Exercise 29, § 5

Let $G$ be a locally compact, $\sigma$-compact, topological group, or else a complete metrizable group whose topology has a countable base. Let $H$ be a closed normal subgroup of $G$ and let $A$ be a closed subgroup of $G$. Then the quotient group $A/(A \cap H)$ is isomorphic to $AH/H$ if and only if $AH$ is a closed subgroup of $G$ [to show that the condition is necessary, note that $A/(A \cap H)$ is complete, by Chapter III, § 4, no. 6, Proposition 13, and Chapter IX, § 3, no. 1, Proposition 4; to show that the condition is sufficient, use Exercise 28].