Closed subgroup of finite index

Definition
A subgroup of a topological group (or more generally any of the variations of topological group that involve a group structure and a topological space structure, including left-topological group, right-topological group, semitopological group, quasitopological group, or paratopological group) is termed a closed subgroup of finite index or open subgroup of finite index if it satisfies the following equivalent conditions:


 * 1) It is a defining ingredient::closed subgroup that is also a defining ingredient::subgroup of finite index in the whole group
 * 2) It is an defining ingredient::open subgroup that is also a defining ingredient::subgroup of finite index in the whole group

Equivalence of definitions

 * (1) implies (2): closed subgroup of finite index implies open
 * (2) implies (1): follows from open subgroup implies closed