An Introduction to the Theory of Quasi-uniform Spaces

If the symmetry condition in the definition of a pseudometric is deleted, the notion of a quasi-pseudometric is obtained. Asymmetric distance functions already occur in the work of Hausdorff in the beginning of the twentieth century when in his book on set-theory [27] he discusses what is now called the Hausdorff metric of a metric space. A family of pseudo-metrics on a set generates a uniformity. Similarly, a family of quasi-pseudometrics on a set generates a quasi-uniformity [60]. In 1937 Weil published his booklet on (entourage) uniformities, which is now usually considered as the beginning of the modern theory of uniformities. Three years later Tukey suggested an approach to uniformities via uniform coverings. The study of quasi-uniformities started in 1948 with Nachbin’s investigations (recorded in [57]) on uniform preordered spaces, that is, those topological preordered spaces whose preorder is given by the intersection of the entourages of a (filter) quasiuniformity U and whose topology is induced by the associated sup-uniformity U ∨ U−1. He proved that the topological ordered spaces of this kind can be characterized by the property that they admit T2-ordered compactifications. The filter U−1 of inverse relations of a quasi-uniformity U is also a quasi-uniformity. Similarly, each quasi-pseudometric has an obvious conjugate by interchanging the order of points. Hence quasi-uniformities and quasi-metrics naturally yield bitopological spaces (in the sense of Kelly [35]), that is, sets endowed with two topologies. The term quasi-proximity first appeared in the articles of Pervin and Steiner [59, 66]. The connection between totally bounded quasi-uniformities and quasi-proximities generalizes the well-known correspondence between totally bounded uniformities and proximities. The work of Fox, Junnila and Kofner showed that in the class of T1-spaces the concepts of a γ-space (= a topological space admitting a local quasi-uniformity with a countable base), a quasi-pseudometrizable space and a non-archimedeanly quasi-pseudometrizable space are all distinct [22, 36] and that the fine quasi-uniformity of metrizable and suborderable (= generalized ordered) spaces has a base consisting of transitive entourages [33, 38]. Császár [4] developed the theory of the bicompletion for quasi-uniform spaces that was later popularized by Fletcher, Lindgren and Salbany [17, 63]. It satisfactorily generalizes the theory of the completion from the metric and uniform setting to the asymmetric context. Since its underlying idea is that of a complete uniformity however, many further attempts have been made to find other theories of asymmetric completions (see for instance the work of Deák and Doitchinov [6, 11]). Brümmer (see e.g. [3]) was first to consider explicitly the class of all functorial quasi-uniformities on topological spaces, although some basic work on canonical quasi-uniformities was done at about the same time by Fletcher and Lindgren. In the last decade the interest in the study of quasi-uniform function spaces and hyperspaces increased considerably. Partially this fact is explained by their applications in theoretical computer science (see the work of Smyth and Sünderhauf [65, 67]). In the spirit of Nachbin several authors tried to develop with the help of quasi-pseudometrics a common generalization of the two well-established theories of metric spaces and partially ordered sets that would unify common classical results like fixed point theorems and completions. In connection with such investigations also the idea to replace the reals in the definition of a quasi-pseudometric by some more general structure became popular (e.g. in the studies on Kopperman’s continuity spaces, e.g. [39]). In these theories a quasi-uniformity is understood as a kind of generalized quasi-pseudometric. Quasi-uniform structures were also investigated in various kinds of topological algebraic structures (see for instance the recent work of Romaguera and his collaborators [55, 25]). In particular the study of paratopological groups with the help of quasi-uniformities is well known. Furthermore the study of asymmetric norms naturally leads to a theory of asymmetric functional analysis. Various researchers attempted more or less successfully to extend important classical results about quasi-uniformities to fuzzy mathematics. Based on Lowen’s theory of approach spaces [53], the concept of an approach quasi-uniformity was introduced and investigated. In recent years more and more results about quasi-uniformities were also generalized to a pointfree setting (for instance in the work of Picado and his colleagues [15]).