Closed subgroup of semitopological group
This article defines a property that can be evaluated for a subgroup of a semitopological group
WARNING: POTENTIAL TERMINOLOGICAL CONFUSION: Please don't confuse this with homomorphism-closed subgroup
For a semitopological group
- The quotient topology on its coset space (left or right) is a -topology, viz, points are closed in the quotient
- The subgroup is closed as a subset of the topological space
For an algebraic group
Further information: closed subgroup of algebraic group
Relation with other properties
This property of a subgroup in a topological group is transitive
A closed subgroup of a closed subgroup is closed -- this follows from the topological fact that a closed subset of a closed subset is closed.
Intermediate subgroup condition
This property satisfies the topological intermediate subgroup condition: in other words, if a subgroup satisfies the property in the whole group, it also satisfies the property in every intermediate subgroup
A closed subgroup of a group is also closed in any intermediate subgroup; this again follows from the corresponding topological fact for closed subsets.
If is a closed subgroup of and is any subgroup, then is closed in . This again follows from corresponding facts for closed subsets.