# The descriptive look at the size of subsets of groups

## TL;DR

This paper investigates the Borel complexity of various subsets of countable groups and applies these findings to the Stone-ech compactification, revealing the topological complexity of the minimal ideal's closure.

## Contribution

It provides a detailed analysis of the Borel complexity of subset families in countable groups and characterizes the topological type of the minimal ideal's closure in ech compactification.

## Key findings

- Closure of the minimal ideal in ech is of type F_{sigmadelta}
- Classifies subsets of groups by size and their Borel complexity
- Connects subset properties to topological structure of ech compactification

## Abstract

We explore the Borel complexity of some basic families of subsets of a countable group (large, small, thin, sparse and other) defined by the size of their elements. Applying the obtained results to the Stone-\v{C}ech compactification $\beta G$ of $G$, we prove, in particular, that the closure of the minimal ideal of $\beta G$ is of type $F_{\sigma\delta}$.