Volume 18 , Issue 4 , PP: 192-203, 2022 | Cite this article as | XML | Html | PDF | Full Length Article
Jili Basumatary 1 * , Bhimraj Basumatary 2 , Said Broumi 3
Doi: https://doi.org/10.54216/IJNS.180418
Let X be a finite set having n elements. The formula for giving the number of topologies T(n) is still not obtained.
If the number of elements n of a finite set is small, we can compute it by hand. However, the difficulty
of finding the number of the topology increases when n becomes large. A topology describes how elements of
a set are spatially related to each other, and the same set can have different topologies. Studying this particular
area is also a highly valued part of the topology, and this is one of the fascinating and challenging research
areas. Note that the explicit formula for finding the number of topologies is undetermined till now, and many
researchers are researching this particular area. This paper is towards the formulae for finding the number of
neutrosophic clopen topological spaces having two, three, four, and five open sets. In addition, some properties
related to formulae are determined.
Combinatorics , Neutrosophic Set , Neutrosophic Clopen Topological Space , Number of Neutrosophic
Clopen Topological Space
[1] F. Smarandache, Neutrosophic set - a generalization of the intuitionistic fuzzy set, International Journal
of Pure and Applied Mathematics, vol. 24, no. 3, pp. 287–297, 2005.
[2] K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy sets and systems, vol. 20, no. 1, pp. 87–96, 1986.
[3] A. A. Salama and S. A. Alblowi, Neutrosophic Set and Neutrosophic Topological Spaces, IOSR Journal
of Mathematics, vol. 3, no. 4, pp. 31–35, 2012.
[4] D. Coker, An introduction to intuitionistic fuzzy topological spaces, Fuzzy Sets and Systems, vol. 88, no.
1, pp. 81–89, 1997.
[5] H. Wang, F. Smarandache, Y. Q. Zhang and R. Sunderraman, Single valued neutrosophic set, Multispace
and Multistructure, vol. 4, pp. 410–413, 2010.
[6] F. G. Lupi´a˜nez, On neutrosophic topology, The international Journal of systems and Cybernatics, vol. 37,
no. 6, pp. 797–800, 2008.
[7] A. A. Salama, S. Broumi and S. A. Alblowi, Introduction to Neutrosophic Topological Spatial Region,
Possible Application to GIS Topological rules, International Journal of Information Engineering and
Electronic Business, vol. 6, no. 6, pp. 15–21, 2014.
[8] P. Iswarya and K. Bageerathi, On Neutosophic Semi-Open sets in Neutrosophic Topological Spaces,
International Journal of Mathematics Trends and Technology, vol. 37, no. 3, pp. 214-223, 2016.
[9] R. Dhavaseelan and S. Jafari, Generalized neutrosophic closed sets, New Trends in Neutrosophic Theory
and Applications, vol. II, pp. 261–273, 2017.
[10] V. K. Shanthi, S. Chandrasekar and K. S. Begam, Neutrosophic Generalized Semi Closed Sets in Neutrosophic
Topological Spaces, International Journal of Reasearch in Advent Technology, vol. 6, no. 7, pp.
1739–1743, 2018.
[11] S. Saranya, and M. Vigneshwaran, C# application to deal with neutrosophic α-closed sets, Journal of
Advanced Research in Dynamical and Control Systems, vol. 11, no. 1 (special issue), pp. 1347–1355,
2019.
[12] A. A. Salama, F. Samarandache and V. Kroumov, Neutrosophic Closed Set and Neutrosophic Continuous
Functions, Neutrosophic Sets and Systems, vol. 4, pp. 4–8, 2014.
[13] H. Sharp Jr., Cardinality of finite topologies, Journal of Combinatorial Theory, vol. 5, no. 1, pp. 82–86,
1968.
[14] R. Stanley, On the number of open sets of finite topologies, Journal of Combinatorial Theory, vol. 10, pp.
75–79, 1971.
[15] G. A. Kamel, Partial chain topologies on finite sets, Computational and Applied Mathematics Journal,
vol. 1, pp. 174–179, 2015.
[16] M. Benoumhani, The number of Topologies on a Finite Set, Journal of Integer Sequences, vol. 9, Article
06.2.6, 2006.
[17] V. Krishnamurty, On the number of topologies on a finite set, The American Mathematical Monthly, vol.
73, no. 2, pp. 154–157, 1966.
[18] M. Benoumhani and A. Jaballah, Finite fuzzy topological spaces, Fuzzy Sets and Systems, vol. 321, pp.
101–114, 2017.
[19] A. Ahmad and A. A. Fora, The number of fuzzy clopen sets in fuzzy topological space, Journal of
Mathematical Sciences and Applications, vol. 5, no. 1, pp. 24–26, 2017.
[20] M. O. Francis and A. O. Adenji, On number of k-element in open and clopen topological topological
Space with corresponding graph n ≤ 4, Academic Journal of Applied Mathematical Sciences, vol. 5, pp.
130–135, 2009.
[21] C. L. Chang, Fuzzy Topological spaces, Journal of mathematical analysis and applications, vol. 24, no.
1, pp. 182–190, 1968.
[22] L. A. Zadeh, Fuzzy Sets, Information and Control, vol. 8, pp. 338–353, 1965.
