Countable uniformly discrete sets in functionally balanced groups

Abstract

We prove that every countable left uniformly discrete subset of a Hausdorff functionally balanced topological group is right thin. As applications, we answer two questions of Bouziad and Troallic: every left precompact subset of a Hausdorff functionally balanced group is right precompact, and every Hausdorff $\omega$-narrow functionally balanced group is a SIN group.