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Math Help - Metric Spaces

  1. #1
    Junior Member tedii's Avatar
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    Metric Spaces

    Let X be an infinite set. For p\in X,q\in X define,

    d(p,q)=\left\{\begin{array}{cc}1(p\not=q)\\0(p=q)\  end{array}\right.

    Prove that this is a metric. Which subsets of the resulting metric space are open? Which are closed? Which are compact?

    Proving it's metric is easy no problem there.

    The rest I just want to make sure I'm thinking of this correctly.

    Every subset of the metric space is finite, therefore every set is closed, compact, and not open.

    Does this sound right?
    Last edited by tedii; September 21st 2009 at 03:53 PM.
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  3. #3
    Junior Member tedii's Avatar
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    Quote Originally Posted by Plato View Post
    On that page it says that \{x_0\}\mbox{ if }0<r\leq1.

    What about when r=1 this still does not contain X? Why is that?
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    Quote Originally Posted by tedii View Post
    On that page it says that \{x_0\}\mbox{ if }0<r\leq1.
    What about when r=1 this still does not contain X? Why is that?
    Can there be a~\&~b such that D(a,b)>1???
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  5. #5
    Junior Member tedii's Avatar
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    No, but r can be and it's just at length r=1 that I don't understand why that doesn't contain X. Is it because it's an open ball?
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