I am trying to show that every metrizable locally compact topological group admits a complete metric generating the topology of the group

  • 2
    $\begingroup$ Could you please give some background or motivation (why do you want to know? why do you think it's true? for which examples do you know it's true? etc) $\endgroup$ – Yemon Choi Aug 15 '11 at 20:41
  • $\begingroup$ suggestions: Choose one side, say left. Show that the left uniformity is complete. Show that a left-invariant metric exists. Relate these two. $\endgroup$ – Gerald Edgar Aug 15 '11 at 21:24
  • $\begingroup$ If you make the completion of the metric space (taking the Cauchy sequences) this is coherent by algebraic group operations $\endgroup$ – Buschi Sergio Aug 15 '11 at 21:56

Every second countable, locally compact group admits a metric which is left-invariant, generates the topology, and is proper (i.e. closed balls are compact). See Theorem 4.5 in http://arxiv.org/pdf/math/0606794

Such a metric is clearly complete.

  • $\begingroup$ However, the solution for the homework-type problem proposed was known long before 2006. $\endgroup$ – Gerald Edgar Aug 16 '11 at 12:06
  • 2
    $\begingroup$ @Gerald: Indeed, Yves Cornulier just pointed out to me this paper: Struble, Raimond A. Metrics in locally compact groups. Compositio Mathematica, 28 no. 3 (1974), p. 217-222 numdam.org/numdam-bin/fitem?id=CM_1974__28_3_217_0 $\endgroup$ – Alain Valette Aug 16 '11 at 14:36
  • $\begingroup$ A related result from 1951: "Invariant Metrics in Groups" by V.L. Klee Jr., in which it is shown that every Abelian, completely metrizable group admits a complete invariant metric. $\endgroup$ – Alex M. Jan 25 '18 at 15:46
  • $\begingroup$ @AlainValette I cannot see the link anymore...to the numdam paper, do you know where it has traveled? Merci en avance :) $\endgroup$ – AIM_BLB Jan 22 at 14:37

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.