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Math Help - Isolated point

  1. #1
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    Isolated point

    Let a be an element of A. Prove that A is an isolated point of A iff there exists an epsilon neighborhood V(a) such that V(a)intersectA={a}



    A point is an isolated point if it is not a limit point.
    Let a be an element of A.
    Let be an isolated point. We want to show V(a)intersectA={a}.
    Since a is not a limit point, we say x=liman satisfying an=x
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  2. #2
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    Quote Originally Posted by kathrynmath View Post
    Let a be an element of A. Prove that A is an isolated point of A iff there exists an epsilon neighborhood V(a) such that V(a)intersectA={a}
    A point is an isolated point if it is not a limit point.
    To say that x\notin A^{\prime}, x is not a limit point of A, then there is a neighborhood V(x) that does not any point of A\setminus \{x\}.
    Therefore, if x\in A then V(x)\cap A=\{x\}, so it is isolated.
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  3. #3
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    I don't get where A' comes from?
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