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Limit Point of a Set

  1. Jul 30, 2009 #1
    I've been reading Shilov's book and the definition of a limit point is as follows: x is a limit point of A if every neighborhood of x (any open ball centered at x with arbitrary radius r) contains at least one point y distinct from x which belongs to A.

    I feel that from this definition a point at the center of the set would be a limit point. If that is the case then from what I understand the set B of all limit points of A is a superset of A.

    However there is an exercise which says find a set A that is not empty and the set of limit points of A is empty. What could I be missing here?
     
  2. jcsd
  3. Jul 30, 2009 #2
    What do you mean by 'the center of the set'?
    As for your exercise, think about the integers or the naturals.
     
  4. Jul 30, 2009 #3
    If by "a point in the center of the set", you mean, "a point that has a neighborhood of points also in the set", then that's fine. Note that if you mean a geometric center, you are assuming something that a topological set does not necessarily have, a metric, which is a function assigning the distance between two elements of the space (some topological spaces are not even metrizable!).
     
  5. Aug 10, 2009 #4
    a set with the discrete topology for instance.
     
  6. Aug 10, 2009 #5
    Reread the definition and put a huge emphasis on the word "distinct".

    Maybe consider what the limit points of the following sets in R:

    The empty set (even though the problem says it isn't a solution)
    R itself.
    The subset of the integers
    The closed unit interval [0, 1]
    The open unit interval (0, 1)
    The half open unit intervals (0, 1] and [0, 1).
    Singleton sets {0}, {1}, {e}, etc.
    The subset of the rationals
     
  7. Aug 11, 2009 #6
    a set with the discrete topology has no limit points.

    what about the converse? If a topological space has no limit points is it discrete?
     
  8. Aug 11, 2009 #7
    Note that the limit point does not need to be an element of the set. For example. consider the following union of intervals on the real line: (0, 1) U (1, 2). The number 1 is a limit point of this set even though it isn't an element of the set. 0 and 2 are also limit points of this set, and they lie outside of the set as well (without being in a gap). -1 is not a limit point of the set (why?).
     
  9. Dec 20, 2009 #8
    geometrical concept of a limit point of a set is that it is a very nearest point to that set ,means attached with that set or in other words attached with elements of that set .
    haider_uop99@yahoo.com (pakistan)
     
  10. Dec 20, 2009 #9
    1 is limit point of A= (0,1)U(1,2) , because any open interval containing 1 contains infinite points of A .
     
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