Complexity Spaces: Lifting & Directedness

The theory of complexity spaces has been introduced in [Sch95] as part of the development of a topological foundation for Complexity Analysis. The topological study of these spaces has been continued in the context of the theory of upper weightable spaces ([Sch96]), while the specific properties of total boundedness and Smyth completeness have been analyzed in [RS96]. Here we introduce a technique of “lifting”, which allows one to extend an upper weightable space, and hence a complexity space, by a maximum. This leads to a characterization of the upper weightable spaces as the weightable spaces which have a weightable directed extension. We motivate the property of directedness from a complexity theoretic point of view, which leads to the study of the particular class of weightable directed spaces. Weightable directed spaces are shown to be non metrizable and their weighting functions are analyzed. These weighting functions are shown to be upper weightings among which there is a “fading” weighting. Finally we show that the process of lifting of a weighted directed space of fading weight, in particular of a complexity space, does not essentially alter the topology of the original space. AMS Subject Classification: 54E15, 54E35