Let $G$ be a Lie group and $H\le G$ be a countable and closed subgroup. Can we conclude that $H$ must be discrete?
Suppose $H$ is not discrete, then $e$, as well as any element in $H$, is a limit point. I have a feeling that $H$ would be uncountable but cannot come up with a proof.
The condition "Lie group" might be too strong than needed (perhaps locally compact Hausdorff is enough?) but I am not sure.
If $G$ is a Lie group then its closed subgroup $H$ is a submanifold. If $H$ is countable, then it must be of zero dimensional and thus discrete. Let me know if this argument is wrong. I am not sure about the more general case, though.