On the Borel Complexity of Characterized Subgroups

2014/12/09 by Dikranjan, Dikran, Impieri, Daniele
#22C05 #FOS: Mathematics #General Topology (math.GN)

paper · doi:10.48550/arxiv.1412.2949

Abstract

In a compact abelian group X, a characterized subgroup is a subgroup H such that there exists a sequence of characters \vs=(v_n) of X such that H=\x∈ X:v_n(x)→ 0 in \T\. Gabriyelyan proved for X=\T, that \x∈\T:n!x→ 0 in \T\ is not an F_σ-set. In this paper, we give a complete description of the F_σ-subgroups of \T characterized by sequences of integers \vs=(v_n) such that v_n|v_n+1 for all n∈\N (we show that these are exactly the countable characterized subgroups). Moreover in the general setting of compact metrizable abelian groups, we give a new point of view to study the Borel complexity of characterized subgroups in terms of appropriate test-topologies in the whole group.

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