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

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.

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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.Conflicts of Interest

The authors declare no conflicts of interest.

[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. |

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