Topologies for the digital spaces Z 2 and Z 3

We show that there are only two topologies in Z2 and five topologies in Z3 whose connected sets are connected in the intuitive sense. Both topologies for Z2 are well known (e.g., one is presented in D. Marcus, F. Wyse et al., Amer. Math. Monthly 77 (1979) 1119, and the second in E. Khalimsky et al., Topology and its Applications 36 (1990) 1) and found applications in computer graphics and computer vision (e.g., A. Rosenfeld, Amer. Math. Monthly 77 (1979) pp. 621, and T.Y. Kong et al., Amer. Math. Monthly 98 (1991) 901). Two of the five topologies for Z3 are products of the topologies known from Z1 and Z2. The remaining three topologies for Z3 are also generated from the two topologies for Z2 however, they are not product topologies.