The Tamano-Dowker type theorems for nearness frames
type: Item
instance of: scholarly article
title: The Tamano-Dowker type theorems for nearness frames
zbMATH Open document ID: 0849.18004
DOI: 10.1016/0022-4049(94)00049-O
author: T. A. Dube
published in: Journal of Pure and Applied Algebra
publication date: 1995-06-08T00:00:00Z
review text: A nearness frame \((L, \mu)\) is a frame \(L\) together with a compatible nearness \(\mu\) on \(L\). This means \(L\) is a regular frame. In this paper the author introduces notions of regularity and normality for nearness frames, defines the coproduct of two nearness frames and gives necessary and sufficient conditions in order that a nearness frame \((L, \mu)\) is normal. Let \((o[0,1], \text{Cov} (o[0,1]))\) be the nearness frame with the frame \(o[0,1]\) of all open sets of the unit interval \([0,1]\) and the nearness \(\text{Cov} (o[0,1])\) consisting of all covers of \(o[0,1]\). Then \((L, \mu)\) is normal iff the coproduct of \((L, \mu)\) and \((o[0,1], \text{Cov} (o[0,1]))\) is normal. The author gives other equivalent conditions expressed by the normality of the coproduct of \((L, \mu)\) and \((M, \nu)\) for some nearness frame \((M, \nu)\). These results are the localic version of the Tamano-Dowker type theorems for nearness spaces as obtained by \textit{H. L. Bentley}. The proofs are accomplished in the realm of the frame theory without using points. As an application it is shown that for a Dowker space \(X\) the coproduct of \((oX, \text{Cov} (oX))\) and \((o[0,1], \text{Cov} (o[0,1]))\) is a normal nearness frame, while the coproduct of \(oX\) and \(o[0,1]\) is not a normal frame.
Mathematics Subject Classification ID: 18B30
Mathematics Subject Classification ID: 54E17
Mathematics Subject Classification ID: 54D15
zbMATH DE Number: 762335
zbMATH Keywords: nearness frame
zbMATH Keywords: frame
zbMATH Keywords: regularity
zbMATH Keywords: normality
zbMATH Keywords: coproduct
zbMATH Keywords: nearness spaces
zbMATH Keywords: Dowker space
reviewed by: Ryosuke Nakagawa
MaRDI profile type: Publication
cites work: Samuel compactification and completion of uniform frames
cites work: Paracompact spaces
cites work: Q4113080
cites work: Q3965241
cites work: Q3820652
cites work: Q3679869
cites work: Q3825733
full work available at URL: 0022-4049(94)00049-o
OpenAlex ID: W2027319495
description: scientific article; zbMATH DE number 762335