On Interval-Valued Fuzzy Metric Spaces - Semantic Scholar

International Journal of Fuzzy Systems, Vol. 14, No. 1, March 2012

35

On Interval-Valued Fuzzy Metric Spaces Yonghong Shen, Haifeng Li, and Faxing Wang Abstract1

non-negative, upper semi-continuous, normal and convex fuzzy number to measure the distance between two In this paper, following the ideas of (continuous) points in a fuzzy metric space. In [5], Abu Osman det-norm and interval numbers, a concept of (continu- fined the fuzzy metric between two fuzzy sets by means ous) interval-valued t-norm is proposed. Based on the of their membership functions. Obviously, the above nointerval-valued fuzzy set and the continuous inter- tions are different from that proposed by Kramosil and val-valued t-norm, we propose a notion of inter- Michalek. Later, some problems related to the fuzzy val-valued fuzzy metric space, which is a generaliza- metric space has been widely and deeply investigated by tion of fuzzy metric space in the sense of George and many authors [6-9] from different points of view. With Veeramani [Fuzzy Sets and Systems 64 (1994): the help of continuous t -norm, Grabiec [9] redefined 395-399]. Meanwhile, we show that each metric in- the fuzzy metric introduced by Kramosil and Michalek duces an interval-valued fuzzy metric in certain con- and extended the well-known fixed point theorems of ditions. Finally, we define a Hausdorff topology on an Banach and Edelstein to fuzzy metric spaces. In [7, 8], in interval-valued fuzzy metric space and generalize order to introduce a Hausdor topology on the fuzzy metsome well-known conclusions of general metric ric space, George and Veeramani modified the definition spaces. of the fuzzy metric space given by Grabiec and proved some known results of metric spaces including Baire’s Keywords: Interval numbers, Interval-valued fuzzy sets, theorem for fuzzy metric spaces. Furthermore, a necesInterval-valued t-norm, Interval-valued fuzzy metric sary and sufficient condition for the fuzzy metric space spaces, Topology, Complete. to be complete is presented, and the uniform limit theorem is generalized to fuzzy metric spaces. In [6], Chak1. Introduction rabarty defined some concepts associated with fuzzy topology and presented some properties for fuzzy metric Fuzzy metric is regarded as a generalization of usual spaces in the sense of Abu Osman. metric and it plays an important role in the study of As a natural generalization of the fuzzy metric space, fuzzy topology. In 1975, Kramosil and Michalek [1] first in recent years, the concept of intuitionistic fuzzy metric introduced the notion of fuzzy metric inspired by statis- space was proposed and studied by many authors tical metric spaces. Afterwards, Erceg [2] presented two [10-18]. We noticed that this space is constructed on the kinds of notions of pseudo-metric and metric on the basis of continuous t -norm. Meanwhile, some main fuzzy set as another generalization of usual metric in results of fuzzy metric spaces are extended in previous 1979. In 1982, Deng [3] extended pseudo metric to fuzzy literature. In particular, Park [10] first defined the notion pseudo metric and introduced the concept of pseudo of intuitionistic fuzzy metric space and generalized some metric space. Soon after, the concept of fuzzy metric well-known results of fuzzy metric spaces, such as space was formally proposed by Kaleva and Seikkala [4], Baire's theorem, uniform limit theorem, etc.. and by Abu Osman [5], respectively, by using two difThe concept of interval-valued fuzzy set was introferent ways. In [4], Kaleva and Seikkala made use of a duced by Zadeh in 1975 [19]. An interval-valued fuzzy set is characterized by an interval-valued membership function, and it is taken as another generalization of the Corresponding Author: Yonghong Shen is with the School of Mathematics and Statistics, Tianshui Normal University, Tianshui, fuzzy set. In 2009, Li [20] introduced three kinds of distances between two interval-valued fuzzy sets (or num741001, P.R. China. E-mail: [email protected] bers) defined on real line . Moreover, he noted that Haifeng Li is with the School of Physics and Information Sciences, each kind of distance is a metric on the corresponding Tianshui Normal University, Tianshui, 741001, P.R. China sets and each interval-valued fuzzy numbers metric E-mail: [email protected] Faxing Wang is with the Tongda College of Nanjing University of space is complete. Obviously, the idea of above defined Posts and Telecommunications, Nanjing, 210046, P.R. China metric space is different from that of George and VeeraE-mail: [email protected] mani [7]. It should be noted that George and Veeramani Manuscript received 7 Sep. 2010; revised 14 Feb. 2011; accepted 24 applied fuzzy set to express the uncertainty of the disFeb. 2012.

