Almost maximally almost-periodic group topologies determined by T-sequences

摘要

A sequence {a_n} in a group G is a T-sequence if there is a Hausdorff group topology τ on G such that a_n →~τ 0. In this paper, we provide several sufficient conditions for a sequence in an abelian group to be a T-sequence, and investigate special sequences in the Pruefer groups Z(p~∞). We show that for p ≠ 2, there is a Hausdorff group topology τ on Z(p~∞) that is determined by a T-sequence, which is close to being maximally almost-periodic—in other words, the von Neumann radical n(Z(p~∞), τ) is a non-trivial finite subgroup. In particular, n(n(Z(p~∞), τ)) is not contained in n(Z(p~∞), τ). We also prove that the direct sum of any infinite family of finite abelian groups admits a group topology determined by a T-sequence with non-trivial finite von Neumann radical.