Fuzzy Topology On Fuzzy Sets: Regularity and Separation Axioms

Transcription

1 wwwaasrcorg/aasrj American Academic & Scholarl Research Journal Vol 4, No 2, March 212 Fuzz Topolog n Fuzz Sets: Regularit and Separation Aioms AKandil 1, S Saleh 2 and MM Yakout 3 1 Mathematics Department, Facult of Science, Helwan Universit, Cairo, Egpt drali_kandil@ahoocom 2 Mathematics Department, Facult of Education-Zabid, Hodeidah Universit, Yemen, S_wosabi@ahoocom 3 Mathematics Department, Facult of Education, Ain Shams Universit, Cairo, Egpt mmakout@ahoocom Abstract: In this paper, separation and regularit aioms in fuzz topolog on fuzz set are defined and studied We investigate some of its characterizations and discuss certain relationship among them with some necessar countereamples Moreover some of their basic properties are eamined In addition, goodness and hereditar properties are discussed 1 Introduction: The notion of fuzz topolog on fuzz sets was introduced b Chakrabort and Ahsanullah [1] as one of treatments of the problem which ma be called the subspace problem in fuzz topological spaces ne of the advantages of defining topolog on a fuzz set lies in the fact that subspace topologies can now be developed on fuzz subsets of a fuzz set Later Chaudhur and Das [2] studied several fundamental properties of such fuzz topologies The concept of separation aioms is one of most important concepts in topolog In fuzz setting, it had been studied b man authors such as [3,5,6,7,1]However, the separation and regularit aioms has not et been studied in the new setting, onl in [2] the introduced the concept of Hausdorff, regular and normal spaces The object of the present paper is to introduce a set of new regularit and separation aioms which are called (FR i,i =,1,2,3) and (FT i,i=,1,2,3,4) b using quasi-coincident and neighborhood sstem ur work organized as follows, In section 1 We give some preliminar concepts, investigating some of new results in the new setting In section 2 We give the definition of regularit aioms (FR i ; i=,1,2,3) and some characteristics theorems are proved Net the separation aioms (FT i ; i =,1,2,3,4) are introduced, investigating man of its properties in section 3 Finall, in section 4 We eamine the hereditar and good etension propert in the sense of Lowen [9] 2 Definitions and Notations Throughout this paper, denotes a non-empt set, the smbol will denote the closed unit interval and a fuzz set of is a function with domain and values in A fuzz point is a fuzz set such if for all and if We write if The famil of all fuzz points of will be denoted b If then is said to be a fuzz subset of and denoted b The famil of all fuzz subsets of will denoted b ie The set is said to be the support of If, the fuzz subset of which assigns will be denoted b If then denotes the characteristic function of on 11 Definition If Then denotes the characteristic function of referred to In general a fuzz subset of is called a maimal if ie if, then If, then the complement of B referred to A, denoted b and defined b, Let Then are said to be quasi-coincident referred to, denoted b iff there eists such that If is not quasi-coincident with referred to then we denoted for this b Now one can easil prove the following proposition as in [1] 12 Proposition Let and Then: 1), 2), 3), 4), 5) U G,

2 wwwaasrcorg/aasrj American Academic & Scholarl Research Journal Vol 4, No 2, March 212 6) 7), for some, 8), 9), 1) 13 Lemma Let and Then: i) ii) Proof bvious Now we recall the basic definition of fuzz topolog on fuzz set as in [1] 14 Definition Let be a fuzz subset of A collection of fuzz subsets of ie satisfing the following conditions: i),, ii), iii) is called a fuzz topolog on The pair is called a fuzz topological space, members of will be called a fuzz open sets and their complements referred to are called a fuzz closed sets of The famil of all fuzz closed sets in will be denoted b Note: Unless otherwise mentioned b fuzz topological spaces we shall mean it in the sense of the above definition and will denote a fuzz topological 15 Definition A fuzz topological space is called a full stratified if each fuzz subset in the form is in for all 16 Definition Let be a fuzz topological space, Then an fuzz set contains is called a neighborhood (nbd, for short) of in The set of all neighborhoods of will be denoted b, In general for an, denotes a fuzz open subset of A contains 17 Definition Let be a fuzz topological space, Then the closure(interior) of is defined b: i) A ii) respectivel 18 Proposition Let be a fuzz topological space, and Then we have: i) ii) there eists such that iii), for all iv), for all Proof Stratiforward In the following we recall the concept of the strong -cut of an fuzz subset of as in [1] 19 Definition: For an We define, 11 Proposition Let, and is a finite inde set Then we have: i), ii) B using the Lemma (13), it is eas to prove the following theorem

