# On Products of Closed Subsets in Free Groups

## TL;DR

This paper explores the properties of closed subsets in free groups, providing examples where their product isn't closed, and characterizes conditions under which products remain closed in the profinite topology.

## Contribution

It introduces new examples of non-closed products and offers a characterization of closed subsets whose products with finitely generated subgroups are also closed.

## Key findings

- Examples of closed subsets with non-closed products in free groups
- Characterization of subsets with closed products in the profinite topology
- Conditions ensuring closure of products with finitely generated subgroups

## Abstract

We present examples of closed subsets of a free group such that their product is not closed in the profinite topology. We discuss how to characterize a subset of a free group which is closed in the profinite topology and its product with any finitely generated subgroup of a free group is also closed in the profinite topology.