# Precompact groups and convergence

## TL;DR

This paper investigates the properties of precompact group topologies, demonstrating conditions under which they are metrizable or Fréchet, and constructs examples and counterexamples related to these topological properties.

## Contribution

It shows that certain natural constructions yield metrizable groups, explores set-theoretic consistency results, and provides counterexamples to existing conjectures in the theory of topological groups.

## Key findings

- Natural constructions of precompact topologies are metrizable
- Consistency results for sequential precompact topologies being Fréchet
- Counterexample to a conjecture of D. Shakhmatov

## Abstract

We consider precompact sequential and Fr\'echet group topologies and show that some natural constructions of such topologies always result in metrizable groups answering a question of D.~Dikranjan et al. We show that it is consistent that all sequential precompact topologies on countable groups are Fr\'echet (or even metrizable). For some classes of groups (for example boolean) extra set-theoretic assumptions may be omitted (although in this case such groups do not have to be metrizable). We also build (using $\diamondsuit$) an example of a countably compact Fr\'echet group that is not $\alpha_3$ and obtain a counterexample to a conjecture of D.~Shakhmatov as a corollary.