An [formula omitted]-digitization of Hausdorff spaces by using a connectedness graph of the Marcus–Wyse topology

Abstract

The study of 2D digital spaces plays an important role in both topology and digital geometry. To propose a certain method of digitizing subspaces of the 2D Euclidean space (or Hausdorff space, denoted by R2), the present paper follows a Marcus–Wyse (M-, for short) topological approach because the M-topology was developed for studying digital spaces in Z2, where Z2 is the set of points in R2 with integer coordinates. Hence the present paper uses several tools associated with M-topology, e.g. an M-localized neighborhood of a point p∈Z2, a topological graph (or a connectedness graph) induced by the M-topology (or M-connectedness graph), a new type of lattice-based connectedness graph homomorphism (or lattice-based M-adjacency map, LMA-map for brevity) which are substantially helpful to MA-digitize subspaces of R2, where “MA” means the M-adjacency (see Definition 10 and Theorem 3.9 of the present paper). Besides, the paper proposes an algorithm supporting an MA-digitization of subspaces of R2. Furthermore, to investigate a relation between subspaces of R2 and their corresponding MA-digitized spaces, and to classify subspaces of R2 associated with the M-topology, the paper uses both the first homotopy group (or the fundamental group) in algebraic topology and an MA-fundamental group.