Closed subsets which are not retracts in the Bohr topology
Every Abelian group \(G\) can be equipped with its largest totally bounded group topology. Also known as the Bohr topology of the (discrete) group \(G\), this can be realized as the topology induced on \(G\) by the group \(\text{Hom}(G,\mathbb T)\) of homomorphisms from \(G\) into the usual circle group \(\mathbb T\). Since \(\text{Hom}(G,\mathbb T)\) separates the points of \(G\), the Bohr topology is a Hausdorff group topology and hence a Tychonoff topology. Let \(G^{\sharp}\) denote \(G\) with the Bohr topology. \textit{E. E. van Douwen} [Topology Appl. 34, 69-91 (1990; Zbl 0696.22003)] initiated the formal and extensive study of these groups as topological spaces and he posed a number of questions that have inspired decisively the interest of many workers on this area of research. One of van Douwen's questions was whether every closed subset of a countable group of the form \(G^{\sharp}\) is a retract. This question was solved in the negative by \textit{H. Gladdines} [Topology Appl. 67, 81-84 (1995; Zbl 0838.54024)] taking for \(G\) the countable Boolean group \(\bigoplus_{\omega}\{0,1\}\). In the present paper, the author extends Gladdines' result to arbitrary countably infinite Abelian groups. More precisely, the following is proved: If \(G\) is a countably infinite Abelian group, then there is a closed \(F\subset G\) such that in the Bohr topology, \(F\) is not a retract and \(F\) has exactly one accumulation point. An essential ingredient in the proof is a result by \textit{K. Kunen} and \textit{W. Rudin} [Math. Proc. Cambridge Philos. Soc. 126 (1), 117-137 (1999; Zbl 0915.43003)] which establishes the existence of certain conveniently placed infinite discrete subsets in the Bohr topology.
scientific article; zbMATH DE number 1901918