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Construct compact set of R with countable limit points

  1. Oct 7, 2004 #1
    Construct a compact set of real numbers whose limit points form a
    countable set.
     
  2. jcsd
  3. Oct 7, 2004 #2

    arildno

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    Shouldn't a single point be enough?
     
  4. Oct 7, 2004 #3

    arildno

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    Note:
    I may have forgotten the precise definition of "the limit point".
    You might instead look at a convergent sequence in R; that is a compact set, with one limit point.
     
  5. Oct 7, 2004 #4
    for example {(0, 1/n) : n=1,2,3,......} is compact but the only limit point is 0. Still I need countable limit points.
     
  6. Oct 7, 2004 #5
    Note: A single point has no limit point, since
    a limit point of a set A is a point p such that for any neighborhood of p
    (ie Ball(p,r) , where p is the origin and r=radius can take any value >0)
    there exists a q≠p where q belongs in B(p,r) and q belongs to A.
     
  7. Oct 8, 2004 #6

    matt grime

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    You can construct a set with one limit point. Now you can make one with two limit points, 3 limit points, indeed any number of limit points countable or otherwise.
     
  8. Oct 8, 2004 #7

    arildno

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    Yeah, I kind of remembered that a bit late...:redface:

    Finite sets are countable.
     
    Last edited: Oct 8, 2004
  9. Oct 8, 2004 #8

    HallsofIvy

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    But the original post probably meant "countably infinite".
     
  10. Oct 8, 2004 #9
    Taking A={0, 1/n + 1/m | n,m >=1 in N}. Thus the limit points are 1/n which are countable.
    Since the set is closed and bouned then it is compact. (theorem)
    Or
    It can prove by definition that A is compact, which is what I did since I forgot to use the theorem above which would have made life easier :)
     
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