Set-Theoretical Algebraic Approaches to Connectivityin Continuous or Digital Spaces

  • Authors:

  • Christian Ronse

  • Affiliations:

  • LSIIT–UPRES-A 7005, Université Louis Pasteur, Département d‘Informatique, 7 rue René Descartes, F-67000 Strasbourg, France. www: http://dpt-info.u-strasbg.fr/∼ronsel ...

  • Venue:
  • Journal of Mathematical Imaging and Vision
  • Year:
  • 1998

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Abstract

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