M. Isabel Garrido

Generation of the uniformly continuous functions

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).

The Samuel realcompactification of a metric space

Abstract In this paper we introduce a realcompactification for any metric space ( X , d ) , defined by means of the family of all its real-valued uniformly continuous functions. We call it the Samuel realcompactification, according to the well known Samuel compactification associated to the family of all the bounded real-valued uniformly continuous functions. Among many other things, we study the corresponding problem of the Samuel realcompactness for metric spaces. At this respect, we prove that a result of Katětov–Shirota type occurs in this context, where the completeness property is replaced by Bourbaki-completeness (a notion recently introduced by the authors) and the closed discrete subspaces are replaced by the uniformly discrete ones. More precisely, we see that a metric space ( X , d ) is Samuel realcompact iff it is Bourbaki-complete and every uniformly discrete subspace of X has non-measurable cardinal. As a consequence, we derive that a normed space is Samuel realcompact iff it has finite dimension. And this means in particular that realcompactness and Samuel realcompactness can be very far apart. The paper also contains results relating this realcompactification with the so-called Lipschitz realcompactification (also studied here), with the classical Hewitt–Nachbin realcompactification and with the completion of the initial metric space.