Remarks on topological spaces on $ {\mathbb Z}^n $ which are related to the Khalimsky $ n $-dimensional space

Sang-Eon Han; Saeid Jafari; Jeong Min Kang; Sik Lee; Sang-Eon Han; Saeid Jafari; Jeong Min Kang; Sik Lee

doi:10.3934/math.2022072

AIMS Mathematics

2022, Volume 7, Issue 1: 1224-1240. doi: 10.3934/math.2022072

Previous Article Next Article

Research article

Remarks on topological spaces on $ {\mathbb Z}^n $ which are related to the Khalimsky $ n $-dimensional space

1.
Department of Mathematics Education, Institute of Pure and Applied Mathematics, Jeonbuk National University, Jeonju-City Jeonbuk 54896, Republic of Korea
2.
College of Vestsjaelland South, Herrestraede 11 and Mathematical and Physical Science Foundation, 4200 Slagelse, Denmark
3.
Mathematics, School of Liberal, Arts Education, University of Seoul, Seoul 02504, Republic of Korea
4.
Department of Mathematics Education, Chonnam National University, Gwangju-City 61186, Republic of Korea

Received: 03 August 2021 Accepted: 30 September 2021 Published: 21 October 2021
MSC : 54A05, 54J05, 54F05, 54C08, 54F65, 68U05

The present paper intensively studies various properties of certain topologies on the set of integers $ {\mathbb Z} $ (resp. $ {\mathbb Z}^n $) which are either homeomorphic or not homeomorphic to the typical Khalimsky line topology (resp. $ n $-dimensional Khalimsky topology). This finding plays a crucial role in addressing some problems which remain open in the field of digital topology.
- Khalimsky topology,
- $ T_{\frac{1}{2}} $-separation axiom,
- Alexandroff topology,
- quasi-discrete,
- digital topology
Citation: Sang-Eon Han, Saeid Jafari, Jeong Min Kang, Sik Lee. Remarks on topological spaces on $ {\mathbb Z}^n $ which are related to the Khalimsky $ n $-dimensional space[J]. AIMS Mathematics, 2022, 7(1): 1224-1240. doi: 10.3934/math.2022072

Related Papers:

Abstract

The present paper intensively studies various properties of certain topologies on the set of integers $ {\mathbb Z} $ (resp. $ {\mathbb Z}^n $) which are either homeomorphic or not homeomorphic to the typical Khalimsky line topology (resp. $ n $-dimensional Khalimsky topology). This finding plays a crucial role in addressing some problems which remain open in the field of digital topology.

References

[1]	P. S. Alexandorff, Uber die Metrisation der im Kleinen kompakten topologischen R$\ddot{a}$ume, Math. Ann., 92 (1924), 294–301. doi: 10.1007/BF01448011. doi: 10.1007/BF01448011
[2]	P. Alexandorff, Diskrete R$\ddot{a}$ume, Mat. Sb., 2 (1937), 501–518.
[3]	V. A. Chatyrko, S. E. Han, Y. Hattori, Some remarks concerning semi-$T_{\frac{1}{2}}$ spaces, Filomat, 28 (2014), 21–25.
[4]	W. Dunham, $T_{\frac{1}{2}}$-spaces, Kyungpook Math. J., 17 (1977), 161–169.
[5]	S. E. Han, Covering rough set structures for a locally finite covering approximation space, Inf. Sci., 480 (2019), 420–437. doi: 10.1016/j.ins.2018.12.049. doi: 10.1016/j.ins.2018.12.049
[6]	S. E. Han, Roughness measures of locally finite covering rough sets, Int. J. Approx. Reason., 105 (2019), 368–385. doi: 10.1016/j.ijar.2018.12.003. doi: 10.1016/j.ijar.2018.12.003
[7]	S. E. Han, Digital topological rough set structures and topological operators, Topol. Appl., 301 (2021), 107507, 1–19. doi: 10.1016/j.topol.2020.107507.
[8]	S. E. Han, S. Jafari, J. M. Kang, Topologies on ${\mathbb Z}^n$ that are not homeomorphic to the $n$-dimensional Khalimsky topological space, Mathematics, 7 (2019), 1072. doi: 10.3390/math711072. doi: 10.3390/math711072
[9]	S. E. Han, A. Sostak, A compression of digital images derived from a Khalimsky topological structure, Comput. Appl. Math., 32 (2013), 521–536. doi: 10.1007/s40314-013-0034-6. doi: 10.1007/s40314-013-0034-6
[10]	J. M. Kang, S. E. Han, Compression of Khalimsky topological spaces, Filomat, 26 (2012), 1101–1114. doi: 10.2298/FIL1206101K. doi: 10.2298/FIL1206101K
[11]	E. D. Khalimsky, Applications of connected ordered topological spaces in topology, Conf. Math. Dep. Povolosia, 1970.
[12]	E. Khalimsky, R. Kopperman, P. R. Meyer, Computer graphics and connected topologies on finite ordered sets, Topol. Appl., 36 (1990), 1–17. doi: 10.1016/0166-8641(90)90031-V. doi: 10.1016/0166-8641(90)90031-V
[13]	C. O. Kiselman, Digital Geometry and Mathematical Morphology, Lecture Notes, Uppsala University, Department of Mathematics, 2002.
[14]	N. Levine, Semi-open sets and semi-continuity in topological spaces, Am. Math. Mon., 70 (1963), 36–41. doi: 10.2307/2312781. doi: 10.2307/2312781
[15]	J. J. Li, Topological properties of approximation spaces and their applications, Math. Pract. Theor., 39 (2009), 145–151. doi: 10.1360/972009-1650. doi: 10.1360/972009-1650
[16]	E. F. Lashin, A. M. Kozae, A. A. Abo Khadra, T. Medhat, Rough set theory for topolgoical spaces, Int. J. Approx. Reason., 40 (2005), 35–43. doi: 10.1016/j.ijar.2004.11.007. doi: 10.1016/j.ijar.2004.11.007
[17]	J. R. Munkres, Topology: A First Course, Prentice-Hall Inc., 1975. doi: 10.2307/3615551.

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)