Hi!
I have one interesting question. Is it possible to find a special metric that a space R is not complete by this metric? I now that the space R is complete with usual Euclidean metric but I don't now about other metrics in this space.
Put $\displaystyle d(x,y)=|e^x-e^y|$. It's a metric on $\displaystyle \mathbb R$, which is topologically equivalent to the usual metric, but the sequence $\displaystyle \{-n\}$ is a Cauchy sequence which is not convergent.