Countable covers and uniform closure

Abstract

We present, in a unified way, several results of uniform approximation for real-valued continuous and uniformly continuous functions on a space X . We obtain all of them by applying a general method of proof that involves a certain kind of countable covers of X , the so-called 2-finite covers. For instance, if X is endowed with the weak uniformity given by a vector lattice F of real-valued functions on X containing all the real constant functions then, using that method, we characterize the uniform density of F only in terms of the family F , thus improving a previous result in this line.