Same open sets + same bounded sets => same Cauchy sequences?

  • #1
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Let ##d_1## and ##d_2## be two metrics on the same set ##X##. Suppose that a set is open with respect to ##d_1## if and only if it is open with respect to ##d_2##, and a set is bounded with respect to ##d_1## it and only if it is bounded with respect to ##d_2##. (In technical language, ##d_1## and ##d_2## induce the same topology and the same bornology.) My question is, does this imply that a sequence is Cauchy with respect to ##d_1## if and only it is Cauchy with respect to ##d_2##?

If not, does anyone know of an example of two metrics which share the same open sets and the same bounded sets, but have different collections of Cauchy sequences?
 
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  • #2
mathman
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Set of all reciprocal of integers between 0 and 1. Let ##d_1## be defined as ##d_1(x,y)=1## if ##x\ne y##. Let ##d_2## be defined as ##d_2(x,y)=|x-y|## if ##x\ne y##. Then ##(\frac{1}{2},\frac{1}{3},...)## will be a Cauchy sequence for ##d_2##, but not for ##d_1##..
 
  • #3
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Thanks, that makes sense.
 

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