Cut topology generated by generalized triangular fuzzy numbers

Abstract

In this study, we propose a new topology called the cut topology. We first develop a synthetic base by using the 𝛼 cut family of the generalized triangular fuzzy number. Then, we generate the cut topology by the synthetic base and give subspace cut topology on a subset of any set. Moreover, the fundamental concepts such as interior, closure, and limit points of a set in the cut topology are discussed. Finally, neighborhood, continuity, and homeomorphism are ana-lyzed with examples.

Keywords

• Cut Topology
• Fuzzy Set
• Generalized Triangular Fuzzy Number

References

1. References
2. Atanassov KT. Intuitionistic fuzzy sets. Fuzzy Sets Syst 1986;20:8796. Buckley JJ. Eslami, E. An Introduction to Fuzzy Logic and Fuzzy Sets. New York: Physica-Verlag Heidelberg; 2002.
3. Chang CL. Fuzzy topological spaces. J Math Anal Appl 1968;24:182190.
4. Chang SSL, Zadeh LA. On fuzzy mapping and control. IEEE Trans Syst Man Cybern 1972;2:3034.
5. Chen S. Ranking fuzzy numbers with maximizing set and minimizing set. Fuzzy Sets Syst 1985;17:113129.
6. Chen CT. Extensions of the TOPSIS for group decision-making under fuzzy environment. Fuzzy Sets Syst 2000;114:19.
7. Chen SM, Chen JH. Fuzzy risk analysis based on ranking generalized fuzzy numbers with different heights and different spreads. Expert Sys Appl 2009;36:68336842.
8. Chen SH, Hsieh CH. Representation, ranking, distance, and similarity of L-R type fuzzy number and application. Aust J Intell Process Syst 2000;6:217229.

9. Dutta P, Dash SR. Medical decision making via the arithmetic of generalized triangular fuzzy numbers. Open Cybern Syst J 2018;12:119.
10. Font JJ, Miralles, A, Sanchis M. On the fuzzy number space with the level convergence topology. J Funct Spaces Appl 2012;2012;326417.
11. Klir GJ, Yuan B. Fuzzy Set and Fuzzy Logic: Theory and Applications. New Jersey: Upper Saddle River, Prentice Hall Publishers; 1995.
12. Lowen. R. Fuzzy topological spaces and fuzzy compactness. J Math Anal Appl 1979;56:621633.
13. Manes EG, Dubois D, Prade H. Review: Didier Dubois and Henri Prade, Fuzzy sets and systems: Theory and applications. Bullet Am Math Soc 1982;7:603612.
14. Munkers JR. Topology. 2nd ed. Upper Saddle River, NJ; Prentice Hall; 2000.
15. Onasanya BO, Hoshoka-Mayerova S. Some Topological and algebraic properties of alpha level subsets’ topology of a fuzzy subset. An Sti Univ Ovid Co-Mat 2018;26:213227.
16. Padmapriya R, Thangavelu P. Topologies generated by the α cuts of a fuzzy set. Int J Math Arch 2014;5:156165.
17. Padmapriya R, Thangavelu P. Topologies generated by the fuzzy numbers. Int J Eng Trends Technol 2017;43:243246.
18. Sutherland WA. Introduction to Metric and Topological Spaces. Oxford: Clarendon Press; 1975.
19. Yanardağ Y. Cut topology generated by generalized triangular fuzzy numbers. [In Turkish]. Master’s Thesis. Kilis: Kilis 7 Aralık University, Graduate School of Natural and Applied Science; 2020.
20. Zadeh LA. Fuzzy sets. Inform Control 1965;8:338353.
21. Zadeh LA. Fuzzy logic. IEEE Comput 1988;21:8393.
22. Zhang DA. Natural topology for fuzzy numbers. J Math Anal Appl 2001;264:344353.