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Limit point and isolated point

  1. Feb 26, 2013 #1
    Would it be correct to say that out of the following two statements, exactly one is always true and one is always false?

    1) x is a limit point of S, where S is a subset of ℝ
    2) x is an isolated point of S, where S is a subset of ℝ

    In other words, every point is either a limit point of a set or an isolated point of that set.

    Also, for a point to be a limit point/isolated point of a set, does it have to be in the set?

    Thanks!

    BiP
     
  2. jcsd
  3. Feb 26, 2013 #2

    jbunniii

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    Every point in ##S## is either an isolated point of ##S## or a limit point of ##S##. The two characterizations are mutually exclusive: a point in ##S## is an isolated point if and only if it is not a limit point of ##S##.

    ##S## need not contain all of its limit points. ##S## is closed if and only if it does contain them all.

    Isolated points of ##S## are always contained in ##S##.
     
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