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Journal of Mathematics and Statistics 6 (4): 409-415, 2010 ISSN 1549-3644 © 2010 Science Publications Smooth Neighborhood Structures in a Smooth Topological Spaces A.A. Ramadan and M.A. Abdel-Sattar Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt Abstract: Problem Statement: Various concepts related to a smooth topological spaces have been introduced and relations among them studied by several authors (Chattopadhyay, Ramadan, etc). Conclusion/Recommendations: In this study, we presented the notions of three sorts of neighborhood structures of a smooth topological spaces and give some of their properties which are results by Ying extended to smooth topological spaces. Key words: Fuzzy smooth topology, smooth neighborhood structures m(x λ , ∪i∈Γ A i ) = ∨ i∈Γ m(x λ ,A i ) INTRODUCTION Šostak (1985) introduced the fuzzy topology as an extension of Chang (1968) fuzzy topology. It has been developed in many directions (Ramadan, 1992; Chattopadhyay and Samanta, 1993; EL Gayyar et al., 1994; Höhle and Rodabaugh, 1998; Kubiak and Šostak, 1997; Demirici, 1997; Ramadan et al., 2001; 2009; Abdel-Sattar, 2006). Ying (1994) studied the theory of neighborhood systems in fuzzy topology with the method used to develop fuzzifying topology (Ying, 1991) by treating the membership relation as a fuzzy relation. In this study, we generate the structures of neighborhood systems in a smooth topology with the method used in (Ying, 1991), by using fuzzy sets and fuzzy points. Obviously, we have the following properties: (1) m(x, A) = A(x) (2) m(xλ, A) = 1 iff xλ ∈ A, m(xλ, A) = 0 iff λ = 1 and A(x) = 0 (3) m(x λ , È iÎΓ A i ) = Ú iÎΓ m(x λ , A i ) , (generalized multiple choice principles) Definition 3: Ying (1991) let (X,τ) be a fuzzy topological space (fts, for short), let e be a fuzzy point in X and let A be a fuzzy subset of X. Then the degree to which A is a neighborhood of e is defined by: N e (A) = sup{m(e,B) : B ∈ τ, B ⊆ A} Notions and preliminaries: The class of all fuzzy sets on a universal set X will be denote by LX, where L is the special lattice and L = ([0,1], ≤). Also, L0 = (0,1] and L1 = [0, 1). Thus N e ∈ LL is called the fuzzy neighborhood system of e in (X, τ). X Definition 4: Ying (1991) let (X, τ) be a fts, e a fuzzy point in X and A a fuzzy subset of X. Then the degree to which e is an adherent point of A is given as: Definition 1: Pu and Liu (1980) a fuzzy set in X is called a fuzzy point iff it takes the value 0 for all y ∈ X, except one, say x ∈ X. If its value at x is λ (0 < λ≤1) we denote this fuzzy point by xλ, where the point x is called its support. The fuzzy point is said to be contained in a fuzzy set A, or belong to A, denoted by xλ∈ A, iff λ≤A(x). Evidently, every fuzzy set A can be expressed as the union of all