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J. Bangladesh Acad. Sci., Vol. 40, No. 2, 117-124, 2016 ON R0 SPACE IN L-TOPOLOGICAL SPACES RAFIQUL ISLAM AND M. S. HOSSAIN1 Department of Mathematics, Pabna University of Science and Technology, Pabna, Bangladesh ABSTRACT R0 space in L-topological spaces are defined and studied. The authors give eight definitions of R0 space in L-topological spaces and discuss certain relationship among them. They showed that all of these satisfy ‘good extension’ property. Moreover, some of their other properties are obtained. Key words: L-fuzzy set, L-fuzzy topology INTRDUCTION The concept of R0-property first defined by Shanin (1943) and there after Naimpally (1967), Dude (1974), Hutton (1975 a, b), Dorsett (1978), Ali (1990) and Caldas (2000). Khedr (2001) and Ekici (2005) defined and studied many characterizations of R0properties. Later, this concept was generalized to ‘fuzzy R0-propertise’ by Zhang (2008), Keskin (2009) and Roy (2010) and many other fuzzy topologists. In this paper the workers defined possible eight notions of R0 space in L-topological spaces and they showed that this space possesses many nice properties which are hereditary productive and projective. Definition: Let X be a non-empty set and I = [0, 1]. A fuzzy set in X is a function which assigns to each element , a degree of membership, (Zadeh 1965). Definition: Let X be a non-empty set and L be a complete distributive lattice with 0 and 1. An L-fuzzy set in X is a function which assigns to each element , a degree of membership, (Goguen 1967). Definition: Let complement of in Definition: If then the authors refer to 1967). 1 be an L-fuzzy set in (Goguen 1967). .Then and is an L-fuzzy sets in defined by as a constant L-fuzzy sets and denoted it by Department of Mathematics, University of Rajshahi, Rajshahi-6205, Bangladesh. is called the itself (Goguen 118 ISLAM AND HOSSAIN In particular, these are the constant L-fuzzy sets 0 and 1. Definition: An L-fuzzy point function in where Definition: An L-fuzzy point if and only if and al. 1997). is a special L-fuzzy sets with membership (Liu et al.1997). is said to belong to an L-fuzzy set . That is in (Liu et Definition: Let , be a non-empty set and be the collection of all mappings from into , i. e. the class of all fuzzy sets in . A fuzzy topology on is defined as a family of members of , satisfying the following conditions: if for each then if then . The pair is called a fuzzy topological space (fts, in short) and the members of are called -open (or simply open) fuzzy sets. A fuzzy set is called a -closed (or simply closed) fuzzy set if (Chang 1968). Definition: Let X be a non-empty set and L be a complete distributive lattice with 0 and 1. Suppose that be the sub collection of all mappings from to .Then is called L-topology on if it satisfies the following conditions: (i) (ii) If (iii) If of then for each then . Then the pair is called an L-topological space (lts, for short) and the members are called open L-fuzzy sets. An L-fuzzy sets is called a closed L-fuzzy set if (Jin-xuan and Bai-lin 1998). Definition: Let by and defined as The interior of al.1997). be an L-fuzzy set in lts . Then the closure of is denoted . written is defined by (Liu et Definition: An L-fuzzy singleton in is an L-fuzzy set in which is zero everywhere except at one point say , where it takes a value say with and . The authors denote it by and iff (Zadeh 1965). Definition: An L-fuzzy singleton is said to be quasi-coincident (q-coincident, in short) with an L-fuzzy set in , denoted by iff . Similarly, an L-fuzzy set in is said to be q-coincident with an L-fuzzy set in , denoted by ON R0 SPACE IN L-TOPOLOGICAL SPACES if and only if for all , where not q-coincident with an L-fuzzy set in Definition: Let is a fuzzy set in 119 for some denote an L-fuzzy set (Liu et al.1997). . Therefore iff in is said to be be a function and be fuzzy set in which membership function is defined by if . Then the image (Chang 1968). Definition: Let be a real-valued function on an