# 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

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topComfort, 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 -

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## Citations in EuDML Documents

top- 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)
- Jorge Galindo, Salvador Hernández, The concept of boundedness and the Bohr compactification of a MAP Abelian group
- S. S. Gabriyelyan, On characterized subgroups of Abelian topological groups $X$ and the group of all $X$-valued null sequences
- William Wistar Comfort, S. U. Raczkowski, F. Javier Trigos-Arrieta, The dual group of a dense subgroup

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