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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]

Contents

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

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.