# The intersection of open normal subgroups in a compact, totally disconnected topological group is trivial.

I am currently doing self-study on profinite groups and I'm stuck trying to prove the following lemma.

If a topological group $G$ is compact and totally disconnected, then the open normal subgroups of $G$ intersect in the trivial subgroup $\{\,1_{G}\,\}$.

While I hope that mere hints will allow me to see how to prove this, I think I may need someone to just spell-it-out for me. I've been racking my brain as to how I could prove this and I've made little progress. Either way, any help would be much appreciated.

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– Martin Brandenburg Mar 19 '13 at 0:26
@MartinBrandenburg I have read that post several times, and while I see that it is related to my question, I still don't see how it helps me. – David K. Mar 19 '13 at 0:30
It gives you immediately the answer since Hausdorff spaces are accessible. – Damien L Mar 19 '13 at 1:05

Hint: In a compact totally disconnected group $G$, the family of open invariant subgroups forms a local base at the neutral element of $G$.