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Math Help - Are equivalent metrics comparable on compact sets?

  1. #1
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    Are equivalent metrics comparable on compact sets?

    Suppose I have a set X equipped with two equivalent metrics, d_1 and d_2, meaning that the metrics induce the same topology on X. I know that d_1 and d_2 need not be comparable on all of X, however, is it true that they will be comparable on compact subsets of X?
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  2. #2
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    No, they aren't.
    Consider the cantor space C=\{0,1\}^{\mathbb{N}} , which is the product space of the discrete space {0,1}.
    It is compact and induced by two incomparable metrices
    d_{1}(x,y)=2^{-n}
    d_{2}(x,y)=3^{-n}
    , where n\in \mathbb{N} is the smallest i , such that x_{i} \neq y_{i}

    Then there is no constant K such that
    d_{1}(x,y)< K\cdot d_{2}(x,y) for all x,y.
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  3. #3
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    Thanks for your reply. May I also ask, what about in the situation where the distance functions d_1 and d_2 are induced by a Riemannian metric on a smooth manifold? In this case the topologies induced by d_1 and d_2 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?
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