Compactness of Fuzzy Sets

Transcription

1 Compactess of uzzy Sets Amai E. Kadhm Departmet of Egieerig Programs, Uiversity College of Madeat Al-Elem, Baghdad, Iraq. Abstract The objective of this paper is to study the compactess of fuzzy sets i fuzzy topological spaces, especially the coectio betwee compact, closed ad bouded fuzzy sets. 1-Itroductio: The cocept of fuzzy sets was itroduced iitially by Zadeh i Sice the, this cocept is used i topology ad some braches of aalysis, may authors have expesively developed the theory of fuzzy sets ad its applicatio [1]. Chag, C. L. i 1968 used the fuzzy set theory for defiig ad itroducig fuzzy topological spaces, while Wog, C. K. i 1973, discussed the coverig properties of fuzzy topological spaces, [2]. Ercey, M. A. i 1979, studied fuzzy metric spaces ad its coectio with statistical metric spaces, Mig P. P. ad Mig L. Y. i 1980, used fuzzy topology to defie eighborhood structure of fuzzy poit ad Moore- Smith covergece, Zike Deg i 1982, studied the fuzzy poit ad discussed the fuzzy metric spaces with the metric defied betwee two fuzzy poits,[ 3]. The mai objective of this paper is to study the relatioship betwee closed ad bouded fuzzy sets ad the compactess of such fuzzy sets. 2- Basic Cocepts i uzzy Topology [3], [4], [5]: Chag C. L. i 1968 itroduced the otio of fuzzy topological spaces, which is ao-empty set X together with a family of fuzzy sets i X which is closed uder arbitrary uio ad fiite itersectio, as it is give precisely i the ext defiitio. We start first with the obvious defiitio of fuzzy topological spaces.

2 Defiitio (3.1),[3][7]: A family T of fuzzy sets of X is called a fuzzy topology for X if ad oly if it satisfyig the followig coditios: (a), 1 X T.where is the empty fuzzy set with membership fuctio 0 ad 1x is the uiversal set with membership fuctio 1. (b) If A, B T, the A B T. (c) If A i T, i J, where J is ay idex set, the A T. ii T is called fuzzy topology for X, ad the pair (X, T ) is a fuzzy topological space. Defiitio (3.2),[3]: A fuzzy set A X* is said to be a ope fuzzy set if A T ad is said to c be closed fuzzy set if A T. X* is the set of all closed ad bouded fuzzy subsets of X Defiitio (3.3), [2][4]: Let (X, T ) be a fuzzy topological space. A family of fuzzy subsets A of I X is said to be a cover of a fuzzy set B i X if ad oly if B A. If each member of the cover is a member of T, the this cover is said to be a ope cover of B. A subcover of B is a subfamily of the cover which is also a cover. Defiitio (3.4),[5]: A fuzzy topological space is compact if ad oly if each ope cover of the space has a fiite subcover. Defiitio (3.5),[5]: A fuzzy poit x r of fuzzy set A i a fuzzy topological space (X, T ) is said to be fuzzy -cluster poit (fuzzy -cluster poit) of a fuzzy set A i X if ad oly if fuzzy closure (respectively, iterior of i the closure) of every ope q- eighborhood of x r is q-coicidet with A 3- Compactess uzzy Set: The compactess set is oe of the fudametal aspects i topological space, i geeral, ad of fuzzy set i particular, therefore, several approaches are proposed to study this subject. Hece, i this sectio, we will give oe of such approaches as a theorem. Also, we will stad ad preset some of the basic ideas for the costructio ad the proof of the completeess of fuzzy set, where the followig abbreviatio is used, X* is the set of all closed ad bouded fuzzy subsets of X. Theorem(4.1): If a fuzzy set is oempty ad is bouded below, the a ifimum exist. Let A be a oempty fuzzy set which is bouded below. Deote by C1 the fuzzy the set of all real umbers which are lower bouds of A, ad let C2 cosist of all other real umbers. We may the show that [C1, C2] is a Dedekid. 1- The ature of the defiitio of C2 is a assurace that each real umber is i C1 or C2. 2- Sice A is bouded below, C1 is o empty ad sice A is ot empty, ot all real umbers are i C1. Hece C2 is ot empty. 3- Let c1 be ay member of C1, ad let C be ay umber such C < C1.it follows that C is less tha a lower

