Download Smooth Neighborhood Structures in a Smooth Topological Spaces

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
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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.
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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}
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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:
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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
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J. Math. & Stat., 6 (4): 409-415, 2010
Hence τN (A) ≥ r1 . It is a contradiction.
REFERENCES
CONCLUSION
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(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
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