# On proximal fineness of topological groups in their right uniformity

## TL;DR

This paper investigates the property of proximal fineness in topological groups under their right uniformity, proving that certain functionally generated groups are proximally fine, while others are not, depending on their permutation group structure.

## Contribution

It establishes conditions under which topological groups are proximally fine and provides examples of permutation groups that are not proximally fine.

## Key findings

- Functionally generated groups by precompact subsets are proximally fine.
- Certain permutation groups on natural numbers are not proximally fine.
- Proximal fineness depends on the group's uniform structure and generation.

## Abstract

A uniform space $X$ is said to be proximally fine if every proximally continuous map on $X$ into a uniform is uniformly continuous. We supply a proof that every topological group which is functionnaly generated by its precompact subsets is proximally fine with respect to its right uniformity. On the other hand, we show that there are various permutation groups $G$ on the integers $\mathbb N$ that are not proximally fine with respect to the topology generated by the sets $\{g\in G: g(A)\subset B\}$, $A,B\subset \mathbb N$.