Non metrizable topologies on Z with countable dual group

TL;DR

This paper constructs two distinct families of non-metrizable topologies on the integers with countable dual groups, exploring their properties and duality relations, especially focusing on locally quasi-convex and complete topologies.

Contribution

It introduces new non-metrizable topologies on Z with countable duals, linked to D-sequences, and analyzes their duality and topological properties.

Findings

Two families of non-metrizable topologies on Z are constructed.

Dual groups are isomorphic to infinite torsion subgroups of the circle.

Complete topologies are not locally quasi-convex.

Abstract

In this paper we give two families of non-metrizable topologies on the group of the integers having a countable dual group which is isomorphic to a infinite torsion subgroup of the unit circle in the complex plane. Both families are related to $D$-sequences, which are sequences of natural numbers such that each term divides the following. The first family consists of locally quasi-convex group topologies. The second consists of complete topologies which are not locally quasi-convex. In order to study the dual groups for both families we need to make numerical considerations of independent interest.