Take the 2-minute tour ×  So the following statement seems to be
obvious but I don't see how to prove it:  Q: How does one prove
that a closed totally disconnected subgroup of a connected real
Lie group is discrete?  Note that it is essential to take into
account the group structure since the Cantor set is a closed
totally disconnected subset of $\mathbf{R}$ which is not
discrete.  share|improve this question Belated comment: adding a
tag 'lie-groups' would be appropriate here? –  Jim Humphreys Apr
16 '11 at 17:50 and what if we consider a topological group
which is not locally euclidean? –  Giuseppe Tortorella Apr 16
'11 at 18:12 @Giuseppe: If the group isn't locally Euclidean,
then the subgroup needn't be discrete. For example, let $G$ be a
direct product of infinitely many circle groups. The circle
group has a subgroup of order $2$, so $G$ has a subgroup that is
a direct product of infinitely many $2$-element discrete groups.
This subgroup is closed and totally disconnected but not
discrete. –  Stephen S A