Left and right uniform structures on functionally balanced groups

Let G be a Hausdorff topological group. It is shown that there is a class C of subspaces of G, containing all (but not only) precompact subsets of G, for which the following result holds: e that for every real-valued discontinuous function on G there is a set A ∈ C such that the restriction mapping f | A has no continuous extension to G; then the following are equivalent:(i) ft and right uniform structures of G are equivalent, left uniformly continuous bounded real-valued function on G is right uniformly continuous, ery countable set A ⊂ G and every neighborhood V of the unit e of G, there is a neighborhood U of e in G such that A U ⊂ V A . consequence, it is proved that items (i), (ii) and (iii) are equivalent for every inframetrizable group. These results generalize earlier ones established by Itzkowitz, Rothman, Strassberg and Wu, by Milnes and by Pestov for locally compact groups, by Protasov for almost metrizable groups, and by Troallic for groups that are quasi-k-spaces.