Topological factor

Definition
A closed subgroup $$H$$ of a topological group $$G$$ is termed a topological factor if the following equivalent conditions are satisfied:


 * The quotient bundle $$p:G \to G/H$$ is a trivial fiber bundle
 * There exists a subset $$K$$ of $$G$$ (not necessarily a subgroup) such that the map $$H \times K \to G$$ given by $$(h,k) \mapsto hk$$, is a homeomorphism of topological spaces (this forces $$K$$ to intersect every right coset of $$H$$ at one point).

Stronger properties

 * Topological direct factor
 * Topologically split normal subgroup

Metaproperties
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.

If $$H$$ is a topological factor of $$G$$, and $$M$$ is any intermediate subgroup, then $$H$$ is also a topological factor of $$M$$.

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.