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Finding cluster points of a set?

  1. Sep 22, 2009 #1
    1. The problem statement, all variables and given/known data

    What are the cluster points for the set

    S = {all (1/n, 1/m) with n = 1, 2, ..., m = 1, 2, ...}

    2. Relevant equations

    A point p is a cluster point for a set S if every neighborhood about p contains infinitely many points of the set S.

    3. The attempt at a solution

    The graph of the set is the open square formed by the points (0,0) (0,1) (1,0) (1,1)?

    The book says cluster points include those such as (0, 1/n) for each n and others on the horizontal axis, as well as the origin. I don't really understand why though...
     
  2. jcsd
  3. Sep 22, 2009 #2

    Dick

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    No, the graph isn't the open square. It's discrete points in the open square. All of the cluster points are on the boundary of the open square. Do you understand why (1,0) is a cluster point?
     
  4. Sep 22, 2009 #3

    lurflurf

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    Think about when neighborhoods of points in your set pile up.
     
  5. Sep 22, 2009 #4
    The points become closer together near the vertical and horizontal axes. I think that's why the cluster points are on the vertical and horizontal axes. Or, they are of the form (0, 1/m) (1/n, 0) and (0, 0). But why is there a cluster point at, say, (1, 1)?
     
  6. Sep 23, 2009 #5

    Dick

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    There isn't a cluster point at (1,1). The nearest point to (1,1) is (1/2,1/2), isn't it? Hardly a cluster point.
     
  7. Sep 23, 2009 #6
    Ok, thank you so much!
     
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