U(X) as a Ring for Metric Spaces X

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 ⊂ 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 δ > 0 and an infinite subset F ⊂ A such that d(a, x) ≥ δ for every a∈F, x∈X\{a}.