Open subgroups of quotient topological groups

Question

Let $G$ be a topological abelian group and $H$ a closed subgroup of $G$. Is it true that an open subgroup of $G/H$ has the form $K/H$ where $K$ is an open subgroup of $G$ containing $H$?

Answer 1

There is more we can say. We have the following correspondence theorem for groups:

If $G$ is a group and $N$ is a normal subgroup (which we'll express by $N\trianglelefteq G$), then there is an inclusion-preserving bijection $$\mathcal H_N(G)\to\mathcal H(G/N) \\ H\mapsto H/N$$ from the set of subgroups of $G$ containing $N$ to the set of subgroups of $G/N$.

Since $G\to G/N$ is an open map, this also restricts to a bijection between the open subgroups containing $N$ and the open subgroups of $G/N$.