M. Tkachenko

R-factorizable groups and subgroups of Lindelöf P-groups

Abstract The main subject of our study are P-groups, that is, the topological groups whose Gδ-sets are open. We establish several elementary properties of P-groups and then prove that a P-group is R -factorizable iff it is pseudo-ω1-compact iff it is ω-stable. This characterization is used to show that direct products of R -factorizable P-groups as well as continuous homomorphic images of R -factorizable P-groups are R -factorizable. A special emphasis is placed on the study of subgroups of Lindelof P-groups. The concept of stability is applied to prove that if G is a dense subgroup of a direct product of Lindelof Σ-groups, then every continuous homomorphic image of G is R -factorizable and perfectly κ-normal.

Varieties generated by countably compact Abelian groups

We prove under the assumption of Martins Axiom that every precompact Abelian group of size ≤ 2 N 0 belongs to the smallest class of groups that contains all Abelian countably compact groups and is closed under direct products, taking closed subgroups and continuous isomorphic images.

Reflexivity in precompact groups and extensions

We establish some general principles and find some counter-examples concerning the Pontryagin reflexivity of precompact groups and P-groups. We prove in particular that:; (1) A precompact Abelian group G of bounded order is reflexive if the dual group G has no infinite compact subsets and every compact subset of G is contained in a compact subgroup of G.; (2) Any extension of a reflexive P-group by another reflexive P-group is again reflexive.; We show on the other hand that an extension of a compact group by a reflexive omega-bounded group (even dual to a reflexive P-group) can fail to be reflexive.; We also show that the P-modification of a reflexive sigma-compact group can be non-reflexive (even if, as proved in [20], the P-modification of a locally compact Abelian group is always reflexive)

Topological Features of Topological Groups

Our aim is to give a relatively concise description of the state-of-the-art in the theory of topological groups up to the moment. The paper we present is addressed primarily to the General Topology-inclined reader, and this partly explains the choice of the title. The second reason for speaking of topological features of topological groups is that we focus our attention on topological ideas and methods in the area and almost completely omit the very rich and profound algebraic part of the theory of locally compact groups (except for a brief discussion in Sections 2.4 and 2.5). Neither do we have any intention of presenting material concerning the representation theory of (locally) compact groups — the book [We4] by Weil and Section 5 of [Pon4] by Pontryagin are recommended in this respect.

Provisional solution to a Comfort–van Mill problem

We prove under the assumption of Martins Axiom that an abstract Abelian group G of non-measurable cardinality is the intersection of countably compact subgroups of its Bohr compactification bG. This result is used to show that weakly free countably compact topological groups do not exist, thus answering a question posed by Comfort and van Mill in 1988. In fact, we show under MA that a free (P,CC)-group over a space X exists iff X is empty, where P and CC are the classes of pseudocompact and countably compact topological groups, respectively. On the other hand, we prove the existence of a weakly free (P,CC)-group over an arbitrary space X and show that our construction of such a group is functorial. Similar results remain valid in the Abelian case.