Connected implies no proper open subgroup
Statement for semitopological groups
Statement for topological groups
- Open subgroup implies closed
- Closed subgroup of finite index implies open
- Compact implies every open subgroup has finite index
The converse is not true for all groups. See no proper open subgroup not implies connected.
However, the converse is true in some contexts:
- It is true for algebraic groups, i.e.,it is true if the topology is a Zariski topology. See equivalence of definitions of connected algebraic group.
- It is true for all locally connected topological groups. In particular, it is true for Lie groups. See equivalence of definitions of connected Lie group.
- Open subgroup implies closed (this is true in both left-topological groups and right-topological groups)
By Fact (1), a proper open subgroup is a nonempty subset that is both open and closed (note that it is nonempty because it is a subgroup). The existence of such a subset contradicts connectedness.