3 boud of A ad hece is also a lower boud of A.therefore c is a member of C1. This implies that all members of C2 exceed all members of C1. 4- Suppose C2 to have a least member. Deote it by c. the c is ot a lower boud of A, ad, as a cosequece, a umber x of A exists such that x < C. Also, betwee x ad C exists aother real umber y, x < y < C, sice y is less tha c, it is ecessarily i C1. Also, sice y exceeds x; a member of A, it is ot a lower boud ad so is ot i C1.the cotradictio implies that C2 does ot possess a least member. Theorem (4.2): The ifimum of a oempty fuzzy set A is either a member of A or a cluster poit of A. Let b if A, be a fuzzy set with membership fuctio b (x) if A (x), x X It is the miimum fuzzy poit of A, if b is a member of A While if b is ot a member of A, the correspodig to each positive umber > 0, the deleted fuzzy eighborhood * N (b, ) cotais a poit a of A. Hece b is a cluster poit of A. Theorem (4.3): A oempty closed fuzzy set A, which is bouded below possesses a miimum poit. By theorem (4.1) the ifimum of the fuzzy set A exists a fuzzy poit b which is either a poit of A or a cluster poit of A. I the latter case it also follows that b is i A, sice A is closed. Hece the ifimum of A belogs to A ad is, of course, the miimum member of A. Theorem (4.4): Ay family of disjoit fuzzy itervals is coutable. Let { I } be a sequece of fuzzy itervals hece, for all (0, 1], let I [ I, I I] ad hece, their is a equivalet two sequece of ofuzzy itervals { I } ad { I } which are coutable hece { I } is coutable. Similarly, usig the idea of -level sets, oe ca state ad prove the followig theorem: Theorem (4.5): Ay oempty ope fuzzy set is the uio of a uique coutable collectio of disjoit ope itervals. 4- The Mai Results: Deotig that member of the family which correspods to the iteger by A, we may represet the sequece by A 1, A 2,, A, ; or by the symbol {A }, If a sequece of fuzzy sets of poits has the property that A +1 A for each, the the sequece is referred to as decreasig or ested, cocerig a ested sequece of sets each of which is closed, we state the followig. Theorem (5.1):

4 If { } is a ested sequece of o empty, closed, ad bouded fuzzy set, the the fuzzy set empty, where 1 1 (x) (x). is o By theorem (4.3) each of the fuzzy set possesses a maximum poit. Let X X max, for each (x) max (x), from the hypothesis 1 1, (x) < (x). It follows that 1 X, ad for each positive iteger q it is the case that X, for all q. Deote the fuzzy set of umbers X by T, because T 1 ad 1 is bouded, it follows that T is bouded ad, particular, is bouded below. Deotig the ifimum of T by c. We shall show that c, for all. Let q deote a arbitrary positive iteger, ad cosider the subset of T defied by T q { X : q}, sice c is the ifimum of T, it is also the ifimum of T q, by theorem (4.2) it follows that c T q. Also, sice: q X q T q (x). q, we have T (x) q Now, cosider the followig remarks: Remarks (5.1): A. Let =, for all positive itegers where is the empty fuzzy set with membership fuctio 0. The is closed ad bouded, ad { } is ested. However, the fuzzy set 1 1, with membership fuctio (x) (x) is empty sice the empty. are ot o B. Let {(x, (x),where xx is the uiversal set with membership fuctio 1.} here the sequece { } is ested ad is oempty ad closed for each. however, each is ubouded. It is easy to see that 1 (x) (x), with membership fuctio is empty. 1 C. Let (0, 1/], the sequece { } is obviously ested, each is oempty ad bouded but ot closed. The set membership fuctio (x) (x), seems to be empty. 1, with 1 D. Let [2, 2+1]. Here each is oempty closed, ad bouded, but the sequece { } is ot ested. The set 1 fuctio (x) Therem (5.2):, with membership (x) is empty. 1 If L * is ay ope coverig of a fuzzy set A, the there exists a

5 coutable subfamily of L which also covers the fuzzy set A. Let a deote ay member of the give fuzzy set A. The a fuzzy set G a of L exists such that a G a. urther, sice G a is ope, a eighborhood N(a, ) exists such that N(a, ) G a. Now, let r 1 ad r 2 desigate two ratioal umbers with the property that a < r 1 < a < r 2 < a+. It is the the case that the iterval I a (r 1, r 2 ) is such that a I a ad I q G a. Hece, i this maer we may associate with each member a of the fuzzy set A a ope iterval I a with ratioal ed poits. Sice the fuzzy set of all possible itervals with ratioal ed poits is coutable, if follows that the fuzzy set B {I a : a A } is also coutable. Each iterval I a is cotaied i at least oe of the ope fuzzy sets of L ad deote oe such by G a. I this way a subfamily L of L is costructed with the property that with each iterval I a of the coutable fuzzy set B is associated exactly oe member G a of L. Cosequetly L is coutable ad, moreover, covers A sice for each member a of A we have a I a ad I a G a. Theorem (5.3): Let X lr be the uiversal set, the a fuzzy subset of X is closed ad bouded the it is compact. Let A X be a closed ad bouded fuzzy subset of X ad a family of a fuzzy sets A is a cover of a fuzzy set A if ad oly if B { A A A} It is a ope cover if ad oly if each member of A is a ope fuzzy set A. Because of the (Lidelof theorem i fuzzy sets), we may assume, without loss of geerality, that B be a coutable ad thereby deote its members by G 1, G,, G, ad defie the fuzzy set: 2 K K Gi i1 ad L A y which is s also a fuzzy set of X with membership fuctio, for ay idex J L G j (x) sup ij = Mi { A (x) L (x) for 1, 2, Gi (x), x X, y K (x) }, ad observe that, because of the theorems o uios ad itersectio of closed ad ope fuzzy set, the fuzzy set K are ope ad the fuzzy set L closed for all values of. further, it is the case that: K K 1, the K1 (x), x X ad from this follows L (x) L 1 L, the, x X, for all. K (x) L1 (x) Assume ow that oe of the fuzzy sets L is empty fuzzy set, the: ad hece: L (x) 1, sice L A L (x) < A (x), x X 1 < A (x), ad hece A is bouded