© 2012 TFSA

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tance between two points in a fuzzy metric space, and then proposed the concept of the continuous t-norm, which generalizes the triangle inequality of general metric spaces. Therefore, motivated by the idea of fuzzy metric space, we put forward the concept of continuous interval-valued t -norm and define an interval-valued fuzzy metric space. Furthermore, we discuss some topological properties of this kind of metric space.

2. Interval numbers and interval-valued t -norm

mapping A : X → [ I ] is called an interval-valued fuzzy set on X . All interval-valued fuzzy sets on X denoted by IVF ( X ) .

A ∈ IVF ( X ) , let A( x) = [ A− ( x), A+ ( x)] , A− ( x) ≤ A+ ( x) for all x ∈ X , then the ordinary fuzzy sets A− : X → I and A+ : X → I are called the lower fuzzy set and upper fuzzy set of A , respectively. Especially, A is called a degenerate fuzzy set if A− ( x) = A+ ( x) for all x ∈ X . If

For completeness and clarity, in this section, some related concepts and conclusions from [21, 22] are summarized below. Moreover, the concept of an interval-valued t -norm is presented as a generalization of the ordinary t -norm proposed by Kelment and Schweizer, etc. [23, 24]. Meantime, the convergence of sequence of interval numbers is defined for the purpose of introducing the notion of continuous interval-valued t -norm. Let I be a closed unit interval, i.e., I = [0,1] . We denote all interval numbers on the interval I as [ I ] , and which is represented as follows: [ I ] = {a = [a − , a + ] | 0 ≤ a − ≤ a + ≤ 1} .

(O4) A = B iff A− ( x) = B − ( x) and A+ ( x) = B + ( x) for all x ∈ X ; (O5) ( A ∩ B )( x) = A( x) ∧ B( x)

eral operations. (O1) a ∧ b = [a − ∧ b − , a + ∧ b + ] ;

(T1) Commutativity: a *I b = b *I a ,

Based on the relevant operations of interval numbers, we introduce the following operations including equality, intersection, union and complement of interval-valued fuzzy sets. For any A, B ∈ IVF ( X ), x ∈ X , define

= [ A− ( x) ∧ B − ( x), A+ ( x) ∧ B + ( x)] ; (O6) ( A ∪ B )( x) = A( x) ∨ B( x) = [ A− ( x) ∨ B − ( x), A+ ( x) ∨ B + ( x)] ; (O7) Ac ( x) = ( A( x))c = 1 − A( x) − + Especially, for a given a ∈ [ I ] , if a = a , then the = [1 − A+ ( x),1 − A− ( x)] . interval number a will degenerate into an ordinary real Now, we extend t -norm to interval-valued t -norm number a on [0,1] . Conversely, for all a ∈ I , it can for the definition of interval-valued metric space. be written as a = [a, a ] , namely, a ∈ [ I ] . Definition 2: An interval-valued t -norm is a binary opFor the need of narrative below, we write eration on [ I ] , i.e., an operation * :[ I ] × [ I ] → [ I ] I ( I ] = [ I ] − {0} , ( I ) = [ I ] − {0, 1} . such that for all a , b , c ∈ [ I ] the following four condiFor every a , b ∈ [ I ] , we introduce the following sev- tions are satisfied:

(O2) a ∨ b = [a − ∨ b − , a + ∨ b + ] ; (O3) a c = 1 − a = [1 − a + ,1 − a − ] ; In addition, the order relations ≤, = and < in [ I ] are introduced as follows: (R1) a ≤ b iff a − ≤ b − and a + ≤ b + ; (R2) a = b iff a − = b − and a + = b + ; (R3) a < b iff a ≤ b and a ≠ b . Obviously, ([ I ], ≤) is a partial ordered set. And it is not difficult to check that the system ([ I ], ≤, ∨, ∧) constitutes a complete lattice with a minimal element 0 = [0, 0] and a maximal element 1 = [1,1] . Next, we will recall some relevant notions of the interval-valued fuzzy set. Definition 1: Let X be an ordinary nonempty set. The

