Some betweenness relation topologies induced by simplicial complexes

Abstract

This article aims to create an approximation space from any simplicial complex by representing a finite simplicial complex as a union of its components. These components are arranged into levels beginning with the highest-dimensional simplices. The universal set of the approximation space is comprised of a collection of all vertices, edges, faces, and tetrahedrons, and so on. Moreover, new types of upper and lower approximations in terms of a betweenness relation will be defined. A betweenness relation means that an element lies between two elements: an upper bound and a lower bound. In this work, based on Zhang et al.'s concept, a betweenness relation on any simplicial complex, which produces a set of order relations, is established and some of its topologies are studied.