Characterization of Topological Fuzzy Sets in Hausdorff Spaces
محورهای موضوعی : Transactions on Fuzzy Sets and Systems
Benard Okelo
1
(Department of Pure and Applied Mathematics, Jaramogi Oginga Odinga University of Science and Technology, Bondo, Kenya.)
Allan Onyango
2
(Department of Pure and Applied Mathematics, Jaramogi Oginga Odinga University of Science and Technology, Bondo, Kenya.)
کلید واژه: fuzzy set, Fuzzy topological space, Fuzzy Hausdorff space, Topological Data point,
چکیده مقاله :
In this paper, we have characterized big data fuzzy sets and shown that topological data points form singleton fuzzy sets which are closed. Besides, fuzzy sets of topological data points are compact and have at least one closed point. We have also shown that the fuzzy set of all condensation points of a fuzzy Hausdor space is in nite and the cardinality of a topological data fuzzy set is also in nite and arbitrarily distributed in fuzzy Hausdor spaces.
In this paper, we have characterized big data fuzzy sets and shown that topological data points form singleton fuzzy sets which are closed. Besides, fuzzy sets of topological data points are compact and have at least one closed point. We have also shown that the fuzzy set of all condensation points of a fuzzy Hausdor space is in nite and the cardinality of a topological data fuzzy set is also in nite and arbitrarily distributed in fuzzy Hausdor spaces.
[1] K. P. Adlassnig, A survey on medical diagnosis and fuzzy subsets, Springer Verlag, New York, (1982).
[2] R. C. Berkan and S. L. Trubatch, Fuzzy Systems Design Principles: Building Fuzzy IF-THEN Rule Bases, New York, NY: IEEE Press, (1997).
[3] G. Carlsson, Topology and data, Bull. Amer. Math. Soc., 46 (2009), 255-308.
[4] R. W. David, Topological Spaces, Springer Verlag, New York, (2008).
[5] D. Dubois and H. Prade, Towards fuzzy di erential calculus: Part 1 integration of fuzzy mappings; Part 2: integration of fuzzy intervals; Part 3: Di erentiation, Fuzzy Set Syst., 8 (1982), 1-233.
[6] S. Gottwald, Foundations of a theory for fuzzy sets: 40 years of development, Int. J. Gen. Syst., 37 (2008), 69-82.
[7] M. Inuiguchi, H. Ichihashi and Y. Kume, A solution algorithm for fuzzy linear programming with piecewise linear membership functions, Fuzzy Set Syst., 34 (1990), 5-31.
[8] J. Kacprzyk, A. Wilbik and S. Zadrzny, Linguistic summarization of time series using quanti er driven
aggregation, Fuzzy Set Syst., 159 (2008), 1485-1499.
[9] P. E. Kloeden, Chaotic iterations of fuzzy sets, Fuzzy Sets and Systems, 42 (1991), 37-42.
[10] W. A. Lodwick and K. D. Jamison, Interval-valued probability in the ananlysis of problems containing a mixture of possibilistic, probabilistic and interval uncertainty, Fuzzy Set Syst., 159 (2008), 2845-2858.
[11] T. Maeda, On characterization of fuzzy vectors and its application to fuzzy mathematical programming problems, Fuzzy Set Syst., 159 (2008), 3333-3346.
[12] E. H. Mamdani and S. Assilian, An experiment in linguistic synthesis with a fuzzy logic controller, Int. J. Man. Mach. Stud., 7 (1975), 1-13.
[13] W. Pedrycz and P. Rai, Collaborative clustering with the use of Fuzzy C-Means and its quanti cation. Fuzzy Set Syst., 159 (2008), 2399-2427.
[14] D. Ruan, A critical study of widely used fuzzy implication operators and their inference on the inference rules in fuzzy expert systems. Ph.D. Thesis, Gent, (1990).
[15] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.
[16] H. J. Zimmermann, Fuzzy set theory, Comp. Stat., 2 (2010), 317-332.
[17] H. J. Zimmermann, Fuzzy set theory- and its applications, 4th Ed., Springer Dordrechtg, New York, (2001).
