N. Th. Varopoulos [Proc. Camb. Philos. Soc. 60, 465–516 (1964; Zbl 0161.11103)] defined an -group to be a topological group in which the topology is the intersection of a decreasing sequence of Hausdorff locally compact group topologies.
J. B. Reade [Proc. Camb. Philos. Soc. 61, 69–74 (1965; Zbl 0136.29604)] subsequently proved that given two different topologies on an Abelian group the character groups satisfy the relation
Since LCA groups were known to satisfy the stronger (in the absence of the continuum hypothesis) relation
Reade conjectured that this relation should also hold for -groups.
The purpose of the present paper is to prove that conjecture. When are two different topologies on an Abelian group such that is separable and is metrizable (in the compact-open topology) the author manages to find a subset of which is compact as a subset of (the compact group of all characters of with pointwise convergence) yet has a subset whose closure in has cardinality . is actually an -set, a set of interpolation for the almost periodic functions, in the group . The proof of this fact is based on Rosenthal’s -theorem and proves inequality (1) for the above groups (, since is separable). The general case follows after appropriately combining the structure theory of groups developed in N. Th. Varopoulos [loc. cit], L. J. Sulley [J. Lond. Math. Soc., II. Ser. 5, 629–637 (1972; Zbl 0243.22002)] and R. Venkataraman [Monatsh. Math. 100, 47–66 (1985; Zbl 0563.43005)], see also the author and the reviewer [Fund. Math. 159, 195–218 (1999; Zbl 0934.22008)].