Valdivia compact Abelian groups

TL;DR

This paper characterizes compact connected Abelian groups that are topological retracts of certain complex classes of compact spaces, showing they are products of metric groups, thus extending previous classifications.

Contribution

It proves that such groups are necessarily products of metric groups if they are retracts of spaces in class R, expanding the understanding of their structure.

Findings

Compact connected Abelian groups retracting from class R are products of metric groups.

Class R is larger than Valdivia compact spaces.

Completes classification of certain Abelian groups outside class R.

Abstract

Let R denote the smallest class of compact spaces containing all metric compacta and closed under limits of continuous inverse sequences of retractions. Class R is striclty larger than the class of Valdivia compact spaces. We show that every compact connected Abelian group which is a topological retract of a space from class R is necessarily isomorphic to a product of metric groups. This completes the result of V. Uspenskij and the author, where a compact connected Abelian group outside class R has been described.