(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.