Many applications of digital image processing now deal with three-and higher-dimensional images.One way to represent n-dimensional digital images is to use the specialization graphs of subspaces of the Alexandroff topological space Z n (where Z denotes the integers with the Khalimsky line topology). In this paper the dimension of any such graph is defined in three ways, and the equivalence of the three definitions is established. Two of the definitions have a geometric basis and are closely related to the topological definition of inductive dimension; the third extends the Alexandroff dimension to graphs. Diagrams are given of graphs that are dimensionally correct discrete models of Euclidean spaces, n-dimensional spheres, a projective plane and a torus. New characterizations of n-dimensional (digital) surfaces are presented. Finally, the local structure of the space Z n is analyzed, and it is shown that Z n is an n-dimensional surface for all n > 1.

In this paper we develop some combinatorial models for continuous spaces. In this spirit we study the approximations of continuous spaces by graphs, molecular spaces and coordinate matrices. We define the dimension on a discrete space by means of axioms, and the axioms are based on an obvious geometrical background. This work presents some discrete models of n-dimensional Euclidean spaces, n-dimensional spheres, a torus and a projective plane. It explains how to construct new discrete spaces and describes in this connection several three-dimensional closed surfaces with some topological singularities It also analyzes the topology of (3+1)-spacetime. We are also discussing the question by R. Sorkin [19] about how to derive the system of simplicial complexes from a system of open covering of a topological space S. Introduction.A number of workers have been unhappy about applications of the continuum picture of space and spacetime. They have believed that the breakdown of the functional integral at the Plank length shows not merely the failure of the classical field equations but also indicates that a differential manifold upon which they are built should be replaced by some finite theory. This was certainly one of the motivations behind Penrose invention [15] of spin networks and recent works by Finkelstain on a novel spacetime microstructure [1]. Isham, Kubyshin and Renteln [2] introduce a quantum theory on the set

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