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ISSN XXXX XXXX © 2016 IJESC Research Article Volume 6 Issue No. 8 Fuzzy Topological Spaces and Fuzzy Multifunctions Sentamilselv i.M 1 , Mary sophia.S2 Assistant Professor1 , Research Scholar2 Depart ment of Mathemat ics Vivekanandha College of Arts & Science for Wo men (Autonomous), Namakkal, Tamilnadu, India Abstract: In this paper, we introduced several fuzzy topological concepts and examine their properties. Our basic tool in that process is the notion of the fuzzy neighbourhood of a point and also introduce the notion of fuzzy mu ltifunctions, linear fuzzy mult ifunctions and study their properties. Keywords: fu zzy topological space, fuzzy boundary, connected, fuzzy mu lt ifunction, fu zzy upper semicontinuous, fuzzy lower semicontinuous, linear fuzzy mu ltifunctions, fuzzy balanced, fu zzy absorbing . All elements of - open sets). INTRODUCTION The fundamental concept of a fuzzy set was first introduced by Zadeh. Since then work has been done by many authors, in several directions, which has resulted in the formation of a new mathematical field called “fuzzy mathematics”. The theory of fuzzy topological spaces is a branch of such mathematics.Chang was the first to introduce the notion of fuzzy topology. Several others continued the work in this area. We remark that general point set topology can be regarded as a special case of fuzzy topology, where all membership functions are just characteristic functions.In this dissertation entitled, ‘Fuzzy Topological S paces and Fuzzy Multifunctions ’, we introduced several fuzzy topological concepts and examine their properties. Our basic tool in that process is the notion of the fuzzy neighbourhood of a point and also introduce the notion of fuzzy mu ltifunctions, linear fuzzy mu ltifunctions and study their properties. briefly PRELIMINARIES In this chapter, we discussed some basic defin itions which are needed for the further definition. Clearly then 1. 2. are set to be open fuzzy set (or Then an element is said to be a closed fuzzy set (or b riefly - closed ) if and only if is -open. 2.3 Definiti on A map fro m closure operator if for all Kuratowski axio ms: (i) (ii) into (i.e., idempotent), the is said to be a it satisfies the four closure operator is (iii) (iv) An equivalent way to define is is always is the following: -closed and -closed. 2.4 Definiti on A fuzzy topology on this is the collection 2.1 Definiti on Let be a set. A fuzzy set in is an element in that is a function from into Actually this is the membership function of a fuzzy set of 2.2 Definiti on A fuzzy topology to be a subset of such that and let Then if and only if there is a is called an interior such that The least upper bound of all interior fuzzy sets of the interior of and it is denoted by is -open). is called (clearly 2.5 Definiti on (1) (2) If (3) If Then the pair Now let fuzzy set of then for all The fuzzy boundary of then denoted by This is defined to be the infrimu m o f all -closed sets with the property for all for wh ich we have is called a fuzzy topol ogical space. International Journal of Engineering Science and Computing, August 2016 2161 http://ijesc.org/ Clearly, is -closed and Hence, indeed is a Hausdorff fu zzy topological space. 2.6 Definiti on If a map continuous) Hence the proof and are two fuzzy topological space, is said to be fuzzy continuous ( or briefly if for all 3.3 Definiti on Let be an fuzzy topological space. We define the following notions. 2.7 Definiti on (i) A fuzzy set in an fu zzy topological space is said to be a neighborhood of the points if and only if there is a such that and We will denote a neighborhood of by then it is called an open neighborhood of 3. A fuzzy set is said to be fuzzy dense ( dense) if and only if A fuzzy set is said to be fuzzy boundary ( -boundary) if and only if A fuzzy set is said to be fuzzy nowhere dense ( -nowhere dense) if and only if is F-boundary. (ii) (iii) If FUZZY TOPOLOGICAL SPACE In this chapter, we discussed about some theorems on fuzzy topological space. 3.4 Theorem Let and be an fuzzy topological space. Then 3.1 Definiti on An fuzzy topological space is said to be Haus dorff if and only if for all such that is - boundary if and only if (ii) is - boundary if and only if there exist Proof and (i) 3.2 Theorem If space, (i) is a Hausdorff fu zzy -open bijection, then is an is also a Hausdorff Proof So Given that, is a Hausdorff fu zzy topological space, an fuzzy topological space. And is an -open bijection. is also a , since . is -boundary. (ii) Now suppose that Fro m part (i) we have that is -boundary. But So Since is closed, and (3.1) we know Since by hypothesis, is a Hausdorff fuzzy topological space, (3.2) Fro m (3.1) and (3.2), we conclude that there exists such that and But Now, since is an -open map and both belong to So wh ich imp lies that Therefore, by part (i) of this theorem, Also Finally, (see which means that Next, let and Let that fu zzy But Let proved we can show that topological space. To prove that, fu zzy topological space Hausdorff fu zzy topological space. we Theorem:2.6), topological an fuzzy topological space. And As and we finally deduce that is -boundary. Hence the proof International Journal of Engineering Science and Computing, August 2016 2162 http://ijesc.org/ 3.5 Theorem If Let and be fu zzy topological space. Then -boundary sets are stable under -continuous surjections If, in addition, is -closed then the family of -nowhere dense sets is also stable. So is -nowhere dense, then ( . But Hence Proof Which means (theorem 2.8 (i)) that Given that, and be fuzzy topological space. Then -boundary sets are stable under -continuous surjections If, in addition, is -closed To prove that, the family of also stable. Therefore is -boundary. is an - nowhere dense