3 wwwaasrcorg/aasrj American Academic & Scholarl Research Journal Vol 4, No 2, March Theorem a) Let and be a topological space on Then the following structures: and, are fuzz topologies on generated b b) Let ( A, ) be a fuzz topological space on Then the following structures: S ( B) : B and, i) A ii) ] B S( : [ B, are ordinar topologies on generated b 112 Proposition Let, be a topological space on and a fuzz topological space on Then: i) ii) and then, iii) and Proof Straightforward 113 Definition Let A topological space (, τ ) is said to be an s-topological space on iff, τ contains for all The following eample shows the eistence of s-topological space and shows that a topological space (,τ) need not be s-topological space and shows that the famil, need not be a topological space on 114 Eample Let and Then we have: i) and ii) is a topolog on but not s-topolog iii) is s-topolog on iv) is not a topolog on 115 Definition Let be an s-topological space on Then a fuzz subset of is said to be a fuzz lower semi-continuous function on if The set of all fuzz subsets of which is lower semi-continuous functions on will be denoted b, 116 Proposition Let be an s-topological Then the famil is a fuzz topolog on and, ) is called an induced fuzz topological space b Now one can easil prove the following lemma 117 Lemma Let be an s-topological space on, and Then we have: i) iff, ii) iff iii) for all, iv), v) 118 Definition A fuzz topological space is said to be a weakl induced iff, for ever for all ie iff ever element in is a fuzz lower semi-continuous function from to Note: ne of the advantages of defining topolog on a fuzz set lies in the fact that subspace topologies can now be developed on fuzz subsets of a fuzz set as follows: 119 Definition[1,2] Let be a fuzz topological space and Then the famil is a fuzz topolog on and is called a fuzz subspace of

4 wwwaasrcorg/aasrj American Academic & Scholarl Research Journal Vol 4, No 2, March 212 Note: If is a maimal subset of, then is called a maimal subspace of 12 Proposition[13,15] Let be the maimal subspace of a FTS Then: i) is closed in if and onl if, where ii) For ever, we have, where, are closures of in and, respectivel 2 Fuzz regularit aioms 21 Definition A fuzz topological space is said to be: i) FR -space iff with implies, ii) FR 1 -space iff with implies there eist and such that iii) FR 2 -space iff ( and with implies there eist and such that iv) FR 3 -space iff with implies there eist and such that Note: FR 2 (resp FR 3 ) spaces are those which are called fuzz regular (resp fuzz normal) spaces and was introduced in [2] as an etension of its original concept in [8] In the following we introduce some properties of FR 22 Theorem Let be a fuzz topological spaces, and Then the following statements are equivalent: 1) is a FR -space, 2) 3) 4) implies there eists such that 5) 6) implies Proof 1) 2) Let B (ii) of Proposition (18) we have 2) 3) is obvious 3) 4) Let (b (6) of Proposition (12)) 4) 5) Let there eists such that 5) 6) and 6) 1) are obvious 23 Theorem The following implications hold: Proof i) Let ( A, ) be a FR 3, FR and let G, G Then from (5) of the above theorem we have G Since ( A, ) is FR 3, then there eist, G such that G Take, =, then G and hence ( A, ) is FR 2 - ii) Let ( A, ) be a FR 2 space and let Then there eist and such that Take iii) Let ( A, ) be a FR 1 space, eist, such that Proposition (12)) we get 24 Corollar X Hence ( L, ) is a FR 1 and so there (b (iii) of Proposition (18)) So b (6) of Hence ( A, ) is X

