Topologies on Abelian Groups

Abstract

A filter on an abelian group G is called a T-filter if there exists a Hausdorff group topology under which converges to zero. G{} will denote the group G with the largest topology among those making converge to zero. This method of defining a group topology is completely equivalent to the definition of an abstract group by defining relations. We shall obtain characterizations of T-filters and of T-sequences; among these, we shall pay particular attention to T-sequences on the integers. The method of T-sequences will be used to construct a series of counterexamples for several open problems in topological algebra. For instance there exists, on every infinite abelian group, a topology distinguishing between sequentiality and the Frechet-Urysohn property (this solves a problem posed by V.I. Malykhin); we also find a topology on the group of integers admitting no nontrivial continuous character, thus solving a problem of Nienhuys. We show also that on every infinite abelian group there exists a free ultrafilter which is not a T-ultrafilter.