L-topological space. If is open for every real , then is called lower-semi continuous function (lsc, in short) (Liu et al.1997). Definition: Let from into and be two L-topological space and . Then is called (i) Continuous iff for each open L-fuzzy set (ii) Open iff . for each open L-fuzzy set (iii) Closed iff . is s-closed for each (iv) Homeomorphism iff al.1997). is bijective and both be a mapping . where is closed L-fuzzy set in and are continuous (Liu et Definition: Let be a nonempty set and be a topology on . Let be the set of all lower semi continuous (lsc) functions from to (with usual topology). Thus for each . It can be shown that is a L-topology on . Let “P” be the property of a topological space and LP be its L-topological analogue. Then LP is called a “good extension” of P “if the statement has P iff has LP” holds good for every topological space (Liu et al. 1997). Definition: Let be a family of L-topological space. Then the space is called the product lts of the family where denote the usual product L-topologies of the families of L-topologies on (Li 1998). R0 SPACES IN L-TOPOLOGY The authors now give the following definitions of R0 L-topological spaces. Definition: An lts (a) if is called whenever with then 120 ISLAM AND HOSSAIN such that (b) . if whenever then with such that (c) . if for any pair of distinct L-fuzzy points with then such that (d) whenever . if for all pairs of distinct L-fuzzy singletons whenever (e) with then and such that . if for any pair of distinct L-fuzzy points with (f) then whenever such that . if for any pair of distinct L-fuzzy points with whenever then such that . (g) if whenever then (h) with such that if whenever such that . with then . “GOOD EXTENSION”, HEREDITARY PROPERTY AND PRODUCT L-TOPOLOGY Now all the definitions , and are ‘good extensions’ of Theorem: Let , as shown below: be a topological space. Then is iff is , for Proof: Let the topological space fuzzy topological is with . Let , then it is clear that and , with . Since , for any is , then there exist . Now consider the characteristics function with . Thus Conversely, let . Choose be , and characteristic function is , The workers shall prove that the . Choose with see that be . , the workers shall prove that , with . Also it is clear that , then . The authors is such that is , but we know that the . Since Again since ON R0 SPACE IN L-TOPOLOGICAL SPACES is 121 lower semi continuous function then , . Hence and from above, we get is . Similarly the authors showed that are extension’ property. also hold ‘good 122 ISLAM AND HOSSAIN Theorem: Let be an lts, and is , then (a) is . (b) is (c) is is . (d) is is . (e) is (f) is (g) is (h) is is . is . is is is . . . Proof: The authors proved only (b). Suppose is L-topological space and . The workers shall prove is . Let , , and with . Then with as . Consider be the extension function of on the set X, then it is clear that . Since is . Then such that . For , we find such that as . Hence it is clear that the subspace is . Similarly, (a), (c), (d), (e), (f), (g), (h) can be proved. Theorem: Given product of L-topological space is where be a family of L-topological space. Then the is iff each coordinate space . Proof: Let each coordinate space authors show that the product space is with Choose be . Suppose . Then the , , and , but they have , then there exist at least one such that and , . Since is for each , then such that . Now take where and for , then such that . Hence the product L-topological space is . Conversely, let the product of L-topological space is . The workers shall prove each coordinate space is also . Choose and with . Now, construct such that where for and . Then . Further, let be a projection map from into . Now, the workers observe that ON R0 SPACE IN L-TOPOLOGICAL SPACES Then Then that 123 , i.e for , with . Since the product space is . such that . Now choose any L-fuzzy point in . a basic open L-fuzzy set such that which implies or that and hence and . Further, , since for and hence from , . Thus we have . Now consider then with . This showing that is . Moreover one can verify that is is is is is is is is is is is is is Hence, is the . workers see that properties are productive and projective. MAPPING IN L-TOOLOGICAL SPACES The present authors show that and continuous maps for Theorem: Let and property is preserved under one-one, onto . be two L-topological space and be one-one, onto, L-continuous and L-open map, then(a) is is (b) is (c) is is . (d) is is . (e) is is is . . . 