On a duality between metrics and Σ-proximities

In studies of discrete structures, functions are frequently used that express the proximity of objects but do not belong to the family of metrics. We consider a class of such functions that is characterized by a normalization condition and an inequality that! plays the same role as the triangle inequality does for metrics. We show that the introduced functions, named Sigma-proximities ( "sigma-proximities"), are in a definite sense dual to metrics. there exists a natural one-to-one correspondence between metrics and Sigma-proximities defined on the same finite set; in contrast to metrics, Sigma-proximities measure comparative proximity; the closer the objects, the greater the Sigma-proximity; diagonal entries of the Sigma-prozimity matrix characterize the centrality of objects. The results are extended to the case of arbitrary infinite sets of objects.