# Topological factor

This article defines a property that can be evaluated for a subgroup of a semitopological group

BEWARE!This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

## Definition

A closed subgroup of a topological group is termed a **topological factor** if the following equivalent conditions are satisfied:

- The quotient bundle is a trivial fiber bundle
- There exists a subset of (not necessarily a subgroup) such that the map given by , is a homeomorphism of topological spaces (this forces to intersect every right coset of at one point).

## Relation with other properties

### Stronger properties

## Metaproperties

### Transitivity

*This property of a subgroup in a topological group is transitive*

A topological factor of a topological factor is a topological factor. This follows from the fact that a trivial bundle over a trivial bundle is a trivial bundle, or equivalently, from associativity of the product topology.

### 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*

If is a topological factor of , and is any intermediate subgroup, then is also a topological factor of .

## Facts

A topological factor need not be a direct factor or split, and conversely, a direct factor need not be a topological factor. Essentially, the set-theoretic complement that we construct for a topological factor need not be a subgroup.