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Thread: Criterion for closed

  1. #1
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    Criterion for closed

    I have this theorem.

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

    (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 $\displaystyle D$ is closed.

    So $\displaystyle Y-D$ is open in $\displaystyle Y$.

    By using similar theorem for open sets.

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

    But trying to get back to $\displaystyle D$ by taking complements, I get

    $\displaystyle D = X-E \cup X-Y$

    Which doesn't seem to help me at all.

    Also using $\displaystyle D$ 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 $\displaystyle X$ is a metric space and $\displaystyle Y \subset X$. $\displaystyle D \subset Y$ is closed in $\displaystyle Y$ iff $\displaystyle D = C \cap Y$ for some closed set $\displaystyle C$ in $\displaystyle X$.
    The closure of $\displaystyle Y$, $\displaystyle \overline{Y}~,$ in $\displaystyle X$ is the 'smallest' closed set containing $\displaystyle Y$.

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