**Motivation:** It is obvious that for a finite index subgroup $H$ of a group $G$, there exists a normal subgroup $K$ of $G$, $K\subset H$, with $|G/K|<\infty$.

**Our question:** Let $G$ be a topological group and $H$ be a closed but not necessarily normal subgroup of $G$ such that the quotient topological space $G/H$ is a compact space. Does there exist a closed subgroup $K\subset H$ which is normal in $G$ such that $G/K$ is a compact topological group?

**Remark:** One can consider a similar problem in the context of rings and algebras: see this follow-up question.

(a)ask the "remark" in separate question (in which case I could answer this one with a cw answer to make the question settled) or(b)change the current "remark" into the main question (changing the title in particular) giving the original question and its easy examples only as context. I think (a) is a better solution (since these are quite drastic changes). $\endgroup$ – YCor Mar 19 at 20:22