Mikhail Tkachenko

Topological groups and related structures

[i]Topological Groups and Related Structures[/i] provides an extensive overview of techniques and results in the topological theory of topological groups. This overview goes sufficiently deep and is detailed enough to become a useful tool for both researchers and students. This book presents a large amount of material, both classic and recent (on occasion, unpublished) about the relations of Algebra and Topology. It therefore belongs to the area called Topological Algebra. More specifically, the objects of the study are subtle and sometimes unexpected phenomena that occur when the continuity meets and properly feeds an algebraic operation. Such a combination gives rise to many classic structures, including topological groups and semigroups, paratopological groups, etc. Special emphasis is given to tracing the influence of compactness and its generalizations on the properties of an algebraic operation, causing on occasion the automatic continuity of the operation. The main scope of the book, however, is outside of the locally compact structures, thus distinguishing the monograph from a series of more traditional textbooks. The book is unique in that it presents very important material, dispersed in hundreds of research articles, not covered by any monograph in existence. The reader is gently introduced to an amazing world at the interface of Algebra, Topology, and Set Theory. He/she will find that the way to the frontier of the knowledge is quite short — almost every section of the book contains several intriguing open problems whose solutions can contribute significantly to the area. – Contents : – Introduction to Topological Groups and Semigroups; – Right Topological and Semitopological Groups; – Topological Groups: Basic Constructions; – Some Special Classes of Topological Groups; – Cardinal Invariants of Topological Groups; – Moscow Topological Groups and Completions of Groups; – Free Topological Groups; – Factorizable Topological Groups; – Compactness and its Generalizations in Topological Groups; – Actions of Topological Groups on Topological Spaces.

Group reflection and precompact paratopological groups

Abstract We construct a precompact completely regular paratopological Abelian group G of size (2ω)+ such that all subsets of G of cardinality ≤ 2ω are closed. This shows that Protasov’s theorem on non-closed discrete subsets of precompact topological groups cannot be extended to paratopological groups. We also prove that the group reflection of the product of an arbitrary family of paratopological (even semitopological) groups is topologically isomorphic to the product of the group reflections of the factors, and that the group reflection, H*, of a dense subgroup G of a paratopological group G is topologically isomorphic to a subgroup of G*.

Nondiscrete P-Groups can be reflexive

Abstract We present the first examples of nondiscrete reflexive P -groups (topological groups in which countable intersections of open sets are open) as well as of noncompact reflexive ω -bounded groups (precompact groups in which the closure of every countable set is compact). Our main result implies that every product of discrete Abelian groups equipped with the P -modified topology is reflexive. Taking uncountably many nontrivial factors, we thus answer a question posed by P. Nickolas and solve a problem raised by Ardanza-Trevijano, Chasco, Dominguez, and Tkachenko. New examples of non-reflexive P -groups are also given which are based on a further development of Leptins technique going back to 1955.

LOCAL COMPACTNESS IN FREE TOPOLOGICAL GROUPS

We show that the subspace An(X) of the free Abelian topological group A(X) on a Tychonoff space X is locally compact for each n ∈ ω iff A2(X) is locally compact iff F2(X) is locally compact iff X is the topological sum of a compact space and a discrete space. It is also proved that the subspace Fn(X) of the free topological group F (X) is locally compact for each n ∈ ω iff F4(X) is locally compact iff Fn(X) has pointwise countable type for each n ∈ ω iff F4(X) has pointwise countable type iff X is either compact or discrete, thus refining a result by Pestov and Yamada. We further show that An(X) has pointwise countable type for each n ∈ ω iff A2(X) has pointwise countable type iff F2(X) has pointwise countable type iff there exists a compact set C of ∗The second author wishes to thank the first author, and his department, for hospitality extended during the course of this work, and acknowledges the financial support of a Quality Fund grant from the Institute for Mathematical Modelling and Computational Systems (IMMaCS) at the University of Wollongong. †AMS classification numbers: 22A05, 54H11, 54A25, 54D30, 54D45

Paratopological and Semitopological Groups Versus Topological Groups

We present a survey on paratopological and semitopological groups relating these classes with the class of topological groups. A special attention is given to compactness-type properties in paratopological and semitopological groups which often imply automatic continuity of inversion or multiplication (or both) in these classes of objects.

Dieudonné Completion and PT-Groups

We consider the classes of PT-groups, strong PT-groups, completion friendly groups, and Moscow groups introduced by Arhangel’skii for the study of the Dieudonné completion of topological groups. We show that every subgroup H of a Lindelöf P-group is a PT-group, and that H is a strong PT-group iff it is

Cardinal Invariants of Topological Groups

{\mathbb R}

Unions of chains in dyadic compact spaces and topological groups

-factorizable. Assuming CH, we prove that every ω-narrow P-group is a PT-group. Several results regarding products of PT-groups and

Introduction to Topological Groups and Semigroups

{\mathbb R}

Topological groups: Basic constructions

-factorizable groups are established as well. We prove that the product of a Lindelöf group and an arbitrary subgroup of a Lindelöf Σ-group is completion friendly, and the same conclusion is valid for the product of an