van Douwen's problems related to the Bohr topology

We comment van Douwen's problems on the Bohr topology of the abelian groups raised in his paper (The maximal totally bounded group topology on G and the biggest minimal G-space for Abelian groups G) as well as the steps in the solution of some of them. New solutions to two of the resolved problems are also given.Comment: 14 page

Metrization criteria for compact groups in terms of their dense subgroups

According to Comfort, Raczkowski and Trigos-Arrieta, a dense subgroup D of a compact abelian group G determines G if the restriction homomorphism G^ --> D^ of the dual groups is a topological isomorphism. We introduce four conditions on D that are necessary for it to determine G and we resolve the following question: If one of these conditions holds for every dense (or G_\delta-dense) subgroup D of G, must G be metrizable? In particular, we prove (in ZFC) that a compact abelian group determined by all its G_\delta-dense subgroups is metrizable, thereby resolving a question by Hernandez, Macario and Trigos-Arrieta. (Under the additional assumption of the Continuum Hypothesis CH, the same statement was proved recently by Bruguera, Chasco, Dominguez, Tkachenko and Trigos-Arrieta.) As a tool, we develop a machinery for building G_\delta-dense subgroups without uncountable compact subsets in compact groups of weight \omega_1 (in ZFC). The construction is delicate, as these subgroups must have non-trivial convergent sequences in some models of ZFC.Comment: The exposition has substantially improved. Remarks 5.6 and 5.7 are new. Three references adde

Characterizing Subgroups of Compact Abelian Groups

We prove that every countable subgroup of a compact metrizable abelian group has a characterizing set. As an application, we answer several questions on maximally almost periodic (MAP) groups and give a characterization of the class of (necessarily MAP) abelian topological groups whose Bohr topology has countable pseudocharacter.Comment: 12 page

The Markov-Zariski topology of an abelian group

According to Markov, a subset of an abelian group G of the form {x in G: nx=a}, for some integer n and some element a of G, is an elementary algebraic set; finite unions of elementary algebraic sets are called algebraic sets. We prove that a subset of an abelian group G is algebraic if and only if it is closed in every precompact (=totally bounded) Hausdorff group topology on G. The family of all algebraic subsets of an abelian group G forms the family of closed subsets of a unique Noetherian T_1 topology on G called the Zariski, or verbal, topology of G. We investigate the properties of this topology. In particular, we show that the Zariski topology is always hereditarily separable and Frechet-Urysohn. For a countable family F of subsets of an abelian group G of cardinality at most the continuum, we construct a precompact metric group topology T on G such that the T-closure of each member of F coincides with its Zariski closure. As an application, we provide a characterization of the subsets of G that are dense in some Hausdorff group topology on G, and we show that such a topology, if it exists, can always be chosen so that it is precompact and metric. This provides a partial answer to a long-standing problem of Markov

On the Borel Complexity of Characterized Subgroups

In a compact abelian group $X$, a characterized subgroup is a subgroup $H$ such that there exists a sequence of characters \vs=(v_n) of $X$ such that H=\{x\in X:v_n(x)\to 0 \text{ in } \T\}. Gabriyelyan proved for X=\T, that \{x\in\T:n!x\to 0 \text{ in }\T\} is not an $F_\sigma$-set. In this paper, we give a complete description of the $F_\sigma$-subgroups of \T characterized by sequences of integers \vs=(v_n) such that $v_n|v_{n+1}$ for all $n\in\N$ (we show that these are exactly the countable characterized subgroups). Moreover in the general setting of compact metrizable abelian groups, we give a new point of view to study the Borel complexity of characterized subgroups in terms of appropriate test-topologies in the whole group

Topics in uniform continuity

This paper collects results and open problems concerning several classes of functions that generalize uniform continuity in various ways, including those metric spaces (generalizing Atsuji spaces) where all continuous functions have the property of being close to uniformly continuous

Hewitt-Marczewski-Pondiczery type theorem for abelian groups and Markov's potential density

For an uncountable cardinal \tau and a subset S of an abelian group G, the following conditions are equivalent: (i) |{ns:s\in S}|\ge \tau for all integers n\ge 1; (ii) there exists a group homomorphism \pi:G\to T^{2^\tau} such that \pi(S) is dense in T^{2^\tau}. Moreover, if |G|\le 2^{2^\tau}, then the following item can be added to this list: (iii) there exists an isomorphism \pi:G\to G' between G and a subgroup G' of T^{2^\tau} such that \pi(S) is dense in T^{2^\tau}. We prove that the following conditions are equivalent for an uncountable subset S of an abelian group G that is either (almost) torsion-free or divisible: (a) S is T-dense in G for some Hausdorff group topology T on G; (b) S is T-dense in some precompact Hausdorff group topology T on G; (c) |{ns:s\in S}|\ge \min{\tau:|G|\le 2^{2^\tau}} for every integer n\ge 1. This partially resolves a question of Markov going back to 1946

Entropy in Topological Groups, Part 2

Entropy was introduced first in thermodynamics and statistical mechanics, as well as information theory. In the last sixty years entropy made its way also in topology, ergodic theory, as well as other branches of mathematics as algebra, geometry and number theory where dynamical systems appear in one way or another. Roughly speaking, entropy is a non-negative real number or infinity assigned to a selfmap T of a space X, where the space X can be a topological or uniform space, a measure space, an abstract or topological group (or vector space) or just a set. The selfmap T can be, respectively, a (uniformly) continuous selfmap, a measure preserving transformation, a (continuous) endomorphism, etc. Depending on each choice, one may have a topological entropy, uniform entropy, measure entropy, algebraic entropy, etc. Topics for discussion: (a) the connection between these entropies with particular emphasis on the case of topological groups; (b) a unified (categorical) approach to entropy based on appropriate functors to the category of normed semigroups; (c) the connection of entropy to other well-known functions (e.g., the scale function of Georege Willis, the Mahler measure and the related Lehmer problem in number theory, etc); (d) entropy of semigroup actions (in place of selfmaps)