Estructuras de grupo topol\'ogico en (Z,+)

TL;DR

This paper explores various topological group structures on the integers, presenting new and known results about their complexity, including families of precompact, metrizable, and complete topologies, and discusses open questions about their cardinalities.

Contribution

It introduces new families of topological group structures on the integers, expanding understanding of their complexity and properties.

Findings

A family of 2^cont precompact Hausdorff topologies.

A wide family of metrizable non-precompact, non-complete topologies.

A wide family of complete non-metrizable topologies.

Abstract

In this work we have considered the complexity of the different structures as topological group on Z. We collect some new results, as well as some known results on the group of the integers in order to present: -A family of $2^\cont$ precompact Hausdorff. -A wide family of metrizable non-precompact, non-complete group topologies. -A wide family of complete non-metrizable topologies. We leave some open questions - for instance the problem of the cardinality of the families above.