Computation on metric spaces via domain theory
type: Item
instance of: scholarly article
title: Computation on metric spaces via domain theory
zbMATH Open document ID: 0922.54025
DOI: 10.1016/S0166-8641(97)00152-1
published in: Topology and its Applications
publication date: 1999-07-20T00:00:00Z
review text: It is well known that partially ordered structures such as the set of all closed subintervals of a closed real interval (viewed as a partially ordered set ordered by reverse inclusion) can be useful for the study of continuous and computable functions on the closed interval. In this article the author tries to identify essential mathematical features of such computational examples and to indicate how the general constructions developed in domain theory provide an appropriate computational framework for some topological spaces. He mainly concentrates on recent approaches to realizing (or embedding) a Polish space as the set of maximal points of a continuous domain, that is, he discusses in some detail how metric spaces arise as the set of maximal points of an \(\omega\)-continuous domain and he explains his characterization of such spaces as the class of Polish spaces, where the maximal points are endowed with the relative Scott topology. Such realizations then provide a convenient framework in which to model certain computational algorithms on the space. Moreover, he argues that if in this way a continuous domain \(P\) is appropriately associated with a topological space, then one has at hand via the probabilistic power domain of \(P\) an effective tool for the study of a variety of problems associated with integration and measure theory on the space.
Mathematics Subject Classification ID: 54E35
Mathematics Subject Classification ID: 06B35
Mathematics Subject Classification ID: 28C15
Mathematics Subject Classification ID: 54F05
zbMATH DE Number: 1308000
zbMATH Keywords: Polish space
zbMATH Keywords: continuous domain
zbMATH Keywords: Borel measures
zbMATH Keywords: set of maximal points of a domain
author: Jimmie D. Lawson
reviewed by: Hans Peter Künzi
MaRDI profile type: Publication
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full work available at URL: s0166-8641(97)00152-1
OpenAlex ID: W2051598506
description: scientific article; zbMATH DE number 1308000