Closed subgroup of finite index

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

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 closed subgroup that is also a subgroup of finite index in the whole group
2. It is an open subgroup that is also a 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