Working Paper Article Version 1 This version is not peer-reviewed

Completeness in Quasi-Pseudometric Spaces

Version 1 : Received: 14 July 2020 / Approved: 15 July 2020 / Online: 15 July 2020 (08:49:38 CEST)

A peer-reviewed article of this Preprint also exists.

Cobzas, Ş. Completeness in Quasi-Pseudometric Spaces—A Survey. Mathematics 2020, 8, 1279. Cobzas, Ş. Completeness in Quasi-Pseudometric Spaces—A Survey. Mathematics 2020, 8, 1279.

Journal reference: Mathematics 2020, 8, 1279
DOI: 10.3390/math8081279

Abstract

The aim of this paper is to discuss 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. A discussion on nets defined over ordered or pre-ordered directed sets is also included.

Subject Areas

quasi-pseudometric space; Cauchy sequence; Cauchy net; Cauchy filter; completeness

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