fuzzy points which belong to A. ad(e,A) = inf c (1 − N e (B) ) B⊆ A where, Ac is the complement of A. Definition 5: Ramadan (1992) A smooth topological space (sts, for short) is an ordered pair (X, τ), where X is a non-empty set and τ: LX → L is a mapping satisfying the following properties: Definition 2: Ying (1991) Let X be a non-empty set. Let xλ be a fuzzy point in X and let A be a fuzzy subset of X. Then the degree to which xλ belongs to A is: Corresponding Author: A.A. Ramadan, Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt 409 J. Math. & Stat., 6 (4): 409-415, 2010 (O1) τ(1) = τ(0) = 1 Remark 2: (O2) For all A1 , A 2 ,∈ LX , τ(A1 ∩ A 2 ) ≥ τ(A1 ) ∧ τ(A 2 ) (O3) ∀I, τ (∪ i∈Γ A i ) ≥ ∧i∈Γ τ(A i ) • Definition 6: EL Gayyar et al. (1994) let (X,τ) be a sts and α∈Lo. Then the family: τα = {A ∈ LX : τ(A) ≥ α} . which is clearly a fuzzy topology Chang (1968) sense. • Definition 7: Demirici (1997) Let (X,τ) be a sts and A∈LX. Then the τ-smooth interior of A, denoted by: Proposition 1: Let (X, τ) be a sts and A ∈ LX. Then a mapping NA: LX → LLX is the smooth nbd system of A with respect to the st τ iff: A 0 = ∪{B ∈ LX : τ(B) > 0,B ⊆ A} Remark 1: Demirici (1997) let τ be a Chang’s fuzzy topology (CFT, for short) on the non-empty set X. Then the smooth topology and smooth cotopology τs, τs*: LX→L, defined by: 1, τ (A) = 0, sup{τ(C) : A ⊆ C ⊆ B}, N A (B) 0, if A ⊆ B if A ⊆/ B Proof: if A ∈ τ if A ∉ τ (1) Suppose that the mapping NA: LX → LLX is the smooth nbd systems of A with respect to the st τ. Consider the following two cases: and τs (A) = τ(A ) for each A∈LX, identify the CFT τ * The real number NA(B) is called the degree of nbdness of the fuzzy set B to the fuzzy set A. If the smooth nbd system of a fuzzy set A has the following property: NA(LX) ⊆ {0, 1}, then NA is called the fuzzy nbd system of A We say that the family (NA)α = {B: NA(B)>α} is a fuzzy nbd system of A for each α∈[0,1) and (NA)α is called the strong α -level fuzzy nbd of A c and corresponding fuzzy cotopology for it. Thus the τs smooth interior of A is: • For the case A⊄B, suppose that NA(B) > 0. From Definition 1, there exists C∈τα such that A⊆C⊆B, i.e., A⊆B, a contradiction. Thus NA(B) = 0 • For the case A ⊆ B. We may have NA(B) = 0 or NA(B) > 0. If NA(B) = 0, then it is obvious that NA(B) = 0 ≤ sup{τ (C): A ⊆C⊆B}, if sup{τ (C): A⊆C ⊆B} = λ > 0, then ∃C∈LX such that τ (C) >0 and A⊆C⊆B: We obtain NA(B) > 0, a contradiction A 0 = ∪{B ∈ LX : τs (B) > 0, B ⊆ A} = ∪ {B ∈ LX : B ∈ τ, B ⊆ A} This show that Ao is exactly the interior of A with respect to τ in Chang (1968) sense. Lemma 1: Ramadan (1992) supα∈L sup{A(x)∧B(x): A(x)≥α} = supα∈L sup{α∧B(x): A(x)≥α}. Therefore: N A (B) = 0 = sup{τ(C) : A ⊆ C ⊆ B} Smooth neighborhood systems of a fuzzy set: Here, we build a smooth neighborhood systems of a fuzzy set