L - ZHR UNIVERSIT - GZ DENSHIP OF GRDUTE STUDIES & SCIENTIFIC RESERCH On Tplgical Structures and Fuzzy Sets y Nashaat hmed Saleem Raab Supervised by Dr. Mhammed Jamal Iqelan Thesis Submitted in Partial Fulfillment f the Requirements f the Degree f Master f Science in Mathematics Department f Mathematics Faculty f Science l-zhar University - Gaza 2013
TO Spirit f my father, llah's mercy My mther My wife My sns My brthers and sisters My belved and glrius teachers I
cknwledgement Firstly, all gratitude and thanks t llah wh helped me t prduce this wrk. I wuld like t epress my gratitude and appreciatin t Dr. Mhammed Iqelan fr his cnstructive and fruitful help thrughut writing this thesis. Special thanks t all staff members f Department f Mathematics, and t all master degree students whm shared me the last years. My deep thanks t my family especially my mther, my wife, my brthers and sisters. Finally, thanks t all peple wh helped me t achieve this wrk. II
bstract In this thesis, we shall use sme classes f near pen sets t get several results in tplgical spaces and Fuzzy Set Thery. New cncepts n fuzzy sets and fuzzy relatins in prduct tplgical spaces will be intrduced. III
Preface Mathematics is the ldest and mst imprtant science inspired by the human, and sme scientists prefer t lk at mathematics as the lgical study f frm, rder and quantity, this view has changed when the tplgy is appeared, since it eamines the study f spaces away frm the frms and dimensins. The tplgy is ne f the mst imprtant branches f mdern mathematics. It is interested in the study f structures, cmpnents and the characteristics f all the different spaces, s that these characteristics remain similar under the cntinuus restructuring peratins withut leaving hles in the transitin frm ne space t anther and vice versa. In 1965 L.. Zadeh published his famus paper Fuzzy sets [29] prviding a new mathematical tl which enables t describe and handle vague r ambiguus ntins. Fuzzy set allws n sharply defined bundaries because f a generalizatin f a characteristic functin with range {0,1} t a membership functin with range [0,1]. lgebraic prperties f fuzzy sets have been studying [7, 9, 10, 11, 20, 21]. Pawlak in 1982 [23] intrduced a mathematical tl that supprts the uncertainty reasning in apprimatin space which generated by equivalence relatins. ut b Khadra [17] intrduced these ideas in terms f cncepts in tplgy. These ideas give a mathematical tl called the rugh membership functin. Mrever, Iqelan [10] intrduced anther mathematical tl called near rugh membership functin using sme classes f near pen sets. This thesis includes fur chapters: Chapter One: In the first chapter, we intrduced sme basic cncepts t be used thrugh this thesis. These cncepts are cnsidered as a fundamental cncepts in tplgy and fuzzy set thery. IV
Chapter Tw: In this chapter, we intrduced sme prperties f sme near pen sets in tplgical spaces. ls we presented sme relatins generated via tplgical spaces using sme classes f near pen sets. Chapter Three: In the cntet f Chapter 3, we intrduced sme results n rugh membership functin, near rugh membership functin and fuzzy sets in tplgical spaces. Chapter Fur: Thrugh this chapter, we intrduced sme results n fuzzy sets and fuzzy relatins in prduct tplgical spaces. V
Cntents cknwledgement bstract Preface Cntents II III IV VI CHPTER 1 Intrductin 1 1.1 Fundamental cncepts f tplgical spaces 2 1.2 Fundamental ntins in fuzzy set and fuzzy relatin 7 CHPTER 2 Generating relatins via tplgical spaces using sme classes f near pen sets 14 2.1 Near pen sets in tplgical spaces 15 2.2 Generating relatins via tplgical spaces 20 CHPTER 3 Fuzzy sets in tplgical spaces 30 3.1 Membership functin 31 3.2 Fuzzy sets in tplgical spaces 44 CHPTER 4 Fuzzy sets in Prduct tplgical spaces 54 4.1 4.2 inary prduct in tplgical spaces Fuzzy sets in prduct tplgical spaces 55 63 Cnclusin 69 References 71 VI
CHPTER 1 INTRODUCTION 1
Chapter 1 Intrductin The purpse f this thesis is t put a starting pint fr the applicatins f abstract tplgical thery int fuzzy set thery. Thrugh this thesis, we shall integrate sme ideas in terms f cncepts in tplgy and fuzzy set thery. Sectin 1.1 cntains fundamental cncepts f tplgical spaces, Sectin 1.2 t gives an accunt n fuzzy set thery. 1.1 Fundamental cncepts f tplgical spaces Sme f the basic cncepts in tplgy which are useful fr ur study are given thrugh this sectin. Definitin 1.1.1 [15]. Let be a nnempty set and be a cllectin f subsets f. The cllectin is said t be a tplgy fr if satisfies the fllwing three cnditins: i) and. ii) is clsed under arbitrary unin. iii) is clsed under finite intersectin. If is a tplgy n, then the pair (, ) is called a tplgical space. The elements f are called pints f the space, the subsets f belnging t are called the -pen (simply, pen) sets in the space. The cmplements f the subsets f belnging t are called the -clsed (simply, clsed) sets in the space, and the * family f all -clsed subsets f is dented by. In a tplgical space (, ), a subset f is called clpen subset f if it is pen and clsed. 2
Definitin 1.1.2 [17]. tplgical space (, ) is called a quasi-discrete (partitin space) if every member f is clpen subset f. Definitin 1.1.3 [15]. family is called a base fr a tplgy n if every nnempty pen subset f can be represented as a unin f subfamily f. Clearly, a tplgical space can have many bases. Since each unin f members f belngs t, a base f is cmpletely determined by. Definitin 1.1.4 [15]. family S is called a subbase fr a tplgy n if the family f all finite intersectins f S is a base fr. Definitin 1.1.5 [12]. Let (, ) be a tplgical space and. The clsure f (briefly cl()) dented by and is defined by Evidently, clsed if and nly if F : F and F *. is the smallest clsed subset f which cntains. Nte that, is. Definitin 1.1.6 [12]. Let (, ) be a tplgical space and. The interir f (briefly int()) dented by Evidently, pen if and nly if and is defined by G : G and G. is the largest pen subset f which cntained in. Nte that, is. Definitin 1.1.7 [15]. Let (, ) be a tplgical space and. The bundary f (briefly N()) dented by b and is defined by b. 3
Definitin 1.1.8 [12]. Let (, ) be a tplgical space and. The eterir f dented by ET ( ) and is defined by ET ( ). Definitin 1.1.9 [12]. Let (, ) be a tplgical space and. The pint is called a cluster pint f if fr each G such that G, we have G { }. The set f all cluster pints f is called the derived set f and it is dented by. We shall recall sme definitins abut sme classes f near pen sets which are essential fr ur present study. Sme frms f near pen sets are intrduced in the fllwing definitin. Definitin 1.1.10. Let (, ) be a tplgical space. Then the subset is called: i) Regular-pen [25] (briefly r-pen) if ii) Semi-pen [18] (briefly s-pen) if.. iii) Pre-pen [20] (briefly p-pen) if. iv) -pen [8] if. v) -pen [21] if. vi) -pen [1] if. The cmplement f an r-pen (resp. s-pen, p-pen, -pen, -pen and - pen) set is called r-clsed (resp. s-clsed, p- clsed, - clsed, - clsed and - clsed) set. The family f all r-pen (resp. s-pen, p-pen, -pen, -pen and - pen) sets is dented by RO( ) (resp. SO( ), PO ( ), O( ), O( ) (r ) and O ( )). 4
Remark 1.1.1 [1, 8]. 1) In the tplgical spaces, the unin f -pen (resp. the intersectin f -clsed) sets is -pen set (resp. -clsed set), where { s, p,,, }. In general, this remark is nt true in the case f =r as shwn in Eample 1.1.1. 2) The intersectin f -pen (resp. the unin f -clsed) even tw sets need nt be - pen (resp. -clsed). Eample 1.1.1. Let { a, b, c, d} and,,{ a},{ b, d},{ a, b, d}. Then RO( ),,{ a},{ b, d}. Nte that, the sets {a} and {b,d} are r-pen sets, but a, set. b d ={a,b,d} is nt r-pen Remark 1.1.2 [2]. In a tplgical space (, ), the implicatins between and the families f near pen sets are given in the fllwing diagram. Remark 1.1.3 [2]. In a tplgical space (, ), the implicatins between near clsed sets are given in the fllwing diagram. * and the families f Definitin 1.1.11 [3]. Let (, ) be a tplgical space and int ( ) ) dented by fr all r, s, p,,, and is defined by. The -interir f (briefly Evidently, G : G, G is a -pen set. is the largest -pen set which cntained in, where s, p,,, 5.
