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A theorem on cardinal numbers associated with Abelian groups. (English) Zbl 1037.22001

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 1 2 on an Abelian group the character groups (G, i ) satisfy the relation

(G, 2 ) (G, 1 ) 2 1 ·

Since LCA groups were known to satisfy the stronger (in the absence of the continuum hypothesis) relation

(G, 2 ) (G, 1 ) 2 𝔠 ,(1)

Reade conjectured that this relation should also hold for -groups.

The purpose of the present paper is to prove that conjecture. When 1 2 are two different topologies on an Abelian group such that (G, 1 ) is separable and (G, 2 ) is metrizable (in the compact-open topology) the author manages to find a subset K of (G, 2 ) which is compact as a subset of G d (the compact group of all characters of G with pointwise convergence) yet has a subset B whose closure in G d has cardinality 2 𝔠 . B is actually an I 0 -set, a set of interpolation for the almost periodic functions, in the group (G, 1 ) . The proof of this fact is based on Rosenthal’s 1 -theorem and proves inequality (1) for the above groups (|(G, 1 ) |𝔠, since (G, 1 ) 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)].

MSC:
22A05Structure of general topological groups