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 subset of 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 of A such that d(a,x) >= delta for every a is an element of F; x is an element of X \ {a}.