The Bohr compactification, modulo a metrizable subgroup

W. Comfort; F. Trigos-Arrieta; S. Wu

Fundamenta Mathematicae (1993)

  • Volume: 143, Issue: 2, page 119-136
  • ISSN: 0016-2736

Abstract

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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.

How to cite

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Comfort, W., Trigos-Arrieta, F., and Wu, S.. "The Bohr compactification, modulo a metrizable subgroup." Fundamenta Mathematicae 143.2 (1993): 119-136. <http://eudml.org/doc/211996>.

@article{Comfort1993,
abstract = {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.},
author = {Comfort, W., Trigos-Arrieta, F., Wu, S.},
journal = {Fundamenta Mathematicae},
keywords = {maximally almost periodic Abelian group; totally bounded group topology; Bohr compactification; locally compact Abelian group},
language = {eng},
number = {2},
pages = {119-136},
title = {The Bohr compactification, modulo a metrizable subgroup},
url = {http://eudml.org/doc/211996},
volume = {143},
year = {1993},
}

TY - JOUR
AU - Comfort, W.
AU - Trigos-Arrieta, F.
AU - Wu, S.
TI - The Bohr compactification, modulo a metrizable subgroup
JO - Fundamenta Mathematicae
PY - 1993
VL - 143
IS - 2
SP - 119
EP - 136
AB - 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.
LA - eng
KW - maximally almost periodic Abelian group; totally bounded group topology; Bohr compactification; locally compact Abelian group
UR - http://eudml.org/doc/211996
ER -

References

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Citations in EuDML Documents

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  1. W. Comfort, F. Trigos-Arrieta, Ta-Sun Wu, Correction to the paper “The Bohr compactification, modulo a metrizable subgroup” (Fund. Math. 143 (1993), 119–136)
  2. Jorge Galindo, Salvador Hernández, The concept of boundedness and the Bohr compactification of a MAP Abelian group
  3. S. S. Gabriyelyan, On characterized subgroups of Abelian topological groups X and the group of all X -valued null sequences
  4. William Wistar Comfort, S. U. Raczkowski, F. Javier Trigos-Arrieta, The dual group of a dense subgroup

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