On topology induced by Bohr compactification (Q1125513)

This paper contains some results on the Bohr topology of a discrete Abelian group. To the reviewer's knowledge, only the last one (Proposition 3) is new. Let us first recall that the Bohr topology of a locally compact abelian (LCA) group \(G\) is the topology that \(G\) inherits from its Bohr compactification. This topology coincides with the weak topology induced by the set of all continuous characters of \(G\). The first part of the paper is devoted to proving that the Bohr topology of a discrete Abelian group is zero-dimensional. This fact was first proved by \textit{E. K. van Douwen} [Topology Appl. 34, 69-91 (1990; Zbl 0696.22003)] and by \textit{W. W. Comfort} and \textit{F. J. Trigos-Arrieta} [General topology and applications, Lect. Notes Pure Appl. Math. 134, 25-33 (1991; Zbl 0777.22002)] independently. It has subsequently been improved in several directions. \textit{D. Shakhmatov} [Topology Appl. 36, 181-204 (1990; Zbl 0709.22001)] has shown that the Bohr topology of a discrete Abelian group is even strongly zero-dimensional, in the sense that its covering dimension is zero and \textit{S. Hernández} [Topology Appl. 86, 63-67 (1998; Zbl 0935.22006)] has extended this result to arbitrary LCA groups: if \(G^{+}\) denotes an LCA group endowed with its Bohr topology, then the covering dimensions of \(G\) and \(G^{+}\) are the same. The last section of the paper proves then that the Bohr compactification of a discrete group \(G\) is either zero-dimensional or infinite dimensional. The main ingredient of the proof is the structure of torsion Abelian groups, which is applied to the character group of \(G\).