Topologies on groups determined by discrete subsets

Let G be an infinite group. Given a filter F on G, let T [ F ] denote the largest left invariant topology on G in which F converges to the identity. In this paper, we study the topology T [ F ] in case when F contains the Fréchet filter and there is M : G → F such that all the subsets x M ( x ) , where x ∈ G , are pairwise disjoint. We show that T [ F ] possesses interesting extremal properties. We consider also the question whether T [ F ] can be a group topology.