A coarse convergence group need not be precompact
A completion functor for Cauchy groups.
A countably cellular topological group all of whose countable subsets are closed need not be -factorizable
We construct a Hausdorff topological group such that is a precalibre of (hence, has countable cellularity), all countable subsets of are closed and -embedded in , but is not -factorizable. This solves Problem 8.6.3 from the book “Topological Groups and Related Structures" (2008) in the negative.
A density property of the tori and duality
A Dowker group
A noncommutative Orlicz-Pettis type theorem
A non-standard metric in the group of reals
A note on embeddings manifolds into topological groups preserving dimensions
A quest for nice kernels of neighbourhood assignments
Given a topological property (or a class) , the class dual to (with respect to neighbourhood assignments) consists of spaces such that for any neighbourhood assignment there is with and . The spaces from are called dually . We continue the study of this duality which constitutes a development of an idea of E. van Douwen used to define -spaces. We prove a number of results on duals of some general classes of spaces establishing, in particular, that any generalized ordered space...
A relatively free topological group that is not varietal free
Answering a 1982 question of Sidney A. Morris, we construct a topological group G and a subspace X such that (i) G is algebraically free over X, (ii) G is relatively free over X, that is, every continuous mapping from X to G extends to a unique continuous endomorphism of G, and (iii) G is not a varietal free topological group on X in any variety of topological groups.
A solenoidal and monothetic minimally almost periodic group
A solution to Comfort's question on the countable compactness of powers of a topological group
In 1990, Comfort asked Question 477 in the survey book “Open Problems in Topology”: Is there, for every (not necessarily infinite) cardinal number , a topological group G such that is countably compact for all cardinals γ < α, but is not countably compact? Hart and van Mill showed in 1991 that α = 2 answers this question affirmatively under . Recently, Tomita showed that every finite cardinal answers Comfort’s question in the affirmative, also from . However, the question has remained...
A Structure On A Set Of Subsets
A survey on just-non- groups.
A Vanishing Theorem for Group Compactifications.
Abelian pro-countable groups and orbit equivalence relations
We study a class of abelian groups that can be defined as Polish pro-countable groups, as non-archimedean groups with a compatible two-sided invariant metric or as quasi-countable groups, i.e., closed subdirect products of countable discrete groups, endowed with the product topology. We show that for every non-locally compact, abelian quasi-countable group G there exists a closed L ≤ G and a closed, non-locally compact K ≤ G/L which is a direct product of discrete countable groups....
Actions of a locally compact group with zero II.
Algebraic obstructions to sequential convergence in Hausdorff abelian groups.
Almost all submaximal groups are paracompact and σ-discrete
We prove that any topological group of a non-measurable cardinality is hereditarily paracompact and strongly σ-discrete as soon as it is submaximal. Consequently, such a group is zero-dimensional. Examples of uncountable maximal separable spaces are constructed in ZFC.
An Example of a Q-minimal Precompact Topological Group Containing a Nonminimal Closed Normal Subgroup.