On Stoltenberg's quasi-uniform completion

TL;DR

This paper introduces a new, well-behaved completion method for quasi-uniform spaces that generalizes existing theories and aligns with classical uniform space completions, emphasizing the concept of cuts of nets.

Contribution

It presents a novel completion theory for quasi-uniform spaces that extends and unifies previous approaches, incorporating the idea of cuts of nets.

Findings

The new completion coincides with classical uniform space completion.

The theory generalizes Doitchinov's and Stoltenberg's approaches.

It introduces the concept of cuts of nets for quasi-uniform spaces.

Abstract

In this paper, we give a new completion for quasi-uniform spaces which generalizes the completion theories of Doitchinov [8] and Stoltenberg [20]. The presented completion theory is very well-behaved and extends the completion theory of uniform spaces in a natural way. That is, the definition of Cauchy net and the constructed completion coincide with the classical in the case of uniform spaces. The main contribution this completion theory makes is the notion of the cut of nets which generalize the idea of Doitchinov for the notion of-Cauchy net [1]