Open subgroup

This article defines a property that can be evaluated for a subgroup of a semitopological group

Definition

A subgroup of a topological group is termed an open subgroup if it is an open subset in the subspace topology.

Relation with other properties

Stronger properties

• Closed subgroup of finite index

Weaker properties

• Closed subgroup

Facts

• All cosets of an open subgroup are open. Thus, in a connected topological group, there cannot exist any proper open subgroup.
• In a compact group, any open subgroup must have finite index