Is the subgroup generated by two closed subgroups closed?

Question

Let $G$ be a topological group and $H_1, H_2$ be closed subgroups. Can we conclude that $\langle H_1, H_2 \rangle$ is closed?

My attempt is to show that $\overline{\langle H_1, H_2 \rangle} \subseteq \langle \overline{H}_1, \overline{H}_2 \rangle$. But I doubt that this result doesn't hold.

Answer 1

Hint: In $\mathbb{R}$ as an additive subgroup, consider the two subgroups $\mathbb{Z}$ and $\mathbb{Z}\sqrt{2}$.