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Math Help - Prove that set of all limit points is closed?

  1. #1
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    Prove that set of all limit points is closed?

    Hey guys,

    If A is a subset of R, let L be the set of all limit points. We want to show that L is closed. Here's how I was thinking about proving it:

    To show that L is closed, we can show that the complement of L (call it c(L)) is open. c(L) is open if we can put an epsilon-neighborhood around each of the isolated points of L and always have it included in L. I'm not exactly sure where to go from here.

    Any hints would be much appreciated.

    Thanks,
    Mariogs
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  2. #2
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    Re: Prove that set of all limit points is closed?

    Quote Originally Posted by mariogs379 View Post
    If A is a subset of R, let L be the set of all limit points. We want to show that L is closed. Here's how I was thinking about proving it: To show that L is closed
    Surely you mean that L is the set of all limit points of A~?
    That is usually called the derived set of A and is noted by A'
    Now if t\notin L then t is not a limit point of A.
    So there is open set Q_t such that O_t\cap (A\setminus\{t\})=\emptyset.
    Can finish?
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  3. #3
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    Re: Prove that set of all limit points is closed?

    Yeah, that's what I meant. So Qt must be open and, because it is the complement of L, L must be closed. Yeah?
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