Cross Sections and Homeomorphism Classes of Abelian Groups Equipped with the Bohr Topology

A closed subgroup of a topological group is a if there is a continuous cross section from to — that is, a continuous function Γ such that ∘ Γ = id (with : → the natural homomorphism).The symbol denotes an Abelian group with its Bohr topology, i.e., the topology induced by .A topological group is (#) [resp., (#)] if is a ccs-subgroup [resp., is a retract] in every group of the form containing as a (necessarily closed) subgroup. One then writes ∈ (#) [resp., ∈ (#)].Theorem 1. Every ccs-subgroup of a group of the form is a retract of (and is homeomorphic to () × ); hence (#) ⊆ (#).Theorem 2. ∈ (#) [resp., ∈ (#)] iff is a ccs-subgroup of its divisible hull (()) [resp., is a retract of (())].Theorem 3. (a) Every cyclic group is in (#).(b) The classes (#) and (#) are closed under finite products.(c) Not every Abelian group is in (#).The following corollary to Theorem 3 answer a question of KUNEN:Corollary. The spaces ((ℤ)) and (((ℤ)/ℤ) × ℤ) are homeomorphic.