Topological adjacency relations on Zn

AbstractFor which adjacency relations (i.e., irreflexive symmetric binary relations) α on Zn does there exist a topology τ on Zn such that the τ-connected sets are exactly the α-path-connected subsets of Zn? If such a topology exists then we say that the relation α is topological.Let l1 and l∞, respectively, denote the 4- and the 8-adjacency relations on Z2 and the analogs of these two relations on Zn (for any positive integer n). Consider adjacency relations α on Zn such that 1.For x,y∈Zn,xl1y⇒xαy⇒xl∞y.2.For all x∈Zn, the set {x}∪{y|xαy} is l1-path-connected. Among the uncountably many adjacency relations α satisfying conditions 1 and 2 above, Eckhardt and Latecki showed that there are (up to isomorphism) just two topological relations on Z2, and essentially showed that there are just four topological relations on Z3.We show in this paper that for any positive integer n there are only finitely many topological adjacency relations on Zn that satisfy conditions 1 and 2, and we relate the problem of finding these relations to the problem of finding all sets of vertices of an n-cube such that no two vertices in the set are the endpoints of an edge of the n-cube. From our main theorems we deduce the above-mentioned results of Latecki and Eckhardt, and also deduce that there are (again, up to isomorphism) exactly 16 topological adjacency relations on Z4 that satisfy conditions 1 and 2