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Math Help - Criterion for closed

  1. #1
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    Criterion for closed

    I have this theorem.

    If X is a metric space and Y \subset X. D \subset Y is closed in Y iff D = C \cap Y for some closed set C in X.

    (I have proved a very similar theorem for open sets.)

    However I cant seem to make any progress for the above theorem. (In either direction)

    Going forward, we have D is closed.

    So Y-D is open in Y.

    By using similar theorem for open sets.

    Y-D = E \cap Y for some open set E in X.

    But trying to get back to D by taking complements, I get

    D = X-E \cup X-Y

    Which doesn't seem to help me at all.

    Also using D has all its limit points doesn't give me a lot to work with either.

    This leads me to believe that the theorem is not true.

    Is there something I am not considering? Even going backwards, I can't figure it out.

    Thank you for your help.
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  2. #2
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    Re: Criterion for closed

    Quote Originally Posted by Sheld View Post
    I have this theorem.
    If X is a metric space and Y \subset X. D \subset Y is closed in Y iff D = C \cap Y for some closed set C in X.
    The closure of Y, \overline{Y}~, in X is the 'smallest' closed set containing Y.

    If D\subseteq Y then \overline{D}\subseteq\overline{Y}.
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