## Totally bounded subgroup. [closed]

Is it true to say every totally bounded subgroup of a complete group is finite?

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I assume that "totally bounded" and complete" are with respect to the (left, say) uniform structure on the topological group $G$. In particular, a counterexample is given by any metrizable infinite compact group, e.g. the circle group $S^1$. – Pete L. Clark Jun 30 2010 at 19:15
The OP has been here recently and has not responded to Pete's comment above. This seems to me to indicate a lack of interest in clarifying or improving the question, which really is too elementary as it stands (assuming Pete's interpretation is right, as seems likely). Hence I am voting to close. – Harald Hanche-Olsen Jun 30 2010 at 20:39