Remark 1.1.4 [3]. In a tplgical space (, ), the implicatins between the interir and -interir f a subset f are given in the fllwing diagram. Definitin 1.1.12 [3]. cl Let (, ) be a tplgical space and ( )) dented by fr all r, s, p,,, and is defined by. The -clsure f (briefly Evidently, H : H, H is a -clsed set. is the smallest -clsed set which cntains, where s, p,,,. Remark 1.1.5 [3]. In a tplgical space (, ), the implicatins between the clsure and -clsure f a subset f are given in the fllwing diagram. Definitin 1.1.13 [10]. Let (, ) be a tplgical space and b N ( )) dented by fr all r, s, p,,,. The -bundary f (briefly and is defined by b. Definitin 1.1.14 [10]. Let (, ) be a tplgical space and by ( ) ET fr all r, s, p,,, and is defined by ET ( ).. The -eterir f dented 6
Definitin 1.1.15 [12]. Let 1 and 2 be tw tplgies n a nnempty set. Then 1 is said t be weaker (r carser) than 2 if 1 2. In this case we als say that 2 is strnger (r finer) than 1. Definitin 1.1.16 [12]. tplgical space (, ) is said t be a T space if given any tw distinct pints in, there eists an pen set which cntains ne f them but nt the ther. Definitin 1.1.17 [12]. tplgical space (, ) is said t be a T1 -space if fr each, y, y, there eist tw pen sets U, V such that U, y U and y V, V. Definitin 1.1.18 [12]. tplgical space (, ) is said t be a T 2 - space (r Hausdrff space) if fr each, y, y, there eist tw disint pen sets U, V such that U and y V. Definitin 1.1.19 [12]. tplgical space (, ) is said t be regular at a pint if fr every clsed subset F f nt cntaining, there eist tw disint pen sets U, V such that U and F V. tplgical space (, ) is said t be regular space if it is regular at each f its pints. 1.2 Fundamental ntins in fuzzy set and fuzzy relatin In 1965, L.. Zadeh published his famus paper Fuzzy set [29] prviding a new mathematical tl which enables t describe vague r ambiguus ntins. Fuzzy set allws n sharply defined bundaries because f a generalizatin f a characteristic functin with range 0,1 t a membership functin with range [0,1]. 7
Let be a classical set f bects, called the universe, whse generic elements are referred. Membership in a classical subset f is ften viewed as a characteristic functin frm t 0,1 such that 1, iff, ( ) 0, iff. The set {0,1} is called a valuatin set. If the valuatin set is allwed t be the real interval 0,1, then is called a fuzzy subset and it is dented by [29]. Definitin 1.2.1 [28]. The fuzzy subset f is a set f rdered pairs each cnsists f the element and the assciated membership functin, that is is cmpletely characterized by the set f pairs (, ( )) :. mre cnvenient ntin fr a fuzzy subset was prpsed in [5, 16, 24]. Eample 1.2.1. In a space a, b, c, d, e, let ( a,0.3),( b,0.7),( c,0.9),( d,0),( e,0.2) fuzzy subset f. Hence be a ( a ) 0.3, ( b) 0.7, ( c) 0.9, ( d ) 0, and ( e ) 0.2. In the fllwing we shall intrduce sme simple peratins n fuzzy sets. Definitin 1.2.2 [14]. Let and be tw fuzzy subsets f. We say that is included in (briefly ) if ( ) ( ) fr all. Definitin 1.2.3 [14]. (briefly Let and be tw fuzzy subsets f. We say that and are equal ) if ( ) ( ) fr all. 8
Definitin 1.2.4 [14]. Let and be tw fuzzy subsets f. We say that and are cmplementary (briefly ( ) c ) if ( ) 1 ( ) fr all. One bviusly always has ( ) Definitin 1.2.4. c c. The fllwing eample illustrates Eample 1.2.2. Let be a fuzzy subset f such that a, b, c, d ( b,0.73),( c,0),( d,0.8). Then c ( ) ( a,0.9),( b,0.27),( c,1),( d,0.2). and ( a,0.1), Definitin 1.2.5 [14]. Let, and C be fuzzy subsets f. The intersectin is the largest fuzzy subset cntained at the same time in and. That is, if C, then ( ) min ( ), ( ) fr all. C Definitin 1.2.6 [14]. Let, and C be fuzzy subsets f. The unin is the smallest fuzzy subset that cntains bth and. That is, if C, then ( ) ma ( ), ( ) fr all. C The fllwing eample illustrates Definitin 1.2.5 and Definitin 1.2.6. Eample 1.2.3. Let a, b, c, d, ( a,0.2),( b,0.7),( c,1),( d,0) ( c,0.4),( d,1). Then and ( a,0.5),( b,0.3), ( a,0.2),( b,0.3),( c,0.4),( d,0) ( a,0.5),( b,0.7),( c,1),( d,1). 9, and
It is clearly that, and. Similarly fr. Definitin 1.2.7 [14]. Let and be tw fuzzy subsets f. The difference ( ) c. is defined by Nte that, ust as in classical sets, the membership value f any element in the null set is zer, and the membership value f any element in the whle set is ne. The apprpriate ntatin fr these ideas is as fllws: fr all, ( ) 0 and ( ) 1. The peratins n classical sets als hld fr fuzzy sets, ecept fr the ecluded middle aims [25]. The ecluded middle aims etended fr fuzzy sets are epressed by where is a fuzzy subset f a space. ( ) c and ( ) c, De Mrgan s principles fr classical sets als hld fr fuzzy sets, as dented by these epressins [25]: ( ) c ( ) c ( ) c, ( ) c ( ) c ( ) c. Fuzzy relatins als map elements f ne space say t anther space say, thrugh the prduct f the tw spaces. Hwever, the strength f the relatin between rdered pairs f the tw spaces is nt measured with the characteristic functin, but rather with a membership functin epressing varius degrees f strength f the relatin n the unit interval 0,1. Hence a fuzzy relatin R is a mapping frm the prduct space t the unit interval 0,1. Definitin 1.2.8 [26]. fuzzy relatin R frm a set t a set (r between and ) is a fuzzy set in the prduct, and it is characterized by a membership functin R where R : 0,1. If, then R is knwn as a fuzzy relatin n. 10
The fuzzy relatin frm t can be epressed in the frm Eample 1.2.4. R ((, y ), (, y )), where and y. R n eample f a fuzzy relatin frm t, where a, b, c, d, e is R (( a, d ),0.2),(( a, e),0),(( b, d ),0.7),(( b, e),0.3),(( c, d ),0),(( c, e),0.8). Definitin 1.2.9 [26]. R in Let and y y y 1,, m 1, 2,, n be tw finite sets. The fuzzy relatin can be epressed by an m n matri as the fllwing: R y y y 1 2 n (, y ) (, y ) (, y ) 1 R 1 1 R 1 2 R 1 n (, y ) (, y ) (, y ) 2 R 2 1 R 2 2 R 2 n (, y ) 3 R 3 1. (, y ) (, y ) m R m 1 R m n Eample 1.2.5. matri. We can write the fuzzy relatin R given in Eample 1.2.4 by the fllwing R d e a b 0.2 0 0.7 0.3. c 0 0.8 11
Since the fuzzy relatins frm t are fuzzy subsets f the prduct, we can use the peratins f fuzzy sets n the fuzzy relatins. Definitin 1.2.10 [6]. Let R be a fuzzy relatin in and S be a fuzzy relatin in Z. The cmpsitin f R and S (briefly RS ) is the fuzzy relatin in Z defined by {((, ), RS (, )): RS (, ) ma min R (, ), S (, ) } RS z z z y y z, where, y and z Z. y Eample 1.2.6. Cnsider the fllwing tw fuzzy matrices f tw fuzzy relatins R and S Z, where ={a,b,c}, ={m,n,v,w} and Z={h,k,u}. R m n v w a b 0.1 0.2 0.0 1.0 0.3 0.3 0.0 0.2, c 0.8 0.9 1.0 0.4 S h k u m 0.9 0.0 0.3 n 0.2 1.0 0.8. v 0.8 0.0 0.7 w 0.4 0.2 0.3 12
Then the fuzzy matri f RS is RS h k u a b 0.4 0.2 0.3 0.3 0.3 0.3. c 0.8 0.9 0.8 13
CHPTER 2 GENERTING RELTIONS VI TOPOLOGICL SPCES USING SOME CLSSES OF NER OPEN SETS 14
Chapter 2 Generating relatins via tplgical spaces using sme classes f near pen sets 2.1 Near pen sets in tplgical spaces In this sectin, we intrduce sme results n sme classes f near pen sets in tplgical spaces. Lemma 2.1.1 [12]. Let be a subset f in a tplgical space (, ) and. Then if and nly if every pen set cntaining meets. Lemma 2.1.2. Let be an s-pen (r-clsed) subset f in a tplgical space (, ). Then. Prf. Let be an s-pen subset f. Then. Thus. Since,. Hence. Similarly, we can prve this lemma if is r-clsed. Prpsitin 2.1.1. Let be an s-pen (r-clsed) subset f in a tplgical space (, ) and. Then if and nly if every pen set cntaining meets. Prf. Hence Let be an s-pen subset f. Then by Lemma 2.1.2, we have iff, iff every pen set cntaining intersects 15 by Lemma 2.1.1..
