On convergent sequences in dual groups
M. TkachenkoRevista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas
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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 quotient group of G is countable. Finally, 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 Theorem 1.6 in [J.E. Hart and K. Kunen, Limits in function spaces and compact groups, Topol. Appl. 151 (2005), 157-168]. As a consequence, we obtain an example of a precompact reflexive abelian group which is of the first Baire category.
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