Asymmetry and duality in topology
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 one on the underlying set (for example, the Alexandroff dual or the de Groot dual). Section 5 discusses the relationship between this bitopology-based duality theory and one based on Nachbin's topological ordered spaces. Finally, Section 6 develops a bitopological compactification theory. This highly interesting and easy-to-read article is recommendable not only to topologists, but also to any mathematician interested in general mathematical structures.
scientific article; zbMATH DE number 823468
scientific article