Some density results for uniformly continuous functions

Abstract

Let X be a set and F a family of real-valued functions (not necessarily bounded) on X. We denote by μFX the space X endowed with the weak uniformity generated by F. and by U(μFX) the collection of uniformly continuous functions to the real line R.

In this note we study necessarily and sufficient conditions in order that the family F, be uniformly dense in U(μFX). Firstly, we give a ore direct proof of a result by Hager involving an external condition over F given in terms of composition with the uniformly continuous and real-valued functions defined on subsets of Rn. From this external condition we can derive as easy corollaries most of the results already known in this context. In the second part of this note we obtain an internal necessary and sufficient condition of uniform density set by means of certain covers of X by cozero-sets of functions in F.