in a sts and we give some of its properties. For a mapping M: LX→ LLX and A∈ LX, α∈ [0; 1); let us define the family MAα = {B ∈ LX: MA(B) > α}; which will play an important role in this part. Now suppose that NA(B) = λ > 0. For an arbitrary 0 < ε ≤λ, we have NA(B) = λ-ε, i.e., B ∈ NAλ−ε. Since the mapping: NA: LX → LLX is a smooth nbd system of A, ∃C∈LX such that C∈τλ−ε and A⊆C⊆B, i.e., sup{τ(C): A⊆C⊆B}>λ-ε. Since ε> 0 is arbitrary we have: Definition 8: Let (X,τ) be a sts and A∈ LX: Then a mapping N A : LX → LL is called the smooth neighborhood (nbd, for short) of A with respect to the st τ iff for each α∈[0,1): X sup{τ(C) : A ⊆ C ⊆ B} ≥ λ = N A (B) On the other hand, let sup{τ (C): A⊆C⊆B}=γ >0. Then for every 0<ε≤γ, ∃C∈ LX such that τ(C)>γ-ε and A⊆C⊆B. Therefore B∈NAγ-ε, i.e., NA(B)>γ-ε. Since ε is an arbitrary we have: N αA = {B ∈ LX : (∃C ∈ τα )(A ⊆ C ⊆ B)} where, τα = {A∈LX: τ(A) > α} the strong α- level of τ. 410 J. Math. & Stat., 6 (4): 409-415, 2010 N A (B) ≥ γ = sup{τ(C) : A ⊆ C ⊆ B} Thus, sup{τ (C): A⊆C⊆B}≤ sup{NA(C)∧NC(B)}. Hence: Hence the inequality follows: N A (B) ≤ sup {N A (C) ∧ N C (B)} α (2) For α∈ [0, 1), let B ∈ N A , i.e., NA(B)>α: Then A⊆ C⊆ B we can write α < NA(B)= sup{τ (C): A⊆C⊆B},i.e., ∃C∈ LX such that τ(C)>α, A ⊆ C ⊆ B. Then we have: Smooth neighborhood systems of a fuzzy points: Definition 9: Let (X,τ) be a sts, e a fuzzy point in X and A be a fuzzy subset of X. Then the degree to which A is a NBD of e is defined by: N αA ⊆ {B ∈ LX :(∃C ∈ τα )(A ⊆ C ⊆ B)} By the same way we can show that: {B ∈ LX :(∃C ∈ τα )(A ⊆ C ⊆ B)} ⊆ N Aα sup {m(e,B) ∧ τ(B) : τ(B) > 0}, if m(e,A) > 0 N e (A) = B ⊆ A otherwise 0, N αA = {B ∈ LX :(∃C ∈ τα )(A ⊆ C ⊆ B)} Thus N e ∈ LL is called the smooth NBD system of e in (X, τ). Hence: X Remark 3: In Proposition 3, the fuzzy subsets A of X can be replaced by the fuzzy points on X, that is, by the special fuzzy subsets e, in this case: sup{τ(C) : e ∈ C ⊆ A}, N e (A) 0, Remark 4: It is clear that when a fuzzy point e∈ B∈LX, then m(e, B) = 1 and if e ∈ A if e ∉ A sup {τ(B) : e ∈ B ⊆ A}, if e ∈ A N e (A) = B⊆ A if e ∉ A 0, Proposition 2: Let (X,τ) be a sts and A∈LX. If the mapping N A : LX → LL is the smooth nbd system of A with respect to the st τ, then the following properties hold: X is the NBD systems in the sense of Demirici (1997) Remark 5: For any crisp point x in X, we have: (N1) N 0 (0) = N 1 (1) = 1 and N A (B) > 0 ⇒ A ⊆ B N x (A) = sup{B(x) ∧ τ(B) : τ(B) > 0},B(x) ≠ 0. B⊆ A (N2) If A1 ⊆ A and B ⊆ B1 , then N A (B) ≤ N A1 (B1 ) (N3) N A (B1 ) ∧ N A (B2 ) ≤ N A (B1 ∩ B2 ) Proposition 3: The NBD systems N e of e in sts can be constructed from the cuts τα, α∈(0,1], by using the equality: (N4) N A (B) ≤ sup A ⊆ C ⊆ B{N A (C) ∧ N C (B)}, ∀A,B,C ∈ LX Proof: (N1) and (N2) follows directly from Definition 1 and Proposition 