Complete metric space

**aliceinwonderland** · January 16th 2009, 12:39 PM

(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.

**aliceinwonderland** · January 16th 2009, 11:26 PM

Originally Posted by aliceinwonderland

(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) $f<img src=$ - \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.

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