Connectivity Spaces

Abstract

Connectedness is a fundamental property of objects and systems. It is usually viewed as inherently topological, and hence treated as derived property of sets in (generalized) topological spaces. There have been several independent attempts, however, to axiomatize connectedness either directly or in the context of axiom systems describing separation. In this review-like contribution we attempt to link these theories together. We find that despite differences in formalism and language they are largely equivalent. Taken together the available literature provides a coherent mathematical framework that is not only interesting in its own right but may also be of use in several areas of computer science from image analysis to combinatorial optimization

Oriented components and their separations

“…This result directly generalizes the 1-1 correspondence between connectivity spaces and symmetric separations satisfying the same axioms [26]. We will return to this point in section 4.5.…”

“…The cost function f : X → R together with the topological structure on X implied by the search spaces give rise to "fitness landscapes" [30], in which concepts such as local minima, saddle points, and basins of attraction are well defined. These notions are inherently topological in nature and require only the definition of connected sets on X [9,12,26]. Search operators are not necessarily symmetric, however.…”

“…Property (SR0), on the other hand, appeared in [26] as a condition for the existence of a bijection between connected components and symmetric separations.…”

“…Ronse characterized the separations that are equivalent to abstract connectivities [20]. These results were generalized further in [26]. Since connectedness is an intrinsically symmetric notion, the theory requires that separations satisfy a symmetry axiom.…”

“…On the other hand, reaches are rather natural generalizations of connectivity structures, and the latter have been shown to be equivalent to certain generalized proximities in [20,26]. It is the purpose of this contribution to show that there is also a 1-1 correspondence of reachability structures general enough to encompass chemical reaction networks, and thus directed hypergraphs, and a suitable class of generalized separations or, equivalently, proximities.…”