On 2005, F. Smarandache generalized the Atanassov's intuitionistic fuzzy setsto neutrosophic sets. Also, this author and some co-workers introduced the notion of interval neutrosophic set, which is an instance of neutrosophic set and studied various properties. The notion of neutro-sophic topology on the non-standard interval is also due to Smarandache. We study in this paper relations between interval neutrosophic sets and topology.
Purpose -Recently, Smarandache generalized the Atanassov's intuitionistic fuzzy sets (IFSs) and other kinds of sets to neutrosophic sets (NSs). Also, this author defined the notion of neutrosophic topology on the non-standard interval. One can expect some relation between the intuitionistic fuzzy topology (IFT) on an IFS and the neutrosophic topology. This paper aims to show that this is false. Design/methodology/approach -The possible relation between the IFT and the neutrosophic topology is studied. Findings -Relations on neutrosophic topology and IFT are found. Research limitations/implications -Clearly, this paper is confined to IFSs and NSs. Practical implications -The main applications are in the mathematical field. Originality/value -The paper shows original results on fuzzy sets and topology.
We show here some of our results on intuitionistic fuzzy topological spaces. In 1983, K.T. Atanassov proposed a generalization of the notion of fuzzy set: the concept of intuitionistic fuzzy set [1]. Some basic results on intuitionistic fuzzy sets were publised in [2,3], and the book [4] provides a comprehensive coverage of virtually all results until 1999 in the area of the theory and applications of intuitionistic fuzzy set. D. Ç oker and M. Demirci [9] defined and studied the basic concept of intuitionistic fuzzy point, later D. Ç oker [7, 8] constructed the fundamental theory on intuitionistic fuzzy topological spaces, and D. Ç oker and others mathematicians [6, 10-27] studied compactness, connectedness, continuity, separation, convergence and paracompactness in intuitionistic fuzzy topological spaces. Finally, G.-J Wang and Y.Y. He [31] showed that every intuitionistic fuzzy set may be regarded as an L-fuzzy set for some appropriate lattice L. Nevertheless, the results obtained by the above authors are not redundants with other for the ordinary fuzzy sense. Recently, Smarandache [30] defined and studied neutrosophic sets (NSs) which generalize IFSs. This author defined also the notion of neutrosophic topology [28]. In [27] we proved that neutrosophic topology does not generalize the concept of intuitionistic fuzzy topology.
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