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Weakly induced modifications of \(I\)-fuzzy topologies. (English) Zbl 1121.54017

This paper extends the literature concerning the categorical structures of fuzzy topological spaces as originally introduced by [C. L. Chang, J. Math. Anal. Appl. 24, 182–190 (1968; Zbl 0167.51001)] and later on modified by several authors. In this paper the authors use the definition of an I-fuzzy topology as a mapping from the class of Zadeh’s fuzzy sets to the unit interval [0,1] satisfying properties that are analogous to the ones defining U. Höhle’s fuzzifying topology [ibid. 78, 659-673 (1980; Zbl 0462.54002)]. They study weakly induced I-fuzzy topological spaces and weakly induced modifications of I-fuzzy topologies from a categorical point of view. It is shown that the category of weakly induced I-fuzzy topological spaces constitutes a reflective and coreflective full subcategory of the category of I-fuzzy topological spaces. Further on the authors study the relationships between several categories of fuzzy topological spaces. Among them the category of stratified Chang-Goguen topological spaces.
Reviewer: E. Kerre (Gent)

MSC:

54A40 Fuzzy topology
03E72 Theory of fuzzy sets, etc.
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References:

[1] Adámek, J.; Herrlich, H.; Strecker, G. E., Abstract and Concrete Categories. Abstract and Concrete Categories, Pure and Applied Mathematics (New York), xiv+482 (1990), New York: John Wiley & Sons, New York · Zbl 0695.18001
[2] Chang, C. L., Fuzzy topological spaces, Journal of Mathematical Analysis and Applications, 24, 1, 182-190 (1968) · Zbl 0167.51001 · doi:10.1016/0022-247X(68)90057-7
[3] Fang, J.; Yue, Y., Base and subbase in -fuzzy topological spaces, Journal of Mathematical Research and Exposition, 26, 1, 89-95 (2006) · Zbl 1101.54005
[4] Höhle, U., Upper semicontinuous fuzzy sets and applications, Journal of Mathematical Analysis and Applications, 78, 2, 659-673 (1980) · Zbl 0462.54002
[5] Höhle, U.; Šostak, A. P.; Höhle, U.; Rodabaugh, S. E., Axiomatic foundations of fixed-basis fuzzy topology, Mathematics of Fuzzy Sets-Logic, Topology, and Measure Theory. Mathematics of Fuzzy Sets-Logic, Topology, and Measure Theory, Handb. Fuzzy Sets Ser., 3, 123-272 (1999), Massachusetts: Kluwer Academic, Massachusetts · Zbl 0977.54006
[6] Kubiak, T., On fuzzy topologies, M.S. thesis (1985), Poznan: Adam Mickiewicz University, Poznan
[7] Kubiak, T.; Rodabaugh, S. E.; Klement, E. P.; Höhle, U., The topological modification of the -fuzzy unit interval, Applications of Category Theory to Fuzzy Subsets (Linz, 1989). Applications of Category Theory to Fuzzy Subsets (Linz, 1989), Theory Decis. Lib. Ser. B Math. Statist. Methods, 14, 275-305 (1992), Dordrecht: Kluwer Academic, Dordrecht · Zbl 0766.54006
[8] Liu, Y.-M.; Luo, M.-K., Fuzzy Topology. Fuzzy Topology, Advances in Fuzzy Systems—Applications and Theory, 9, x+353 (1997), New Jersey: World Scientific, New Jersey · Zbl 0906.54006
[9] Rodabaugh, S. E.; Höhle, U.; Rodabaugh, S. E., Categorical foundations of variable-basis fuzzy topology, Mathematics of Fuzzy Sets-Logic, Topology, and Measure Theory. Mathematics of Fuzzy Sets-Logic, Topology, and Measure Theory, Handb. Fuzzy Sets Ser., 3, 273-388 (1999), Massachusetts: Kluwer Academic, Massachusetts · Zbl 0968.54003
[10] Rodabaugh, S. E.; Höhle, U.; Rodabaugh, S. E., Powerset operator foundations for poslat fuzzy set theories and topologies, Mathematics of Fuzzy Sets-Logic, Topology, and Measure Theory. Mathematics of Fuzzy Sets-Logic, Topology, and Measure Theory, Handb. Fuzzy Sets Ser., 3, 91-116 (1999), Massachusetts: Kluwer Academic, Massachusetts · Zbl 0974.03047
[11] Šostak, A. P., On a fuzzy topological structure, Rendiconti del Circolo Matematico di Palermo. Serie II. Supplemento, 11, 89-103 (1987) (1985) · Zbl 0638.54007
[12] Warner, M. W., Fuzzy topology with respect to continuous lattices, Fuzzy Sets and Systems, 35, 1, 85-91 (1990) · Zbl 0707.54004 · doi:10.1016/0165-0114(90)90020-7
[13] Ying, M. S., A new approach for fuzzy topology. I, Fuzzy Sets and Systems, 39, 3, 303-321 (1991) · Zbl 0718.54017 · doi:10.1016/0165-0114(91)90100-5
[14] Ying, M. S., A new approach for fuzzy topology. III, Fuzzy Sets and Systems, 55, 2, 193-207 (1993) · Zbl 0785.54013 · doi:10.1016/0165-0114(93)90132-2
[15] Yue, Y.; Fang, J., Generated -fuzzy topological spaces, Fuzzy Sets and Systems, 154, 1, 103-117 (2005) · Zbl 1079.54011 · doi:10.1016/j.fss.2005.03.003
[16] Zhang, D., -fuzzifying topologies as -topologies, Fuzzy Sets and Systems, 125, 2, 135-144 (2002) · Zbl 0992.54008 · doi:10.1016/S0165-0114(00)00126-3
[17] Zhang, D.; Liu, Y., Weakly induced modifications of -fuzzy topological spaces, Acta Mathematica Sinica, 36, 1, 68-73 (1993) · Zbl 0783.54009
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