A solution to the completion problem of quasi-uniform spaces

TL;DR

This paper introduces a new method called $ au$-completion for completing $T_0$ quasi-uniform spaces, extending classical and existing completions using nets, Mac Neille's cut, and directed sets.

Contribution

It presents a novel $ au$-completion procedure for quasi-uniform spaces, unifying and extending classical completions with new properties.

Findings

Every $T_0$ quasi-uniform space has a $ au$-completion.

The $ au$-complement coincides with classical completions in uniform spaces.

It extends Doitcinov's completion for quiet spaces.

Abstract

We give a new completion for the quasi-uniform spaces. We call the whole procedure {\it $\tau$-completion} and the new space {\it $\tau$-complement of the given}. The basic result is that every $T_{_0}$ quasi-uniform space has a $\tau$-completion. The $\tau$-complement has some \textquotedblleft crucial\textquotedblright properties, for instance, it coincides with the classical one in the case of uniform space or it extends the {\it Doitcinov's completion for the quiet spaces}. We use nets and from one point of view the technique of the construction may be considered as a combination of the {\it Mac Neille's cut} and of the completion of partially ordered sets via {\it directed subsets}.