Oriented Surfaces in Digital Spaces

We define a digital space to be a pair consisting of an arbitrary nonempty set V and a symmetric binary relation π on V with respect to which V is connected. Our intent is to investigate the notion of an oriented surface in this general environment. Our terminology reflects this: we refer to elements of V as spels (short for "spatial elements"), to artibrary nonempty subsets of π as surfaces, and we define the notions of the interior and the exterior of a surface. We introduce the notion of a near-Jordan surface, its interior and exterior partition V. We call a symmetric binary relation on V that contains π a spel-adjacency. For spel-adjacencies κ and λ, we call a surface κλ-Jordan if it is near-Jordan, its interior is κ-connected, and its exterior is λ-connected. We prove a number of results which characterize κλ-Jordan surfaces in general digital spaces and in binary pictures (in which there is an assignment of a 1 or a 0 to elements of V).