Almost fixed point property for digital spaces associated with Marcus-Wyse topological spaces

Volume 10, Issue 1, pp 34--47 http://dx.doi.org/10.22436/jnsa.010.01.04

Publication Date: January 26, 2017 Submission Date: April 07, 2016

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Authors

Sang-Eon Han - Department of Mathematics Education, Institute of Pure and Applied Mathematics, Chonbuk National University, 54896, Jeonju-City Jeonbuk, Republic of Korea.

Abstract

The present paper studies almost fixed point property for digital spaces whose structures are induced by Marcus-Wyse (M-, for brevity) topology. In this paper we mainly deal with spaces \(X\) which are connected M-topological spaces with M-adjacency (MA-spaces or M-topological graphs for short) whose cardinalities are greater than 1. Let MAC be a category whose objects, denoted by Ob(MAC), are MA-spaces and morphisms are MA-maps between MA-spaces (for more details, see Section 3), and MTC a category of M-topological spaces as Ob(MTC) and M-continuous maps as morphisms of MTC (for more details, see Section 3). We prove that whereas any MA-space does not have the fixed point property (FPP for short) for any MA-maps, a bounded simple MA-path has the almost fixed point property (AFPP for short). Finally, we refer the topological invariant of the FPP for M-topological spaces from the viewpoint of MTC.

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Keywords

• Digital topology
• fixed point property
• Marcus-Wyse topology
• MA-map
• MA-isomorphism
• MA-homotopy
• MA-space
• MA-contractibility
• M-topological graph
• almost fixed point property.

MSC

•  54A10
•  54C05
•  54C08
•  54F65

References

• [1] P. Alexandroff, Diskrete Räume, Mat. Sb., 2 (1937), 501–518.
  • Google Scholar


• [2] L. Boxer , A classical construction for the digital fundamental group, J. Math. Imaging Vision, 10 (1999), 51–62.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [3] R. F. Brown, The Lefschetz fixed point theorem, Scott, Foresman and Co., , Glenview, Ill.-London (1971)
  • Google Scholar


• [4] V. A. Chatyrko, S.-E. Han, Y. Hattori, Some remarks concerning semi-\(T_{1/2}\) spaces , Filomat, 28 (2014), 21–25.
  • View Article
  • MathSciNet
  • MATH


• [5] O. Ege, I. Karaca, Lefschetz fixed point theorem for digital images, Fixed Point Theorey Appl., 2013 (2013 ), 13 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [6] S.-E. Han, On the classication of the digital images up to a digital homotopy equivalence, J. Comput. Commun. Res., 10 (2000), 194–207.
  • Google Scholar


• [7] S.-E. Han, Non-product property of the digital fundamental group, Inform. Sci., 171 (2005), 73–91.
  • View Article
  • MathSciNet
  • Google Scholar


• [8] S.-E. Han, On the simplicial complex stemmed from a digital graph, Honam Math. J., 27 (2005), 115–129.
  • MathSciNet
  • Google Scholar
  • MATH


• [9] S.-E. Han, Equivalent \((k_0, k_1)\)-covering and generalized digital lifting, Inform. Sci., 178 (2008), 550–561.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [10] S.-E. Han, The k-homotopic thinning and a torus-like digital image in \(Z^n\), J. Math. Imaging Vision, 31 (2008), 1–16.
  • View Article
  • MathSciNet
  • Google Scholar


• [11] S.-E. Han, KD-\((k_0, k_1)\)-homotopy equivalence and its applications, J. Korean Math. Soc., 47 (2010), 1031–1054.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [12] S.-E. Han, Fixed point theorems for digital images, Honam Math. J., 37 (2015), 595–608.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [13] S.-E. Han, Generalizations of continuity of maps and homeomorphisms for studying 2D digital topological spaces and their applications, Topology Appl., 196 (2015), 468–482.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [14] S.-E. Han, Banach fixed point theorem from the viewpoint of digital topology, J. Nonlinear Sci. Appl., 9 (2016), 895–905.
  • MathSciNet
  • Google Scholar
  • MATH


• [15] S.-E. Han, Contractibility and fixed point property: the case of Khalimsky topological spaces, Fixed Point Theory Appl., 2016 (2016 ), 20 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [16] S.-E. Han, W.-J. Chun, Classification of spaces in terms of both a digitization and a Marcus Wyse topological structure, Honam Math. J., 33 (2011), 575–589.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [17] S.-E. Han, B. G. Park, Digital graph \((k_0, k_1)\)-homotopy equivalence and its applications, http://atlas-conferences. com/c/a/k/b/35.htm, (2003)
  • Google Scholar


• [18] S.-E. Han, B. G. Park, Digital graph \((k_0, k_1)\)-isomorphism and its applications, http://atlas-conferences.com/c/a/ k/b/36.htm, (2003)
  • Google Scholar


• [19] S.-E. Han, W. Yao, Homotopy based on Marcus-Wyse topology and its applications, Topology Appl., 201 (2016), 358–371.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [20] G. T. Herman, Oriented surfaces in digital spaces, CVGIP: Graph. Models Image Proc., 55 (1993), 381–396.
  • View Article
  • Google Scholar


• [21] E. Khalimsky , Motion, deformation and homotopy in finite spaces, Man and CyberneticsProceedings of the IEEE International Conference on Systems, , (1987), 227–234.
  • Google Scholar


• [22] E. Khalimsky, R. Kopperman, P. R. Meyer, Computer graphics and connected topologies on finite ordered sets, Topology Appl., 36 (1990), 1–17.
  • View Article
  • Google Scholar
  • MATH


• [23] T. Y. Kong, A. Rosenfeld, Topological algorithms for digital image processing, Elsevier Science, Amsterdam (1996)
  • Google Scholar


• [24] S. Lefschetz, Topology , Amer. Math. Soc. , (1930), New York


• [25] S. Lefschetz , On the fixed point formula, Ann. of Math., 38 (1937), 819–822.
  • View Article
  • MathSciNet


• [26] A. Rosenfeld , Digital topology, Amer. Math. Monthly, 86 (1979), 621–630.
  • Google Scholar


• [27] A. Rosenfeld, Continuous functions on digital pictures, Pattern Recogn. Lett., 4 (1986), 177–184.
  • View Article
  • Google Scholar


• [28] J. Šlapal, Topological structuring of the digital plane, Discrete Math. Theor. Comput. Sci., 15 (2013), 165–176.
  • MathSciNet
  • Google Scholar
  • MATH


• [29] F. Wyse, D. Marcus, Solution to problem 5712, Amer. Math. Monthly, 77 (1970), 1119.
  • Google Scholar