No proper open subgroup not implies connected

Statement

It is possible to have a topological group $G$ with no proper open subgroup but such that $G$ is not a connected topological group.

Related facts

• Connected implies no proper open subgroup
• Locally connected and no proper open subgroup implies connected
• Open subgroup implies closed
• Closed subgroup of finite index implies open
• Subgroup of finite index need not be closed in T0 topological group
• Subgroup of finite index need not be closed in algebraic group

Proof

The additive group of rational numbers is an example. The group is a totally disconnected group but it has no proper open subgroup.