(T2) Associativity: a *I (b *I c ) = (a *I b ) *I c , (T3) Monotonicity: a *I b ≤ a *I c whenever

b ≤c , (T4) Boundary condition; a *I 1 = a , a *I = [a − , a + ]*I [0,1] = [0, a + ] . Example 1: (i) a *I b = [a − ⋅ b − , a + ⋅ b + ] ; (ⅱ) a *I b = [a − ∧ b − , a + ∧ b + ] . It is easy to see that several additional useful properties follow immediately for an interval-valued t -norm *I on [ I ] . Remark 1: (i) Directly from Definition 2 we can deduce that, for all a ∈ [ I ] , each inteval-valued t -norm *I satisfies the following additional boundary conditions:

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Yonghong Shen, Haifeng Li, and Faxing Wang: On Interval-Valued Fuzzy Metric Spaces

metric space using the previous definitions. Definition 5: The triple ( X , M ,*I ) is said to be an

0 *I a = a *I 0 = 0, I *I a = [0,1]*I [a − , a + ] = 0, 1 *I a = 1. (ii) In essence, according to commutativity (T1), it can easily be verified that the monotonicity of an interval-valued t -norm *I in its second component described by (T3) is equivalent to the (joint) monotonicity in both components, namely, a *I b ≤ c *I d when-

interval-valued fuzzy metric space if X is an arbitrary set, *I is a continuous interval-valued t -norm on [ I ] and M is an interval-valued fuzzy set on X 2 × (0, ∞) satisfying the following conditions: (C1) M ( x, y, t ) > 0; (C2) M ( x, y, t ) = 1 if and only if x = y; (C3) M ( x, y, t ) = M ( y, x, t );

ever a ≤ c and b ≤ d . Remark 2: For all a ∈ [ I ] , it is obvious that 0 ≤ a ≤ 1 . (C4) M ( x, y, t ) *I M ( y, z , s ) ≤ M ( x, z , t + s ); (C5) M ( x, y, ⋅) : (0, ∞) → ( I ] is continuous; As a consequence of (ii) we obtain for every a , b ∈ [ I ] , (C6) lim M ( x, y, t ) = 1 , where x, y, z ∈ X and

0 = a *I 0 ≤ a *I b ≤ a *I 1 = a ,

t →∞

t, s > 0 .

0 = a *I 0 ≤ 0 *I b ≤ a *I b = b . Hence 0 = a *I 0 ≤ a *I b ≤ a ∧ b . Definition 3: Let {an } = {[an− , an+ ]}(n ∈

+

) be a se-

−

quence of interval numbers in [ I ] , a = [a , a + ] ∈ [ I ] , if lim an− = a − and lim an+ = a + , then we call the sen →∞

n →∞

quence {an } is convergent to a , and which is denoted by lim an = a . n →∞

In general, most of interval-valued t-norms we will consider are continuous, so we will define the continous interval-valued t -norm using the idea of continuous t -norm [23, 25]. Definition 4: An interval-valued t -norm *I is continuous if and only if it is continuous in its first component, i.e., for each b ∈ [ I ] , if lim an = a , then n →∞

lim (an *I b ) = (lim an *I b ) = a *I b , n →∞

n →∞

where {an } ⊆ [ I ] , a ∈ [ I ] . Remark 3: Similar to the properties of continuous t -norm, if *I is a continuous interval-valued t -norm, the following conclusions can be obtained: (i) For any r1 , r2 ∈ [ I ] with r1 > r2 , there exists a

r3 ∈ [ I ] such that r1 *I r3 ≥ r2 , (ii) For any r4 ∈ [ I ] , there exists a r5 ∈ [ I ] such that r5 *I r5 ≥ r4 . 3. Interval-valued fuzzy metric spaces As a natural generalization of the fuzzy metric space in the sense of George and Veeramani [7, 8], in this section, we put forward the concept of interval-valued fuzzy

