K. Fan’s theorem in fuzzifying topology. (English) Zbl 1072.54006

The authors give a new definition of connectivity in a fuzzifying topological space and generalize a known theorem of K. Fan to their setting [Fan’s theorem says that a topological space \(X\) is connected iff for any \(a\), \(b\) of \(X\), there is a finite subset \(\{x_1,x_2,\dots, x_n\}\) of \(X\) such that \(x_1= a\), \(x_2= b\) and \(N(x_i)\cap N(x_{i+1})\neq\phi\), for \(i= 1,2,\dots, n-1\), where \(N: X\to\wp(X)\) such that \(N(x)\) is neighbourhood of \(x\) for all \(x\in X\)].


54A40 Fuzzy topology
03B50 Many-valued logic
54D05 Connected and locally connected spaces (general aspects)
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[1] Chang, C. L., Fuzzy topological spaces, J. Math. Anal. Appl., 24, 182-190 (1968) · Zbl 0167.51001
[2] Chattopadhyay, K. C.; Hazra, R. N.; Samanta, S. K., Gradation of openness: fuzzy topology, Fuzzy Sets and Systems, 49, 242-273 (1992) · Zbl 0762.54004
[4] Höhle, U., Upper semicontinuous fuzzy sets and applications, J. Math. Anal. Appl., 78, 659-673 (1980) · Zbl 0462.54002
[6] Liu, Y. M.; Luo, M. K., Fuzzy Topology (1998), World Scientific Publication: World Scientific Publication Singapore, (English)
[7] Lowen, R., Fuzzy neighborhood systems spaces, Fuzzy Sets and Systems, 7, 165-189 (1982) · Zbl 0487.54008
[8] Pu, B. M.; Liu, Y. M., Fuzzy topology (I), neighborhood structure of a fuzzy point and Moore Smith convergence, J. Math. Anal. Appl., 76, 571-599 (1980) · Zbl 0447.54006
[9] Ramadan, A. A., Smooth topological spaces, Fuzzy Sets and Systems, 48, 371-375 (1992) · Zbl 0783.54007
[10] Rodabaugh, S. E., Categorical Frameworks for Stone Representation Theories, (Rodabaugh, S. E.; etal., Applications of Category Theory to Fuzzy Subsets (1992), Kluwer Academic Publishers: Kluwer Academic Publishers Netherlands), 177-231 · Zbl 0789.18005
[12] Šostak, A. P., On some modifications of fuzzy topologies, Matimatiki Vesnik, 41, 20-37 (1989)
[13] Hazra, R. N.; Samanta, S. K.; Chattopadhyay, K. C., Fuzzy topology redefined, Fuzzy Sets and Systems, 45, 79-82 (1992) · Zbl 0756.54002
[15] Ying, M. S., A new approach to fuzzy topology (I), Fuzzy Sets and Systems, 39, 303-321 (1991) · Zbl 0718.54017
[16] Ying, M. S., A new approach to fuzzy topology (II), Fuzzy Sets and Systems, 47, 221-232 (1992) · Zbl 0752.54002
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