Similarly, we can prve this prpsitin if is r-clsed. Prpsitin 2.1.2. Let be an s-pen (r-clsed) subset f in a tplgical space (, ). Then N ( ) N ( ). Prf. Hence Let be an r-clsed subset f. Then by Lemma 2.1.2, we have. N ( ) ( ) ( ) N ( ). Similarly, we can prve this prpsitin if is s-pen. Lemma 2.1.3 [12]. Let be a subset f in a tplgical space (, ) and. Then if and nly if there eists an pen set G such that G and G. Lemma 2.1.4. Let be an s-clsed (r-pen) subset f in a tplgical space (, ) and. Then. Prf. Let be an s-clsed subset f. Then. Thus. ut, since. Hence. Similarly, we can prve this lemma if is r-pen. Prpsitin 2.1.3. Let be an s-clsed (r-pen) subset f in a tplgical space (, ) and in.. Then if and nly if there eists an pen set cntaining cntained 16
Prf. Hence Let be an s-clsed subset f. Then by Lemma 2.1.4, we have iff,. iff there eists an pen set cntaining cntained in by Lemma 2.1.3. Similarly, we can prve this prpsitin if is r-pen. Prpsitin 2.1.4. Let be an s-clsed (r-pen) subset f in a tplgical space (, ). Then N ( ) N ( ) Prf. Let be an r-pen subset f. Then by Lemma 2.1.4, we have Hence N ( ) ( ) ( ) N ( ). Similarly, we can prve this prpsitin if is s-clsed. Lemma 2.1.5 [10]. Let be a subset f in a tplgical space (, ) and. Then if and nly if fr each -pen set G cntaining, we have G, where r, p, s,,,. Prf. We shall prve this lemma in the case f prved similarly. Let that and the ther cases can be. Suppse t cntrary that G is a -pen set such G and G. Then G and G. ut G is a - clsed set cntaining. Hence, which is a cntradictin. Thus G. Cnversely suppse that fr each -pen set G cntaining, G. Suppse t cntrary that, then there eists -clsed set H such that H and 17
H. Hence H is a -pen set cntaining, and ( H ), which is a cntradictin. Thus. Lemma 2.1.6 [10]. Let be a subset f in a tplgical space (, ) and. Then if and nly if there eists a -pen set G such that G, r, p, s,,,. where Prpsitin 2.1.5. Let be an s-pen subset f in a tplgical space (, ) and, then each pen set cntaining intersects, where s, p,,,.. If Prf. Let be an s-pen subset f and, where s, p,,,. Since fr each s, p,,,, then. Since is s-pen, then by Prpsitin 2.1.1, we get every pen set cntaining intersects. Prpsitin 2.1.6. Let be an r-clsed subset f in a tplgical space (, ) and. Then if and nly if each pen set cntaining intersects s, p,,,., where Prf. Let be an r-clsed subset f. Then s it is a -clsed set s, p,,, iff. Thus, where s, p,,,,. ut. Hence is a clsed set, and iff each pen set cntaining intersects by Lemma 2.1.1. Prpsitin 2.1.7. Then Let be an r-clsed subset f in a tplgical space (, ) and. if and nly if each -pen set cntaining intersects fr all s, p,,,. 18
Prf. Let be an r-clsed subset f. Then s it is a -clsed set fr all s, p,,,. Thus. ut is a clsed set, and. Hence iff, iff each -pen set cntaining intersects by Lemma 2.1.5. Prpsitin 2.1.8. Let be an s-clsed subset f in a tplgical space (, ) and. If, then there eists a -pen set G such that G, where s, p,,,. Prf. We shall prve this prpsitin in the case f and the ther cases can be prved similarly. Let be an s-clsed subset f. Then. ut is pen set, and s it is -pen set. Thus. Nw if, then. Hence by Lemma 2.1.6, there eists an -pen set G such that G. Prpsitin 2.1.9. Let be an r-pen subset f in a tplgical space (, ) and. Then if and nly if there eists an pen set G such that G, where s, p,,, Prf. We shall prve this prpsitin in the case f can be prved similarly. Let be an r-pen subset f and. ut iff, is pen set, and s it is -pen set. Thus and the ther cases. Then. Then iff there eists an pen set G such that G by Lemma 2.1.3. 19
Prpsitin 2.1.10. Then Let be an r-pen subset f in a tplgical space (, ) and. if and nly if there eists a -pen set G such that G, where s, p,,,. Prf. We shall prve this prpsitin in the case f =s and the ther cases can be prved similarly. Let be an r-pen subset f. Then pen set, and s it is s-pen set. Thus s iff, s. Then. ut iff there eists an s-pen set G such that G by Lemma 2.1.6. is 2.2 Generating relatins via tplgical spaces In this sectin we intrduce sme definitins f generating relatins n tplgical spaces using sme classes f near pen sets. Definitin 2.2.1 [22]. Let (, ) be a tplgical space. Then the relatin n generating by is dented by R and it is defined by Eample 2.2.1. Then Let R (, y ) : cl y.,, a n a, b, c. The family f clsed sets is,, b, c cl a, cl b b, c and cl c b, c. Hence accrding t Definitin 2.2.1, we get R ( a, a),( b, a),( c, a),( b, b),( c, b),( b, c),( c, c).. Definitin 2.2.2. Let (, ) be a tplgical space. Then the relatin n generating by the class O ( ) is dented by R and it is defined by (, ) :, where r p s R y cl y 20,,,,,.
