Does there exist a metric $d$ on $\Bbb Q$ which is equivalent to the usual metric on $\Bbb Q$ such that $(\Bbb Q,d)$ is complete?

I have a confusion regarding that. Because I know that equivalence of metrics doesn't preserve completeness for arbitrary metric spaces. Please help me in this regard.

Thank you very much.

  • $\begingroup$ Do you know the Baire category theorem? $\endgroup$ – bof Oct 7 '18 at 4:39
  • $\begingroup$ Yeah. Oh! Every complete metric space is of second category but Q is of first category. $\endgroup$ – Dbchatto67 Oct 7 '18 at 4:40
  • $\begingroup$ By the way, what do you mean by "equivalent metrics"? The usage of this term is not consistent. Do you mean "metrics that induce the same topology"? $\endgroup$ – bof Oct 7 '18 at 4:40
  • $\begingroup$ I am a foolish guy. $\endgroup$ – Dbchatto67 Oct 7 '18 at 4:40
  • $\begingroup$ Yeah @bof I mean this. $\endgroup$ – Dbchatto67 Oct 7 '18 at 4:41

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