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

Definition

For a semitopological group

A subgroup of a semitopological group is termed a closed subgroup if it satisfies the following equivalent conditions:

• The quotient topology on its coset space (left or right) is a $T_1$-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

Metaproperties

Transitivity

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 $H$ is a closed subgroup of $G$ and $K$ is any subgroup, then $H \cap K$ is closed in $K$. This again follows from corresponding facts for closed subsets.