Axiomatic Digital Topology

As explained in the lecture "Introduction", topological notions of connectedness and of the boundary of a subset are important for image analysis. The widely used concept of m, nadjacencies defines these notions not satisfactorily: different adjacencies for the foreground and background are only applicable for binary images. A subset has two boundaries: the inner and the outer ones. They do not satisfy the Jordan Curve Theorem. A digital topological space must be locally finite which means that each space element must have a finite neighborhood, while in the classical Hausdorff-topology each neighborhood must be infinite. A space with infinite neighborhoods cannot be explicitly represented in a computer.