Asymmetry and duality in topology
type: Item
instance of: scholarly article
title: Asymmetry and duality in topology
zbMATH Open document ID: 0858.54001
DOI: 10.1016/0166-8641(95)00116-X
author: Ralph Kopperman
published in: Topology and its Applications
publication date: 1996-01-16T00:00:00Z
review text: Many mathematical structures come in symmetric and asymmetric versions. Classical examples include commutative and noncommutative algebraic structures, as well as symmetric preorders (equivalence relations) and asymmetric such (partial orders). In these cases there is always a duality available, whose use simplifies their study, and which reduces to the identity in the symmetric case. Also, in each of these cases, there are many asymmetric examples. This survey article is devoted to the study of asymmetry and duality with respect to topological spaces. First of all, a topological space \((X,\tau)\) is weakly symmetric if \(x\in\text{cl} \{y\}\) implies \(y\in\text{cl} \{x\}\). Asymmetric topology is the study of spaces which are not weakly symmetric. Recent applications to computer science involve asymmetric spaces (Scott topology of denotational semantics, Khalimsky line, digital \(k\)-space). The notion of duality becomes apparent if we consider the unit interval with the upper topology. This space is \(T_0\) but fails to be \(T_1\). Nonetheless, this space has a straightforward dual, the lower topology, and the usual topology is the join of the upper and the lower topology. Thus both the usual topology and the usual order are encoded into the upper topology provided one can find a way to construct its dual from it. One purpose of this paper is to discuss this construction. Constructions of dual spaces are useful in the context of quasi-uniformities, quasi-proximities, Topological ordered spaces (a Tikhonov space equipped with a partial order closed in its square) and bitopological spaces. Sections 2 and 3 of this paper investigate the relationship between duality and bitopology. In Section 3, compactness is studied for bitopological spaces. In that case, compactness breaks into two different notions. Section 4 applies the bitopological theory to topological spaces by studying ideas of topological duality, that is, ways to associate with a given topology a second …
Mathematics Subject Classification ID: 54-02
Mathematics Subject Classification ID: 54A05
Mathematics Subject Classification ID: 54D15
Mathematics Subject Classification ID: 54D30
Mathematics Subject Classification ID: 54D35
Mathematics Subject Classification ID: 54E05
Mathematics Subject Classification ID: 54E55
Mathematics Subject Classification ID: 06B35
Mathematics Subject Classification ID: 54B30
Mathematics Subject Classification ID: 54E15
Mathematics Subject Classification ID: 54F05
Mathematics Subject Classification ID: 54D10
zbMATH DE Number: 823468
zbMATH Keywords: skew compactification
zbMATH Keywords: bitopological spaces
zbMATH Keywords: Alexandroff dual
zbMATH Keywords: de Groot dual
zbMATH Keywords: Nachbin's topological ordered spaces
reviewed by: Maximilian Ganster
MaRDI profile type: Publication
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full work available at URL: 0166-8641(95)00116-x
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