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The maximal totally bounded group topology on G and the biggest minimal G-space, for Abelian groups G. (English) Zbl 0696.22003
Let G be an abstract Abelian group. \(G^{\#}\) denotes the group G with the topology it inherits from bG, its Bohr compactification. This is the maximal totally bounded group topology on G and the major portion of the paper is devoted to the investigation of the topological group \(G^{\#}\). For example, it is shown that \(G^{\#}\) is 0-dimensional. It is shown further that every infinite subset A of \(G^{\#}\) has a relatively discrete subset D with \(| D| =| A|\) that is N- embedded in \(G^{\#}\) and is I-embedded in bG where N and I denote the natural numbers and the closed unit interval respectively. This implies that no nontrivial sequence in \(G^{\#}\) converges to a point in bG. The results on \(G^{\#}\) are then applied to gain information about BG. This is a compact space on which G acts and is, in a certain sense, unique. It is referred to here as the biggest minimal G-space and coincides with what is referred to in [R. Ellis, Lectures on Topological Dynamics (Benjamin, New York, 1969; Zbl 0193.515)] as a universal minimal set.
Reviewer: K.D.Magill, jun

22A05 Structure of general topological groups
54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
54H10 Topological representations of algebraic systems
54B99 Basic constructions in general topology
54C20 Extension of maps
54G99 Peculiar topological spaces
20K45 Topological methods for abelian groups
Full Text: DOI
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