3. (N3) Suppose that NA(B1) = α1>0 and NA(B2)>α2>0. Then for a fixed ε>0 such that: ε≤ α1∧α2 ⇒ NA(B1) > α1-ε ≥ 0 and NA(B2) >α2-ε≥0. From Definition 1, it is clear that there exists C1, C2∈LX such that: N e (A) = sup{[N*e (A)]α ∧ α} α>0 where, [N*e (A)]α = sup{m(e, B) : B ⊆ A, B ∈ τα }, is the NBD systems in the sense of (Ying, 1994; Theorem 1). τ(C1 ) > α1 − ε, τ(C2 ) > α 2 − ε and Proof: By using Definition 9, we have: A ⊆ C1 ⊆ B1 ,A ⊆ C2 ⊆ B2 N e (A) = sup B⊆ A {m(e, B) ∧ τ(B) : τ(B) > 0} Therefore, τ(C1∩C2) ≥ τ(C1)∧τ(C2) > (α1-ε)∧(α2-ε) = (α1∧α2)-ε and A⊆ C1∩C2 ⊆ B1∩B2. Thus NA(B1∩B2)≥(α1∧α2)-ε: Since ε is arbitrary, we find that NA(B1∩B2) ≥ NA(B1)∧NA(B2). (N4) NA(B)= sup{τ(C): A⊆C⊆B}. From Proposition 3, we obtain τ(C)≤ NA(C) and τ(C)≤ NC(B). = sup α > 0 sup B⊆ A {m(e,B) ∧ α : τ(B) ≥ α} = sup α > 0 {sup B⊆ A {B(x) : τ(B) ≥ α} ∧ α} = sup α > 0 {sup B⊆ A {m(e, B) : B ∈ τα } ∧ α} = sup α > 0 {[N*e (A)]α ∧ α} 411 J. Math. & Stat., 6 (4): 409-415, 2010 Remark 6: For any crisp point x in X; we have: (O2): τ(A ∩ B) = inf e {m(e, A ∩ B) ∧ N e (A ∩ B)} N x (A) = sup{[N*x (A)]α ∧ α} α >0 ≥ inf e {m(e, A ∩ B) ∧ (min(N e (A), N e (B)))} where, [N (A)] = sup B⊆ A {B(x) :B ∈ τα } . = inf e {min(m(e, A), m(e, B)) ∧ (min(N e (A), N e (B)))} Theorem 1: Let (X, τ) be a sts and e a fuzzy point of X. If the mapping Ne: LX → L is the smooth NBD systems of e with respect to τ, then the following properties hold: = min(τ(A), τ(B)) * x α = min{inf e {m(e, A) ∧ N e (A)},inf e {m(e, B) ∧ N e (B)}) (O3): τ(∪ i∈I A i ) = inf e {m(e, ∪ i∈I A i ) ∧ N e (∪ i∈I A i )} = inf e {supi∈I m(e, Ai ) ∧ N e (∪ i∈I A i )} (N1)N e (A) ≤ m(e,B) (N2) If A ⊆ Band A, B ∈ L , then N e (A) ≤ N e (B) ≥ inf e {supi∈I m(e,A i ) ∧ inf i∈I N e (A i )} (N3) For all A, B ∈ L , N e (A ∩ B) ≥ N e (A) ∧ N e (B) ≥ inf e {infi∈I m(e,A i ) ∧ inf i∈I N e (A i )} x x = infi∈I inf e {m(e,A i ) ∧ N e (A i )} α (N4)N e (A) ≤ sup α > 0 {sup B⊆ A {[N (B) ∧ α : * e = infi∈I τ(A i ) for allfuzzy point d, m(d,B) ≤ N (B)}} * d Now, we show that the mapping Ne: LX → L which satisfies the conditions (N1)-(N4) is exactly the smooth NBD systems of e for the sts (X, τ): Let the mapping Me: LX → L be the smooth NBD systems of e of the sts (X,τ). Then applying (N1) we have: Proof: (N1) and (N2) follows directly from Remark 2. (N3): min(N e (A), N e (B)) = min(sup C ⊆ A {m(e,C) ∧ τ(C) : τ(C) > 0},sup D ⊆ B M e (A) = sup B ⊆ A {m(e,B) ∧ τ(B)} {m(e, D) ∧ τ(D) : τ(D) > 0}) = sup B⊆ A {m(e,B) ∧ inf p (m(p,B) ∧ N p (B))} = sup D ⊆ B sup C ⊆ A {min(m(e,C), m(e,D)) ∧ τ(C) ∧ τ(D) : τ(C), τ(D) > 0} Since: = sup D ⊆ B sup C ⊆ A {min(m(e,C ∩ D) ∧ τ(C) ∧ τ(D) : τ(C), inf (m(p, B) ∧ N p (B)) = inf m(p, B) ∧ inf N p (B) ≤ inf N p (B) ≤ N e (B) τ(D) > 0} p ≤ sup D ⊆ B sup C ⊆ A {min(m(e,C ∩ D) ∧ τ(C ∩ D) : p p p Thus: τ(C), τ(D) > 0} M e (A) ≤ sup{m(e, B) ∧ N e (B)} ≤ N