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Let $G$ be a topological group. How can we prove that if $H$ is a normal subgroup of $G$, then $\overline H$ is a normal subgroup of $G$ also?

First of all, we have to prove that $\overline H$ is a subgroup of $G$, that is easy, I'm having problems to prove that $\overline H$ is normal.

Thanks

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up vote 2 down vote accepted

Since $H$ is invariant under conjugation and the topology is invariant under conjugation, the result follows.

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can you write the details please? –  user42912 Nov 23 '12 at 7:21
    
what do you mean by the topology is invariant under conjugation? –  user42912 Nov 23 '12 at 7:37
    
@Rafael A conjugate of an open set is also an open set. –  Ted Nov 23 '12 at 9:05
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