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Math Help - Complete metric space

  1. #1
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    Complete metric space

    (a)Give an example of two metric spaces (X_{1}, d_{1}) and (X_{2}, d_{2}) which are topologically equivalent and for which (X_{1}, d_{1}) is complete and (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 (X_{1}, d_{1}) and (X_{2}, d_{2}) which are topologically equivalent and for which (X_{1}, d_{1}) is complete and (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) - \pi /2, \pi /2) \rightarrow R" alt="f- \pi /2, \pi /2) \rightarrow R" />, given by f(x) = tan(x),
    By f, (- \pi /2,  \pi /2) is topologically equivalent to R with a usual metric d.
    R is complete with a usual metric d, but (- \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
    d(x,y) \leq d'(x,y),
    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; January 16th 2009 at 11:54 PM.
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