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Two approaches to providing an abstract theoretical foundation to the geometry of digital spaces are presented and illustrated. They are critically compared and the possibility of combining them into a single theory is discussed. Such a theory allows us to state and prove results regarding geometrical concepts as they occur in a digital environment independently of the specifics of that environment. In particular, versions of the Jordan Curve Theorem are discussed in this general digital setting.

An Abstract Theoretical Foundation of the Geometry of Digital Spaces