Lawson topology of the space of formal balls and the hyperbolic topology
type: Item
instance of: scholarly article
title: Lawson topology of the space of formal balls and the hyperbolic topology
zbMATH Open document ID: 1151.54024
author: Hideki Tsuiki
author: Yasunao Hattori
published in: Theoretical Computer Science
publication date: 2008-10-21T00:00:00Z
review text: Let \(\mathbb R_+\) denote the set of non-negative real numbers. From a metric space \((X,d)\) one can construct a partially ordered set \((B^+X,\sqsubseteq)\) where \(B^+X=X\times \mathbb R_+,\) and \((x,r)\sqsubseteq (y,s)\) if \(d(x,y)\leq r-s.\) An element of \(B^+X\) is called a formal ball in \((X,d)\). In this article the authors also consider formal balls with negative radii and study the partially ordered set \(BX=X\times\mathbb R\) of such generalized formal balls with the same order relation. In particular they investigate when the Lawson topology (defined through the order relation \(\sqsubseteq\)) and the product topology (of the metric topology and the Euclidian topology) coincide on \(B^+X\), or on \(BX.\) For instance they show that the two topologies coincide on \(BX\) if \((X,d)\) is a totally bounded metric space. They also introduce and investigate the hyperbolic topology of a metric space \((X,d)\), which is generated by the sets \(\{y:d(a,y)-d(b,y)<s\}\) for \(a,b\in X\) and \(-d(a,b)<s\), and which in general is different from the metric topology. In particular they prove that the hyperbolic topology and the metric topology coincide on \(X\) if and only if the Lawson topology and the product topology coincide on \(BX.\)
reviewed by: Hans Peter Künzi
Mathematics Subject Classification ID: 54E35
zbMATH DE Number: 5354953
zbMATH Keywords: metric space
zbMATH Keywords: formal ball
zbMATH Keywords: Lawson topology
zbMATH Keywords: Scott topology
zbMATH Keywords: normed linear space
zbMATH Keywords: totally bounded metric
zbMATH Keywords: hyperbolic topology
MaRDI profile type: Publication
Wikidata QID: Q56830052
full work available at URL: j.tcs.2008.06.034
OpenAlex ID: W2165060927
cites work: A computational model for metric spaces
cites work: Domain representability of metric spaces
cites work: Q3213662
cites work: Q3994935
cites work: Q4855460
cites work: Continuous Lattices and Domains
cites work: Approximation of metric spaces by partial metric spaces
cites work: Asymmetry and duality in topology
cites work: Spaces of maximal points
cites work: An Introduction to Banach Space Theory
cites work: Partial Metric Topology
cites work: Q3368439
cites work: Embedding metric spaces into CPO's
DOI: 10.1016/J.TCS.2008.06.034
Recommended article: The topology of balls and Gromov hyperbolicity of Riemann surfaces
Recommended article: The topology of balls in Riemannian surfaces and Gromov hyperbolicity
Recommended article: Q3399023
Recommended article: The Lawson homology for Fulton-MacPherson configuration spaces
Recommended article: Q4990314
Recommended article: On the computational content of the Lawson topology
Recommended article: Q4675442
Recommended article: Homotopical and topological rigidity of hypersurfaces of spherical space forms
Recommended article: Topology theorems formulated from hyperbolic geometry considerations
Recommended article: Q3432064
description: scientific article; zbMATH DE number 5354953