1. The problem statement, all variables and given/known data Give an example of two metric spaces (X1, d1) and (X2, d2) which are topologically equivalent and for which (X1, d1) is complete and (X2, d2) is not. 2. The attempt at a solution The open unit disc and R2. They are homeomorphic, but there are Cauchy sequences in the disc which will converge to limits outside of the disc, so it's not complete.