On strict extensions of nearness spaces
type: Item
instance of: scholarly article
title: On strict extensions of nearness spaces
zbMATH Open document ID: 0974.54016
DOI: 10.1023/A:1008614929530
author: C. A. Hendrie
author: Pieter Collins
published in: Applied Categorical Structures
publication date: 2001-12-16T00:00:00Z
review text: Let \((X,\mu)\) be a nearness space with underlying \(T_1\)-topology \(\tau_\mu\), and let \(X^*\) be any collection of round Cauchy filters on \(X\) in the sense of \textit{D. Harris} [Structures in topology, Mem. Am. Math. Soc. 115 (1971; Zbl 0239.54020)] containing all neighbourhood filters of \(\tau_\mu\). It is shown that there is a natural nearness structure \(\mu^*\) on \(X^*\) which has the following properties: (i) the underlying topology of \((X^*,\mu^*)\) is \(T_1\); (ii) \((X,\mu)\) can be embedded into \((X^*,\mu^*)\) by identifying points in \(X\) with its neighbourhood filters; (iii) \((X^*,\mu^*)\) is a strict extension of \((X,\mu)\). It turns out that if \(X^*\) is the collection of all round Cauchy filters on \(X\), then \((X^*,\mu^*)\) is isomorphic to the strict completion of \((X,\mu)\) introduced by \textit{H. Herrlich} [General Topol. Appl. 4, 191-212 (1974; Zbl 0288.54004)]. Moreover, if \(X^*\) is the collection of all Morita-generated filters on \((X,\mu)\), then \((X^*,\mu^*)\) is isomorphic to the simple extension defined by \textit{K. Morita} in [Proc. Japan Acad. 27, 65-72 (1951; Zbl 0042.41203); No. 3, 130-137; No. 4, 166-171 (1951; Zbl 0043.16504); 632-636 (1951; Zbl 0045.11702)]. Using this approach, the completeness concepts for nearness spaces introduced by K. Morita, H. Herrlich, and J. W. Carlson are compared. They are shown to be equivalent in regular nearness spaces.
Mathematics Subject Classification ID: 54E17
Mathematics Subject Classification ID: 54D35
Mathematics Subject Classification ID: 54E15
zbMATH DE Number: 1563359
zbMATH Keywords: round Cauchy filter
zbMATH Keywords: strict extension
zbMATH Keywords: completeness
reviewed by: Harald Brandenburg
MaRDI profile type: Publication
full work available at URL: a:1008614929530
OpenAlex ID: W31173289
description: scientific article; zbMATH DE number 1563359