Eample 2.2.2. Let a, b, c, d and,, a, b, d, a, b, d { }. Then { }. RO( ),,{ d},{ a, b}, and RC ( ),,{ a, b, c},{ c, d} Thus r,,, cl c c and cl d c, d cl a cl b a b c Hence accrding t Definitin 2.2.2, we get r r r. R ( a, a),( b, a),( c, a),( a, b),( b, b),( c, b),( c, c),( c, d ),( d, d ). r Prpsitin 2.2.1. Let (, ) be a tplgical space. Then the implicatins between R and where r, s, p,,, are given in the fllwing diagram. R, Prf. y using Remark 1.1.5, then fr any y, we have Hence accrding t Definitin 2.2.1 and Definitin 2.2.2, we get. 21
Definitin 2.2.3 [19]. relatin R n a set is said t be an equivalence relatin if it satisfies the fllwing cnditins: i) (, ) R, (refleive). ii) If (, y ) R, then ( y, ) R (symmetric). iii) If (, y ) R and ( y, z ) R, then (, z ) R (transitive). Prpsitin 2.2.2. Let (, ) be a tplgical space. Then R and fr all r, s, p,,,. R are refleive relatins n Prf. y using Definitin 1.1.5 and Definitin 1.1.12, we have cl and cl fr all, r, s, p,,, Then by Definitin 2.2.1 and Definitin 2.2.2, we get Hence R and (, ). R and (, ) R fr all. R are refleive relatins n fr all r, s, p,,,. Prpsitin 2.2.3. Let (, ) be a tplgical space. Then R and fr all s, p,,,. R are transitive relatins n Prf. y Let (, ), (, ) y y z R. Then by Definitin 2.2.1, we have cl y cl z. Thus cl y cl z and s cl z (, z ) R. Therefre R is transitive relatin n. and. gain by Definitin 2.2.1, we get Similarly, we can prve R is transitive relatin n fr all s, p,,,. Eample 2.2.3. Let a, b, c, d,,, a, b, d, a, b, d be a tplgy n. Then { } RO( ),,{ a},{ b, d}, 22
{ }, { } { } { { b, c, d }}, and O( ),,{ a},{ a, b, d},{ b, d} SO( ),,{ a},{ a, c},{ a, b, d},{ b, d},{ b, c, d}, PO ( ),,{ a},{ b},{ d},{ a, b},{ a, d},{ a, b, c},{ a, b, d},{ a, c, d},{ b, d}, O ( ),,{ a},{ b},{ d },{ a, b},{ a, c},{ a, d },{ a, b, c},{ a, b, d },{ a, c, d },{ b, d } { O ( ),,{ a},{ b},{ d},{ a, b},{ a, c},{ a, d},{ a, b, c},{ a, b, d},{ a, c, d},{ b, c},{ b, d } Thus { b, c, d },{ c, d } { } }. RC ( ),,{ b, c, d},{ a, c}, { }, C ( ),,{ b, c, d},{ c},{ a, c} { } SC ( ),,{ b, c, d},{ b, d},{ c},{ a, c},{ a}, { } PC ( ),,{ b, c, d},{ a, c, d},{ a, b, c},{ c, d},{ b, c},{ d},{ c},{ b},{ a, c}, { C ( ),,{ b, c, d },{ a, c, d },{ a, b, c},{ c, d},{ b, d},{ b, c},{ d},{ c},{ b},{ a, c} { {} a }, and C ( ),,{ b, c, d },{ a, c, d },{ a, b, c},{ c, d },{ b, d },{ b, c},{ d },{ c},{ b},{ a, d },{ a, c} Hence } { a},{ a, b}.,, cl b cl b cl b b c d cl a cl a cl a a c r, cl c cl c cl c c r 23 r,,, cl d cl d cl d b c d cl a a, cl b b, d, cl c c, cl d b, d, s s s s cl a a, c, cl b b, cl c c, cl d d, p p p p r,,, cl a cl a a, cl b cl b b, cl c cl c c, and. cl d cl d d ccrding t Definitin 2.2.1 and Definitin 2.2.2, we get R R r R ( a, a),( c, a),( b, b),( c, b),( d, b),( c, c),( b, d ),( c, d ),( d, d ), R ( a, a),( b, b),( d, b),( c, c),( b, d ),( d, d ), s R ( a, a),( c, a),( b, b),( c, c),( d, d ), and p R R ( a, a),( b, b),( c, c),( d, d ). Then R, R are refleive and transitive relatins n fr all s, p,,,.
Lemma 2.2.1 [19]. Let (, ) be a tplgical space. Then (, ) is a T1 space if and nly if is a clsed subset f,. Prpsitin 2.2.4. Prf. Let (, ) be a tplgical space. Then (, ) is a T1 space if and nly if (, ) :, fr all s, p,,, R R. Let (, ) be a T1 space. Then by Lemma 2.2.1, we have is a clsed subset f,. Thus cl. ut cl cl and s, p,,,. Hence cl. Therefre (, ) :, fr all s, p,,, R R. Cnversely, let R R (, ) :. Then cl, cl fr all fr all. Thus is a clsed set. Hence by Lemma 2.2.1, we have (, ) is a T1 space. Prpsitin 2.2.5. Let (, ) be a tplgical space. If (, ) is a T1 space, then R and are equivalence relatins n fr all s, p,,,. R Prf. y using Prpsitin 2.2.4, the prf is bvius. Prpsitin 2.2.6. Every finite regular tplgical space is a quasi-discrete tplgical space. Prf. Let (, ) be a finite regular tplgical space and G Since is regular, then. Then G. fr each G, there eists H such that G H and H. 24
Thus G H and H G G. Then ( H ) G. G ut ( H ) G, hence G Since is finite, then G H G H G. is pen. Thus G is pen set, and s G is clsed. Therefre (, ) is a quasi-discrete tplgical space. Definitin 2.2.4 [15]. The family f all subsets f is a tplgy n called the discrete tplgy and it is dented by. Prpsitin 2.2.7. Let (, ) be a quasi-discrete tplgical space. Then i) JO ( ) ii) ( ) fr all J R S,,. JO fr all J P,,. Prf. i) Let G SO( ), then G Thus G G. ut G G G ut SO( ). Then SO( ) Similarly, we can prve O ( ). Since is quasi-discrete, then G G., then G G, that is G. Hence SO( ).... Since is quasi-discrete, then * Nw, let G G and G G. Thus G RO( ). Hence RO( ). ut RO( ). Then RO( ). ii) Let G. Since is quasi-discrete, then G G. Hence G G, since G G. That is G PO ( ). Thus PO ( ). ut PO ( ). Then PO ( ). Similarly, we can prve O( ) O ( ). 25
Lemma 2.2.2 [15]. Let (, ) be a regular tplgical space. Then fr any tw pints, y either cl y r cl cl cl y. Prpsitin 2.2.8., y Let (, ) be a finite regular tplgical space. Then fr any tw pints either fr all r, s, p,,,. r cl cl y, cl cl y Prf. Let (, ) be a finite regular tplgical space. Then by Prpsitin 2.2.6, is quasi-discrete. Thus by Prpsitin 2.2.7 part (i), we have Hence fr any pint RO( ) SO( ) O( )., we have r cl cl cl cl. s Since is regular, then by Lemma 2.2.2, we have fr any tw pints, y cl y r cl cl y. cl Hence fr any tw pints, y either r cl cl y, where r, s, cl cl y ls by using Prpsitin 2.2.7 part (ii), we have Hence fr any pint PO ( ) O( ) O ( )., we have Therefre fr any tw pints, y p cl cl cl. either. r cl cl y, where p,, cl cl y. either 26
Prpsitin 2.2.9. Let (, ) be a regular tplgical space. Then R is symmetric relatin n. Prf. Let (, ) be a regular tplgical space and (, y ). Then cl y. ut cl. Thus cl cl y. Since is regular, then by Lemma 2.2.2, we have cl cl y. Hence y cl, and s ( y, ) relatin n. R R. Therefre R is symmetric Prpsitin 2.2.10. Let (, ) be a finite regular tplgical space. Then R is symmetric relatin n fr all r, s, p,,,. Prf. We shall prf this prpsitin in the case f r, and the ther cases can be prved similarly. Let (, ) be a finite regular tplgical space and (, y ) Rr. Then r cl y. ut cl. Thus cl cl y. Hence by Prpsitin 2.2.8, r r we have cl cl y. Then y cl. Thus ( y, ) Rr. Therefre R r is r r r r symmetric relatin n. Prpsitin 2.2.11. Let (, ) be a finite regular tplgical space. Then R and R are equivalence relatins fr all s, p,,,. Prf. y using Prpsitin 2.2.2, Prpsitin 2.2.3, Prpsitin 2.2.9 and Prpsitin 2.2.10, the prf is bvius. Definitin 2.2.5. each pint tplgical space (, ) is said t be a -regular tplgical space, if fr and -clsed set F des nt cntain, there eist tw disint -pen sets G and H such that G and F H, where r, s, p,,,. 27
