Topological properties of closed digital spaces: One method of constructing digital models of closed continuous surfaces by using covers

This paper studies properties of closed digital n-dimensional spaces, which are digital models of continuous n-dimensional closed surfaces. We show that the minimal number of points in a closed digital n-dimensional space is 2n + 2 points. A closed digital n-dimensional space with 2n + 2 points is the minimal n-dimensional sphere, which is the join of n + 1 copies of the 0-dimensional sphere. We prove that a closed digital n-dimensional space cannot contain a closed digital n-dimensional subspace, which is different from the space itself. We introduce the general definition of a closed digital space and prove that a closed digital space is necessarily a closed digital n-dimensional space. Finally, we present conditions which guarantee that every digitization process preserves important topological and geometric properties of continuous closed 2-surfaces. These conditions also allow us to determine the correct digitization resolution for a given closed 2-surface.