Topological adjacency relations on Z(n)
For which adjacency relations (i.e., irreflexive symmetric binary relations) alpha on Z(n) does there exist a topology tau on Z(n) such that the tau-connected sets are exactly the alpha-path-connected subsets of Z(n)? If such a topology exists then we say that the relation alpha is topological. Let l(1) and l(infinity), respectively, denote the 4- and the 8-adjacency relations on Z(2) and the analogs of these two relations on Z(n) (for any positive integer n). Consider adjacency relations alpha on Z(n) such that 1. For x, y is an element of Z(n), x l(1) y double right arrow x alpha y double right arrow x l(infinity) y 2. For all x is an element of Z(n), the set (x) boolean OR (y\ x alpha y) is l(1)-path-connected. Among the uncountably many adjacency relations alpha satisfying conditions 1 and 2 above, Eckhardt and Latecki showed that there are (up to isomorphism) just two topological relations on Z(2), and essentially showed that there are just four topological relations on Z(3). We show in this paper that for any positive integer n there are only finitely many topological adjacency relations on Z(n) that satisfy conditions I 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 Z(4) that satisfy conditions I and 2. (C) 2002 Elsevier Science B.V. All rights reserved.
adjacency digital topology multidimensional 4-dimensional