In the above definition, M = [ M − , M + ] is called an interval-valued fuzzy metric on X . The functions M − ( x, y, t ) and M + ( x, y, t ) denote the lower nearness degree and the upper nearness degree between x and y with respect to t , respectively. Besides, it should be pointed out that the condition (C5 ) implies the functions M − ( x, y, ⋅) and M + ( x, y, ⋅) are continuous from (0, ∞) to (0,1] satisfying

M − ( x, y, t ) ≤ M + ( x, y, t ) for all t > 0 . The combination of conditions (C3 ) and (C6 ) shows that the lower nearness degree and upper nearness degree are non-decreasing and tend to gradually 1, respectively. Note that an interval-valued fuzzy metric space will degenerate into an ordinary fuzzy metric space if M − ( x, y, t ) = M + ( x, y, t ) for all t > 0 . Example 2: Let X = . Define a *I b = [a − ⋅ b − , a + ⋅ b + ] and the interval-valued fuzzy metric function is taken as −k⋅ x− y

− x− y

M ( x, y, t ) =[M − ( x, y , t ), M + ( x, y, t )] = [e t , e t ] (k ≥ 1) . For any x, y ∈ X and t ∈ (0, ∞) . Then ( X , M ,*I ) is an interval-valued fuzzy metric space. Proof: (a) Obviously, M ( x, y, t ) > 0 for all x, y ∈ X , t > 0. (b) M ( x, y, t ) = 1 if and only if x = y . (c) M ( x, y, t ) = M ( y, x, t ) . (d) Now we start to prove M ( x, y, t ) *I M ( y, z , s) ≤ M ( x, z , t + s ) . For all t , s > 0 , since

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t+s t+s ) x− y +( ) y − z , we have t t x−z x− y y−z ≤ + , t+s t s k⋅ x−z k⋅ x− y k⋅ y−z ≤ + (k ≥ 1) . t+s t s

x−z ≤(

Therefore, we can obtain x− z

x− y

x− z

k⋅ x− z

k⋅ x− y

k⋅ x− z

Essentially, in the above example,

and x − y

can be replaced by any metric space X and an ordinary metric or distance d ( x, y ) , respectively. Furthermore, the conclusion still holds if we replace the interval-valued t -norm with a *I b = [a − ∧ b − , a + ∧ b + ]. Next, we show that every metric can induce an interval-valued fuzzy metric. Example 3: (Inuced interval-valued fuzzy metric) Let ( X , d ) be a general metric space. Define the inter-

≤e ⋅e , e ≤e ⋅e s . − − Thus M ( x, y, t ) ⋅ M ( x, y, s) ≤ M − ( x, y, t + s ) and val-valued t -norm a *I b = [a − ⋅ b − , a + ⋅ b + ] and inM + ( x , y , t ) ⋅ M + ( x, y , s ) ≤ M + ( x , y , t + s ) , i.e., terval-valued fuzzy metric M ( x, y, t ) = [ M − ( x, y, t ), M + ( x, y, t )] M ( x, y, t ) *I M ( y, z , s) ≤ M ( x, z , t + s ) . ht n ht n (e) M ( x, y, ⋅) : (0, ∞) → ( I ] is continuous. =[ n , n ] ht + ld ( x, y ) ht + md ( x, y ) − x− y −k ⋅ x − y = lim e =1 (f) lim e for all For all h, l , m, n ∈ + and l ≥ m . Then ( X , M ,* ) I t →∞ t →∞ t t is an interval-valued fuzzy metric space. x, y ∈ X . Remark 5: Note that the above example holds even if the Hence ( X , M ,*I ) is an interval-valued fuzzy metric interval-valued t -norm a *I b =[a− ∧b−, a+ ∧b+ ] . Espespace. Note: It is clear that this example is a direct extension of cially, if h = n = 1 and l ≥ m , we then obtain e

t +s

t

s

t+s

t

example 2.7 in [7]. In this example, k ≥ 1 is a fixed constant, if k > 1 , then ( X , M ,*I ) is a classical in-

terval-valued fuzzy metric space. In particular, if k = 1 , then ( X , M ,*I ) becomes a fuzzy metric space given by George and Veeramani [7]. Remark 4: In the example 2, we take

