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

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 Prüfer 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 ) ) ⊊ 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.