Pairings of binary reflexive relational structures

Abstract

The main purpose of this thesis is to study the interplay between relational structures and topology, and to portray pairings in terms of some finite poset models and order preserving maps. We show the interrelations between the categories of topological spaces, closure spaces and relational structures. We study the 4-point non-Hausdorff model S4 weakly homotopy equivalent to the circle s'. We study pairings of some objects in the category of relational structures, similar to the multiplication of Hopf spaces in topology. The multiplication S4 x S4 ---7 S4 fails to be order preserving for posets. Nevertheless, applying a single barycentric subdivision on S4 to get Ss, an 8-point model of the circle enables us to define an order preserving poset map Ss x Ss ---7 S4' Restricted to the axes, this map yields weak homotopy equivalences Ss ---7 S4' Hence it is a pairing. Further, using the non-Hausdorff join Ss ® Ss, we obtain a version of the Hopf map Ss ® Ss ---7 §S4. This model of the Hopf map is in fact a map of non-Hausdorff double mapping cylinders.