Limits in function spaces and compact groups

Abstract

For B an infinite subset of ω and X a topological group, let C B X be the set of all x ∈ X such that 〈 x n : n ∈ B 〉 converges to 1. C B T always has measure 0 in the circle group T . If F is a filter of infinite sets, let D F X = ⋃ { C B X : B ∈ F } . Then C B X and D F X are subgroups of X when X is Abelian. We show that there is a filter F such that D F T has measure 0 but is not contained in any C B T . In contrast, for any compact metric group X, there is a filter G such that D G X = X ; this follows from a more general result in this paper on limits in function spaces. Also, we show that some of the properties of D F X , for arbitrary compact groups X, are determined by the special cases X = T or X = T ω .