TITLE: Component of identity
QUESTION [6 upvotes]: Could you please help me to solve this one: 
The connected component of the identity of a topological group is a normal subgroup? I also need a hint to show  path-connected components are normal subgroups. I am not familiar with deep properties of topological groups other than the definition.

REPLY [4 votes]: For proving that $H$ is normal, let's observe that $gHg^{-1}$ is a connected subset containing $e$. Since $H$ is the connected component of $e$, then $gHg^{-1}\subseteq H$. The proof for the path component of $e$ is similar.