1Department of Mathematics Wesleyan University Middletown, Connecticut 06459 U.S.A. 2Department of Mathematics California State University Bakersfield, California 93311-1099 U.S.A. 3Department of Mathematics Case Western Reserve University Cleveland, Ohio 44106-7058 U.S.A.
Fundamenta Mathematicae, Tome 143 (1993) no. 2, pp. 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. Comfort
1
;
F. Javier Trigos-Arrieta
2
;
Ta-Sun Wu
3
1
Department of Mathematics Wesleyan University Middletown, Connecticut 06459 U.S.A.
2
Department of Mathematics California State University Bakersfield, California 93311-1099 U.S.A.
3
Department of Mathematics Case Western Reserve University Cleveland, Ohio 44106-7058 U.S.A.
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title = {The {Bohr} compactification, modulo a metrizable subgroup},
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W. W. Comfort; F. Javier Trigos-Arrieta; Ta-Sun Wu. The Bohr compactification, modulo a metrizable subgroup. Fundamenta Mathematicae, Tome 143 (1993) no. 2, pp. 119-136. doi: 10.4064/fm-143-2-119-136
