A metric and an ordering on sets

Beginning with sets of arbitrary elements, concepts of distance and betweenness of sets are defined. Since betweenness as defined is not transitive, an investigation is made of the conditions which ensure desirable regularity. It is found that a straight line or linear array of sets is a generalization of nested sets (Guttman scales). Close relationships among the notions of distance, betweenness, and linear arrays are demonstrated. Parallel and perpendicular arrays, dimensions, and multidimensional spaces are characterized.