set. Hence the proof -nowhere dense sets is 3.6 Definiti on First we will show that for If Fro m the definition of fu zzy interiors we have , then is said to be dense in if and only if 3.7 Definiti on An fuzzy topological space -compact for = if and only if for all exists a neighborhood But because is -continuous and is said to be locally of such that there is -compact. ,we have that 3.8 Theorem So is If -open in is a locally space, -compact fuzzy topological is a Hausdorff fuzzy topological space and Now d irectly fro m the definit ion of the fu zzy interior is an -continuous and -open surjection then is also locally -co mpact. we deduct that Proof Applying in both sides and since Given that, for all topological space, (fro m the surjective of ), space and We deduct that is a locally -compact fuzzy is a Hausdorff fu zzy topological is an -continuous and -open surjection. as claimed. To prove that, is also locally -co mpact. Now let be an -boundary set. Let We know fro m theorem 3.4 (i) that and such that which means that Since by defin ition Let be a neighborhood of Since is -open, then such that is -compact. is a neighborhood of Also So But (3.3) , So But since is -continuous, Hence again by theorem 3.4 (i), we conclude that is is - boundary. Finally, assume that in addit ion is -compact and so it is also -closed. (3.4) -closed. Then ( International Journal of Engineering Science and Computing, August 2016 Hence But again by the -continuity of 2163 we have that http://ijesc.org/ (3.5) Therefore, fro m (3.3),(3.4) and (3.5) above, Hence we is disconnected, which contradicts the hypothesis of the proposition. conclude that So Hence the proof is -co mpact. Since FUZZY MULTIFUNCTIONS 4. was arbitrary, we conclude that is indeed locally -compact. In this chapter, we discussed about some theorems on fuzzy mu ltifunctions and linear fuzzy mu lt ifunctions. Hence the proof 4.1 Definiti on 3.9 Definiti on Let An fuzzy topological space such that and an fuzzy topological space. Let is said to be connected if and only if there do not exist -open sets, A subset is to be connected if it is connected as a fuzzy topological be an ordinary topological space and a mapping such that for every we say that i) is a fuzzy multifunction ( -mult ifunction). every neighborhood neighborhood of 3.10 Theorem and be fuzzy topological space, is -continuous and is connected then ii) is also is said to be every connected. or there exists such that a for all i.e., -lower semicontinuous if and only if for such that neighborhood of there exists a such that for all . Proof Given that, is and be fuzzy topological space, -continuous and 4.2 Theorem is connected. is an To prove that, is also connected. Suppose not. Then we can find -upper semicontinuous mult ifunction if and only if for all is open in such that Proof and Then since to is a fu zzy set. Then is said to be -upper semicontinuous if and only if for subspace of If be is -continuous, both and belong Also we have that First Let suppose that is -upper semicontinuous. Then fro m definit ion 3.1(i) We know that there exists a neighborhood for all So claimed. of Which means that such that is open as The other direction is just the definition of semicontinuity. -upper Hence the proof And 4.3 Definiti on An fuzzy topological space regular if and only if for set such that that International Journal of Engineering Science and Computing, August 2016 2164 is said to be quasi a fuzzy point and (i.e., and -closed there exist and http://ijesc.org/ such 4.4 Definiti on So we have that for A fuzzy multifunction and only if when neighborhood (i.e., of ( ) wh ich imp lies that is said to be closed if and is indeed fuzzy convex. there is a such that Hence the proof (i.e., for which if 4.7 Definiti on 4.5 Definiti on A fuzzy mult ifunction A fuzzy set is said to be (i) fuzzy bal anced if and only if is said to be linear if and only if : for (ii) i) ii) iii) fuzzy absorbing if and only if for all If then 4.8 Definiti on iv) If A fuzzy set then is said to be fuzzy cone if and only if for every If, in addition, is fuzzy convex, then it is called a fuzzy convex c one. 4.9 Theorem 4.6 Theorem If If is a linear fu zzy mult ifunction with the property that for convex then then if is a cone and mu ltifunction then is a linear fuzzy is a fuzzy cone. is Proof is fu zzy convex. Given that, Proof Let and is a cone and We have is a linear fuzzy mu ltifunction. To prove that, Let For any is a fu zzy cone. We have ]( ) But since by hypothesis So Hence is a cone in for all is indeed a fuzzy cone. Hence the proof for all International Journal of Engineering Science and Computing, August 2016 2165 http://ijesc.org/ CONCLUS ION In this dissertation, we introduced several fuzzy topological concepts and examine their properties and also introduced the notion of fuzzy multifunctions and linear fuzzy mu ltifunctions. We discussed about some properties of fuzzy mu ltifunctions and linear fuzzy mult ifunctions. In Chapter III, the concept of fuzzy topological spaces. In Chapter II, we have discussed some basic definitions of fuzzy topological spaces. In chapter I, the definit ions which are necessary for above were given. BIB LIOGRAPHY 1. K. AZAD, On fu zzy semicountinuity, fu zzy almost continuity and fuzzy weak continuity, J. Math. Anal. Appl. 82 (1981), 14-32. 2. C. CHANG, Fu zzy topological spaces, J. Math. Anal. Appl. 24 (1968), 191-201. 3. A. KATSARAS AND D. LIU, Fu zzy vector spaces and fuzzy topological vector spaces, J. Math. Anal. Appl. 58 (1977), 135-146. 4. NIKOLAOS S. PAPAGEORGIOU, fu zzy topology and fuzzy mu ltifunctions, J. Math. Anal. Appl.109(1985), 397-425.. 5. C. WONG, Fuzzy points and topology, J. Math. Anal. Appl. 46 (1974), 314-328. International Journal of Engineering Science and Computing, August 2016 2166 http://ijesc.org/