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1993 | 143 | 2 | 119-136
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The Bohr compactification, modulo a metrizable subgroup

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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.
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  • Department of Mathematics, Case Western Reserve University, Cleveland, Ohio 44106-7058, U.S.A
  • Department of Mathematics, California State University, Bakersfield, California 93311-1099, U.S.A.
  • Department of Mathematics, Wesleyan University, Middletown, Connecticut 06459, U.S.A.
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