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Limit points

  1. Jun 20, 2011 #1
    Is it true that finite sets don't have limit points?
     
  2. jcsd
  3. Jun 20, 2011 #2
    it depends on what topology u use on the unversal set
     
  4. Jun 20, 2011 #3
    The real numbers and the Euclidean metric.
     
  5. Jun 20, 2011 #4

    HallsofIvy

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    Yes, in any metric space, a finite set has no limit points. A point p is a limit point of set A if and only if, for any [itex]\delta> 0[/itex], there exist a point, q, of A other than p such that [itex]d(p,q)< \delta[/itex]. If A is a finite set, then there exist a "shortest" distance between points: M= min(d(p,q)) where the minimum is take over all pairs of points in A. Taking [itex]\delta[/itex] to be smaller than M shows that A cannot have any limit points.
     
  6. Jun 20, 2011 #5
    with the eucliden metric on R we can deduct the standard topology on R which is a Hausdorff space so the set of limit points is close in R
     
  7. Jun 20, 2011 #6
    Not true in general for topological spaces. For example, for any set X with the indiscrete topology [itex]\{\emptyset,X\}[/itex], every point is a limit point of every set with at least two elements.

    But it's true for every T1 topological space. This means that for any two distinct points x and y, there is an open set containing x but not y.
     
  8. Jun 21, 2011 #7
    Thanks!
     
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