The dimension of an LCA group in its Bohr topology
In the useful terminology proposed by \textit{F. J. Trigos-Arrieta} [J. Pure Appl. Algebra 70, No.~1/2, 199-210 (1991; Zbl 0724.22003)] an LCA group \(G\) ``respects (a topological property) \(\mathbb{P}\)'' provided: A subset \(F\) of \(G\) has \(\mathbb{P}\) in the topology of \(G\) iff \(F\) has \(\mathbb{P}\) in the topology of \(G^+\). (Here \(G^+\) denotes \(G\) with the topology inherited from its Bohr compactification, i.e., in the topology induced on \(G\) by the set of all \(G\)-continuous homomorphisms from \(G\) to \(\mathbb{T}\).) Among the properties known to be respected by each LCA group \(G\) are: compactness, \(\sigma\)-compactness, the Lindelöf property, pseudocompactness, and functional boundedness. Further, an LCA group \(G\) is realcompact iff some \(\sigma\)-compact clopen subgroup \(H\) has \(|G/H|\) not Ulam-measurable, a condition equivalent to the condition that \(G^+\) is realcompact (equivalently, topologically complete). In the present paper the author, responding to a question posed by M. G. Tkachenko, shows for each LCA group that the covering dimension dim satisfies \(\dim(G) = \dim(G^+)\). This result may be read together with the earlier achievement of \textit{D. B. Shakhmatov} [Quest. Answers Gen. Topology 8, No. 1, 101-128 (1990; Zbl 0715.54023)]: The Čech-Lebesgue dimension function dim satisfies \(\dim(G) = 0\) for every precompact group \(G\) with a clopen basis; in particular, \(\dim(G^+)= 0\) for \(G\) discrete Abelian.
scientific article; zbMATH DE number 1307980
scientific article