an:04140061
Zbl 0696.22003
van Douwen, Eric K.
The maximal totally bounded group topology on G and the biggest minimal G-space, for Abelian groups G
EN
Topology Appl. 34, No. 1, 69-91 (1990).
00173083
1990
j
22A05 54D35 54H10 54B99 54C20 54G99 20K45
Bohr compactification; maximal totally bounded group topology; topological group; biggest minimal G-space
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 [\textit{R. Ellis}, Lectures on Topological Dynamics (Benjamin, New York, 1969; Zbl 0193.515)] as a universal minimal set.
K.D.Magill, jun
Zbl 0193.515