5 wwwaasrcorg/aasrj American Academic & Scholarl Research Journal Vol 4, No 2, March 212 Let be a fuzz topological Then is a FR 1 if and onl if with implies there eist such that Proof Follows from the above implication and from (2) of Theorem (22) 25 Lemma Let be a topological space, Then we have: i) for all, ii) Proof bvious 26 Theorem Let be a topological space on Then is FR if and onl if is a R - Proof Let be a FR -space Then (b the above lemma) implies (b (iii) of Proposition (18) ) above lemma) Hence is a - Conversel, let be a R -space, ince Then, when then clearl (since is R i e ) Hence is a FR - 27 Theorem Let be a topological If is -space, then is FR 1 - Proof Let be a R 1 -space,, FP( with Then either or (, ) (a) If, then either or i) If, then there eist such that Now we take and, then Hence is FR 1 ii) If, this case is ecluded (since is (b) If, then we take, to be the required neighborhoods Hence is a FR 1-28 Theorem Let be a to o ogica s ace if and onl if for all such (b the Proof Let be a FR 2, and Then there eist N( ), G N( ) such that G implies that Conversel, let, be such that Then ie ( ), so there eists such that = (b hpothesis) and Hence is a FR 2-29 Theorem Let be a to o ogica s ace is (normal) if and onl if there eists s c t at Proof The proof is analogous to the above proof 3 Fuzz separation aioms 31 Definition A fuzz topological space is said to be: i) FT -space iff with implies there eists such that or there eists such that ii) FT 1 -space iff with implies there eist such that and there eists such that iii) FT 2 -space iff with implies thereeist and such that iv) FT 3 -space iff it is FR 2 and FT 1 - v) FT 4 -space iff it is FR 3 and FT 1-32 Theorem ) be a fuzz topological Then ( A, ) is FT if and onl if ( implies or )

6 wwwaasrcorg/aasrj American Academic & Scholarl Research Journal Vol 4, No 2, March 212 Proof ) be a FT and, FP( with (18))Conversel, let is a FT 33 Theorem or there eists or such that Then there eists such that (b (iii) of Proposition or Hence ( A, ) Let be a topological Then ( A, ) is FT if and onl if is T - Proof ) be a FT -space and Then, in particular implies there eists such that or there eists such that Take, then or take, then Hence is a T - Conversel, let be a T and, FP( with (, A() ) If, then there eists such that Then either or or there eists such that Now take, then or take, then Hence ( A, ) is a FT - If, A(), then we take and (b(i) of Proposition (112) ) to be the required neighborhoods Hence ( A, ) is a - 34 Theorem ) be a fuzz topological spaces If ( A, ) is FT, then is a T - Proof The proof is analogous to that of necessit of the above theorem 35 Theorem ) be a full stratified fuzz topological Then ( A, ) is FT if and onl if is a T - Proof Necessit, follows from the above theorem Conversel, let be a T -space and, FP( Then either or (, A() ) If, then there eists such that or there eists such that Now take, then or take, then Hence ( A, ) is a FT - If (, A() ), then Take or (since ( A, ) is full stratified) to be the required neighborhoods Hence ( A, ) is a - 36 Theorem ) be a full stratified fuzz topological If is at -space, then ( A, ) is a FT - Proof Let be a T and,, FP( Then either or (, A() ) If, then there eists such that or there eists such that Now take, then or take, then Hence ( A, ) is FT If, A(), then take or (since ( A, ) is full stratified) to be the required neighborhoods Hence ( A, ) is a - 37 Theorem ) be a full stratified and weakl induced fuzz topological Then ist -space if and onl if ( A, ) is FT Proof Necessit, follows from the above theorem Conversel, let ( A, ) be FT and, then Since ( A, ) is FT, then there eists such that or there eists such that Now take or take (since ( A, ) is weakl induced), then it is eas to see that or Hence is T - In the following theorems we stud some properties of FT1 spaces 38 Theorem ) be a fuzz topological Then the following statements are equivalent: i) ( A, ) is a FT 1 space,