124 ISLAM AND HOSSAIN (f) is (g) is (h) is is . is . is Proof: Suppose . Let is onto, one. is , such that and . . The authors shall prove that is with . Since and as is one- Now and Since is L-continuous then and , . Since is , then such that . Now . Since is L-open, . Now, it is clear that such that . Hence, it is clear that the L-topological space is . Similarly (a), (c), (d), (e), (f), (g), (h) can be proved. Theorem: Let and be two L-topological spaces and be one-one, L-continuous and L-open map, then(a) is (b) is (c) is is . (d) is is . (e) is (f) is (g) is (h) is Proof: Suppose . Let is one-one map then is . is . is . is is is is . . . . The workers shall prove that and with , . is . Since ON R0 SPACE IN L-TOPOLOGICAL SPACES Now 125 as is one-one And So, the authors have L-open map. , with Since is such . This implies that and as is L-continuous and . Now it is clear that such that , . Hence the L-topological space . Similarly (a), (c), (d), (e), (f), (g), (h) can be proved. , as is that is REFERENCES Ali, D. M., Wuyts, P. and A.K. Srivastava. 1990. On the R0-property in fuzzy topology, Fuzzy Sets and Systems 38: 97-113. Chang, C.L. 1968. Fuzzy topological spaces. J. Math. Anal Appl. 24: 182-192. Caldas, M., Jafari, S. and T. Noiri. 2000. Characterization of pre R0 and R1 topological spaces. Topology proceeding 25: 17-30. Dorsett, C. 1978. R0 and R1 topological spaces. Math. Vesnik 2: 112-117. Dude, K.K. 1974. A note on R0- topological spaces. Math.Vesnik 11: 203-208. Ekici, E. 2005. On R spaces. Int. J. Pure. Appl. Math. 25: 163-172. Goguen, J.A. 1967. L-fuzzy sets. J. Math. Anal. Appl. 18: 145-174. Hutton, B. 1975. Normality in fuzzy topological spaces. J. Math. Anal. Appl. 11:74-79. Hutton, B. 1975. Products of fuzzy topological spaces. J. Math. Anal. Appl. 11: 59-67. Jin-xuan, F. and Ren Bai-lin.1998. A set of new separation axioms in L-fuzzy topological spaces. Fuzzy Sets and Systems 96: 359-366. Keskin, A. and T. Noiri. 2009. On – R0 and – R1 spaces. Miskole Mathematics Notes 10(2): 137143. Khedr, F.M., Zeyada, F.M. and Sayed. 2001. On separation axioms in fuzzyfing topology. Fuzzy Sets and Systems 119: 439-458. Li, S.G. 1998. Remarks on product of L-fuzzy topological spaces. Fuzzy Sets and Systems 94: 121124. Liu Ying-Ming and Luo Mao-Kang. 1997. Fuzzy Topology. Copyright © by World Scientific Publishing Co. Pte. Ltd. Lowen, R. 1976. Fuzzy topological spaces and fuzzy compactness. J. Math. Anal. Appl. 56: 621633. Naimpally, S.A. 1967. On R0-topological spaces. Ann.Univ. Sci. Budapest. Eotuos. Sect. Math. 10: 53-54. Nouh, A.A. 2002. C-closed sets in L-fuzzy topological space and some of its applications.Turk J. Math. 26: 245-261. Pu Ming, Pao, Ming and Liu Ying. 1980. Fuzzy topology I. Neighborhood Structure of a fuzzy point and Moore-Smith convergence. J. Math.Anal. Appl. 76: 571-599. Pu Ming, Pao, Ming and Liu Ying. 1980. Fuzzy topology. II. Product and quotient spaces. J. Math. Anal. Appl. 77: 20-37. 126 ISLAM AND HOSSAIN Roy, B. and M.N. Mukherjee. 2010. A unified theory for R0, R1 and certain other separation properties and their variant form. Bol. Soc. Paran. Math. 28: 15-24. Shanin, N.A. 1943. On separation in topological spaces; C. R. (Doklady) Acad. Sci. URSS (N. S.) 38: 110-113. Zadeh, L.A. 1965. Fuzzy sets. Information and Control 8: 338-353. Zhang, J., Wang, W. and M. Hu. 2008. Topology on the set of R0 semantics for R0 algebra. Soft Comput. 12: 585-591. Zhao Bin. 2005. Inclusive regular separation in L-fuzzy topological spaces. J. of Mathematical Research and Exposition 25(4). (Received revised manuscript on 30 August, 2016)