A Unified Theory (I) for Neighborhood Systems and Basic Concepts on Fuzzifying Topological Spaces

Author(s)
Osama Rashed Sayed

ABSTRACT

This paper considers fuzzifying topologies, a special case of I-fuzzy topologies (bifuzzy topologies), introduced by Ying. It investigates topological notions defined by means of -open sets when these are planted into the frame-work of Ying’s fuzzifying topological spaces (by Lukasiewicz logic in [0, 1]). In this paper we introduce some sorts of operations, called general fuzzifying operations from P(X) to , where (X, τ) is a fuzzifying topological space. By making use of them we contract neighborhood structures, derived sets, closure operations and interior operations.

This paper considers fuzzifying topologies, a special case of I-fuzzy topologies (bifuzzy topologies), introduced by Ying. It investigates topological notions defined by means of -open sets when these are planted into the frame-work of Ying’s fuzzifying topological spaces (by Lukasiewicz logic in [0, 1]). In this paper we introduce some sorts of operations, called general fuzzifying operations from P(X) to , where (X, τ) is a fuzzifying topological space. By making use of them we contract neighborhood structures, derived sets, closure operations and interior operations.

Cite this paper

O. Sayed, "A Unified Theory (I) for Neighborhood Systems and Basic Concepts on Fuzzifying Topological Spaces,"*Applied Mathematics*, Vol. 3 No. 9, 2012, pp. 983-996. doi: 10.4236/am.2012.39146.

O. Sayed, "A Unified Theory (I) for Neighborhood Systems and Basic Concepts on Fuzzifying Topological Spaces,"

References

[1] U. H?hle, “Many Valued Topology and Its Applications,” Kluwer Academic Publishers, Dordrecht, 2001. doi:10.1007/978-1-4615-1617-0

[2] U. H?hle and S. E. Rodabaugh, “Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory,” Handbook of Fuzzy Sets Series, Vol. 3, Kluwer Academic Publishers, Dordrecht, 1999.

[3] U. H?hle, S. E. Rodabaugh and A. ?ostak (Eds.), “Special Issue on Fuzzy Topology,” Fuzzy Sets and Systems, Vol. 73, 1995, pp. 1-183.

[4] T. Kubiak, “On Fuzzy Topologies,” Ph.D. Thesis, Adam Mickiewicz University, Poznan, 1985.

[5] Y. M. Liu and M. K. Luo, “Fuzzy Topology,” World Scientific, Singapore, 1998.

[6] G. J. Wang, “Theory of L-Fuzzy Topological Spaces,” Shanxi Normal University Press, Xi’an, 1988 (in Chinese).

[7] C. L. Chang, “Fuzzy Topological Spaces,” Journal of Mathematical Analysis and Applications, Vol. 24, No. 1, 1968, pp. 182-190. doi:10.1016/0022-247X(68)90057-7

[8] J. A. Goguen, “The Fuzzy Tychonoff Theorem,” Journal of Mathematical Analysis and Applications, Vol. 43, No. 3, 1973, pp. 182-190. doi:10.1016/0022-247X(73)90288-6

[9] J. L. Kelley, “General Topology,” Van Nostrand, New York, 1955.

[10] U. H?hle, “Uppersemicontinuous Fuzzy Sets and Applications,” Journal of Mathematical Analysis and Applications, Vol. 78, No. 2, 1980, pp. 659-673. doi:10.1016/0022-247X(80)90173-0

[11] U. H?hle and A. ?ostak, “Axiomatic Foundations of Fixed-Basis Fuzzy Topology,” In: U. H?hle and S. E. Rodabaugh, Eds., Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, Vol. 3, Kluwer Academic Publishers, Dordrecht, 1999, pp. 123-272.

[12] S. E. Rodabaugh, “Categorical Foundations of VariableBasis Fuzzy Topology,” In: U. H?hle and S. E. Rodabaugh, Eds., Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, Vol. 3, Kluwer Academic Publishers, Dordrecht, 1999, pp. 273-388.

[13] J. B. Rosser and A. R. Turquette, “Many-Valued Logics,” North-Holland, Amsterdam, 1952.

