Two metrics 'd' and 'f' are said to be equivalent on a metric space X, if they have the same set of open sets. This is equivalent to saying every open ball with respect to d contains an open ball with respect to f (different radius) and vice versa. (As every open set is a union of open balls). What I don't quite understand is how these definitions imply that there exist 2 positive constants 'a' and 'b' such that a*d(x,y) <= f(x,y) <= b*d(x,y) for all x,y in X Can someone please shed some light on this or point me in the direction of some related article. PS can someone show me where the "less than equal to" sign is? Thanks.