(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.
(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.
(a) - \pi /2, \pi /2) \rightarrow R" alt="f- \pi /2, \pi /2) \rightarrow R" />, 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.