Regular relations and strictly completely regular ordered spaces.
type: Item
instance of: scholarly article
title: Regular relations and strictly completely regular ordered spaces.
zbMATH Open document ID: 1035.54017
DOI: 10.1016/S0166-8641(03)00108-1
author: Xiaoquan Xu
author: Yingming Liu
published in: Topology and its Applications
publication date: 2004-02-03T00:00:00Z
review text: Zareckiĭ's theorem about regular relations is slightly generalized and an intrinsic characterization of regularity of relations is obtained. With the help of these results the authors characterize strict complete regularity of partially ordered topological spaces by means of a certain regular relation between the closed and the open upper sets. As an application they conclude that a quasicontinuous domain endowed with the Lawson topology is strictly completely regular provided that the Lawson-open lower sets are contained in the lower topology. Finally they present a new proof of the strict Tychonoff embedding theorem for strictly completely regular ordered spaces which makes use of regular relations. Reviewer's remark: A binary relation \(\rho:X\rightharpoonup Y\) is called regular if \(\exists \sigma: Y\rightharpoonup X\) such that \(\rho\circ \sigma \circ \rho =\rho.\) Zareckiĭ's theorem says that a binary relation \(\rho\) on a set \(X\) is regular if and only if the complete lattice of all images of \(\rho\) under the order of set inclusion is completely distributive.
Mathematics Subject Classification ID: 54F05
Mathematics Subject Classification ID: 54D15
Mathematics Subject Classification ID: 03E20
Mathematics Subject Classification ID: 06D10
zbMATH DE Number: 2037224
zbMATH Keywords: regular relation
zbMATH Keywords: complete distributivity
zbMATH Keywords: strict complete regularity
zbMATH Keywords: strictly topological embedding
reviewed by: Hans Peter Künzi
MaRDI profile type: Publication
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full work available at URL: s0166-8641(03)00108-1
OpenAlex ID: W2010724775
description: scientific article; zbMATH DE number 2037224