Topological factor

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

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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).

Relation with other properties

Stronger properties

• Topological direct factor
• Topologically split normal subgroup

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 $H$ 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.