If $H$ is a closed subgroup of $G$ then $H=G$ or $H$ is discrete?

Question

If $G$ is a topological group and $H$ is a closed subgroup, is it the case the $H$ is either discrete or else $H=G$? I see this is true for $G=\mathbb{R}^d$ in Subgroup of $\mathbb{R}$ either dense or has a least positive element?

Does the same hold for general $G$? I'm willing to assume $G$ is locally compact, second-countable, Hausdorff (i.e. a Polish group).

Answer 1

$R$, the real line, is a closed subgroup of $R\times R$, but is not discrete.

Answer 2

Consider the multiplicative group $(\mathbb{C} \setminus \{0\}, \cdot )$, which is locally compact and Hausdorff. Then the torus $( \mathbb{T}, \cdot)$ is a non-discrete proper closed subgroup of $(\mathbb{C} \setminus \{0\}, \cdot )$.