# No proper open subgroup not implies connected

From Groupprops

## Statement

It is possible to have a topological group with no proper open subgroup but such that 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.