Countable covers and uniform closure

Abstract

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