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Math Help - Show that X bar (X closure) is a completion of X.

  1. #1
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    Show that X bar (X closure) is a completion of X.

    (X,dx), a metric space, is embedded in (Y,dy), another metric space. (Y,dy) is complete and (X,dx) has dx as the induced metric. Show that X closure (Xbar) is a completion of X.

    Thanks a lot!
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  2. #2
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    Re: Show that X bar (X closure) is a completion of X.

    Do you know what you have to show?
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    Re: Show that X bar (X closure) is a completion of X.

    I have to show that there is a subspace W of X bar that is dense in X bar and W is isometric to X.
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  4. #4
    Super Member girdav's Avatar
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    Re: Show that X bar (X closure) is a completion of X.

    If dx is the induced metric of dy on X, why won't you try W=X?
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    Re: Show that X bar (X closure) is a completion of X.

    So if I let W=X, then "clearly" W and X are isometric. So all that remains to show is that X (or W) is everywhere dense in X bar. So, if I let a sequence in X be x_n --> x in X bar. Then d (x_n,x) < epsilon. So we can make the distance between them as small as we want. So X is dense in X bar??
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  6. #6
    Super Member girdav's Avatar
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    Re: Show that X bar (X closure) is a completion of X.

    Yes, that's true (and that's why I don't understand exactly the point of the exercise).
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  7. #7
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    Re: Show that X bar (X closure) is a completion of X.

    When are we using the fact that Y is complete?

    I feel like using your suggestion makes the proof almost trivial!
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  8. #8
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    Re: Show that X bar (X closure) is a completion of X.

    To see that \overline{X} is complete (because each closed subset of a complete metric space is complete for the induced metric).
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    Re: Show that X bar (X closure) is a completion of X.

    Yep! I get you. Thank you soooooo much!

    The question was from a past exam. You've been very helpful.
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    Re: Show that X bar (X closure) is a completion of X.

    Oh just btw, you know how in the general proof for completion, we use the metric on the completion as lim n--> inf of d (x_n, y_n). Here we can just use the same metric right? How do we justify the use of the same metric for X bar and X?
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  11. #11
    Super Member girdav's Avatar
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    Re: Show that X bar (X closure) is a completion of X.

    Since X has the induced metric of Y, you can give to \overline{X} the same metric.
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  12. #12
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    Re: Show that X bar (X closure) is a completion of X.

    Because X closure would be the smallest subset containing X, so if X is a subspace of Y, then either X closure is in Y or is Y. So, X bar will also have the same induced metric?
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    Re: Show that X bar (X closure) is a completion of X.

    Anyway, thanks a lot!
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  14. #14
    Super Member girdav's Avatar
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    Re: Show that X bar (X closure) is a completion of X.

    The metric on \overline{X} is the same as the induced metric on X when you take two elements in X. By continuity, we can check that it's the case for all the elements of \overline{X}.
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