e (A) ≤ sup E ⊆ A ∩ B{(m(e,E) ∧ τ(E) : τ(E) > 0,C ∩ D = E} B⊆ A = N e (A ∩ B) (1) On the other hand, using (N4) and Theorem 1, in (Ying, 1994) we may write: (N4) Combining axiom (4) in Theorem 1, in (Ying, 1994) and Proposition 4, (N4) follows. N x λ (A) ≤ sup α > 0 {sup B⊆ A {[N*x λ (B)]α ∧ α :for each fuzzy point d, m(e, B) ≤ N*d (B)}] Theorem 2: Let the mapping Ne: L →L satisfy the conditions (N1)-(N4), then the mapping τ: LX → L defined by: X ≤ sup α > 0 {sup B ⊆ A {[N*x λ (B)]α ∧ α :for each crisp point y, m(y,B) ≤ N*y (B)}] = sup α > 0 {sup B ⊆ A {[N*x λ (B)]α ∧ α : B ∈ τα }} inf {m(e, A) ∧ N e (A)}, if m(e,A) > 0,A ≠ 0 and A ≠ 1 τ(A) = e if A = 0 or A = 1 1, = sup α > 0{sup B⊆ A {min(1,1 − λ + N*x (B)) ∧ α : B ∈ τα }} ≤ supα > 0{sup B⊆ A {min(1,1 − λ + m(x, B)) ∧ α : Where: A∈LX = A st on X, furthermore the mapping Ne = Exactly the smooth nbd systems of e with respect to st τ. B ∈ τα }} = sup α > 0{sup B⊆ A {m(x λ , B) ∧ α : B ∈ τα }} = sup B⊆ A {m(x λ , B) ∧ τ(B)} = M x λ (A) Proof: (O1) Obvious. 412 (2) J. Math. & Stat., 6 (4): 409-415, 2010 Hence, the equality Ne = Me follows at once from (1) and (2). (1) (2) (3) (4) Definition 10: Let (X,τ) be a sts, e a fuzzy point in X and A a fuzzy subset of X. Then the degree to which e is an adherent point of A is given as: N e (A,r) ≤ m(e,A) for each r∈L0 Ne(A, r) ≤ Ne(B, r), if A⊆B N e (A,r) ∧ N e (B,r) ≤ N e (A ∩ B, r) Ne(A, r) ≤ sup{Ne(B, r): B ≤ A, m(d, B) ≤ Nd(B, r); for all fuzzy point d in X} (5) Ne(A, r) ≥ Ne(A, s), if r ≤ s (6) N x (A, r) = min(1,1 − t + N x (A, r)) ad(e,A) = inf c (1 − N e (B) t 1 B⊆ A Proof: (2) and (5) are easily proved. (1) It is proved from the following: where, Ac is the complement of A. Remark 6: For any crisp point x in X, we have: N e (A,r) = sup{m(e, Bi ) : Bi ≤ A, τ(Bi ) ≥ r} = {m(e, ∪Bi ) : ∪ Bi ≤ A, τ(∪Bi ) ≥ r} ad(x, A) = inf c (1 − N x (B)) ≤ m(e,A) B⊆ A Proposition 4: Suppose there exist A, B∈LX and r∈L0 such that: ad(e,A) = inf{[ad(e,A)]α ∨ (1 − α)} N e (A,r) ∧ N e (B, r) > t > N e (A ∩ B,r) α >0 [ad(e, A)]α = inf c (1 − [N*e (B)]α ) B⊆ A Since Ne(A,r)>t C1,C2∈LX with: Proof: Follows from Proposition 4. and Ne(B,r)>t, there exist C1 ⊆ A, τ(C1 ) ≥ r,C2 ⊆ B, τ(C2 ) ≥ r Proposition 5: Such that: N x λ (A) ≤ sup{min(1,1 − λ + [N*x (A)α ]) ∧ α)} α >0 m(e,C1 ) ∧ m(e,C2 ) = m(e,C1 ∩ C2 ) > t Proof: On the other hand, since: N xλ (A) = sup α> 0{[N*x λ (A)]α ∧ α} = sup α> 0{sup B ⊆ A {m(x λ , B) : B ∈ τα } ∧ α} C1 ∩ C2 ⊆ A ∩ B, τ(C1 ∩ C2 ) ≥ r = sup α> 0{sup B⊆ A {min(1,1 − λ + m(x, B)) : B ∈ τα } ∧ α} We have: ≤ sup α > 0{min(1,1 − λ + sup B⊆ A {m(x, B) : B ∈ τα }) ∧ α} = sup α> 0 {min(1,1 − λ + [N*x ,(A)]α ) ∧ α} N e (A ∩ B, r) ≥ m(e,C1 ∩ C 2 ) > t Fuzzy smooth r-neighborhood: Definition 11: Let (X,τ) be a sts, A∈LX, e a fuzzy point in X and r∈L0. Then the degree to which A is a fuzzy smooth