[14] M. S. Ying, “A New Approach for Fuzzy Topology (I),” Fuzzy Sets and Systems, Vol. 39, No. 3, 1991, pp. 303-321. doi:10.1016/0165-0114(91)90100-5

[15] M. S. Ying, “A New Approach for Fuzzy Topology (II),” Fuzzy Sets and Systems, Vol. 47, No. 2, 1992, pp. 221-223. doi:10.1016/0165-0114(92)90181-3

[16] K. M. Abd El-Hakeim, F. M. Zeyada and O. R. Sayed, “ -Continuity and D(c, )-Continuity in Fuzzifying Topology,” The Journal of Fuzzy Mathematics, Vol. 7, No. 3, 1999, pp. 547-558.

[17] K. M. Abd El-Hakeim, F. M. Zeyada and O. R. Sayed, “Pre-Continuity and D(c, P)-Continuity in Fuzzifying Topology,” Fuzzy Sets and Systems, Vol. 119, No. 3, 2001, pp. 459-471. doi:10.1016/S0165-0114(99)00097-4

[18] F. H. Khedr, F. M. Zeyada and O. R. Sayed, “Fuzzy Semi-Continuity and Fuzzy Csemi-Continuity in Fuzzifying Topology,” The Journal of Fuzzy Mathematics, Vol. 7, No. 1, 1999, pp. 105-124.

[19] F. H. Khedr, F. M. Zeyada and O. R. Sayed, “ -Continuity and -Continuity in Fuzzifying Topology,” Fuzzy Sets and Systems, Vol. 116, No. 3, 2000, pp. 325-337. doi:10.1016/S0165-0114(98)00386-8

[20] T. Noiri and O. R. Sayed, “Fuzzy Open Sets and Fuzzy -Continuity in Fuzzifying Topology,” International Journal of Mathematics and Mathematical Sciences, Vol. 31, No. 1, 2002, pp. 51-63. doi:10.1155/S0161171202007755

[21] T. Noiri and O. R. Sayed, “Fuzzy Open Sets and Fuzzy -Continuity in Fuzzifying Topology,” Scientiae Mathematicae Japonicae, Vol. 55, No. 2, 2002, pp. 255263.

[22] S. Kasahara, “Operation-Compact Spaces,” Mathematica Japonica, 24, No. 1, 1979, pp. 97-105.

[23] D. S. Jankovic’, “Properties of -Continuous Functions,” The Proceedings of Fifth Prague Topological Symposium, 1981.

[24] M. E. Abd El-Monsef, F. M. Zeyada and A. S. Mashhour, “Operations on Topologies and Its Applications on Some Types of Covering,” Annales de la Société Scientifique de Bruxelles, Vol. 79, 1983, pp. 155-172.

[25] E. E. Kerre, A. A. Nouh and A. Kandil, “Operations and the Class of Fuzzy Sets on a Universe Endowed with a Fuzzy Topology,” Proceedings of IFSA, Vol. 109-113, Brussels, 1991.

[26] A. Kandil, E. E. Kerre and A. A. Nouh, “Operations and Mappings on Fuzzy Topological Spaces,” Annales de la Société Scientifique de Bruxelles, Vol. 105, No. 4, 1991, pp. 167-188.

[1] U. H?hle, “Many Valued Topology and Its Applications,” Kluwer Academic Publishers, Dordrecht, 2001. doi:10.1007/978-1-4615-1617-0

[2] U. H?hle and S. E. Rodabaugh, “Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory,” Handbook of Fuzzy Sets Series, Vol. 3, Kluwer Academic Publishers, Dordrecht, 1999.

[3] U. H?hle, S. E. Rodabaugh and A. ?ostak (Eds.), “Special Issue on Fuzzy Topology,” Fuzzy Sets and Systems, Vol. 73, 1995, pp. 1-183.

[4] T. Kubiak, “On Fuzzy Topologies,” Ph.D. Thesis, Adam Mickiewicz University, Poznan, 1985.

[5] Y. M. Liu and M. K. Luo, “Fuzzy Topology,” World Scientific, Singapore, 1998.

[6] G. J. Wang, “Theory of L-Fuzzy Topological Spaces,” Shanxi Normal University Press, Xi’an, 1988 (in Chinese).

[7] C. L. Chang, “Fuzzy Topological Spaces,” Journal of Mathematical Analysis and Applications, Vol. 24, No. 1, 1968, pp. 182-190. doi:10.1016/0022-247X(68)90057-7

[8] J. A. Goguen, “The Fuzzy Tychonoff Theorem,” Journal of Mathematical Analysis and Applications, Vol. 43, No. 3, 1973, pp. 182-190. doi:10.1016/0022-247X(73)90288-6

[9] J. L. Kelley, “General Topology,” Van Nostrand, New York, 1955.

