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Are there any metric spaces with no Cauchy sequences?

  1. Oct 1, 2014 #1
    A metric space is considered complete if all Cauchy sequences converge within the metric space. I was just curious if you could have a case of a metric space that doesn't have any Cauchy sequences in it. Wouldn't it be complete by default?

    When trying to think of a space with no cauchy sequences, I invision a space where points are sufficiently far apart from each other. Since all metric spaces are T1,T2,T3, and T4, I get the idea that you can "separate" points, and closed sets in general in a metric space. That's what made me think about a metric space where points are so far apart, that there are no cauchy sequences.
     
  2. jcsd
  3. Oct 1, 2014 #2

    WWGD

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    You can always define Cauchy sequences by repeating a point : {p,p,....,p,.....} . In discrete (and other) spaces , only eventually-constant sequences converge. So the best is to say that sequences without repeated terms may not converge.
     
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