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Thread: Complete metric space

  1. #1
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    Complete metric space

    (a)Give an example of two metric spaces $\displaystyle (X_{1}, d_{1})$ and $\displaystyle (X_{2}, d_{2})$ which are topologically equivalent and for which $\displaystyle (X_{1}, d_{1})$ is complete and $\displaystyle (X_{2}, d_{2})$ is not.

    (b)Give an example of a set X with two equivalent metrics d and d' for which (X, d) is complete and (X, d') is not.
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  2. #2
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    Quote Originally Posted by aliceinwonderland View Post
    (a)Give an example of two metric spaces $\displaystyle (X_{1}, d_{1})$ and $\displaystyle (X_{2}, d_{2})$ which are topologically equivalent and for which $\displaystyle (X_{1}, d_{1})$ is complete and $\displaystyle (X_{2}, d_{2})$ is not.

    (b)Give an example of a set X with two equivalent metrics d and d' for which (X, d) is complete and (X, d') is not.
    (a) $\displaystyle f- \pi /2, \pi /2) \rightarrow R$, given by $\displaystyle f(x) = tan(x)$,
    By f, $\displaystyle (- \pi /2, \pi /2)$ is topologically equivalent to R with a usual metric d.
    R is complete with a usual metric d, but $\displaystyle (- \pi /2, \pi /2)$ is not complete with a usual metric d.

    (b) Consider the set X = {1/k}, k=1,2,3,,,,,n.
    A usual metric d and and a discrete metric d' is equivalent, since
    $\displaystyle d(x,y) \leq d'(x,y)$,
    $\displaystyle d'(x,y) \leq n^{2}d(x,y) $.

    (discrete metric d' : d'(x,x)=0, and d'(x,y)=1 for x not y, x,y in X)

    (X,d) is not complete since 0 is not in X.
    (X, d') is complete since every Cauchy sequence is constant and converge in X.
    Last edited by aliceinwonderland; Jan 16th 2009 at 11:54 PM.
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