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.
