# Complete metric space

• Jan 16th 2009, 12:39 PM
aliceinwonderland
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.
• Jan 16th 2009, 11:26 PM
aliceinwonderland
Quote:

Originally Posted by aliceinwonderland
(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.