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Equivalent metrics

  1. Sep 5, 2010 #1
    Is it true that two metrics d_1 and d_2 on a set X are equivalent iff they have the same convergent sequences (i.e., a sequence that converges in d_1 converges in d_2 and vice versa)?
     
  2. jcsd
  3. Sep 7, 2010 #2
    Yes.

    Engelking, "General Topology", Thm. 4.1.2

    "Two metrics on X are equivalent iff they induce the same convergence."
     
  4. Sep 7, 2010 #3
    Yup, although the correct term is "topologically equivalent", since metrics can also be "uniformly equivalent" or "Lipschitz equivalent" and there might be more such equivalencies.
     
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