The maximal totally bounded group topology on G and the biggest minimal G-space, for Abelian groups G.

*(English)*Zbl 0696.22003Let 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

##### MSC:

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 |

##### Keywords:

Bohr compactification; maximal totally bounded group topology; topological group; biggest minimal G-space
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\textit{E. K. van Douwen}, Topology Appl. 34, No. 1, 69--91 (1990; Zbl 0696.22003)

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##### References:

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