Set-Theoretical Algebraic Approaches to Connectivityin Continuous or Digital Spaces

Abstract

Connectivity has been defined in the framework of topological spaces, but also in graphs; the two types of definitions do not always coincide. Serra gave a set of formal axioms for connectivity, which consists in a list of properties of the family of all connected subsets of a space; this definition includes as particular case connected sets in a topological space or in a graph. He gave an equivalent characterization of connectivity in terms of the properties of the operator associating to a subset and a point of that space, the connected component of that subset containing that point. In this paper we give another family of axioms, equivalent to those of Serra, where connectivity is characterized in terms of separating pairs of sets. In the case of graphs, where connected sets are generated by pairs of end-vertices of edges, this new set of axioms is equivalent to the separation axioms given by Haralick.