# Complete metrics in locally compact topological groups

Hello,

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

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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) –  Yemon Choi Aug 15 '11 at 20:41
suggestions: Choose one side, say left. Show that the left uniformity is complete. Show that a left-invariant metric exists. Relate these two. –  Gerald Edgar Aug 15 '11 at 21:24
If you make the completion of the metric space (taking the Cauchy sequences) this is coherent by algebraic group operations –  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.

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However, the solution for the homework-type problem proposed was known long before 2006. –  Gerald Edgar Aug 16 '11 at 12:06
@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 –  Alain Valette Aug 16 '11 at 14:36