Quasi-uniform completions of partially ordered spaces

Abstract

In 1938 Weil introduced and studied the concept of a uniform space. Although the theory of uniform spaces is a theory in its own right and is independent to the theory of topological spaces, the theory of uniform spaces turned out to be an important tool in the investigation of topological spaces. To a uniform space and a uniformly continuous map one can assign, in a standard way, a topological space and a continuous map. Also, in a uniform space one has the notion of distance between pairs of points. Consequently, Cauchy filters, a generalization of Cauchy (set) sequences, and in turn complete uniform spaces can be defined. Weil showed that every Tychonoff space can be induced by a uniformity and that a space with a topology induced by a uniformity is Tychonoff. Note that a direct method of constructing a completion of a given uniform space is known.