# 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

## Contents

## 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 -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

### Stronger properties

## 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.

Template:Topological transfercondn

If is a closed subgroup of and is any subgroup, then is closed in . This again follows from corresponding facts for closed subsets.