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
✍ Scribed by T.Yung Kong
- Publisher
- Elsevier Science
- Year
- 2003
- Tongue
- English
- Weight
- 270 KB
- Volume
- 305
- Category
- Article
- ISSN
- 0304-3975
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✦ Synopsis
We give a proof of the result stated in the title. Here the concepts of 2n-and (3 n -1)-(dis)connected sets are the natural generalizations to Z n 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 Z n . We are interested in topological spaces (Z n ; ) 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 (Z n ; ) 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 ÿnite open set and a ÿnite 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.