CHARACTERIZATIONS OF (L;M)-FUZZY TOPOLOGY DEGREES

Document Type: Research Paper

Authors

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.

Keywords

References

[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. Garca, T. Kubiak, A.P. Sostak, Ideal-valued topological structures, Fuzzy Sets and
Systems, 161 (2010), 2380{2388.
[6] U. Hohle, Probabilistic metrization of fuzzy uniformities, Fuzzy Sets and Systems, 8 (1982),
63{69.
[7] U. Hohle and A. P. Sostak, Axiomatic Foundations of Fixed-basis Fuzzy Topology, In: U.
Hohle, 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, Poznan, Poland, 1985.
[11] T. Kubiak and A. P. Sostak, A fuzzifi cation 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 defi nition 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 Scientifi c, (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.