Eample 2.2.4. Then Let a, b, c, d and let,, a, b, d, a, b, d be a tplgy n. SO( ),, a, a, c, a, b, d, b, d, b, c, d and SC ( ),, b, c, d, b, d, c, a, c, a. Hence is s-regular, since fr each pint and s-clsed set F where F, there eist tw disint s-pen sets G and H such that G and F H. Prpsitin 2.2.12. Let (, ) be a -regular tplgical space. Then fr each tw pints and y f we have either where s, p,,,., cl cl y r cl cl y Prf. We shall prve this prpsitin in the case f and the ther cases can be prved similarly. Let (, ) be -regular tplgical space and let, y. Suppse that cl cl y. Then either cl y r y cl y. Let cl y y. Since cl is an -clsed set des nt cntain, then there eist tw disint -pen sets G and H such that G and. Then cl H. Hence cl y H. ut H is an -clsed set cntaining Therefre ( ) cl cl y H H. cl cl y. Prpsitin 2.2.13. Let (, ) be a -regular tplgical space. Then, where s, p,,,. R is symmetric relatin n 28
Prf. We shall prve this prpsitin in the case f and the ther cases can be prved similarly. Let (, ) be a -regular tplgical space and (, y ). ut cl. Thus cl cl y cl y R. Then. Since is -regular, then by Prpsitin 2.2.12, we have cl cl y. Hence y cl ( y, ). Therefre R is symmetric relatin n. R, and s Prpsitin 2.2.14. Let (, ) be a -regular tplgical space. Then, where s, p,,,. R is equivalence relatin n Prf. y using Prpsitin 2.2.2, Prpsitin 2.2.3 and Prpsitin 2.2.13, The prf is bvius. 29
CHPTER 3 FUZZ SETS IN TOPOLOGICL SPCES 30
Chapter 3 Fuzzy sets in tplgical spaces In this chapter, we intrduced sme results n rugh membership functin, near rugh membership functin and fuzzy sets in tplgical spaces. 3.1 Membership functin In this sectin, we intrduce sme results n membership functin. Lemma 3.1.1. Let be a subset f in a tplgical space (, ) and. Then if and nly if fr each subset f where, we have. Prf. Let be a subset f and such that. Since each subset f cntaining, then by Lemma 2.1.1, we have Cnversely, suppse that, fr each subset f where. Then fr any pen set G cntaining, we have is pen set fr. G, we have. Since G is pen set, then. G G. Thus G. Hence by Lemma 2.1.1, we have Lemma 3.1.2. Let be a subset f in a tplgical space (, ) and. Then if and nly if there eists a subset f such that and. Prf. Let be a subset f and such that. Then by Lemma 2.1.3, there eists an pen set such that. Hence since is pen. 31
Cnveresly, if there eists a subset G f such that G, then by Lemma 2.1.3, we have since G is pen set. Definitin 3.1.1 [17]. Let be a base f a tplgy n a finite set. Then the rugh membership functin n f a subset where is ( ),, are the members f cntaining. Thrugh this thesis we shall call membership functin. a membership functin instead f rugh In Definitin 3.1.1, since G : G, G :,, the membership functin n f a subset can be defined by where G are the pen sets cntaining. ( ) G,, G Eample 3.1.1. Let 2, 3, 0,1,2, 2,3,4, 3,5 be a base fr a tplgy n a finite set 0,1, 2,3, 4,5, and let 2,4,5 0,1,2. Then 0,1, 2 2, 4,5 1 1 (0), (1), (2) 1, 3 3 2 1 (3) 0, (4), and (5). 3 2 Nte that, the rugh membership functin is a mapping such that where is a tplgy n a finite set and. : 0,1, 32
Lemma 3.1.3 [17]. i) ii) Let be a subset f in a finite tplgical space (, ) and, if and nly if ( ) 1, b if and nly if 0 ( ) 1. Then iii) is eterir pint f (briefly ET ( ) ) if and nly if ( ) 0. Iqelan [10] intrduced the fllwing definitin fr the near rugh (briefly - rugh) membership functin fr a subset where r, p, s,,,. using the classes f -pen sets, Definitin 3.1.2 [10]. Let (, ) be a finite tplgical space. The near rugh (briefly -rugh) membership functin n f a subset, and it is defined by where ( ) : fr all r, p, s,,, : 0,1, 1 if 1 ( ), ( ) min ( ) therwise, is a -pen set,. is Thrugh this thesis we shall call a -membership functin instead f -rugh membership functin, where r, p, s,,,. Lemma 3.1.4 [10]. i) Let be a subset f in a finite tplgical space (, ) and, if and nly if ( ) 1. Then ii), b if and nly if 0 ( ) 1 iii) is -eterir pint f (briefly ET ( ) ) if and nly if ( ) 0, fr all r, p, s,,,. 33
Eample 3.1.2. Let a, b, c, d,,, a, b, d, a, b, d and a, c {. Then we get O ( ),,{ a},{ b},{ d},{ a, b},{ a, d },{ a, b, c},{ a, b, d},{ a, c, d},{ b, d},{ b, c, d }, } { cd, }, O ( ) {,,{ d},{ a, b},{ a, b, d} }, and { O ( ),,{ a},{ b},{ d},{ a, b},{ a, c},{ a, d},{ a, b, c},{ a, b, d},{ a, c, d},{ b, c},{ b, d }, Hence } { b, c, d},{ c, d}. 1 ( a) 1, ( b) 0, ( c), ( d ) 0, 3 1 1 1 ( a), ( b), ( c), ( d ) 0, 3 3 2 ( a) 1, ( b) 0, ( c) 1, and ( d ) 0. Therefre b a, c, ET ( ) b, d, b, a, b, c, ET ( ) d, b a, c,, and ET ( ) b, d. Remark 3.1.1. b Since, then by Lemma 3.1.4, if and nly if ( ) 0. In Eample 3.1.2, we get a, c, a, b, c, and a, c. Prpsitin 3.1.1. Let be an s-pen )r-clsed( subset f in a finite tplgical space (, ) and. Then ( ) 0 if and nly if fr each subset f where ( ) 1, there eists y such that ( y ) 1 and ( y ) 1. Prf. Let be an s-pen subset f and. Hence ( ) 0 iff by Lemma 3.1.3,. Then by Lemma 2.1.2, we have 34
iff, iff fr each subset f and 3.1.1, iff there eists y s.t. y,, we have by Lemma iff fr each subset f and ( ) 1, there eists y s.t. ( y ) 1 and ( y ) 1 by Lemma 3.1.3. Similarly we can prve this prpsitin if is r-clsed. Prpsitin 3.1.2. Let be a subset f in a finite tplgical space (, ) and. Then ( ) 1 if and nly if there eists a subset f s.t. ( ) 1 and fr each y if ( y ) 1, then ( y ) 0. Prf. Let be a subset f and ( ) 1 iff by Lemma 3.1.3,. Then iff there eists a subset f s.t. by Lemma 3.1.2, iff there eists a subset f s.t. ( ) 1 and fr each y if ( y ) 1, then ( y ) 0, by Lemma 3.1.3 and since. Lemma 3.1.5. Let be a subset f in a tplgical space (, ) and. Then if and nly if fr each subset f such that, where p, s,,,., we have Prf. Let be a subset f and such that p, s,,,. Since is a - pen set fr each, where and p, s,,,, then by Lemma 2.1.5, fr each subset f and, we have. Cnversely, suppse that, fr each subset f such that, we have 35
