Polish spaces, computable approximations, and bitopological spaces
type: Item
instance of: scholarly article
title: Polish spaces, computable approximations, and bitopological spaces
zbMATH Open document ID: 0991.54034
DOI: 10.1016/S0166-8641(01)00072-4
author: Bob Flagg
author: Ralph Kopperman
author: Krzysztof Chris Ciesielski
published in: Topology and its Applications
publication date: 2002-06-24T00:00:00Z
review text: The authors solve an open problem by showing that every Polish space can be the maximal point space of some bounded complete continuous domain with a countable base (bounded complete continuous domains are sometimes called continuous Scott domains). The bounded complete domain is called a computational model or domain environment for the given space. The proof depends on an intricate construction valid in any Polish space of a countable family \({\mathcal C}\) of non-empty closed sets that (i) forms a closed subbase for a compact topology \(\tau^*\), (ii) is a basis of (closed) neighborhoods for the original topology \(\tau\), and (iii) given \(C\in{\mathcal C}\) and \(x\notin C\), there exists \(D\in{\mathcal C}\) such that \(C\subseteq \text{int}(D)\) and \(x\notin D\). The authors show that the bitopological space \((X,\tau, \tau^*)\) is pairwise regular and then they use the existence of such a topology \(\tau^*\) to construct the bounded complete domain environment for the space.
Mathematics Subject Classification ID: 54E50
Mathematics Subject Classification ID: 06B35
Mathematics Subject Classification ID: 54E55
Mathematics Subject Classification ID: 06F30
Mathematics Subject Classification ID: 54C25
Mathematics Subject Classification ID: 54D30
Mathematics Subject Classification ID: 54D80
Mathematics Subject Classification ID: 54H12
zbMATH DE Number: 1758562
zbMATH Keywords: directed complete poset (dcpo)
zbMATH Keywords: bounded complete dcpo
zbMATH Keywords: \(\omega\)-continuous
zbMATH Keywords: maximal point space
zbMATH Keywords: Scott topology
zbMATH Keywords: Polish space
zbMATH Keywords: bitopological space
reviewed by: Jimmie D. Lawson
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1758562