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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.
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.
Keywords
quasi-pseudometric space; Cauchy sequence; Cauchy net; Cauchy filter; completeness
Subject
Computer Science and Mathematics, Geometry and Topology
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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