# SSGP topologies on free groups of infinite rank

## TL;DR

This paper proves that free groups with infinitely many generators can be equipped with a Hausdorff topology where every element can be approximated by elements with cyclic subgroups in any neighborhood, answering a longstanding open question.

## Contribution

It establishes the existence of a Hausdorff topology with the small subgroup generating property for free groups of infinite rank, extending previous results.

## Key findings

- Every element can be expressed as a product of elements with cyclic subgroups in any open neighborhood.
- Provides a positive answer to a question of Comfort and Gould for infinite rank free groups.
- The case for finitely generated free groups remains unresolved.

## Abstract

We prove that every free group G with infinitely many generators admits a Hausdorff group topology T with the following property: for every T-open neighbourhood U of the identity of G, each element g in G can be represented as a product g=g_1 g_2 ... g_k such that the cyclic group generated by each g_i is contained in U. In particular, G admits a Hausdorff group topology with the small subgroup generating property of Gould. This provides a positive answer to a question of Comfort and Gould in the case of free groups with infinitely many generators. The case of free groups with finitely many generators remains open.