| 研究実績の概要 |
(1) Let G be an Abelian group which is either torsion or torsion-free or V-free, where V is either a precompact variety of groups or any variety of Abelian groups. We prove that if G admits a pseudocompact group topology, then G admits also a selectively sequentially pseudocompact group topology. This gives a partial answer to a question of Garcia-Ferreira and Tomita.
(2) We give a ZFC example of a Boolean topological group G without non-trivial convergent sequences having the following "selective" compactness property: For each free ultrafilter p on N and every sequence {U_n:n in N} of non-empty open subsets of G one can choose a point x_n in U_n for all n in such a way that the resulting sequence {x_n:n in N} has a p-limit in G, that is, {n in N: x_n in V} belongs to p for every neighbourhood V of x in G. In particular, G is selectively pseudocompact (strongly pseudocompact) but not selectively sequentially pseudocompact. This answers a question of Dorantes-Aldama and Shakhmatov.
(3) We establish a fairly general result involving the Bohr topology on all countable subgroups of a given topological group from which we deduce that many known examples in the literature of pseudocompact Boolean groups are not selectively pseudocompact.
(4) We give a complete characterization of Abelian groups of finite positive divisible rank which admit a group topology having the small subgroup generating property. This finishes the classification of Abelian groups admitting a group topology with the small subgroup generating property, answering a question of Comfort and Gould.
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