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Homework Help: Finite subset of a metric space

  1. Feb 24, 2008 #1
    1. The problem statement, all variables and given/known data

    Let X be an infinite set. For p,q [tex]\in[/tex] X define:
    d(p,q) = {1 if p [tex]\neq[/tex] q; 0 if p = q
    Suppose E is a finite subset of X, find all limit points of E.

    2. Relevant equations

    definition: a point p is a limit point of E if every neighborhood of p contains a point q [tex]\neq[/tex] p s.t. q[tex]\in[/tex]E

    3. The attempt at a solution
    My thoughts were that there are no limit points for any subset of X because for any point p, B(p, 1/n) is empty, [tex]\forall[/tex]n[tex]\in[/tex]N. Therefore, for any point p, there exists at least one neighborhood that contains no points q [tex]\in[/tex] E.

    I feel that this is true, but have I made any incorrect assumptions??
    Thanks a lot for any feedback.
  2. jcsd
  3. Feb 24, 2008 #2


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    B(p,1/n) isn't empty for all n. For instance p is always in B(p,1/n), and if n=1, then B(p,1/n) is in fact all of X.

    You definitely have the right idea, but you have to be a bit more careful.
  4. Feb 24, 2008 #3


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    remember that you have the additional rquirement that q[itex]\ne[/itex] p. I think that is what you meant to say. For all n, B(p, 1/n) contains p, but for some n, B(p, 1/n) does not include any other point of the set.
  5. Feb 25, 2008 #4
    Right, ok. I can fix that detail. Thanks a lot guys.
  6. Sep 1, 2011 #5
    i need the proof for every finite set is a closed set in a metric space ..
    and finite set in a metric space in compact as soon as possible../.
  7. Sep 1, 2011 #6
    You will need to make a new topic for this. And it would also be nice to see your attempt...
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