Filters: Topological congruence relations on groups

A filter F on a group G is a T -filter if there is a Hausdorff group topology τ on G such that F τ −→ 0. This notion can be specialized for sequences, in which case we say that {an} is a T -sequence. In this paper, T -filters and T -sequences are studied. We characterize T -filters in non-abelian groups, show that certain filters can be interpreted as topological extensions of the notion of kernel (i.e., normal subgroup, congruence relation), and provide several sufficient conditions for a sequence in an abelian group to be a T -sequence. As an application, special sequences in the Prüfer groups Z(p) are investigated. We prove that for p 6= 2, there is a Hausdorff group topology τ on Z(p) that is neither maximally nor minimally 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).