Computation on metric spaces via domain theory
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.
scientific article; zbMATH DE number 1308000
scientific article