[10] U. H?hle, “Uppersemicontinuous Fuzzy Sets and Applications,” Journal of Mathematical Analysis and Applications, Vol. 78, No. 2, 1980, pp. 659-673. doi:10.1016/0022-247X(80)90173-0

[11] U. H?hle and A. ?ostak, “Axiomatic Foundations of Fixed-Basis Fuzzy Topology,” In: U. H?hle and S. E. Rodabaugh, Eds., Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, Vol. 3, Kluwer Academic Publishers, Dordrecht, 1999, pp. 123-272.

[12] S. E. Rodabaugh, “Categorical Foundations of VariableBasis Fuzzy Topology,” In: U. H?hle and S. E. Rodabaugh, Eds., Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, Vol. 3, Kluwer Academic Publishers, Dordrecht, 1999, pp. 273-388.

[13] J. B. Rosser and A. R. Turquette, “Many-Valued Logics,” North-Holland, Amsterdam, 1952.

[14] M. S. Ying, “A New Approach for Fuzzy Topology (I),” Fuzzy Sets and Systems, Vol. 39, No. 3, 1991, pp. 303-321. doi:10.1016/0165-0114(91)90100-5

[15] M. S. Ying, “A New Approach for Fuzzy Topology (II),” Fuzzy Sets and Systems, Vol. 47, No. 2, 1992, pp. 221-223. doi:10.1016/0165-0114(92)90181-3

[16] K. M. Abd El-Hakeim, F. M. Zeyada and O. R. Sayed, “ -Continuity and D(c, )-Continuity in Fuzzifying Topology,” The Journal of Fuzzy Mathematics, Vol. 7, No. 3, 1999, pp. 547-558.

[17] K. M. Abd El-Hakeim, F. M. Zeyada and O. R. Sayed, “Pre-Continuity and D(c, P)-Continuity in Fuzzifying Topology,” Fuzzy Sets and Systems, Vol. 119, No. 3, 2001, pp. 459-471. doi:10.1016/S0165-0114(99)00097-4

[18] F. H. Khedr, F. M. Zeyada and O. R. Sayed, “Fuzzy Semi-Continuity and Fuzzy Csemi-Continuity in Fuzzifying Topology,” The Journal of Fuzzy Mathematics, Vol. 7, No. 1, 1999, pp. 105-124.

[19] F. H. Khedr, F. M. Zeyada and O. R. Sayed, “ -Continuity and -Continuity in Fuzzifying Topology,” Fuzzy Sets and Systems, Vol. 116, No. 3, 2000, pp. 325-337. doi:10.1016/S0165-0114(98)00386-8

[20] T. Noiri and O. R. Sayed, “Fuzzy Open Sets and Fuzzy -Continuity in Fuzzifying Topology,” International Journal of Mathematics and Mathematical Sciences, Vol. 31, No. 1, 2002, pp. 51-63. doi:10.1155/S0161171202007755

[21] T. Noiri and O. R. Sayed, “Fuzzy Open Sets and Fuzzy -Continuity in Fuzzifying Topology,” Scientiae Mathematicae Japonicae, Vol. 55, No. 2, 2002, pp. 255263.

[22] S. Kasahara, “Operation-Compact Spaces,” Mathematica Japonica, 24, No. 1, 1979, pp. 97-105.

[23] D. S. Jankovic’, “Properties of -Continuous Functions,” The Proceedings of Fifth Prague Topological Symposium, 1981.

[24] M. E. Abd El-Monsef, F. M. Zeyada and A. S. Mashhour, “Operations on Topologies and Its Applications on Some Types of Covering,” Annales de la Société Scientifique de Bruxelles, Vol. 79, 1983, pp. 155-172.

[25] E. E. Kerre, A. A. Nouh and A. Kandil, “Operations and the Class of Fuzzy Sets on a Universe Endowed with a Fuzzy Topology,” Proceedings of IFSA, Vol. 109-113, Brussels, 1991.

[26] A. Kandil, E. E. Kerre and A. A. Nouh, “Operations and Mappings on Fuzzy Topological Spaces,” Annales de la Société Scientifique de Bruxelles, Vol. 105, No. 4, 1991, pp. 167-188.