An MA-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 R 2 ), the present paper follows a Marcus–Wyse ( M -, for short) topological approach because the M -topology was developed for studying digital spaces in Z 2 , where Z 2 is the set of points in R 2 with integer coordinates. Hence the present paper uses several tools associated with M -topology, e.g. an M -localized neighborhood of a point p ∈ Z 2 , 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, L M A -map for brevity) which are substantially helpful to M A -digitize subspaces of R 2 , where “ M A ” means the M -adjacency (see Definition 10 and Theorem 3.9 of the present paper). Besides, the paper proposes an algorithm supporting an M A -digitization of subspaces of R 2 . Furthermore, to investigate a relation between subspaces of R 2 and their corresponding M A -digitized spaces, and to classify subspaces of R 2 associated with the M -topology, the paper uses both the first homotopy group (or the fundamental group) in algebraic topology and an M A -fundamental group.