When is U(X) a ring?

TL;DR

This paper characterizes when the space of real-valued uniformly continuous functions on a metric space forms a ring, linking this to properties of subsets such as Bourbaki-boundedness and uniform isolation.

Contribution

It provides a necessary and sufficient condition for the uniform continuous functions space to be a ring based on subset properties of the metric space.

Findings

The space is a ring iff every subset is Bourbaki-bounded or contains an infinite uniformly isolated subset.

Connects algebraic structure of function spaces with geometric properties of metric spaces.

Offers a complete characterization of when U(X) forms a ring.

Abstract

In this short paper, we will show that the space of real valued uniformly continuous functions defined on a metric space $(X,d)$ is a ring if and only if every subset $A\subset X$ has one of the following properties: $A$ is Bourbaki-bounded, i.e., every uniformly continuous function on $X$ is bounded on $A$. $A$ contains an infinite uniformly isolated subset, i.e., there exist $\delta>0$ and an infinite subset $F\subset A$ such that $d(a,x)\geq \delta$ for every $a\in F, x\in X$.