### A coarse convergence group need not be precompact

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Given a topological property (or a class) $\mathcal{P}$, the class ${\mathcal{P}}^{*}$ dual to $\mathcal{P}$ (with respect to neighbourhood assignments) consists of spaces $X$ such that for any neighbourhood assignment $\{{O}_{x}:x\in X\}$ there is $Y\subset X$ with $Y\in \mathcal{P}$ and $\bigcup \{{O}_{x}:x\in Y\}=X$. The spaces from ${\mathcal{P}}^{*}$ are called dually $\mathcal{P}$. We continue the study of this duality which constitutes a development of an idea of E. van Douwen used to define $D$-spaces. We prove a number of results on duals of some general classes of spaces establishing, in particular, that any generalized ordered space...

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.

In 1990, Comfort asked Question 477 in the survey book “Open Problems in Topology”: Is there, for every (not necessarily infinite) cardinal number $\alpha \le {2}^{}$, a topological group G such that ${G}^{\gamma}$ is countably compact for all cardinals γ < α, but ${G}^{\alpha}$ is not countably compact? Hart and van Mill showed in 1991 that α = 2 answers this question affirmatively under $M{A}_{countable}$. Recently, Tomita showed that every finite cardinal answers Comfort’s question in the affirmative, also from $M{A}_{countable}$. However, the question has remained...

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....

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.

We generalize to the non-separable context a theorem of Levi characterizing Baire analytic spaces. This allows us to prove a joint-continuity result for non-separable normed groups, previously known only in the separable context.