Normalizer of topological subgroup

Question

Let $G$ be a topological group and $H$ a closed subgroup of $G$. Let $H^o$ be the connected component of $H$ containing the identity.

Now if $h\in H$ is it correct that $hH^oh^{-1}\in H^o$?

Answer 1

The map $g\mapsto hgh^{-1}$ is continuous so the image of $H^0$ under this map must be connected. But clearly $hH^0h^{-1}$ contains $e$ so it must lie in the same connected component as the identity, or $hH^0h^{-1}\subset H^0$.