Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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CONTINUITIES AND HOMEOMORPHISMS IN COMPUTER TOPOLOGY AND THEIR APPLICATIONS

  • Han, Sang-Eon (College of Environmental Science and Engineering Honam University)
  • Published : 2008.07.31
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Abstract

In this paper several continuities and homeomorphisms in computer topology are studied and their applications are investigated in relation to the classification of subs paces of Khalimsky n-dimensional space $({\mathbb{Z}}^n,\;T^n)$. Precisely, the notions of K-$(k_0,\;k_1)$-,$(k_0,\;k_1)$-,KD-$(k_0,\;k_1)$-continuities, and Khalimsky continuity as well as those of K-$(k_0,\;k_1)$-, $(k_0,\;k_1)$-, KD-$(k_0,\;k_1)$-homeomorphisms, and Khalimsky homeomorphism are studied and further, their applications are investigated.

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References

  1. P. Alexandorff, Diskrete Raume, Mat. Sb. 2 (1937), 501-518 https://doi.org/10.1070/SM1967v002n04ABEH002351
  2. L. Boxer, A classical construction for the digital fundamental group, J. Math. Imaging Vision 10 (1999), no. 1, 51-62 https://doi.org/10.1023/A:1008370600456
  3. J. Dontchev and H. Maki, Groups of $\theta$-generalized homeomorphisms and the digital line, Topology Appl. 95 (1999), no. 2, 113-128 https://doi.org/10.1016/S0166-8641(98)00004-2
  4. G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, and D. S. Scott, A Compendium of Continuous Lattices, Springer-Verlag, Berlin-New York, 1980
  5. S. E. Han, Computer topology and its applications, Honam Math. J. 25 (2003), no. 1, 153-162
  6. S. E. Han, Comparison between digital continuity and computer continuity, Honam Math. J. 26 (2004), no. 3, 331-339
  7. S. E. Han, Digital coverings and their applications, Jour. of Applied Mathematics and Computing 18 (2005), no. 1-2, 487-495
  8. S. E. Han, Non-product property of the digital fundamental group, Inform. Sci. 171 (2005), no. 1-3, 73-91 https://doi.org/10.1016/j.ins.2004.03.018
  9. S. E. Han, Connected sum of digital closed surfaces, Inform. Sci. 176 (2006), no. 3, 332-348 https://doi.org/10.1016/j.ins.2004.11.003
  10. S. E. Han, Minimal simple closed 18-surfaces and a topological preservation of 3D surfaces, Inform. Sci. 176 (2006), no. 2, 120-134 https://doi.org/10.1016/j.ins.2005.01.002
  11. S. E. Han, Various continuities of a map f : (X, k, $T^{n}_{X}$) $\rightarrow$ (Y, 2, $T_{Y}$) in computer topology, Honam Math. J. 28 (2006), no. 4, 591-603
  12. S. E. Han, Digital fundamental group and Euler charact eristic of a connected sum of digital closed surfaces, Inform. Sci. 177 (2007), no. 16, 3314-3326 https://doi.org/10.1016/j.ins.2006.12.013
  13. S. E. Han, Strong k-deformation retract and its applications, J. Korean Math. Soc. 44 (2007), no. 6, 1479-1503 https://doi.org/10.4134/JKMS.2007.44.6.1479
  14. S. E. Han, The k-fundamental group of a closed k-surface, Inform. Sci. 177 (2007), no. 18, 3731-3748 https://doi.org/10.1016/j.ins.2007.02.031
  15. S. E. Han, Equivalent ($k_{0}$, $k_{1}$)-covering and generalized digital lifting, Inform. Sci. 178 (2008), no. 2, 550-561 https://doi.org/10.1016/j.ins.2007.02.004
  16. S. E. Han, The k-homotopic thinning and a torus-like digital image in $\mathbb{Z}^{n}$, Journal of Mathematical Imaging and Vision 31 (2008), no. 1, 1-16 https://doi.org/10.1007/s10851-007-0061-2
  17. E. Khalimsky, R. Kopperman, and P. R. Meyer, Computer graphics and connected topologies on finite ordered sets, Topology Appl. 36 (1990), no. 1, 1-17 https://doi.org/10.1016/0166-8641(90)90031-V
  18. T. Y. Kong and A. Rosenfeld, Topological Algorithms for the Digital Image Processing, Elsevier Science, Amsterdam, 1996
  19. E. Melin, Extension of continuous functions in digital spaces with the Khalimsky topology, Topology Appl. 153 (2005), no. 1, 52-65 https://doi.org/10.1016/j.topol.2004.12.004
  20. T. Noiri, On $\delta$-continuous functions, J. Korean Math. Soc. 16 (1980), no. 2, 161-166
  21. A. Rosenfeld, Arcs and curves in digital pictures, J. Assoc. Comput. Mach. 20 (1973), 81-87 https://doi.org/10.1145/321738.321745
  22. J. Slapal, Digital Jordan curves, Topology Appl. 153 (2006), no. 17, 3255-3264 https://doi.org/10.1016/j.topol.2005.10.011

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  20. KD-(k0, k1)-HOMOTOPY EQUIVALENCE AND ITS APPLICATIONS vol.47, pp.5, 2010, https://doi.org/10.4134/JKMS.2010.47.5.1031
  21. A compression of digital images derived from a Khalimsky topological structure vol.32, pp.3, 2013, https://doi.org/10.1007/s40314-013-0034-6
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