Zhong, Y., Shi, F. (2018). CHARACTERIZATIONS OF (L;M)-FUZZY TOPOLOGY DEGREES. Iranian Journal of Fuzzy Systems, 15(4), 129-149. doi: 10.22111/ijfs.2018.4121

Yu Zhong; Fu-Gui Shi. "CHARACTERIZATIONS OF (L;M)-FUZZY TOPOLOGY DEGREES". Iranian Journal of Fuzzy Systems, 15, 4, 2018, 129-149. doi: 10.22111/ijfs.2018.4121

Zhong, Y., Shi, F. (2018). 'CHARACTERIZATIONS OF (L;M)-FUZZY TOPOLOGY DEGREES', Iranian Journal of Fuzzy Systems, 15(4), pp. 129-149. doi: 10.22111/ijfs.2018.4121

Zhong, Y., Shi, F. CHARACTERIZATIONS OF (L;M)-FUZZY TOPOLOGY DEGREES. Iranian Journal of Fuzzy Systems, 2018; 15(4): 129-149. doi: 10.22111/ijfs.2018.4121

^{1}College of Science, North China University of Technology, Beijing, P. R. China

^{2}School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, P.R. China and Beijing Key Laboratory on MCAACI, Beijing Institute of Technology, Beijing, P.R. China.

Abstract

In this paper, characterizations of the degree to which a mapping $\mathcal{T} : L^{X}\longrightarrow M$ is an $(L, M)$-fuzzy topology are studied in detail. What is more, the degree to which an $L$-subset is an $L$-open set with respect to $\mathcal{T}$ is introduced. Based on that, the degrees to which a mapping $f: X\longrightarrow Y$ is continuous, open, closed or a quotient mapping with respect to $\mathcal{T}_{X}$ and $\mathcal{T}_{Y}$ are defined, and their characterizations are given, respectively. Besides, the relationships among the continuity degrees, the openness degrees, the closedness degrees and the quotient degrees of mappings are discussed.

[1] C. L. Chang, Fuzzy topological spaces, Journal of Mathematical Analysis and Applications, 24 (1968), 182{190. [2] G. Gierz, et al., A Compendium of Continuous Lattices, Springer, Berlin, 1980. [3] J. A. Goguen, L-fuzzy sets, Journal of Mathematical Analysis and Applications, 18 (1967), 145{174. [4] J. A. Goguen, The fuzzy tychonoff theorem, Journal of Mathematical Analysis and Applications, 43 (1973), 734{742. [5] J. G. Garca, T. Kubiak, A.P. Sostak, Ideal-valued topological structures, Fuzzy Sets and Systems, 161 (2010), 2380{2388. [6] U. Hohle, Probabilistic metrization of fuzzy uniformities, Fuzzy Sets and Systems, 8 (1982), 63{69. [7] U. Hohle and A. P. Sostak, Axiomatic Foundations of Fixed-basis Fuzzy Topology, In: U. Hohle, S.E. Rodabaugh (Eds.), The Handbooks of Fuzzy Sets Series Mathematics of Fuzzy Sets: Logic Topology and Measure Theory, vol. 3, Kluwer Academic Publishers, Dordrecht, 1999. [8] B. Hutton, Normality in fuzzy topological spaces, Journal of Mathematical Analysis and Applications, 50 (1975), 74{79. [9] J. L. Kelley, General Topology, New York, Springer, 1955. [10] T. Kubiak, On Fuzzy Topologies, PhD thesis, Adam Mickiewicz, Poznan, Poland, 1985. [11] T. Kubiak and A. P. Sostak, A fuzzification of the category of M-valued L-topological spaces, Applied General Topology, 5 (2004), 137{154. [12] C. Y. Liang and F. G. Shi, Degree of continuity for mappings of (L;M)-fuzzy topological spaces, Journal of Intelligent and Fuzzy Systems, 27 (2014), 2665{2677. [13] H. Y. Li and F. G. Shi, Measures of fuzzy compactness in L-fuzzy topological spaces, Computers and Mathematics with Applications, 59 (2010), 941{947. [14] R. Lowen, Fuzzy topological spaces and fuzzy compactness, Journal of Mathematical Analysis and Applications, 56 (1976), 621{633. [15] J. R. Munkres, Topology (second edition), New Jersey, Person Education Inc., 2000. [16] B. Pang, Degrees of continuous mappings, open mappings, and closed mappings in L- fuzzifying topological spaces, Journal of Intelligent and Fuzzy Systems, 27 (2014), 805{816.

[17] S. E. Rodabaugh, Powerset operator based foundation for point-set lattice-theoretic (poslat) fuzzy set theories and topologies, Quaestiones Mathematicae, 20 (1997), 463{530. [18] F. G. Shi, A new definition of fuzzy compactness, Fuzzy Sets and Systems, 158 (2007), 1486{1495. [19] F. G. Shi, (L;M)-fuzzy matroids, Fuzzy Sets and Systems, 160 (2009), 2387{2400. [20] F. G. Shi and C. Y Liang, Measures of compactness in L-fuzzy pretopological spaces, Journal of Intelligent and Fuzzy Systems, 26 (2014), 1557{1561. [21] A. P. Sostak, On a fuzzy topological structure, Rendiconti del Circolo Matematico di Palermo, 2(11) (1985), 89{103. [22] A. P. Sostak, On compactness and connectedness degrees of fuzzy sets in fuzzy topological spaces, in: General Topology and its Relations to Modern Analysis and Algebra, Heldermann Verlag Berlin 1988. [23] A. P. Sostak, Two decades of fuzzy topology: Basis ideas, notions and results, Russian Math Surveys, 44 (1989), 125{186. [24] A. P. Sostak, Fuzzy categories related to algebra and topology, Tatra Mount. Math. Publ, 16 (1999), 159{186. [25] A. P. Sostak, L-valued categories and examples related to algebra and topology, In: categorical structures and their applications, W. Gahler, G. Preuss eds., World Scientific, (2003), 291{ 311. [26] G. J. Wang, Theory of topological molecular lattices, Fuzzy Sets and Systems, 47 (1992), 351{376. [27] M. S. Ying, A new approach for fuzzy topology (I), Fuzzy Sets and Systems, 39 (1991), 303{321. [28] J. Zhang, F. G. Shi and C. Y. Zheng, On L-fuzzy topological spaces, Fuzzy Sets and Systems, 149 (2005), 473{484.