Completeness in quasi-pseudometric spaces

TL;DR

This paper explores the relationships between different notions of completeness in quasi-metric spaces, introducing a new definition of right K-Cauchy nets that aligns with sequential completeness, thereby extending prior results.

Contribution

It proposes a novel definition of right K-Cauchy nets in quasi-metric spaces, establishing their equivalence with sequential completeness and extending existing theoretical frameworks.

Findings

New definition of right K-Cauchy nets in quasi-metric spaces

Equivalence between the new net completeness and sequential completeness

Extension of classical results by Stoltenberg and Gregori & Ferrer

Abstract

The aim of this paper is to discus the relations between various notions of sequential completeness and the corresponding notions of completeness by nets or by filters in the setting of quasi-metric spaces. We propose a new definition of right $K$-Cauchy net in a quasi-metric space for which the corresponding completeness is equivalent to the sequential completeness. In this way we complete some results of R.~A. Stoltenberg, Proc. London Math. Soc. \textbf{17} (1967), 226--240, and V.~Gregori and J.~Ferrer, Proc. Lond. Math. Soc., III Ser., \textbf{49} (1984), 36.