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I have this assertion which looks rather easy (or as always I am missing something): We have $G$ topological group which is zero dimensional, i.e it admits a basis for a topology which consists of clopen sets, then every open nbhd that contains the identity element of G also contains a clopen subgroup.

I naively thought that if I take $\{e\}$, i.e the trivial subgroup, it's obviously closed, so it's also open in this topology, i.e clopen, and it's contained in every nbhd that contains $e$, but isn't it then too trivial.

Missing something right? :-)

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    In every topological group, an open subgroup is also closed. But if in a Hausdorff topological group every closed subgroup is open, then the subgroup $\{e\}$ is open and therefore all singleton sets are open and $G$ is discrete. Just because you have a base consisting of clopen sets doesn't mean that *every* closed set is open! So that explains your mistake...are you also asking for a correct proof?2011-11-25
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    Regarding Mike's answer, you probably want your group to be locally compact as well as zero-dimensional.2011-11-25

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