have,, where p, s,,, G where p, s,,,. Then fr any -pen set G cntaining we. S, G since G G. Hence by Lemma 2.1.5, we have. Lemma 3.1.6. Let be a subset f in a tplgical space (, ) and. Then if and nly if there eists a subset f such that where p, s,,,. and, Prf. Let be a subset f and such that, where p, s,,,. Then by Lemma 2.1.6, there eists a -pen set such that. Hence there eists a subset f such that since. Cnversely, if there eists a subset f such that p, s,,,, then by Lemma 2.1.6, we have fr all p, s,,,. and, since where is a -pen set Prpsitin 3.1.3. Let be an s-pen subset f in a finite tplgical space (, ) and such that ( ) 0. Fr any subset f, if ( ) 1, then there eists y such that ( y ) 1 and ( y ) 1, where s, p,,,. Prf. Let be an s-pen subset f and such that ( ) 0. Then by Lemma 3.1.4, we have every s, p,,,. Hence. Since is s-pen, then. Thus fr. Then by Lemma 3.1.1, we have fr every subset f, if, then. That is there eists y such that y. Hence by Lemma 3.1.3, we have fr every subset, if ( ) 1, then there eists y such that ( y ) 1 and ( y ) 1. 36
Prpsitin 3.1.4. Let be an r-clsed subset f in a finite tplgical space (, ) and. Then ( ) 0 if and nly if fr each subset f and ( ) 1, there eists y such that ( y ) 1 and ( y ) 1, where s, p,,,. Prf. Let be an r-clsed subset f. Then s, p,,,. Hence ( ) 0 iff. Thus by Lemma 3.1.4, where s, p,,,,, where iff, iff fr each subset f and, we have by Lemma 3.1.1, iff fr each subset f and, there eists y s.t. ( y ) 1 and ( y ) 1 by Lemma 3.1.3. Prpsitin 3.1.5. Then Let be an r-clsed subset f in a finite tplgical space (, ) and. if and nly if fr each subset f and ( ) 1, such that ( ) 1 and ( y ) 0, y where s, p,,,. there eists y Prf. Let be an r-clsed subset f. Then s, p,,,. Hence iff,. Thus, where iff fr every subset f and, we have by Lemma 3.1.5, iff there eists y such that ( ) 1 and ( y ) 0 by Lemma 3.1.3 y and Lemma 3.1.4, where s, p,,,. 37
Prpsitin 3.1.6. such that Let be an s-clsed subset f in a finite tplgical space (, ) and. Then there eists a subset f such that ( ) 1 and fr every y, if ( ) 1, then ( y ) 0 y, where s, p,,,. Prf. Thus Let be an s-clsed subset f and such that. Then, where s, p,,,. Hence since.. Then by Lemma 3.1.6, there eists a subset f such that. Therefre by Lemma 3.1.3 and Lemma 3.1.4, there eists a subset f such that ( ) 1 and fr every y, if ( ) 1, then ( y ) 0 y, where s, p,,,. Prpsitin 3.1.7. Let be an r-pen subset f in a finite tplgical space (, ) and. Then ( ) 1 if and nly if there eists a subset f such that ( ) 1 and fr each y, if ( y ) 1, then ( y ) 0, where s, p,,,. Prf. Let be an r-pen subset f and where s, p,,,. Hence. Then. Thus, ( ) 1 iff iff by Lemma 3.1.4,, iff there eists a subset f such that 3.1.2, and by Lemma iff there eists a subset f such that ( ) 1 and fr every y, if ( y ) 1, then ( y ) 0, where s, p,,, by Lemma 3.1.3. 38
Prpsitin 3.1.8. Then Let be an r-pen subset f in a finite tplgical space (, ) and. if and nly if there eists a subset f such that ( ) 1 and fr every y, if ( ) 1, then ( y ) 0, y where s, p,,,. Prf. Let be an r-pen subset f and. Then. Thus, where s, p,,,. Hence iff, iff there eists a subset f such that by Lemma 3.1.6, iff there eists a subset f such that ( ) 1 and fr every y, if ( ) 1, then ( y ) 0 y, where s, p,,, by Lemma 3.1.3 and Lemma 3.1.4. Lemma 3.1.7. Let (, ) be a tplgical space and. Then if and nly if fr each subset f such that, we have { }. Prf. Let (, ) be a tplgical space and such that. Since is pen set fr any subset f, then by Definitin 1.1.9, such that, we have { }. Cnversely, let G be any pen set cntaining. Then { } G. Thus G { }, since G is pen. Hence by Definitin 1.1.9, we have. Prpsitin 3.1.9. Let be a subset f a finite tplgical space (, ) and such that. Then fr each subset f and ( ) 1, there eists y ther than such that ( y ) 1 and ( y ) 0. 39
Prf. Let be a subset f and such that. Then by Lemma 3.1.7, fr each subset f and, ther than such that y { } we have { }. Thus there eists y. Hence by Lemma 3.1.3, ( y ) 1 and ( y ) 0, where y. Definitin 3.1.3. Let (, ) be a tplgical space. The pint p is said t be a -cluster pint f a subset f fr all r, p, s,,, p, we have { p}. if fr each subset f such that The set f all -cluster pints f is called the -derived set f and it is dented by fr all r, p, s,,,. Prpsitin 3.1.10. Let be a subset f in a finite tplgical space (, ) and such that. Then fr each subset f and ( ) 1, there eists y ther than such that ( ) 1 and ( ) 0, y y where r, s, p,,,. Prf. Let be a subset f and fr each subset f and such that. Then by Definitin 3.1.3,, we have { }. Thus there eists y ther than such that y { }. Hence by Lemma 3.1.4, we have ( ) 1 and ( ) 0 y, where r, s, p,,, y. Prpsitin 3.1.11. Let be a subset f in a finite tplgical space (, ) and such that ( ) 1. If there eists y ther than such that ( y ) 1, where are the pen sets cntaining, then and ( y ) 1. 40
Prf. Let be a subset f and such that ( ) 1. Then fr the pen sets cntaining, we have. ssume that there eists y ther than such that ( y ) 1. Then fr the pen sets y cntaining y, we have y. Thus y. Then ( y ) 1. ut G fr each pen set G cntaining, and s y G. Hence each pen set G cntaining cntains a pint ther than frm which is y. Therefre by Definitin 1.1.9, we have and ( y ) 1. Prpsitin 3.1.12. Let be a subset f in a finite tplgical space (, ) and. If there eists y ther than such that ( y ) 0, ( y) ( ) and ( y ) 1, where are the pen sets cntaining, then. Prf. Let be a subset f and. Suppse that, there eists y ther than s.t. ( y ) 0, ( y) ( ) and ( y ) 1, where is any pen set cntaining. Then fr the pen sets y cntaining y, we have y. If, then ( ) ( y), which is a cntradictin, s we have y. Since is y the smallest pen set cntaining, then. Since ( y ) 0, then by Lemma y 3.1.3, we have y. S, ( ) ; that is, there eists z such that z ( y ), where z pen set G cntaining. Then Therefre by Definitin 1.1.9, we have y. Thus z { }. ut G { }, G such that G.. G fr each Prpsitin 3.1.13. Let, be tw disint pen subsets f in a finite tplgical space (, ) and. If ( ) 1 (resp. ( ) 1), then ( ) 0 (resp. ( ) 0 ). 41
