No, they aren't.
Consider the cantor space , which is the product space of the discrete space {0,1}.
It is compact and induced by two incomparable metrices
, where is the smallest i , such that
Then there is no constant K such that
for all x,y.
Suppose I have a set equipped with two equivalent metrics, and , meaning that the metrics induce the same topology on . I know that and need not be comparable on all of , however, is it true that they will be comparable on compact subsets of ?
Thanks for your reply. May I also ask, what about in the situation where the distance functions and are induced by a Riemannian metric on a smooth manifold? In this case the topologies induced by and each agree with the manifold topology, but these distance functions need not be comparable on the entire manifold. However, will they be comparable on compact subsets of the manifold?