On proximal fineness of topological groups in their right uniformity

Ahmed Bouziad

Published 2019-04-29Version 1

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 $\{g\in G: g(A)\subset B\}$, $A,B\subset \mathbb N$.