Totally disconnected group

Remarks

Example: the p-adic integers $\Z_p$ under addition. Each subgroup $p^n \Z_p$ is open and closed, and two distinct elements $x, y$ lie in different cosets of $p^n \Z_p$ for $n$ large, so no connected set contains both. Non-example: $(\R, +)$ with the usual topology is connected, so it is not totally disconnected. Equivalently, $G$ is totally disconnected if and only if its identity component is $\set{e}$, because each connected component is a coset of the identity component.