U(X) as a ring for metric spaces X

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 ? > 0 and an infinite subset F ? A such that d(a,x) ? ? for every a ? F, x ? X n \{a}.