Characterizing sequences for precompact group topologies

Applied general topology, 2005

We study the sequences of integers (un) that converge to 0 in some precompact group topology on Z and the properties of the finest topology with this property when (un) satisfies a linear recurrence relation with bounded coefficients. Some of the results are extended to the case of sequences in arbitrary Abelian groups.

Journal of Mathematical Analysis and Applications, 2005

The Pontryagin-van Kampen (P-vK) duality, defined for topological Abelian groups, is given in terms of the compact-open topology. Polar reflexive spaces, introduced by Köthe, are those locally convex spaces satisfying duality when the dual space is equipped with the precompact-open topology. It is known that the additive groups of polar reflexive spaces satisfy P-vK duality. In this note we consider the duality of topological Abelian groups when the topology of the dual is the precompactopen topology. We characterize the precompact reflexive groups, i.e., topological groups satisfying the group duality defined in terms of the precompact-open topology. As a consequence, we obtain a new characterization of polar reflexive spaces. We also present an example of a space which satisfies P-vK duality and is not polar reflexive. Some of our results respond to questions appearing in the literature.

Journal of Pure and Applied Algebra, 2012

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, and contain no infinite compact subsets. This combination of properties guarantees that all compact subsets of the dual group G ∧ are finite. We also show that many (non-reflexive) precompact Abelian groups are quotients of reflexive precompact Abelian groups. This includes all precompact almost metrizable groups with the Baire property and their products. Finally, given a compact Abelian group G of weight ≥ 2 ω , we find proper dense subgroups H 1 and H 2 of G such that H 1 is reflexive and pseudocompact, while H 2 is non-reflexive and almost metrizable.

Applied General Topology, 2006

We study various weaker versions of metrizability for pseudocompact abelian groups G: singularity (G possesses a compact metrizable subgroup of the form mG, m > 0), almost connectedness (G is metrizable modulo the connected component) and various versions of extremality in the sense of Comfort and co-authors (s-extremal, if G has no proper dense pseudocompact subgroups, r-extremal, if G admits no proper pseudocompact refinement). We introduce also weakly extremal pseudocompact groups (weakening simultaneously s-extremal and r-extremal). It turns out that this "symmetric" version of extremality has nice properties that restore the symmetry, to a certain extent, in the theory of extremal pseudocompact groups obtaining simpler uniform proofs of most of the known results. We characterize doubly extremal pseudocompact groups within the class of s-extremal pseudocompact groups. We give also a criterion for r-extremality for connected pseudocompact groups.

Journal of Group Theory, 2000

We prove that every dense subgroup of a topological abelian group has the same 'convergence dual' as the whole group. By the 'convergence dual' we mean the character group endowed with the continuous convergence structure. We draw as a corollary that the continuous convergence structure on the character group of a precompact group is discrete and therefore a non-compact precompact group is never reflexive in the sense of convergence. We do not know if the same statement holds also for reflexivity in the sense of Pontryagin; at least in the category of metrizable abelian groups it does.

Forum Mathematicum, 2000

We study Pontryagin reflexivity in the class of precompact topological Abelian groups. We find reflexive groups among precompact not pseudocompact and among pseudocompact not compact groups. Making use of Martin's Axiom we give an example of a reflexive countably compact not compact Abelian group. We also prove that every pseudocompact Abelian group is a quotient of a reflexive pseudocompact group with respect to a closed reflexive pseudocompact subgroup.

Journal of Pure and Applied Algebra, 2010

We show that every Abelian group G with r 0 (G) = |G| = |G| ω admits a pseudocompact Hausdorff topological group topology T such that the space (G, T) is Fréchet-Urysohn. We also show that a bounded torsion Abelian group G of exponent n admits a pseudocompact Hausdorff topological group topology making G a Fréchet-Urysohn space if for every prime divisor p of n and every integer k ≥ 0, the Ulm-Kaplansky invariant f p,k of G satisfies (f p,k) ω = f p,k provided that f p,k is infinite and f p,k > f p,i for each i > k. Our approach is based on an appropriate dense embedding of a group G into a Σproduct of circle groups or finite cyclic groups.

Journal of Mathematical Analysis and Applications, 2013

Topology and its Applications, 2009

We show that the existence of a non-metrizable compact subspace of a topological group G often implies that G contains an uncount- able supersequence (a copy of the one-point compactification of an uncountable discrete space). The existence of uncountable superse- quences in a topological group has a non-trivial impact on functionally bounded subsets of the group. For example, if a topological group G contains an uncountable supersequence and K is a closed func- tionally bounded subset of G which does not contain uncountable supersequences, then any subset A of K is functionally bounded in G n (K n A).