7 wwwaasrcorg/aasrj American Academic & Scholarl Research Journal Vol 4, No 2, March 212 ii), FP( with iii), FP( implies and Proof i) ii) is clearl from (iii) of Proposition (18) i) iii) Let there eists such that this implies ( ) A, thus ( ) A is open ie is closed And this is true for ever FP( iii) i) FP( and Then, A Since and Hence ( A, ) is a FT 1-39 Theorem ) be a fuzz topological If ( A, ) is FT1, then is a T 1 Proof ) be a FT 1 and S( Then A( ) A( ) and so A() this implies that S( A ( ) ) S( \{ } so {} is closed for all S( Hence is a T 1 31 Theorem ) be a full stratified fuzz topological Then ( A, ) is FT1 if and onl if is T 1 - Proof Necessit, follows from the above theorem Conversel, the proof is similar to that of Theorem (35) 311 Theorem Let be a topological Then is T 1 if and onl if ( A, ) is FT 1 Proof Necessit, let be at 1 -space and, FP( Then either or (, A()) Now if, then there eists such that and there eists such that, Take and, then and Hence ( A, ) is FT 1 If (, A()) Take and to be the required neighborhoods Hence A, ) is a FT 1 - ( Conversel, let A, ) be a FT 1 -space Then is a T 1 space (b Theorem (31)) But ( (iii) of Proposition (112)) Hence (,) is a T Theorem ) be a full stratified fuzz topological If ist 1 -space, then ( A, ) is FT1 Proof The proof is analogous to that of Theorem (36) 313 Theorem ) be a full stratified and weakl induced fuzz topological Then ist 1 -space if and onl if ( A, ) is FT1 Proof The proof is analogous to the proof of Theorem (37) The following eample shows that the converse of Theorem (34) and Theorem (39) ma not be true in general 314 Eample Let, Take Then is a fuzz topolog on and is a topolog on which is a T1 - But is not a - In fact, but there is no such that and there is no such that 315 Theorem Let be a fuzz topological If is a FT 2, then: for all Proof Let be a FT 2 space, Then for an there eist such that for all ( b (iii) of Proposition (18) ) and so, (b (6) of Proposition (12)) But clearl Hence we get the result 316 Theorem Let be a topological If is T2 -space, then ( A, ) is FT 2 Proof Let (,) be at 2 -space, Then either or (, A() ) If, then there eist and such that Take and (b

8 wwwaasrcorg/aasrj American Academic & Scholarl Research Journal Vol 4, No 2, March 212, then If (, A()), then take A and A to be the required neighborhoods Hence A, ) is a FT 2 - ( Note: The Eample (34) in [7] shows that the converse of the above theorem ma not be true in general, where consider an infinite set and be an infinite maimal subset of The following eample shows that, in general, a T 2 -space need not impl that be a FT Eample Let be an T 2 -space and Then which is a T 2 - But is not a FT Theorem ) be a full stratified fuzz topological If ( S(,[ ] ) is a -space, then ( A, ) is a FT2 Proof The proof is similar to that of Theorem (36) 319 Theorem ) be a full stratified and weakl induced fuzz topological Then is at 2 -space if and onl if ( A, ) is a FT2 - Proof Straightforward 32 Definition[2] A fuzz topological space ( A, ) is said to be a fuzz T -space if with 2 such that and, (where iff ) 321 Theorem ) be a fuzz topological Then our FT2 -space is a fuzz T -space in the sense of the above 2 definition Proof bvious and so is omitted The following eample shows that the converse of the above theorem ma not be true in general 322 Eample Let and Take Then is a fuzz topolog on which is a fuzz T space in the sense of Definition (32) But 2 is not space in our sense In fact but there do not eist an two members of which not quasi-coincident referred to members of containing 6323 Theorem The following implications hold: FT 4 FT 3 FT 2 FT 1 FT Proof i) Let be a FT 4 space and, Then Since is a FR 3, then there eist, such that Now Put, then Hence is a FT 3 - ii) Let be a FT 3 space and Then, Since is FR 2, then there eist and such that Now Put, then Hence is a FT 2 - iii) Let be a FT 2 space and let Then there eist and such that Hence iv) bvious and and (b (iii) of Theorem (18)) is a FT 1 - From the above theorem and Theorem (23), we obtain the following results 324 Corollar The following implications hold: FT 4 FT 3 FT 2 FT 1 FT FR 3 FR FR 2 FR 1 FR an F-