− x− y ⎧1, x = y M − ( x, y , t ) = ⎨ , M + ( x, y , t ) = e . t ⎩0, x ≠ y

M ( x, y, t ) = [ M − ( x, y, t ), M + ( x, y, t )] . t t =[ , ] t + ld ( x, y ) t + md ( x, y ) We call this interval-valued fuzzy metric induced by a metric d the standard interval-valued fuzzy metric. Moreover, if l = m = 1 , it will reduce to the standard fuzzy metric. Theorem 1: Let ( X , M ,*I ) be an interval-valued

fuzzy metric space, for all x, y ∈ X , if s > t > 0 , then By Definition 5, it is easy to verify that the conditions M ( x , y , t ) ≤ M ( x, y , s ) . (C1-C5) are satisfied. However, the condition (C6) cannot be satisfied. So we can say that ( X , M ,*I ) is not Proof: According to Definition 3.1, if s > t > 0 , then we know that an interval-valued fuzzy metric space under this situation. M ( x, y, t ) *I M ( y, y, s − t ) ≤ M ( x, y, s ) . In fact, for every x, y ∈ X and x ≠ y , we can obtain Since M ( y, y, s − t ) = 1 , we have lim M ( x, y, t ) = [lim M − ( x, y, t ),lim M + ( x, y, t )] = [0,1] . t →∞

t →∞

t →∞

The result shows that the lower nearness degree between x and y is always 0 for sufficiently large

M ( x, y, t ) *I M ( y, y, s − t ) = M ( x, y, t ) *I 1

= M ( x, y , t ) positive number t. M − ( x, y, t ) and M + ( x, y, t ) can Therefore, M ( x, y, t ) ≤ M ( x, y, s ) .

also be regarded as the lower limit and upper limit of the probability that the distance between x and y is smaller than t , respectively. Evidently, we know that M − ( x, y, t ) → 1 and M + ( x, y, t ) → 1 as t → ∞ . Therefore, the example indicates that the condition (C6) is nonsuperfluous and the affixation of the condition is also necessary and reasonable on the basis of the original definition of the fuzzy metric space introduced by Kramosil, Grabiec and George, etc. [1, 7, 9].

.

Remark 6: In an interval-valued fuzzy metric space ( X , M ,*I ) , whenever M ( x, y, t ) > 1 − r for all

x, y ∈ X , t > 0 , r ∈ ( I ), there exists a t0 with 0 < t0 < t such that M ( x, y, t0 ) > 1 − r . 4. Topology induced by an interval-valued fuzzy metric space In this section, we define the topology induced by an

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interval-valued fuzzy metric space, and investigate some relevant topological properties. Definition 6: Let ( X , M ,*I ) be an interval-valued

1 − r ≥ 1 − r1 and 1 − r ≥ 1 − r2 . According to Theorem 1. we can obtain

B ( x, r , t ) = { y ∈ X M ( x, y , t ) > 1 − r }

fuzzy metric space. The open ball B( x, r , t ) with cen-

⊂ { y ∈ X M ( x, y, t ) > 1 − r1}

ter x ∈ X and the interval number r , 0 < r < 1 , t > 0 , is defined as follows: B ( x, r , t ) = { y ∈ X M ( x , y , t ) > 1 − r } . Theorem 2: Every open ball B( x, r , t ) is an open set. Proof: Let B( x, r , t ) be an open ball with center x and interval number radius r with respect to t . For any y ∈ B( x, r , t ) , we know that M ( x, y, t ) > 1 − r . Therefore,

there

exists

t0 ∈ (0, t )

a

M ( x, y, t0 ) > 1 − r . Set

such

that

r0 = M ( x, y, t0 ) . Since

r0 > 1 − r , there exists a

s ∈ (I )

such

that

⊂ { y ∈ X M ( x, y, t1 ) > 1 − r1} Analogously,

= B( x, r1 , t1 ) ⊂ A1 B ( x, r , t ) ⊂ B( x, r2 , t2 ) ⊂ A2

.

