The Bohr compactification, modulo a metrizable subgroup

Volume 143 / 1993

W. W. Comfort, F. Javier Trigos-Arrieta, Ta-Sun Wu Fundamenta Mathematicae 143 (1993), 119-136 DOI: 10.4064/fm-143-2-119-136


The authors prove the following result, which generalizes a well-known theorem of I. Glicksberg [G]: If G is a locally compact Abelian group with Bohr compactification bG, and if N is a closed metrizable subgroup of bG, then every A ⊆ G satisfies: A·(N ∩ G) is compact in G if and only if {aN:a ∈ A} is compact in bG/N. Examples are given to show: (a) the asserted equivalence can fail in the absence of the metrizability hypothesis, even when N ∩ G = {1}; (b) the asserted equivalence can hold for suitable G and N with N closed in bG but not metrizable; (c) an Abelian group may admit two topological group topologies U and T, with U totally bounded, T locally compact,U ⊆ T, with U and T sharing the same compact sets, and such that nevertheless U is not the topology inherited from the Bohr compactification of ⟨ G, T⟩. There are applications to topological groups of the form kG for G a totally bounded Abelian group.


  • W. W. ComfortDepartment of Mathematics
    Wesleyan University
    Middletown, Connecticut 06459
  • F. Javier Trigos-ArrietaDepartment of Mathematics
    California State University
    Bakersfield, California 93311-1099
  • Ta-Sun WuDepartment of Mathematics
    Case Western Reserve University
    Cleveland, Ohio 44106-7058

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