Let X be a set and both d and d' are metrics of X. We say that d is topologically equivalent to d' if for any set U it is satisfied that U is an open subset of (X,d) if and only if U is an open subset of (X,d').

Let (X,d) and (X,d') be metric spaces:

a) Prove that if d and d' are strongly equivalent metrics (i.e. if there exist constants a,b>0 such that ad' b) Prove that if d and d' are topologically equivalent, then a set F is a closed subset of (X,d) if and only if F is a closed subset of (X ,d').
c) Show that the previous part is still true if ''closed'' is changed to ''compact''.
d) Show that part (b) is still true if ''topologically equivalent'' is changed to ''equivalents'' and ''closed'' to ''bounded''.
e) Deduce from the previous parts that for any n natural number and p in [1, infinity], the Heine-Borel theorem is valid in R^n with the d_p metric). (Hint: Remember that for any n natural number and p in [1, infinity], d_2 and d_p are equivalent metrics on R^n.