Thus

B( x, r , t ) ⊂ A2 ∩ A2 and hence A2 ∩ A2 ∈ TM . (iii) For all Aγ ∈ TM (γ ∈ Γ ) , Γ is an index set. For every x ∈ ∪ Aγ , therefore, there exist t > 0 and γ ∈Γ

r ∈ (I )

such

that

B ( x, r , t ) ⊂ Aγ 0

.

Thus

B( x, r , t ) ⊂ ∪ Aγ . It means that ∪ Aγ ∈ TM . The γ ∈Γ

γ ∈Γ

r0 > 1 − s > 1 − r . Now for given r0 and s with proof of the theorem is now completed. r0 > 1 − s , there exists a r0 *I r1 ≥ 1 − s

.

B ( y , 1 − r , t − t0 )

Consider ,

we

r1 ∈ ( I ) the will

such

that

open

ball

obtain

that

B ( y, 1 − r1 , t − t0 ) ⊂ B( x, r , t ) . In fact, for every z ∈ B( y, 1 − r1 , t − t0 ) , we have M ( y, z, t − t0 ) > r1 . Therefore,

M ( y, z , t ) ≥ M ( y, z, t − t0 ) *I M ( y, z, t − t0 ) ≥ r0 *I r1 ≥ 1−s

Remark 7: From Theorem 2 and Theorem 3, each interval-valued fuzzy metric M in an interval-valued fuzzy metric space ( X , M ,*I ) can generate a topology TM on X , and which has a base such as the family of open sets of the form {B( x, r , t ) : x ∈ X , r ∈ ( I ) , t >0} .

Theorem 4: Let ( X , M ,*I ) be an interval-valued fuzzy metric space, the topology TM is first countable. 1 Proof: For all x ∈ X , define Β x = {B( x, rn , ) | n = 1, 2, n 3,…}, where rn = {

1 1 , }. n2 n

> 1−r By Theorem 2, Β x is a family of open sets with cenThus z ∈ B ( x, r , t ) and hence ter x . Obviously, Β x constitutes a local base at x . B( y, 1 − r1, t − t0 ) ⊂ B( x, r , t ) . Therefore, the topology TM is first countable. Theorem 3: Let ( X , M ,*I ) be an interval-valued Remark 8: Notice that Β is a nested local base at x . x fuzzy metric space. Define

1 + , by n . such that B( x, r , t ) ⊂ A} Theorem 1, since 1 − rn ≤ 1 − rn +1 , we have Then TM is a topology on X . 1 1 1 B ( x, rn , ) ⊃ B ( x, rn +1 , ) ⊃ B( x, rn +1 , ) Proof: (i) ∅, X ∈ TM ; n n n +1 (ii) For all A1 , A2 ∈ TM , firstly, if A1 ∩ A2 = ∅ , then Hence, it is obvious that Bn ⊃ Bn +1 A1 ∩ A2 = TM . On the other hand, for every Theorem 5: Every interval-valued fuzzy metric space is a Hausdorff space( T2 -space). x ∈ A1 ∩ A2 , there exist t1 > 0 and r1 ∈ ( I ) such that Proof: Let ( X , M ,*I ) be an interval-valued fuzzy B ( x, r , t ) ⊂ A1 . Meanwhile, there also exist t2 > 0 metric space. Suppose that x and y two distinct and r2 ∈ ( I ) such that B ( x, r2 , t2 ) ⊂ A2 . Set points in X . Clearly, we know that 0 < M ( x, y, t ) < 1 r = r1 ∧ r2 and t = min{t1 , t2 } . Clearly, we have for all t > 0 . Put M ( x, y, t ) = r , for some r , TM = { A ⊂ X ∀x ∈ A, ∃t > 0, r ∈ ( I )

