Topologies induced by the representation of a betweenness relation as a family of order relations

Betweenness relations are the mathematical formalization of the geometrical notion of an element being in between other two elements. In this paper, we exploit a well-known result representing a betweenness relation as a family of order relations and analyse the corresponding family of induced (Alexandrov) topologies. In particular, the intersection of this family of topologies is proved to be the anti-discrete topology and a necessary and sufficient condition for the supremum of this family of topologies to be the discrete topology is provided. Interestingly, this condition is proved to hold when dealing with a finite set. We end with a discussion on the relation between the topology induced by an order relation or a metric and the family of topologies induced by the betweenness relation induced by the same order relation or metric.