(a)Give an example of two metric spaces and which are topologically equivalent and for which is complete and 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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- January 16th 2009, 01:39 PMaliceinwonderlandComplete metric space
(a)Give an example of two metric spaces and which are topologically equivalent and for which is complete and 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. - January 17th 2009, 12:26 AMaliceinwonderland
(a) , given by ,

By f, is topologically equivalent to R with a usual metric d.

R is complete with a usual metric d, but 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

,

.

(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.