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The Bohr compactification, modulo a metrizable subgroup

W. Comfort, F. Trigos-Arrieta, S. Wu (1993)

Fundamenta Mathematicae

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...

The concept of boundedness and the Bohr compactification of a MAP Abelian group

Jorge Galindo, Salvador Hernández (1999)

Fundamenta Mathematicae

Let G be a maximally almost periodic (MAP) Abelian group and let ℬ be a boundedness on G in the sense of Vilenkin. We study the relations between ℬ and the Bohr topology of G for some well known groups with boundedness (G,ℬ). As an application, we prove that the Bohr topology of a topological group which is topologically isomorphic to the direct product of a locally convex space and an ℒ ∞ -group, contains “many” discrete C-embedded subsets which are C*-embedded in their Bohr compactification. This...

The Doitchinov Completion of a Regular Paratopological Group

Künzi, Hans-Peter, Romaguera, Salvador, Sipacheva, Ol’ga (1998)

Serdica Mathematical Journal

In memory of Professor D. Doitchinov ∗ This paper was written while the first author was supported by the Swiss National Science Foundation under grants 21–30585.91 and 2000-041745.94/1 and by the Spanish Ministry of Education and Sciences under DGES grant SAB94-0120. The second author was supported under DGES grant PB95-0737. During her stay at the University of Berne the third author was supported by the first author’s grant 2000-041745.94/1 from the Swiss National Science Foundation.We show...

The dual group of a dense subgroup

William Wistar Comfort, S. U. Raczkowski, F. Javier Trigos-Arrieta (2004)

Czechoslovak Mathematical Journal

Throughout this abstract, G is a topological Abelian group and G ^ is the space of continuous homomorphisms from G into the circle group 𝕋 in the compact-open topology. A dense subgroup D of G is said to determine G if the (necessarily continuous) surjective isomorphism G ^ ↠ D ^ given by h ↦ h | D is a homeomorphism, and G is determined if each dense subgroup of G determines G . The principal result in this area, obtained independently by L. Außenhofer and M. J. Chasco, is the following: Every metrizable group is...

The Lévy continuity theorem for nuclear groups

W. Banaszczyk (1999)

Studia Mathematica

Let G be an abelian topological group. The Lévy continuity theorem says that if G is an LCA group, then it has the following property (PL) a sequence of Radon probability measures on G is weakly convergent to a Radon probability measure μ if and only if the corresponding sequence of Fourier transforms is pointwise convergent to the Fourier transform of μ. Boulicaut [Bo] proved that every nuclear locally convex space G has the property (PL). In this paper we prove that the property (PL) is inherited...

The Lindelöf property and pseudo- ℵ 1 -compactness in spaces and topological groups

Constancio Hernández, Mihail G. Tkachenko (2008)

Commentationes Mathematicae Universitatis Carolinae

We introduce and study, following Z. Frol’ık, the class ℬ ( 𝒫 ) of regular P -spaces X such that the product X × Y is pseudo- ℵ 1 -compact, for every regular pseudo- ℵ 1 -compact P -space Y . We show that every pseudo- ℵ 1 -compact space which is locally ℬ ( 𝒫 ) is in ℬ ( 𝒫 ) and that every regular Lindelöf P -space belongs to ℬ ( 𝒫 ) . It is also proved that all pseudo- ℵ 1 -compact P -groups are in ℬ ( 𝒫 ) . The problem of characterization of subgroups of ℝ -factorizable (equivalently, pseudo- ℵ 1 -compact) P -groups is considered as well. We give some necessary...

The Ribes-Zalesskii property of some one relator groups

Gilbert Mantika, Narcisse Temate-Tangang, Daniel Tieudjo (2022)

Archivum Mathematicum

The profinite topology on any abstract group G , is one such that the fundamental system of neighborhoods of the identity is given by all its subgroups of finite index. We say that a group G has the Ribes-Zalesskii property of rank k , or is RZ k with k a natural number, if any product H 1 H 2 ⋯ H k of finitely generated subgroups H 1 , H 2 , ⋯ , H k is closed in the profinite topology on G . And a group is said to have the Ribes-Zalesskii property or is RZ if it is RZ k for any natural number k . In this paper we characterize groups...

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