Closed subgroup of semitopological group

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

Stronger properties

 * Open subgroup

Metaproperties
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.

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.