On F{\o}lner sets in topological groups

TL;DR

This paper extends F{46}lner's criterion to topological groups, characterizing amenability through approximations of actions, and applies this to solve problems related to the von Neumann problem and a question by Rosendal.

Contribution

It introduces a topological version of F{46}lner's criterion and characterizes amenability via action approximations, providing new insights into topological group theory.

Findings

Topological amenability characterized by action approximations

A topological version of Whyte's geometric solution to the von Neumann problem

Affirmative answer to Rosendal's question on topological groups

Abstract

We extend F{\o}lner's amenability criterion to the realm of general topological groups. Building on this, we show that a topological group $G$ is amenable if and only if its left translation action can be approximated in a uniform manner by amenable actions on the set $G$. As applications we obtain a topological version of Whyte's geometric solution to the von Neumann problem and provide an affirmative answer to a question posed by Rosendal.