Let $\cal T$ be a locally compact topology on a group $G$ and $(x,y)\mapsto xy^{-1}$ be continuous at $(1,1)$ and for every $a\in G$, $x\mapsto xa$ be continuous everywhere with respect to $\cal T$. Let $C$ be a connected component of $(G,\cal T)$ containing the neutral element $1$.

If $C$ is a normal(=invariant) subgroup of $G$ then $C$ is the intersection of all open subgroups of $G$. How if $C$ is not a normal subgroup? then is $C$ the intersection of all open subgroups of $G$?