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Math Help - Sets of isolated points

  1. #1
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    Sets of isolated points

    If we have a metric space X, is S, the set of isolated points of X, open?

    My Answer: Yes

    My Logic: Every point of S has a neighborhood, which is empty. The empty set is a subset of every set. Therefore every point of S has a neighborhood that is a set of S.

    I just want to make sure I did not misunderstand anything.
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  2. #2
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    Quote Originally Posted by arsenicbear View Post
    If we have a metric space X, is S, the set of isolated points of X, open?

    My Answer: Yes

    My Logic: Every point of S has a neighborhood, which is empty. The empty set is a subset of every set. Therefore every point of S has a neighborhood that is a set of S.

    I just want to make sure I did not misunderstand anything.


    Almost, but not quite imo: since any point s\in S has an open neighborhood U_s which only contains THAT POINT s (i.e., U_s = U_s\cap X=\{s\}, it follows that any point s\in S has a neighborhood U_s=\{s\}\subset S \Longrightarrow S is open .

    Tonio
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