No abstract

It is well known that a uniformity on a set X may be described in any one of three equivalent ways: as a collection of relations on X, as a collection of pseudometrics on X, or as a collection of covers of X, where each of these collections satisfies a respective set of axioms. A collection of relations on a set X that satisfies all the axioms of a uniformity, in the sense of relations, except possibly the axiom of symmetry, is called an entourage quasi-uniformity on X. Several authors have recently studied quasiuniform spaces (see [4,6,7]). In particular, Pervin [6] proved that every topological space is quasi-uniformizable, and Stoltenberg [7] proved that quasi-uniform spaces, like uniform spaces, have completions. Moreover, Lane [4] investigated the interrelations between bitopological spaces and quasi-uniform spaces, and observed that quasi-uniformities lend themselves naturally to the study of bitopological spaces [4; Theorem 4.2].Thus, owing to current interest in quasi-uniform spaces, it would seem desirable to have characterization of quasi-uniformities that are analogous to those of uniformities which are given in terms of pseudometrics and uniform covers. The purpose of this paper is to present such characterizations.We refer the reader to Lane [4] for the basic notations and terminology used in this article. By v and A we mean the supremum and infimum, respectively.

No abstract

No abstract

No abstract