presentation given at symposium last oct22, 2006 at University of the Philippines Diliman

3. Z 2 and Z 3

5. 2d-Connected M ⊆ Z d 2d-connected if  x,y ∈ M, ∃ 2d-path from x to y in M * analogous for (3 d -1)-connected

6. Digital Topologies on Z d D1 D2 D3 Topologies on Z d satisfying

9. Open Problem

10. Stable Sets of Z 2 d M ⊆ Z 2 d where none of its elements are 2d-neighbors Also called stable subsets of the hypercube Q d (vertices are pairwise nonadjacent under Q d ) S := collection of stable sets of Z 2 d

12. Admissible Labelings of Z 2 d (3) means no two points in dom  are 2d-neighbors (3) and (4) imply that  (x) ≤ d-2  x ∈ dom 

13. Admissible Labelings of Z 2 d  1 ,  2 isomorphic if ∃ isometry Φ of Z 2 d onto Z 2 d s.t. dom  1 = Φ( dom  2 ) and  1 =  2 ( Φ ( dom  1 ) )   (0,1)-admissible labeling if  (x) ∈ { 0,1 } otherwise, non-(0,1)-admissible labeling

14. Correspondence isomorphism classes of admissible labelings of Z 2 d homeomorphism classes of digital topologies on Z d (Kong: using normalized admissible functions and digital strict partial orders) 1-1 correspondence

17. Results for Z 5 Lower bound for number of digital topologies for d=5

19. References U. Eckhart and L. Latecki . Topologies for the Digital Spaces Z2 and Z3. Computer Vision and Image Understanding 90 (2003) 295-312. R. Klette and A. Rosenfeld . Digital Geometry. Morgan Kauffman Publishers. San Francisco, CA. 2004. T.Y. Kong . 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. Theoretical Computer Science 305 (2003) 221-235. T.Y. Kong . Topological adjacency relations on Zn. Theoretical Computer Science 283 (2002) 3-28. T.Y. Kong, R. Kopperman and P.R. Meyer . A Topological Approach to Digital Topology. Amer. Math. Monthly 98 (1991), 901-917. R. Kopperman . The Khalimsky Line as a Foundation for Digital Topology. Shape in Picture (Proceedings of the NATO Advanced Research Workshop held at Driebergen, Spet. 1992). Springer. 1994 V. Kovalevsk y. Axiomatische Digitaltopologie. http://www.bv.inf.tu-dresden.de/AKTUELL/S_KOLL_0406/VORTRAEGE/V1_Kovalevski.ppt. August 2006. G. Malandain . Digital Topology. http://www-sop.inria.fr/epidaure/personnel/malandain/topology/. August 2000. IMAGES: Open Problems by R.Klette . http://www.citr.auckland.ac.nz/dgt/Problem_Files/._Dagstuhl_2004.pdf Tesseract . http://en.wikipedia.org/wiki/Tesseract, http://en.wikipedia.org/wiki/Image:Changingcube.gif Penteract . http://mathworld.wolfram.com/images/eps-gif/HypercubeGraphs_850.gif