A General Framework For Compactness in L-Topological Spaces FULL TEXT

A general framework for the concepts of compactness, countable compactness, and the Lindel¨of property are introduced in L-topological spaces by means of several kinds of open L-sets and their inequalities when L is a complete DeMorgan algebra. The method used in this paper shows that these results are valid for any kind of open L-sets and thus we do not need to repeat it for each kind separately.

Keywords:

L-topological space, Compactness, Countable compactness, Lindel¨of property 2000 AMS Classification: 54 A 40,

PDF

___

Andrijevi´c, D. On b-open sets, Mat. Vesnik 48, 59ˆu-64, 1996.
Azad, K. K. On fuzzy semicontinuity, fuzzy almost continuity and fuzzy weakly continuity, J. Math. Anal. Appl. 82, 14–32, 1981.
Balasub¨asamanian, G. On fuzzy β-compact spaces and fuzzy β-extremally disconnected spaces, Kybernetika 33, 271–277, 1997.
Bin Shahna, A. S. On fuzzy strong semicontinuity and fuzzy precontinuity, Fuzzy Sets and Systems 44, 303–308, 1991.
Chadwick, J. J. A generalized form of compactness in fuzzy topological spaces, J. Math. Anal. Appl. 162, 92–110, 1991.
Chang, C. L. Fuzzy topological spaces, J. Math. Anal. Appl. 24, 39–90, 1968.
Dwinger, P. Characterizations of the complete homomorphic images of a completely dis- tributive complete lattice I, Indagationes Mathematicae (Proceedings) 85, 403–414, 1982.
Gantner, T. E., and Steinlage, R. C. and Warren, R. H. Compactness in fuzzy topological spaces, J. Math. Anal. Appl. 62, 547–562, 1978.
Gierz, G. et al. A Compendium of Continuous Lattices (Springer Verlag, Berlin, 1980).
Hanafy, I. M. βS*-compactness in L-fuzzy topological spaces, J. Nonlinear Sci. Appl. 9, –37, 2009.
Hanafy, I. M. Fuzzy γ-open sets and fuzzy γ-continuity, J. Fuzzy Math. 7, 419–430, 1999.
H¨ohle, U. and Rodabaugh, S. E. S. E. Rodabaugh, Ed. Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory 3(Kluwer Academic Publishers, Boston/Dordrecht/London, ). Li, Z. F. Compactness in fuzzy topological spaces, Chinese Kexue Tongbao 6, 321–323, 1983.
Liu, Y. M. and Luo, M. K. Fuzzy Topology (World Scientific, Singapore, 1997).
Liu, Y. M. Compactness and Tychnoff theorem in fuzzy topological spaces, Acta Mathemat- ica Sinica 24, 260–268, 1981.
Lowen, R. A comparsion of different compactness notions in fuzzy topological spaces, J. Math. Anal. Appl. 64, 446–454, 1978.
Lowen, R. Fuzzy topological spaces and fuzzy compactness, J. Math. Anal. Appl. 56, 621– , 1976.
Rodabaugh, S. E. Point-set lattice-theoretic topology, Fuzzy Sets and Systems 40, 297–345, Shi, F. -G. P -compactness in L-topological spaces, J. Nonlinear Sci. Appl. 2, 225–233, 2009.
Shi, F. -G. A new definition of fuzzy compactness, Fuzzy Sets and Systems 158, 1486–1495,
Shi, F. -G. A new form of fuzzy α-compactness, Mathematica Bohemica 131, 15–28, 2006.
Shi, F. -G. Semicompactness in L-topological spaces, International Journal of Mathematics and Mathematical Sciences 12, 1869–1878, 2005.
Shi, F. -G. Semicompactness in L-topological spaces, Int. J. Math. Math. Sci. 12, 1869–1878,
Shi, F. -G. A new notion of fuzzy compactness in L-topological spaces, Information Sciences , 35–48, 2005.
Shi, F. -G. Countable compactness and the Lindel¨of property of L-fuzzy sets, Iranian Journal of Fuzzy Systems 1, 79–88, 2004.
Shi, F. -G. Theory of Lβ-nested sets and Lα-nested and their applications, Fuzzy Systems and Mathematics (In Chinese) 4, 65–72, 1995.
Singal, M. K. and Prakash, N. Fuzzy preopen sets and fuzzy preseparation axioms, Fuzzy Sets and Systems 44, 273–281, 1991.
Wang, G. -J. Theory of L-Fuzzy Topological Space (in Chinese) (Shaanxi Normal University Press, Xi’an, 1988).
Wang, G. J. A new fuzzy compactness defined by fuzzy nets, J. Math. Anal. Appl. 94, 1–23, Zadeh, L. A. Fuzzy sets, Inform. Control 8, 338–353, 1965.
Zhao, D. S. The N -compactness in L-fuzzy topological spaces, J. Math. Anal. Appl. 128, –70, 1987.