Topological representation of precontact algebras and a connected version of the Stone duality theorem. I. (Q2363289)

This paper deals with a common approach both to the discrete and to the non-discrete region-based theory of space. It presents the proofs of the results announced in the first seven sections of [\textit{G. Dimov} and \textit{D. Vakarelov}, Lect. Notes Comput. Sci. 3929, 1--16 (2006; Zbl 1185.68665)] as well as many new additional results and applications. From the authors' abstract: ``The notions of 2-precontact and 2-contact spaces are introduced. Using them, new representation theorems for precontact and contact algebras are proved. They incorporate and strengthen both the discrete and topological representation theorems from several papers. It is shown that there are bijective correspondences between such kinds of algebras and such kinds of spaces. As applications of the obtained results, we get new connected versions of the Stone Duality Theorems for Boolean algebras and for complete Boolean algebras, as well as a Smirnov-type theorem for a kind of compact \(T_0\)-extensions of compact Hausdorff extremally disconnected spaces. We also introduce the notion of a Stone adjacency space and using it, we prove another representation theorem for precontact algebras. We even obtain a bijective correspondence between the class of all, up to isomorphism, precontact algebras and the class of all, up to isomorphism, Stone adjacency spaces.''