Algebraic models for \(T_1\)-spaces
type: Item
instance of: scholarly article
title: Algebraic models for \(T_1\)-spaces
zbMATH Open document ID: 1216.54002
author: Marcel Erné
published in: Topology and its Applications
publication date: 2011-05-16T00:00:00Z
review text: The author contributes to the theory of computational models of spaces. He proves that the \(T_1\)-spaces are precisely the maximal point spaces of conditionally up-complete algebraic posets with the Scott topology. He also establishes an equivalence between the category of \(T_1\)-spaces with a distinguished base and a certain category of camps. These are conditionally up-complete, algebraic and maximized posets in which every compact element is a meet of maximal elements. They provide essentially unique algebraic ordered models for \(T_1\)-base spaces. A \(T_1\)-space has a damp model (that is, a domain model that is a camp) if and only if it has a base not containing any free filter base. Consequently all completely metrizable spaces and, more generally, all complete Aronszajn spaces have damp models. Moreover, damp models also exist for all Stone spaces. The latter representation gives rise to an equivalence and a duality for so-called Stone base spaces, which extends the classical Stone duality. Furthermore, it yields a purely order-theoretical description of clopen bases for Stone spaces and, algebraically, of finitary meet bases of Boolean lattices in terms of maximal ideals.
Mathematics Subject Classification ID: 54B05
Mathematics Subject Classification ID: 06B35
Mathematics Subject Classification ID: 18B30
Mathematics Subject Classification ID: 54F05
Mathematics Subject Classification ID: 54B30
zbMATH DE Number: 5888199
zbMATH Keywords: algebraic poset
zbMATH Keywords: camp
zbMATH Keywords: damp
zbMATH Keywords: filter-complete base
zbMATH Keywords: maximal point space
zbMATH Keywords: minimal base
zbMATH Keywords: model
zbMATH Keywords: Scott topology
zbMATH Keywords: Stone duality
zbMATH Keywords: \(T_1\)-space
zbMATH Keywords: zero-dimensional
reviewed by: Hans Peter Künzi
MaRDI profile type: Publication
full work available at URL: j.topol.2011.01.014
OpenAlex ID: W2082247837
cites work: Q5767704
cites work: Q5335870
cites work: Domain-representable spaces
cites work: Domain representability and the Choquet game in Moore and BCO-spaces
cites work: Q5331549
cites work: Q4083724
cites work: A computational model for metric spaces
cites work: Q3222241
cites work: General Stone duality.
cites work: Q3595086
cites work: Continuous Lattices and Domains
cites work: Moore-closed spaces, completeness and centered bases
cites work: Spaces of maximal points
cites work: Q4424948
cites work: Ideal models of spaces.
cites work: The regular spaces with countably based models.
cites work: Q5586965
cites work: Characterizations of Developable Topological Spaces
cites work: Open continuous mappings of spaces having bases of countable order
DOI: 10.1016/J.TOPOL.2011.01.014
description: scientific article; zbMATH DE number 5888199