Precompact Fréchet topologies on abelian groups
The authors introduce, for every countable abelian group \(G\), the notion of \(\gamma_G\)-set and show that a necessary and sufficient condition for the existence of a precompact Fréchet non-metrizable topology on \(G\) is the existence of an uncountable \(\gamma_G\)-set which separates points of \(G\). Assuming the existence of an uncountable \(\gamma\)-set, the authors show that every countable abelian group admits a non-metrizable precompact Fréchet topology. With the assumption \({\mathfrak p}>\omega_1\), they also establish the existence of a non-metrizable Fréchet topology on every countable group which admits a non-discrete topology at all. Through further study of the notion of \(\gamma_G\)-set, they show that the minimal size of a subset of the dual group \(G^\ast\) which is not a \(\gamma_G\)-set is the pseudointersection number \({\mathfrak p}\) for any countable abelian group \(G\).
scientific article; zbMATH DE number 6100635
scientific article