The Urysohn, completely Hausdorff and completely regular axioms in $L$-fuzzy topological spaces

Document Type: Research Paper

Authors

1 College of Science, North China University of Technology, No.5 Jinyuanzhuang Road, Shijingshan District, 100144 Beijing, P.R. China

2 School of Mathematics and Statistics, Beijing Institute of Technology, 5 South Zhongguancun Street, Haidian District, 100081 Beijing, P.R. China

Abstract

In this paper, the Urysohn, completely Hausdorff and completely regular axioms in $L$-topological spaces are generalized to $L$-fuzzy topological spaces. Each $L$-fuzzy topological space can be regarded to be Urysohn, completely Hausdorff and completely regular to
some degree. Some properties of them are investigated. The relations among them and $T_2$ in $L$-fuzzy topological spaces are discussed.

Keywords


[1] C. L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl., 24 (1968), 182–190.
[2] S. L. Chen and Z. X.Wu, Urysohn separation property in topological molecular lattices, Fuzzy
Sets Syst., 123 (2001), 177–184.
[3] P. Dwinger, Characterizations of the complete homomorphic images of a completely distributive
complete lattice I, Indagationes Mathematicae(Proceedings), 85 (1982), 403–414.
[4] J. M. Fang, H()-completely Hausdorff axiom on L-topological spaces, Fuzzy Sets Syst., 140
(2003), 475–469.
[5] J. M. Fang and Y. L. Yue, Urysohn closedness on completely distributive lattices, Fuzzy Sets
Syst., 144 (2004), 367–381.
[6] J. M. Fang and Y. L. Yue, Base and subbase in I-fuzzy topological spaces, J. Math. Res.
Exposition, 26 (2006), 89–95.
[7] G. Gierz, et al., A compendium of continuous lattices, Springer Verlag, Berlin, 1980.
[8] B. Hutton, Normality in fuzzy topological spaces, J. Math. Anal. Appl., 50 (1975), 74–79.
[9] U. H¨ohle, Probabilistic metrization of fuzzy uniformities, Fuzzy Sets Syst., 8 (1982), 63–69.
[10] U. H¨ohle and A. P. ˘Sostak, Axiomatic foudations of fixed-basis fuzzy topology, In: U. H¨ohle,
S. E. Rodabaugh(Eds.), Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory,
Handbook Series, vol.3, Kluwer Academic Publishers, Boston, Dordrecht, London, (1999),
123–173.
[11] T. Kubiak, On fuzzy topologies, Ph. D. Thesis, Adam Mickiewicz, Poznan, Poland, 1985.
[12] H. Y. Li and F. G. Shi, Some separation axioms in I-fuzzy topological spaces, Fuzzy Sets
Syst., 159 (2008), 573–587.
[13] S. E. Rodabaugh, The Hausdorff separation axiom for fuzzy topological spaces, Topology
Appl., 11 (1980), 319–334.
[14] F. G. Shi, Pointwise uniformities and pointwise metrics on fuzzy lattices, Chinese Science
Bulletin, 42 (1997), 718–720.
[15] F. G. Shi, Pointwise uniformities in fuzzy set theory, Fuzzy Sets Syst., 98 (1998), 141-146.
[16] F. G. Shi, Fuzzy pointwise complete regularity and imbedding theorem, The Journal of Fuzzy
Mathematics, 7 (1999), 305–310.
[17] F. G. Shi, A new approach to L-T2, L-Urysohn, and L-completely Hausdorff axioms, Fuzzy
Sets Syst., 157 (2006), 794–803.
[18] F. G. Shi, The Urysohn axiom and the completely Hausdorff axiom in L-topological spaces,
Iranian Journal of Fuzzy Systems, 7(1) (2010), 33–45.
[19] F. G. Shi, (L;M)-fuzzy metric spaces, Indian J. Math., 52 (2010), 231-250.
[20] F. G. Shi, Regularity and normality of (L;M)-fuzzy topological spaces, Fuzzy Sets Syst., 182
(2011), 37–52.
[21] A. P. ˇSostak, On a fuzzy topological structure, Suppl. Rend. Circ. Mat. PalermoSer. II, 11
(1985), 89–103.
[22] A. P. ˇSostak, Two decades of fuzzy topology: basic ideas, notions and results, Russian Math.
Surveys, 44 (1989), 125–186.
[23] M. Ying, A new approach to fuzzy topology (I), Fuzzy Sets Syst., 39 (1991), 303–321.

[24] Y. L. Yue and J. M. Fang, Generated I-fuzzy topological spaces, Fuzzy Sets Syst., 154 (2005),
103–117.
[25] Y. L. Yue and J. M. Fang, On separation axioms in I-fuzzy topological spaces, Fuzzy Sets
Syst., 157 (2006), 780–793.