Type: Article

Precompact Fréchet topologies on Abelian groups

M. Hrušák, U.A. Ramos-García;

Abstract:

We study precompact Frechet topologies on countable Abelian groups. For every countable Abelian group G we introduce the notion of a gamma(G)-set and show that there is a precompact Frechet non-metrizable topology on G if and only if there is an uncountable gamma(G)-set that separates points of G. We show that, assuming the existence of an uncountable gamma-set, there is a non-metrizable precompact Frechet topology on every countable Abelian group. and assuming p > omega(1), there is a non-metrizable Frechet topology on every countable group which admits a non-discrete topology at all. We further study the notion of a gamma(G)-set and show that the minimal size of a subset of the dual group G* which is not a gamma(G)-set is the pseudointersection number p for any countable Abelian group G.