Set Bn = B ( x, rn , )(n = 1, 2,3,…) . For all n ∈

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x, y ∈ A . Then x ∈ B( xi , r , t ) and 0 < r < 1 . For each r0 , r < r0 < 1 , there exist a every r1 ∈ ( I ) such that r1 *I r1 ≥ r0 . Now consider open balls y ∈ B( x j , r , t ) for some i, j . Therefore, we can obtain 1 1 M ( x, xi , t ) > 1 − r and M ( y, x j , t ) > 1 − r . Now let B( x, 1 − r1 , ) and B( y, 1 − r1 , ) . Actually, 2 2 α = [ min {M − ( xi , x j , t )}, min {M + ( xi , x j , t )}] . Then 1≤ i , j ≤ n 1≤ i , j ≤ n 1 1 B(x,1−r1, )∩B(x,1−r1, ) =∅. Conversely, if there exists a α > 0 . Thus, we have 2 2 M ( x, y,3t ) ≥ M ( x, xi , t ) *I M ( xi , y j , t ) *I M ( y j , y, t ) 1 1 z ∈ B( x, 1 − r1 , ) ∩ B( x, 1 − r1 , ) , 2 2 ≥ (1 − r ) *I (1 − r ) *I α

then

>1−s

for some s ∈ ( I ) Taking t ′ = 3t , we get M ( x, y, t ′) > 1 − s for all x, y ∈ A . Hence A is IVF-bounded.

t t r = M ( x, y, t ) ≥ M ( x, z , ) *I M ( z , y, ) 2 2 ≥ r1 *I r1 . ≥ r0 > r This leads to a contradiction. Hence ( X , M ,*I ) is a Hausdorff space ( T2 -space) Corollary 1: Every interval-valued fuzzy metric space is a T1 -space. Remark 9: Let ( X , d ) be a metric space. Let

From Theorem 5, Theorem 6 and Remark 10, we can obtain the following corollary. Corollary 2: Every compact set A of an interval-valued fuzzy metric space ( X , M ,*I ) is closed and bounded. Definition 8: An interval-valued fuzzy metric space ( X , M ,*I ) is compact space if X is a compact set.

M ( x, y, t ) = [ M − ( x, y, t ), M + ( x, y, t )] t t =[ , ] (l ≥ m) t + ld ( x, y ) t + md ( x, y ) be an interval-valued fuzzy metric defined on X . Then the topology Td induced by the metric d and the topology TM induced by the interval-valued fuzzy metric M are the same. Definition 7: Let ( X , M ,*I ) be an interval-valued fuzzy metric space. A subset A of X is said to be IVF-bounded if and only if there exist t > 0 and r ∈ ( I ) such that M ( x, y, t ) > 1 − r for all x, y ∈ A . Remark 10: Let ( X , M ,*I ) be an interval-valued fuzzy metric space induced by a metric d on X . Then A ⊂ X is IVF-bounded if and only if it is

Theorem 7: Every closed subset A of a compact interval-valued fuzzy metric space ( X , M ,*I ) is compact.

bounded. Theorem 6: Every compact subset A of an interval-valued fuzzy metric space ( X , M ,*I ) is IVF-bounded. Proof: Given A is a compact subset of X . Set t > 0 and 0 < r < 1 . Obviously, the family of subsets Tx = {B( x, r , t ) x ∈ A} constitutes an open covering of

val-valued fuzzy metric space. A subset A of X is a closed set if and only if A is a compact subset. Proof: It follows from Theorems 5, 7 and Corollary 2. Theorem 8: Let ( X , M ,*I ) be an interval-valued

A . Since

A

is

a

compact

x1 , x2 ,… , xn ∈ A such that

set,

there

exist

Proof: Suppose that the family of open sets T = {Gi } is an arbitrary open covering of A , i.e., A ⊂ ∪ Gi . Then i

X = (∪ Gi ) ∪ A ( A denotes the complementary set of c

c

i

A ). Since A is a closed set, T * = {Gi } ∪ { Ac } constitutes an open covering of X . In addition, since ( X , M ,*I ) is a compact space, there exist Gi1 , Gi2 ,… , Gin ∈ T such that

X = Gi1 ∪ Gi2 ∪…∪ Gin ∪ Ac . Therefore, we have A ⊂ Gi1 ∪ Gi2 ∪…∪ Gin , namely, for arbitrary open covering of A , there is a finite subcovering of it. Hence A is a compact set. Corollary 3: Let ( X , M ,*I ) be a compact inter-

fuzzy metric space and TM be the topology on X induced by the interval-valued fuzzy metric. Then for a sequence {xn } in X , xn → x if and only if

M ( x, xn , t ) → 1 as n → ∞ . A ⊂ ∪ B( x, r , t ) . For Proof: Fix t > 0 , if xn → x , then for all r ∈ ( I ) , i =1 n

41


there exists a n0 ∈

n ≥ n0

.