Prf. Let, be tw disint pen subsets f. Then is clsed set cntaining. Thus. Hence. Nw let such that ( ) 1, then by Lemma 3.1.3. ut is an pen set. Thus Therefre by Lemma 3.1.3, we get ( ) 0.. Hence. We can use Prpsitin 3.1.13 t get the fllwing remark. Remark 3.1.2. (, ), then If and are tw disint pen subsets f in a finite tplgical space. Prpsitin 3.1.14. Let be an s-pen (r-clsed) subset f in a finite tplgical space (, ) and let be any subset f such that. If ( ) 1 (resp. ( ) 0 ) fr sme, then ( ) 0 (resp. ( ) 1). Prf. Let be an s-pen subset f and let be any subset f such that. Since and are tw disint pen subsets f, then by Remark 3.1.2, we get. Hence ( since is s-pen). Nw, let such that ( ) 1. Then by Lemma 3.1.3, we have. Thus. Therefre ( ) 0. Similarly, we can prve this prpsitin if is r-clsed. Prpsitin 3.1.15. Let (, ) be a finite tplgical space. Then (, ) is a T - space if and nly if fr each tw distinct pints, y f, there eists a subset G f such that ( ) 1, ( y ) 1 r ( y ) 1, ( ) 1. G G G G 42
Prf. Let (, ) be a finite tplgical space. If (, ) is a T - space, then fr each tw distinct pints, y f, there eists an pen set G such that G, y G r y G, G. Suppse that G and y G. Then G and y G, since G is pen. Hence by Lemma 3.1.3, we have ( ) 1 and ( y ) 1. G Cnversely, let, y such that y and let G be a subset f such that G ( ) 1 and ( y ) 1. Then by Lemma 3.1.3, we have G G G and y G. Since G is pen set, then (, ) is a T - space. Prpsitin 3.1.16. Let (, ) be a finite tplgical space. Then (, ) is a T1 - space if and nly if fr each tw distinct pints, y f, there eist tw subsets, f such that ( ) 1, ( y ) 1, ( y ) 1 and ( ) 1. Prf. Let (, ) be a finite tplgical space. If (, ) is a T1 - space, then fr each tw distinct pints, y f, there eist tw pen subsets, f such that, y, y and. Then, y, y and, since, are pen sets. Hence by Lemma 3.1.3, we have ( ) 1, ( y) 1, ( y ) 1 and ( ) 1. Cnversely, let, y such that y, and let, be tw subsets f such that ( ) 1, ( y ) 1, ( y ) 1 and ( ) 1. Then by Lemma 3.1.3, we have, y, y and. Since, are tw pen sets, then (, ) is a T1 - space. Prpsitin 3.1.17. Let (, ) be a finite tplgical space. Then (, ) is a T 2 - space (Hausdrff space) if and nly if fr each tw distinct pints, y f there eist tw subsets, f such that ( ) 1, ( y) 1 and fr any pint z f, if ( z ) 1(resp. ( z ) 1), then ( z ) 1 (resp. ( z ) 1). 43
Prf. Let (, ) be a finite tplgical space. If (, ) be a T 2 - space, then fr each tw distinct pints, y f there eist tw disint pen subsets, f such that and y. Then, y and since, are disint and pen sets. Hence by Lemma 3.1.3 and Prpsitin 3.1.13, we have ( ) 1, ( y) 1 and fr any z, if ( z ) 1 (resp. ( z ) 1), then ( z ) 0 (resp. ( z ) 0 ). Cnversely, let, y such that y and let, be tw subsets f such that ( ) 1, ( y) 1 and fr any pint z f, if ( z ) 1 (resp. ( z ) 1), then ( z ) 1 (resp. ( z ) 1). Then by Lemma 3.1.3, we have, y and. Since, are tw pen sets, then (, ) is a T 2 - space. 3.2 Fuzzy sets in tplgical spaces In this sectin we intrduce sme types f fuzzy sets by using - membership functin, where r, p, s,,,. Definitin 3.2.1 [17]. Let be a subset f in a finite tplgical space (, ). The fuzzy set f is dented by and it is defined by (, ( ) :. Definitin 3.2.2 [10]. Let (, ) be a finite tplgical space and. The -fuzzy set f is dented by and it is defined by (, ( ) :, where r, p, s,,,. Eample 3.2.1. Let a, b, c, d,,, a, b, d, a, b, d, and a, c { 44. Then we get O ( ),,{ a},{ b},{ d},{ a, b},{ a, d },{ a, b, c},{ a, b, d},{ a, c, d},{ b, d},{ b, c, d }, } { cd, },
O ( ) {,,{ d},{ a, b},{ a, b, d} }, and { O ( ),,{ a},{ b},{ d},{ a, b},{ a, c},{ a, d},{ a, b, c},{ a, b, d},{ a, c, d},{ b, c},{ b, d }, Hence Therefre } { b, c, d},{ c, d}. 1 1 1 ( a), ( b), ( c), ( d ) 0, 2 2 2 1 1 1 ( a), ( a), ( c), ( d ) 0, 3 3 2 1 ( a) 1, ( b) 0, ( c), ( d ) 0, 3 ( a) 1, ( b) 0, ( c) 1, and ( d ) 0. 1 1 1 a,, b,, c,,( d,0), 2 2 2 a 1 1 1,,,,,,(,0), 3 b 3 c 2 d 1 ( a,1),( b,0), c,,( d,0), and 3 ( a,1),( b,0),( c,1),( d,0). Prpsitin 3.2.1. that Let and be tw subsets f in a finite tplgical space (, ) such. Then (i). (ii) fr all r, p, s,,,. Prf. (i) Let and be tw subsets f such that where Thus. Then ( ) ( ),, are the pen sets cntaining. ( ) ( ),. 45
Hence ( ) ( ),. That is ( ) ( ),. Therefre. (ii) If ( ) 1, we have finishing the prf. S, we have tw cases: Case (1) If ( ) 1 fr sme, where r p s,,,,,, then 1 ( ). Thus there eists a -pen set G such that G. Then there eists a -pen set G such that G, since. Hence 1 ( ). Then ( ) 1. Case (2) If ( ) 1 and ( ) 1, then fr all -pen set cntaining, since. Then fr all -pen set cntaining. Thus min ( ) min ( ) fr all r, p, s,,,. Hence ( ) ( ),. Therefre fr all r, p, s,,,. Eample 3.2.2. Let,, a, b, d, a, b, d be a tplgy n a, b, c, d a, b, a, b, d, C a, c, d, D a and E c, d 46. Then PO ( ),,{ a},{ b},{ d},{ a, b},{ a, d},{ a, b, c},{ a, b, d},{ a, c, d},{ b, d}, SO( ),,{ a},{ a, c},{ a, b, d},{ b, d},{ b, c, d}, and O ( ),,{ a},{ a, b, d},{ b, d}., and let
Hence 1 1 1 ( a,1), b,, c,, d,, 2 2 2 3 ( a,1),( b,1), c,,( d,1), 4 1 3 1 C ( a,1), b,, c,, d,, 2 4 2 1 D ( a,1),( b,0), c,,( d,0), 4 s s s s 1 1 1 ( a,1), b,, c,, d,, 3 3 3 1 ( a,1),( b,1), c,,( d,1), 2 1 1 C ( a,1), b,,( c,1), d,, 2 2 D ( a,1),( b,0),( c,0),( d,0), C p ( a,1),( b,0),( c,1),( d,1), and E 1 1 1 ( a,0), b,,,,,. 3 c 2 d 3 Nte that, D, D and D. s s s Remark 3.2.1. The cnverse f Prpsitin 3.2.1 is nt true in general as the fllwing eample illustrates. Eample 3.2.3. be the tplgical space given in Eample 3.2.2. If a, b Let (, ) and C a c d,,, then 1 1 1 ( a,1), b,, c,, d, 2 2 2 and 1 3 1 C ( a,1), b,, c,, d,. 2 4 2 Hence C, but C. 47
