# A survey on reflexivity of abelian topological groups

@article{Chasco2012ASO,
title={A survey on reflexivity of abelian topological groups},
author={M. J. Chasco and D. Dikranjan and E. Martin-peinador},
journal={Topology and its Applications},
year={2012},
volume={159},
pages={2290-2309}
}
• Published 2012
• Mathematics
• Topology and its Applications
The Pontryagin duality theorem for locally compact abelian groups (briefly, LCA groups) has been the starting point for many different routes of research in Mathematics. From its appearance there was a big interest to extend it in a context broader than LCA groups. Kaplan in the 40ʼs proposed—and it still remains open—the problem of characterization of all abelian topological groups for which the canonical mapping into its bidual is a topological isomorphism, assuming that the dual and the… Expand
16 Citations
A countable free closed non-reflexive subgroup of ℤ^{}
• Mathematics
• 2015
We prove that the group G=Hom(P,Z) of all homomorphisms from the Baer-Specker group P to the group Z of integer numbers endowed with the topology of pointwise convergence contains no infinite compactExpand
One-parameter subgroups of topological abelian groups
Abstract It is proved that, for a wide class of topological abelian groups (locally quasi-convex groups for which the canonical evaluation from the group into its Pontryagin bidual group is onto) theExpand
On reflexive group topologies on abelian groups of finite exponent
• Mathematics
• 2012
The present paper deals with the existence of nondiscrete reflexive topologies on abelian groups of finite exponent, which turns out to be linked with the cardinality of the corresponding group. WeExpand
On refinements of ω-bounded group topologies
Abstract A topological group is ω -bounded if the closure of any countable subset is compact. Clearly, the ω -bounded groups are countably compact and hence, precompact. It has been pointed outExpand
Locally quasi-convex convergence groups
Abstract This paper deals with the extension of Pontryagin reflexivity of the topological groups to the class of convergence groups. First, we give an example of a non-topological convergence on aExpand
Duality properties of bounded torsion topological abelian groups
• Mathematics
• 2017
Abstract Let G be a precompact, bounded torsion abelian group and G p ∧ its dual group endowed with the topology of pointwise convergence. We prove that if G is Baire (resp., pseudocompact), then allExpand
Generalize Heisenberg Groups and Self-Duality
• Mathematics
• 2016
This paper compares two generalizations of Heisenberg groups and studies their connection to one of the major open problems in the field of locally compact abelian groups, namely the description ofExpand
On topological properties of the group of the null sequences valued in an Abelian topological group
Following [23], denote by $\mathfrak{F}_0$ the functor on the category $\mathbf{TAG}$ of all Hausdorff Abelian topological groups and continuous homomorphisms which passes each $X\in \mathbf{TAG}$ toExpand
Kernel and cokernel in the category of augmented involutive stereotype algebras
We prove several properties of kernels and cokernels in the category of augmented involutive stereotype algebras: 1) the morphisms of the augmented involutive stereotype algebras have kernels andExpand
Selectively Pseudocompact Groups without Infinite Separable Pseudocompact Subsets
• Mathematics, Computer Science
• Axioms
• 2018
It is shown that the free precompact Boolean group of a topological sum ⨁ i ∈ I X i , where each space X i is either maximal or discrete, contains no infinite separable pseudocompact subsets. Expand

#### References

SHOWING 1-10 OF 118 REFERENCES
Open subgroups, compact subgroups and Binz-Butzmann reflexivity
• Mathematics
• 1996
A number of attempts to extend Pontryagin duality theory to categories of groups larger than that of locally compact abelian groups have been made using different approaches. The extension to theExpand
Reflexivity of prodiscrete topological groups
• Mathematics
• 2011
Abstract We study the duality properties of two rather different classes of subgroups of direct products of discrete groups ( protodiscrete groups): P - groups , i.e., topological groups such thatExpand
On strongly reflexive topological groups
• Mathematics
• 2001
An Abelian topological group G is strongly reflexive if every closed subgroup and every Hausdorff quotient of G and of its dual group G ⋀ , is reflexive. In this paper we prove the following: theExpand
Two classes of spaces reflexive in the sense of Pontryagin
• Mathematics
• 2003
The Pontryagin-van Kampen duality for locally compact Abelian groups can be generalized in two ways to wider classes of topological Abelian groups: in the first approach the dual group is endowedExpand
DUALITY OF TOTALLY BOUNDED ABELIAN GROUPS
Let G be a totally bounded Abelian (Hausdorff) group and denote by G the group ofcharacters, Le., continuous homomorphisms from G into the usual Tonas T, equipped with operation defined pointwise,Expand
Open subgroups and Pontryagin duality
• Mathematics
• 1994
For an abelian topological group G, let G∧ denote the character group of G. The group G is called reflexive if the evaluation map is a topological isomorphism of G onto G∧∧, and G is called stronglyExpand
$k$-groups and duality
Recall that a function is Ac-continuous if its restriction to each compact subset of its domain is continuous. We call a topological group G a Ac-group if each ¿-continuous homomorphism on G isExpand
Weak and strong topologies in topological abelian group
The main topic of this thesis are the weak and strong topologies on abelian groups. The former notion is generally known in the theory of topological abelian groups; the most common example isExpand
Pontryagin duality in the class of precompact Abelian groups and the Baire property
• Mathematics
• 2011
We present a wide class of reflexive, precompact, non-compact, Abelian topological groups G determined by three requirements. They must have the Baire property, satisfy the open refinement condition,Expand
Final group topologies, Kac-Moody groups and Pontryagin duality
• Mathematics
• 2006
We study final group topologies and their relations to compactness properties. In particular, we are interested in situations where a colimit or direct limit is locally compact, a kω-space, orExpand