### A coarse convergence group need not be precompact

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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....

A new class of $p$-primary abelian groups that are Hausdorff in the $p$-adic topology and that generalize direct sums of cyclic groups are studied. We call this new class of groups almost coproducts of cyclic groups. These groups are defined in terms of a modified axiom 3 system, and it is observed that such groups appear naturally. For example, $V\left(G\right)/G$ is almost a coproduct of finite cyclic groups whenever $G$ is a Hausdorff $p$-primary group and $V\left(G\right)$ is the group of normalized units of the modular group algebra...

The least cardinal λ such that some (equivalently: every) compact group with weight α admits a dense, pseudocompact subgroup of cardinality λ is denoted by m(α). Clearly, $m\left(\alpha \right)\le {2}^{\alpha}$. We show: Theorem 4.12. Let G be Abelian with |G| = γ. If either m(α) ≤ α and m$\left(\alpha \right)\le {r}_{0}\left(G\right)\le \gamma \le {2}^{\alpha}$, or α > ω and ${\alpha}^{\omega}\le {r}_{0}\left(G\right)\le {2}^{\alpha}$, then G admits a pseudocompact group topology of weight α. Theorem 4.15. Every connected, pseudocompact Abelian group G with wG = α ≥ ω satisfies ${r}_{0}\left(G\right)\ge m\left(\alpha \right)$. Theorem 5.2(b). If G is divisible Abelian with ${2}^{{r}_{0}\left(G\right)}\le \gamma $, then G admits at most ${2}^{\gamma}$-many...

We construct in Bell-Kunen’s model: (a) a group maximal topology on a countable infinite Boolean group of weight ${\aleph}_{1}<\u212d$ and (b) a countable irresolvable dense subspace of ${2}^{{\omega}_{1}}$. In this model $\u212d={\aleph}_{{\omega}_{1}}$.