Relative Spaces as a global concept for Topology

Abstract

In the past several mathematical concepts were introduced and studied for describing structures of a topological kind. Among them we mention here the strong topological universe of preuniform convergence spaces in the sense of Preuss, which enables to simultaneously express "topological" and "uniform" aspects. The introduced concept of set-convergences by Wyler considers the convergence of filters to bounded subsets, and therefore it generalizes the well-known classical point-convergences as well as the supertopologies in the sense of Do\^{i}tchin\'{o}v. Nearness, defined by Herrlich contains in particular contiguities and proximities by studying the internal relationship of sets which are being near in some special sense. At last we still mention the concept of so-called b-topological spaces in which hull-operators are defined on bounded subsets of a carrier set generalizing topological closures in a natural way. Now, the question raises whether there exists a suitable concept for a common study of all former cited structures? At this point we introduce the basics of so-called relative spaces, shortly RELspaces with its corresponding relative maps, shortly RELmaps between them. Hence it is possible to embed the mentioned categories into REL, the category of RELspaces and RELmaps, respectively. Now, this fact of unification being established we turn us towards the study of the relationship between suitable RELspaces and the corresponding strict symmetric topological extensions. Thus, our new presented connection generalizes those one which were studied by Bentley, Ivanova and Ivanov, Lodato and Smirnov, respectively.