The Khalimsky topologies are precisely those simply connected topologies on Zn whose connected sets include all 2n-connected sets but no (3n - 1)-disconnected sets

• Authors: T. Yung Kong
• Venue: Theoretical Computer Science - Topology in computer science
• Year: 2003

We give a proof of the result stated in the title. Here the concepts of 2n- and (3n- 1)- (dis)connected sets are the natural generalizations to Zn of the standard concepts of 4- and 8-(dis)connected sets in 2D digital topology.Suppose we have an n-dimensional scanner that digitizes n-dimensional objects to subsets of Zn. We are interested in topological spaces (Zn, τ) that might allow standard concepts and methods of general topology to be directly and usefully applied to good digitizations produced by the scanner. But our result suggests that if a topological space (Zn, τ) is not a Khalimsky space, then it will not satisfy our requirement.Our proof involves some purely discrete arguments and a fact about simply connected polyhedra that is a well-known consequence of the Simplicial Approximation Theorem, but also uses the following fact (which was one of the main results in an earlier paper (in: R.M. Shortt (Ed.), General Topology and Applications: Proc. 1988 Northeast Conf., Marcel Dekker, New York, 1990, pp. 153-164) by the author and Khalimsky): For any T0 topological space in which each point lies in a finite open set and a finite closed set there exists a polyhedron, whose vertices are in 1-1 correspondence with the points of the space, such that the homotopy classes of continuous maps into the topological space from any metric space are in 1-1 correspondence with the homotopy classes of continuous maps from that metric space into the polyhedron.