On Group Topologies Determined by Families of Sets

Let G be an abelian group, and F a downward directed family of subsets of G. In [6], I. Protasov and E. Zelenyuk describe the finest group topology T on G under which F converges to 0; in particular, their description yields a criterion for T to be Hausdorff. They then show that if F is the filter of cofinite subsets of a countable subset X ⊆ G (the Fréchet filter on X), there is a simpler criterion: T is Hausdorff if and only if for every g ∈ G − {0} and positive integer n, there is an S ∈ F such that g does not lie in the n-fold sum n (S ∪ {0} ∪ −S). In this note, their proof is adapted to a larger class of families F. In particular, if X is any infinite subset of G, κ any regular infinite cardinal ≤ card(X), and F the set of complements in X of subsets of cardinality < κ, then the above criterion holds. We also give some negative examples, including a countable downward directed set F (not of the above sort) of subsets of Z which satisfies the “ g / ∈ n (S ∪ {0} ∪−S) ” condition but does not induce a Hausdorff topology. We end with a version of our main result for noncommutative G.