Connected implies no proper open subgroup

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Statement

Statement for semitopological groups

A connected left-topological group has no proper open subgroup. Similarly, a connected right-topological group has no proper open subgroup.

Since semitopological groups are both left-topological and right-topologica, this tells us that a connected semitopological group has no proper open subgroup.

Statement for topological groups

A connected topological group has no proper open subgroup.

Related facts

Similar facts

Converse

The converse is not true for all groups. See no proper open subgroup not implies connected.

However, the converse is true in some contexts:

Facts used

  1. Open subgroup implies closed (this is true in both left-topological groups and right-topological groups)

Proof

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.