$\overline H$ is a normal subgroup of a topological group $G$.

Question

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

Answer 1

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