Prpsitin 3.2.2. Let and be tw subsets f in a finite tplgical space (, ). Then 1) ( ) aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa a 2) ( ) aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa a 3) ( ), r, p, s,,, aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa 4) ( ), r, p, s,,, aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa Prf. Let and be tw subsets f in a finite tplgical space (, ). 1) Let C. Then ( ) min ( ), ( ),. C Nw, since and, then by Prpsitin 3.2.1, we have ( ) and ( ). aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa a aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa a Thus ( ) ( ) and ( ) ( ),. Hence Therefre ( ) ( ) min ( ), ( ),. C ( ) C. 2) Let C. Then aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa ( ) ma ( ), ( ),. C Nw, since and, then by Prpsitin 3.2.1, we have ( ) and ( ). aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa a Thus Hence aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa a ( ) ( ) and ( ) ( ),. ( ) ma ( ), ( ) ( ),. C 48
Therefre ( ) C. aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa The prfs f parts (3) and (4) are similar. Prpsitin 3.2.3. Let be a subset f in a finite tplgical space (, ). Then c c ( ) ( ). aaaaaaaaaaaaaaaaaaaaaaaaaaaaa Prf. Let be a subset f in a finite tplgical space (, ), and let C c ( ). Then ( ) 1 ( ),. C Nw, we have where are the pen sets cntaining. c ( ) c ( ), c ut ( ) ( ) ( ) Thus Hence c. S, ( ) ( ). ( ) c c c Therefre ( ) ( ). ( ) ( ) 1 1 ( ). c ( ) C ( ),. aaaaaaaaaaaaaaaaaaaaaaaaaaaaa Prpsitin 3.2.4. Prf. Let and be tw subsets f in a finite tplgical space (, ). Then ( ). aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa Let and be tw subsets f. Then by Prpsitin 3.2.2 part 1, we have 49
c c ( ) ( ) ( ). aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaa Hence by Prpsitin 3.2.3, we get c ( ) ( ). aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa Remark 3.2.2. c c In general, fr any subset f in a tplgical space (, ), ( ) ( ), where r, p, s,,,. The fllwing eample illustrates Remark 3.2.2. aaaaaaaaaaaaaaaaaaaaaaaa Eample 3.2.4. Let a, b, c, d,,, a, b, d, a, b, d, and C a, c, d C 1 1 ( a,1), b s,,(,1),,, 2 c d 2 and c 1 1 ( ) ( a,0), b,,( c,0), d, s C 4 4. aaaaaaaaaaaaaaaaaaaaaaaa. Then Hence c 1 1 c ( sc ) ( a,0), b,,( c,0), d, s ( C ). 2 2 aaaaaaaaaaaaaaaaaaaaaaaa Definitin 3.2.3 [16]. Let be the fuzzy set f a subset f and 0,1. Then the cut (resp. strng cut) f is dented by (resp. ), and it is defined by the crisp set : ( ) resp. : ( ), where ( ) is the membership functin f t. The fllwing definitins intrduce the cncepts f cut, strng cut, cut and strng cut f fuzzy sets in tplgical spaces using the membership functin and -membership functin, where r, p, s,,, 0,1. 50 and
Definitin 3.2.4. Let be the fuzzy set f a subset f in a finite tplgical space (, ) and 0,1. Then the cut (resp. strng cut) f is dented by resp. Definitin 3.2.5 (, ), and it is defined by the crisp set : ( ) Let be the -fuzzy set f a subset f in a finite tplgical space where r, p, s,,,. Then the cut (resp. strng cut) is dented by resp., and it is defined by the crisp set where 0,1. : ( ) resp. : ( ). resp. : ( ), Eample 3.2.5. Let (, ) be the tplgical space given in Eample 3.2.2, we get Then s 1 1 1 ( a,1), b,, c,, d,, 2 2 2 1 1 1 ( a,1), b,, c,, d,. 3 3 3 1 1 1 1 2 2 2 2 S S a, b, c, d, a, a, a, 1 1 S 3 3 S a, b, c, d, and a. Lemma 3.2.1 [16]. Let and be tw fuzzy subsets f. Then i) iff ii) iff fr each 0,1, fr each 0,1. 51
Prpsitin 3.2.5., Let, be tw subsets f in a finite tplgical space (, ). If then fr all 0,1, we have i) and, ii) and fr all r, p, s,,,, Prf. i) Let, be tw subsets f such that, and let 0,1. Then by Prpsitin 3.2.1, we have. Thus by Lemma 3.2.1, we have and fr all 0,1. ii) The prf is similar t part (i). Remark 3.2.3. In tplgical spaces, the strng 0 cut f is (i.e. 0 1 cut f is ( i.e. 1 ). Mrever, the strng 0 cut f is ) and the (i.e. 0 ) and the 1 cut f is (i.e. 1 ). Lemma 3.2.2 [16]. Let,, C, D and E be fuzzy subsets f such that C, D and E ( ) c i).. Then fr all 0,1 ii) ( C ) and ( D). iii) ( C ) and ( D)., we have iv) (1 ) ( E) ( ) c. Prpsitin 3.2.6. Let, be tw subsets f in a finite tplgical space (, ). Then fr all 0,1, we have i) ( ) and ( ). ii) ( ) and ( ). 52
iii) ( ) and ( ). iv) ( ) and ( ). Where r, p, s,,,. Prf. i) Let and be tw subsets f and 0,1. Since and, then by Prpsitin 3.2.5, we have ( ), ( ), ( ) and ( ). Hence ( ) and ( ) fr all 0,1. The prf f (ii), (iii) and (iv) are similarly as in (i). 53
CHPTER 4 FUZZ SETS IN PRODUCT TOPOLOGICL SPCES 54
Chapter 4 Fuzzy sets in prduct tplgical spaces The aim f this chapter is t intrduce sme results in the prduct tplgy using fuzzy sets and sme classes f near pen sets. 4.1 inary prduct in tplgical spaces In this sectin, we intrduce sme results n prduct tplgy using sme classes f near pen sets. Definitin 4.1.1 [19]. Let and be tw sets. The prduct set f and is dented by and it is defined by ( a, b) : a, b. Each subclass f is a relatin frm t. Eample 4.1.1. Let 1,2 and a, b. Then ( 1, a),(1, b),(2, a),(2, b). Definitin 4.1.2 [13]. be tw tplgical spaces. The prduct tplgy Let (, ) and (, ) n is the tplgy generated by taken the cllectin G H : G, H as a base. The pair, is called the prduct tplgical space. 55
Eample 4.1.2. d. Let a, b, c, d, e,,,,,, and,, Then the base f the prduct tplgy is Hence a b a b ( a, d ),( a, e),( b, d ),( b, e), ( a, d ),( b, d ).,, ( a, d ),( b, d ),( c, d ), ( a, d ),( a, e), ( a, d ), ( b, d ),( b, e), ( b, d ), a d b d c d a d a e a d b d b e b d,, (, ),(, ),(, ), (, ),(, ), (, ), (, ),(, ), (, ), ( a, d ),( a, e),( b, d ),( b, e), ( a, d ),( b, d ), ( a, d ),( a, e),( b, d ), ( a, d ), ( b, d ), ( c, d ),( a, e), ( a, d ),( b, d ),( c, d ),( b, e), ( a, d ),( b, d ),( c, d ),( a, e), ( b, e ), ( a, d a e b d b e a d a e b d b e a d a e b d ),(, ),(, ),(, ), (, ),(, ),(, ),(, ), (, ),(, ),(, ). Prpsitin 4.1.1 [27]. Let, be tw subsets f, in the tplgical spaces (, ), (, ) respectively, and let,. Then be a subset f in the prduct tplgical space. Prf. Let, be tw subsets f, respectively, and Let (, y ) ( ). Then there eists an pen set G in such that (, y ) G. ut G ( H K ), where H, K and. Thus there eists 0 such that Thus H 0 and y K 0, where H and 0 K 0 (, y ) H K H K. 0 0 H and y K. 0 0. Hence Then and y. Thus (, y ). Hence ( ) (1). 56