Topologies for the digital spaces Z2 and Z3

We show that there are only two topologies in Z and five topologies in Z whose connected sets are connected in the intuitive sense. Both topologies for Z are well known (e.g., one is presented in D. Marcus, F. Wyse et al., Amer. Math. Monthly 77, pp. 1119, 1979, and the second in E. Khalimsky et al., Topology and its Applications 36, pp. 1–17, 1990) and found applications in computer graphics and computer vision (e.g. A. Rosenfeld, Amer. Math. Monthly 77, pp. 621–630, 1979, and T. Y. Kong et al., Amer. Math. Monthly 98, pp. 901–917, 1991). Two of the five topologies for Z are products of the topologies known from Z and Z. The remaining three topologies for Z are also generated from the two topologies for Z, however, they are not product topologies.