Honam Mathematical Journal (호남수학학술지)

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STUDY ON TOPOLOGICAL SPACES WITH THE SEMI-T½ SEPARATION AXIOM

  • Han, Sang-Eon (Faculty of Liberal Education, Institute of Pure and Applied Mathematics, Chonbuk National University)
  • Received : 2013.09.26
  • Accepted : 2013.10.16
  • Published : 2013.12.25
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Abstract

The present paper consists of two parts. Since the recent paper [4] proved that an Alexandroff $T_0$-space is a semi-$T_{\frac{1}{2}}$-space, the first part studies semi-open and semi-closed structures of the Khalimsky nD space. The second one focuses on the study of a relation between the LS-property of ($SC^{n_1,l_1}_{k_1}{\times}SC^{n_2,l_2}_{k_2}$, k) relative to the simple closed $k_i$-curves $SC^{n_i,l_i}_{k_i}$, $i{\in}\{1,2\}$ and its normal k-adjacency. In addition, the present paper points out that the main theorems of Boxer and Karaca's paper [3] such as Theorems 4.4 and 4.7 of [3] cannot be new assertions. Indeed, instead they should be attributed to Theorems 4.3 and 4.5, and Example 4.6 of [10].

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References

  1. P. Alexandorff, Diskrete Raume, Mat. Sb. 2 (1937), 501-518.
  2. F. G. Arenas, Alexandroff Spaces, Acta Math. Univ. Comenian., (N.S.) 68 (1999), 501-518.
  3. L. Boxer and I. Karaca, Fundamental groups for digital products, Advances and Applications in Mathematical Sciences, 11(4) (2012), 161-179.
  4. V. A. Chatyrko, S. E. Han, Y. Hattori, Some remarks concerning semi-$T_{\frac{1}{2}}$ spaces, Filomat, in press.
  5. G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove and D. S. Scott, A Compendium of Continuous Lattices, Springer, Berlin-Heidelberg-New York, 1980.
  6. N. Hamllet, A correction to the paper "Semi-open sets and semi-continuity in topological spaces" by Norman Levine, Proceeding of Amer. Math. Soc. 49(2) (1975), 458-460.
  7. S. E. Han, Computer topology and its applications, Honam Math. Jour., 25(1) (2003), 153-162.
  8. S. E. Han, Non-product property of the digital fundamental group, Information Sciences 171(1-3) (2005), 73-91. https://doi.org/10.1016/j.ins.2004.03.018
  9. S. E. Han, The k-homotopic thinning and a torus-like digital image in $Z^{n}$, Journal of Mathematical Imaging and Vision 31(1) (2008), 1-16. https://doi.org/10.1007/s10851-007-0061-2
  10. S. E. Han, Cartesian product of the universal covering property, Acta Applicandae Mathematicae 108 (2009), 363-383. https://doi.org/10.1007/s10440-008-9316-1
  11. S. E. Han, KD-(k0; k1)-homotopy equivalence and its applications, Journal of Korean Mathematical Society 47(5) (2010), 1031-1054. https://doi.org/10.4134/JKMS.2010.47.5.1031
  12. S. E. Han, Ultra regular covering space and its automorphism group, International Journal of Applied Mathematics & Computer Science, 20(4) (2010), 699-710.
  13. E. D. Khalimsky, On topologies of generalized segments, Soviet Math. Dokl., 10 (1969), 1508-1511.
  14. E. Khalimsky, R. Kopperman, P. R. Meyer, Computer graphics and connected topologies on finite ordered sets, Topology and Its Applications, 36(1) (1991), 1-17.
  15. T. Y. Kong, A. Rosenfeld, Topological Algorithms for the Digital Image Processing, Elsevier Science, Amsterdam, (1996).
  16. N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly 70 (1963), 36-41. https://doi.org/10.2307/2312781
  17. D. Marcus, F. Wyse et al., Solution to problem 5712, Am. Math. Monthly 77 (1970), 1119. https://doi.org/10.2307/2316121
  18. A. Rosenfeld, Arcs and curves in digital pictures, Jour. of the ACM, 20 (1973), 81-87. https://doi.org/10.1145/321738.321745

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