When the identity component is only one point

Question

Let $G$ be a topological group. If $H$ is a closed subgroup such that the identity component $H^0$ of $H$ is only the identity element i.e $H^0=\{e\}$. is it true that $H$ is a discrete subgroup of $G$?

Answer 1

It's not true in general. There exist topological groups $G$ homeomorphic to the Cantor set, for example the inverse limit of the sequence of surjective homomorphisms of finite cyclic groups $$\cdots \to \mathbb{Z} / 2^n \mathbb{Z} \to \mathbb{Z} / 2^{n-1} \mathbb{Z} \to \cdots \to \mathbb{Z} / 2^2 \mathbb{Z} \to \mathbb{Z} / 2\mathbb{Z} $$ Then take $H=G$. The components are single points, but $H=G$ is not discrete.

Answer 2

For simple examples, take $H=G$ itself, with $G$ totally disconnected (connected components are singletons) and not discrete.

For example the group $\mathbb{Q}$ under addition.