Generation of the uniformly continuous functions

Abstract

Let X be a set and F a family of real-valued functions on X. We denote by μFX the space X endowed with the weak uniformity given by F. In this paper we provide a method of generating the set U(μFX), of all uniformly continuous real functions on μFX, by means of the family F. In order to do that we need to study the uniform approximation of real uniformly continuous functions on subsets of Rn. As a consequence, we give an internal condition on F in order to be uniformly dense in U(μFX).