On convergent sequences in dual groups

Abstract

We provide some characterizations of precompact abelian groups G whose dual group G∧ p endowed with the pointwise convergence topology on elements of G contains a nontrivial convergent sequence. In the special case of precompact abelian torsion groups G, we characterize the existence of a nontrivial convergent sequence in G∧ p by the following property of G: No infinite Hausdorff quotient group of G is countable. Also, we present an example of a dense subgroup G of the compact metrizable group Z(2)ω such that G is of the first category in itself, has measure zero, but the dual group G∧ p does not contain infinite compact subsets. This complements a result of J.E. Hart and K. Kunen (2005) on convergent sequences in dual groups. Making use of the group G we construct the first example of a precompact Pontryagin reflexive abelian group which is of the first Baire category