The connectivity in Alexandroff topological spaces is equivalent to the path connectivity. This fact gets some specific properties to Z 2 , equipped with the Khalimsky topology. This allows a sufficiently precise description of the curves in Z 2 and permit to prove a digital Jordan curve theorem in Z 2 .2000 AMS Classification: 54D05, 54D10, 68U05, 68U10, 68R10.

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Many applications of digital image processing now deal with threedimensional images (the third dimension can be time or a spatial dimension). In this paper we develop a topological model for digital three space which can be useful in this context. In particular, we prove a digital, three-dimensional, analogue of the Jordan curve theorem. (The Jordan curve theorem states that a simple closed curve separates the real plane into two connected components.) Our theorem here is a digital topological formulation of the Jordan-Brouwer theorem about surfaces that separate three-dimensional space into two connected components.