r-nbd system of e is defined by: It is a contradiction. (4) If τ (B)≥r, then Nd(B,r) = m(d,B); for each fuzzy point d in X. It implies: N e (A,r) N e (A,r) = sup{m(e, B) : τ(B) ≥ r} = sup{m(e,B) : B ⊆ A, τ(B) ≥ r} B⊆ A = sup{N e (B, r) : B ⊆ A, N d (B, r) = m(d,B),for all A mapping Ne: LX × L0→L is called the fuzzy smooth r-nbd system of e. fuzzy point d in X} ≤ sup{N e (B, r) : B ⊆ A, m(d, B) ≤ N d (B, r),for all fuzzy point d in X} Theorem 2: Let (X,τ) be a sts and Ne the fuzzy smooth r-nbd system of e. For A, B∈LX and r, s∈L0, it satisfies the following properties: (6) It proved from: 413 J. Math. & Stat., 6 (4): 409-415, 2010 N x t (A, r) = sup{m(x t , B) : B ⊆ A, τ(B) ≥ r} For each i∈Γ, there exists ri∈L0 with = sup{min(1,1 − t + B(x)) : B ⊆ A, τ(B) ≥ r} = min(1,1 − t + sup{B(x) : B ⊆ A, τ(B) ≥ r}) m(e, Bi) = Ne(Bi, ri); for all fuzzy point e in X = min(1,1 − t + N x (A,r)) Such that: τN (Bi ) ≥ ri > r Theorem 3: Let N e be the fuzzy smooth r-nbd system of e satisfying the above conditions (1)-(5), the function τN : LX → L defined by: From (I), (IV) and (5), we have: m(e, Bi ) = N e (Bi , ri ) ≤ N e (Bi , r) ≤ m(e,Bi ) τN (A) = ∨{r ∈ L 0 : m(e, A) = N e (A,r) for all fuzzy point e in X} It implies m(e, Bi) = Ne(Bi, r): Furthermore: m(e, ∪i∈Γ Bi ) = ∨ i∈Γ m(e,Bi ) has the following properties: = ∨ i∈Γ N e (Bi ,ri ) (1) τN is a st. on X (2) If Ne is the fuzzy nbd systems of e induced by (X,τ), then τN = τ (3) If N e satisfy the conditions (1)-(6), then: ≤ N e (∪i∈Γ Bi ,r) ≤ m(e, ∪i∈Γ Bi ). τN (A) = ∨{r ∈ L 0 : m(x,A) = N x (A, r), ∀x ∈ X} Thus, Ne(∪ i ∈ Γ Bi, r) = m(e, ∪ i ∈ Γ Bi), i.e., τN(∪ i ∈ Γ Bi) ≥ r0. It is a contradiction for the Eq. III. Proof: (1) We will show that τN(B1 ∩ B2)≥τN(B1)∧ τN(B2), for any B1, B2 ∈ LX. Suppose there exist B1, B2∈LX and r∈L0 such that: τN (B1 ∩ B2 ) < r < τ N (B1 ) ∧ τN (B2 ) (2) Suppose there exists A∈LX such that: τN (A) > τ(A) (I) From the Definition of τN, there exists r0∈L0 with m(e, A) = Ne(A, r0) such that: For each i∈{1,2} there exists ri∈L0 with: m(e, Bi) = Ne(Bi, ri); for all fuzzy point e in X τN (A) ≥ r0 > τ(A) (II) Such that: τN (Bi ) ≥ ri > r, Since: m(e, A) = N e (A,r0 ) = sup{m(e, Bi ) : Bi ⊆ A, τ(Bi ) ≥ r0 } From (I), (II) and (5), we have: Then, for each x ∈ X: m(e, Bi ) = N e (Bi , ri ) ≤ N e (Bi r) ≤ m(e,Bi ) (∪Bi )(x) = sup{m(x,Bi ) : Bi ⊆ A} = m(x, A) = A(x) It implies m(e, Bi) = Ne(Bi, r): Furthermore: Thus, A = ∪ Bi. So, τ(A) ≥ r0. It is a contradiction. Suppose there exists A∈LX such that: m(e, B1 ∩ B2 ) = N e (B1 , r) ∧ N e (B2 , r) ≤ N e (B1 ∩ B2 , r) ≤ m(e, B1 ∩ B2 ) τN (A) < τ(A) There exists r1 ∈ L0 such that: Thus, Ne(B1∩B2, r) = m(e,B1∩B2), i.e.,τN(B1∩B2)≥ r. It is a contradiction for the Eq. I. Suppose there exists B = ∪i∈Γ Bi∈LX and r0∈L0 such that: τN (B) < r0 < ∧τN (Bi ) (IV) τN (A) < r1 ≤ τ(A) Since τ (A) ≥ r1, we have: (III) N e (A,r1 ) = sup{m(e, B) : B ⊆ A, τ(B) ≥ r1 } = m(e,A) i∈Γ 414 J. Math. & Stat., 6 (4): 409-415, 2010 Hence τN (A) ≥ r1 . It is a contradiction. REFERENCES CONCLUSION Abdel-Sattar, M.A., 2006. Some structures of L-fuzzy topology. J. Fuzzy Math., 14: 751-766. Chang, C.L., 1968. Fuzzy topological spaces. J. Math. Anal. Appli., 24: 182-190. Chattopadhyay, K.C. and S.K. Samanta, 1993. Fuzzy topology: Fuzzy closure operator, fuzzy compactness and fuzzy connectedness. Fuzzy Sets and Syst., 54: 207-212. DOI: 10.1016/01650114(93)90277-O Demirici, M., 1997. Neighborhood structures of smooth topological spaces. Fuzzy Sets Syst., 92: 123-128. DOI: 10.1016/S0165-0114(96)00132-7 El Gayyar, M.K., E.E. Kerre and A.A. Ramadan, 1994. Almost compactness and near compactness in smooth topological spaces. Fuzzy Sets Syst., 62: 193-202. DOI: 10.1016/0165-0114(94)90059-0 Höhle, U. and S.E. Rodabaugh, 1998. Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory. 1st Edn., Springer, USA., ISBN: 10: 0792383885, pp: 736. Kubiak, T. and A. Šostak, 1997. Lower set-valued fuzzy topologies. Quaest. Math., 20: 423-429. DOI: 10.1080/16073606.1997.9632016 Pu, P.M. and Y.M. Liu, 1980. Fuzzy topology I: Neighborhood structure of a fuzzy point and Moore-Smith convergence. J. Math. Anal. Appli. 76: 571-599. Ramadan, A.A., 1992. Smooth topological spaces. Fuzzy Sets Syst., 48: 371-375. DOI: 10.1016/01650114(92)90352-5 Ramadan, A.A., S.N. El-Deeb and M.A. Abdel-Sattar, 2001. On smooth topological spaces IV. Fuzzy Sets Syst., 119: 473-482. DOI: 10.1016/S01650114(99)00083-4 Ramadan, A.A., M.A. Abdel-Sattar and Y.C. Kim, 2009. The fuzzy neighborhood structures. Int. J. Math. Comput., 3: 146-156. Šostak, A.P., 1985. On a fuzzy topological structures. Suppl. Rend. Circ. Mat Palermo Ser. II, 11: 89-103. Ying, M., 1991. A new approach for fuzzy topology I. Fuzzy Sets and Syst., 39: 303-321. DOI: 10.1016/0165-0114(91)90100-5 Ying, M., 1994. On the method of neighborhood systems in fuzzy topology. Fuzzy Sets Syst., 68: 227-238. DOI: 10.1016/0165-0114(94)90048-5 (3) We only show that m(xt, A) = Nxt (A, r), for all fuzzy point xt in X iff m(x, A) = A(x) = Nx(A, r), ∀x∈ X: (⇒) It is trivial. (⇐) From the condition (6): N x t (A, r) = min(1,1 − t + N x (A, r)) = min(1,1 − t + m(x,A)) = min(1,1 − t + A(x)) = m(x t , A). Example 1: Let X = {a, b} be a set, N a natural number set and B∈LX as follows: B(a) = 0.3, B(b) = 0.4 We define a smooth fuzzy topology: 1, 1 τ(A) = , 2 0, if A = 0 or 1, if A = B, otherwise From Definition 1, Na, Nb: LX × L0 → L as follows: 1, N a (A) = 0.3, 0, 1, N b (A) = 0.4, 0, if A = 1, r ∈ L0 if 1 ≠ A ⊇ B, 0 < r ≤ 1 2 otherwise if A = 1, r ∈ L0 if 1 ≠ A ⊇ B, 0 < r ≤ 1 2 otherwise From Theorem 2 and Theorem 3 (3), we have: 1, if A = 0 or 1, 1 τN (A) = , if A = B, 2 0, otherwise 415