6 It follows the that the sequece { L } { L 1 }, 1, 2, ; i.e., all. L (x) < L (x) 1, x X, for Satisfies the hypotheses of the theorem (5.2) L 1 L 1 L 1 (x) (x) L 1 if L (x), x X Therefore, for some positive iteger q the fuzzy set L q A K q is empty. mi{ A (x) Lq (x), Kq (x) } 0, for ay idex j, j 1, 2,, q Hece, A x X K q q Gi i1, i.e., A (x) < Kq (x) q (x) Gi i1 sup ij Gi (x), A C (sice A (x) C (x) ) A may be covered by the set C. Let B deote the family of ope fuzzy sets of fuzzy poits G, defied by: G G {(x, G (x) ) : x X, (x), for each positive iteger } It is obvious that B is a ope coverig of the fuzzy set of all real umbers ad hece of ay fuzzy set of real umbers suppose ow that A X* is some compact fuzzy set, A T. The, sice ay ope coverig of A possesses a fiite sub family which also covers A. It follows, i particular, that this is true of B. Cosequetly, a fiite collectio of itervals G covers the fuzzy set A, ad, if deotes the maximum subscript for this fiite family, the clearly the ope iterval: 0 G 0 {(x, (x) 0} G 0 covers A. (x)) : x X; G 0 This implies that a compact fuzzy set is bouded. To show that A is ecessarily closed, let c be a real umber ad cosider the family of closed fuzzy sets: (x) {(x, x X, c 1/ (x) c + 1/}, 1, 2, The fuzzy sets H, 1, 2, H (x) (x), 1, 2, The costitute a family of ope fuzzy sets. It is obvious that the set (x) 1 if ij (x) i 1, x X for j ay idex set cosists of the sigle poits, ad, sice c i et i A, it follows that: i.e., A 1

7 X (1) where: X..(2) A (x) < (x) 1 if ij (x) A (x) < (x) Hi i1 sup ij i, x Hi (x), x Thus the fuzzy set A is covered by the family. The compactess of A implies that a fiite subfamily of exists which also covers A. Therefore a positive iteger 1 exists such that each poit of A is cotaied i at least oe of the ope fuzzy sets H 1, H 2,, H, ; the, o poit of A is cotaied i: 1/ 1 1 {(x, 1 1 (x) (x) c + 1/ 1} ) : x X, c ad from this it follows that the poit c is ot a cluster poit of the fuzzy set A. Thus it is proved that ay poit which is ot a poit of the fuzzy set A is also ot a cluster poit of A. All cluster poits of A are, therefore, poits of A itself. Hece A is closed.

8 تراص الوجووعات الضبابية أهاني التفات كاظن قسن علوم هنذسة البراهجيات كلية هذينة العلن الجاهعة بغذاد العراق الوستخلص يهدف هذا البحث الى دراسة تزاص المجمىعات الضبابية في العالقة بين التزاص,انغالق, وتقييد المجمىعات الضبابية. الفضاءات التبىلىجية الضبابية وباألخص 5- Refereces: 1. Dubois, D. ad Prade, H., uzzy Sets ad Systems: Theory ad Applicatios, Academic Press, Ic., (1980). 2. Muir, M.A.K., O Separatio Axioms of uzzy Topological Spaces'', Ph.D. Thesis, College of Educatio, Al-Mustasiriah Uiversity, (2006). 3. adhel,. S., About uzzy ixed Poit Theorem, Ph.D. Thesis, College of Sciece, Al-Nahrai Uiversity, (1998). 4. Joh, D., Itroductio to Real Aalysis'', Joh Wiley ad Sos, (1988). 5. Mary, M.G., urther Results About uzzy Metric Spaces'', M.Sc. Thesis, College of Educatio, Al-Mustasiriah Uiversity, (2004). 6. Burill, C.W. ad Kudse, J.R. Real Variables'', Holt, Riehart ad Wisto, Ic., (1969). 7. Amai, A.K., About the completeess of fuzzy metric spaces, M.SC. thesis,college of sciece, AL-Naharai Uiversity, Baghdad, Iraq