalize the interval case. The counterexample I could think of was: consider X an infinite metric space with the discrete δ-distance. Pick any x in X and consider of the ball centered at x of radius 2. Then I can disconnect this ball by two open sets U and V consisting of union of open balls of radius 1 centered at any other point of the space. Is this correct? Can you give me another example to completely destroy my previous wrong assumption?
     
  6. Oct 9, 2013 #5
    Yes, it's wrong, I got confused.
     
  7. Oct 9, 2013 #6

    Dick

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    Sure. Any subset of a metric space is a metric space, right? Take the subset of the reals X=(-infinity,-1/2]U[1/2,infinity) and think about the open ball around 0 of radius 1. There's also plenty of examples where X is connected and still has disconnected open balls. Can you think of one?
     
  8. Oct 9, 2013 #7

    Office_Shredder

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    Dick, your example doesn't work because 0 isn't a point in the metric space to have an open ball around, but the open ball around -1/2 of radius 2 does work.

    I second the think of a connected metric space which has disconnected open balls call. There are simple subsets of R2 that work.
     
  9. Oct 9, 2013 #8

    Dick

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    Good catch. Thank you!
     
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