9 wwwaasrcorg/aasrj American Academic & Scholarl Research Journal Vol 4, No 2, March Some Properties 41 Theorem Let be an s-topological space on Then: is a -space is a -space, where Proof For Suppose is R - Let and let Then for an Since is R, then for an It follows that = for an It is eas to see that Then and so is FR Conversel, suppose is a FR -space, and let Then It follows that and so Hence is a R - For Suppose is a R 1 -space,, FP( with Then either or ( ) a) If, then either or and so we have to cases: i) If, then there eist such that Put and, then Hence is a FR 1 - ii) If, this case is ecluded (since is b) If,, then we put A, to be the required neighborhoods Hence is a FR 1 Conversel, suppose is a FR 1 and, then Since is FR 1, then, such that Now put ( and (, then remains to prove that Suppose Hence and Now we have two cases If then which contradicts to If we get the same contradiction is a R 1 - For Let be a R 2 -space and with Now for an we have Since is R 2 and so there eists such that It is clear that Conversel, suppose is a FR 2 -space Then we have and so there eists such that Then Hence is a R 2-42 Theorem Let be an s-topological Then is a -space if and onl if is a - space, for,3 Proof For Suppose is a T 1 -space and Then either or (, A()) i) If then there eists such that and there eists such that and so (sa) and (sa), where ii) If (, A() ), then take A, to be the required neighborhoods Hence - Conversel, suppose is a FT 1 -space and Then belong to the range of In particular, we have Hence there eists such that and there eists such that Thus for an we have τ and Hence is a T 1 - The proof for the case is similar to that of the case of the above theorem The proof for the case is similar to the case of the above theorem For The proof follows from the cases of Theorem(41) and from case of the above theorem

10 wwwaasrcorg/aasrj American Academic & Scholarl Research Journal Vol 4, No 2, March 212 Now, in the following theorems we will show that the aioms (, for ) and, for,3) are hereditar referred to the class of maimal subspaces 43 Theorem Let be a fuzz topological space and be a maimal fuzz subset of If is, then the fuzz subspace is, for Proof As a sample we will prove the case Let be a FR 2 -spasce, ) and be a fuzz subset of with Then Since is a maimal fuzz subset of, then and so Now if hence If, then and so So Since is FR 2, then there eist, such that Take and, then (since is a maimal fuzz subset of ) Hence is a FR 2-44 Theorem Let be a fuzz topological space and be a maimal closed fuzz subset of If is, then the fuzz subspace is Proof bvious and so is omitted 45 Theorem Let be a fuzz topological space and be a maimal fuzz subset of If is, then the subspace is, for Proof As a sample we will prove the case Let be a FT, ) with Then Since is FT, then there eists such that or there eists such that Take or Since is a maimal fuzz subset of, then or Hence is a FT - Remark If an fuzz subset of, then the aioms (, for and (, for need not be hereditar properties The following eample shows the above remark 46 Eample Let and Take Then is a fuzz topolog on which is and Now let be an fuzz subset of Then and so is a fuzz subspace of but it is not a - References [1] M K Chakrabort, T M G Ahsanullah, Fuzz topolog on fuzz sets and tolerance topolog, Fuzz Sets and Sstems, 45(1992)13-18 [2] A K Chaudhuri, P Das, Some results on fuzz topolog on fuzz sets, Fuzz Sets and Sstems, 56(1993) [3] S Gangul, SSaha, n separation aioms and Ti-fuzz continuit, Fuzz Sets and Sstems, 16 (1985) [4] W Geping, H Lanfang, n induced fuzz topological spaces, J Math Anal Appl, 18 (1985) [5] B Hutton, Normalit in fuzz topological spaces, J Math Anal Appl, 5 (1975), [6] B Hutton, I Reill, Separation aioms in fuzz topological spaces, Fuzz Sets and Sstems, 3 (1989) [7] A Kandil and A M El-Etrib, n Separation Aioms in Fuzz Topological Spaces, Tamkang Journal of Math, 18(1) (1987) [8] A Kandil, and M E El-Shafee, Regularit aioms in fuzz topological spaces and FR i -proimities, Fuzz sets and Sstems, 27(1988) [9] R Lowen, Comparison of different compactness notions in Fuzz topological spaces and fuzz compactness, J Math Anal Appl, 64 (1978) [1] Y M Liu, M K Luo, Fuzz topolog, World Scientific Publishing, Singapore,1997