Let a be a compact set, then you can define f: a->; R, f(x)=inf|x-Y|, Y∈B, and verify that d(A, B)=inf f(x)=min f(x).

It is not difficult to give an example, but you must find an unbounded closed set.

A={(x，y):x & gt； 0，y & gt= 1/x}

B={(x，y):x & gt； 0，y & lt=- 1/x}

d(A，B)=0