Closed subgroup of finite index
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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 a closed subgroup of finite index or open subgroup of finite index if it satisfies the following equivalent conditions:
- It is a closed subgroup that is also a subgroup of finite index in the whole group
- 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
