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.