Clearly,

such that xn ∈ B ( x, r , t ) for all

M ( xn , x, t ) > 1 − r

,

i.e.,

fuzzy complete, then there exists x ∈ such that M ( xn + p , xn , t ) → 1 as n → ∞ . From this if follow

1 − M ( xn , x, t ) < r . Therefore, M ( x, xn , t ) → 1 as that t t M ( xn , x, t ) = [ , ] → 1(n → ∞) . n → ∞ . On the other hand, if for each t > 0 , t + l xn − x t + m xn − x M ( x, xn , t ) → 1 as n → ∞ , then for r ∈ ( I ) , there Furthermore, we have xn − x → 0 as n → ∞ . So exists a n0 ∈ such that 1 − M ( xn , x, t ) < r for all xn → x in which is not true. The example shows n ≥ n0 . It follows that M ( xn , x, t ) > 1 − r for all that {xn } is a Cauchy sequence, and is not comn ≥ n0 . Thus, xn ∈ B( x, r , t ) for all n ≥ n0 and plete under the previous definition. Whereas, one can see hence xn → x . that Definition 9 guarantees is a complete interDefinition 9: Let ( X , M ,*I ) be an interval-valued val-valued fuzzy metric space. fuzzy metric space. A sequence {xn } in X is called a Theorem 9: Let ( X , M ,*I ) be a compact interCauchy sequence if for all ε > 0 and t > 0 , there exists a n0 ∈ such that M ( xn , xm , t ) > 1 − ε for all

n, m ≥ n0 . Definition 10: An interval-valued fuzzy metric space ( X , M ,*I ) in which every Cauchy sequence is convergent is called a complete interval-valued fuzzy metric space. Remark 11: Similar to the definition of Cauchy sequence in the fuzzy metric space [9], we can also introduce another definition of Cauchy sequence in the interval-valued fuzzy metric space, i.e., a sequence {xn } is a Cauchy sequence if and only if lim M ( xn + p , xn , t ) = 1 for each p > 0 and t > 0 . n →∞

However, we note that with this definition, even to be complete. For example, Let

fails

1 1 xn = 1 + + + 2 3

1 + in ( , M ,*I ) , where n t t M ( x, y , t ) = [ , ](l ≥ m > 0) , t + ld ( x, y ) t + md ( x, y ) d ( x, y ) = x − y is a metric on . For each p > 0 and t > 0 , we can obtain

t t M(xn+p, xn,t) =[ , ] t +l xn+p −xn t +mxn+p −xn

val-valued fuzzy metric space. If every Cauchy sequence {xn } in X has a convergent subsequence, then

( X , M ,*I ) is complete. Proof: Let {xin } be a subsequence of the Cauchy sequence {xn } that converges to x . Now we prove that

xn → x . For any t > 0 and ε ∈ ( I ) . Choose r ∈ ( I ) such that (1 − r ) *I (1 − r ) ≥ 1 − ε . Since {xn } is a Cauchy sequence, there exists

n0 ∈

such that

1 M ( xm , xn , ) > (1 − r ) for all m, n ≥ n0 . Since 2 xin → x , there is a positive integer i p such that 1 M ( xi p , x, ) > (1 − r ) for all n ≥ n0 and i p ≥ n0 , we 2 have

t t M ( xn , x, t ) ≥ M ( xn , xi p , ) *I M ( xi p , x, ) 2 2 > (1 − r ) *I (1 − r ) ≥ 1− ε Therefore xn → x and hence ( X , M ,*I ) is complete. Theorem 10: Let ( X , M ,*I ) be a compact interval-valued fuzzy metric space and let A be a subset of X . M A = M A2 ×(0,∞ ) denotes the restriction of the in-

t t =[ , ] 1 1 1 1 1 1 t +l( + + + ) t +m( + + + ) terval-valued fuzzy metric M on A . Then n+1 n+2 n+ p n+1 n+2 n+ p ( A, M A ,*I ) is complete if and only if A is a closed subset of X . Therefore, lim M ( xn + p , xn , t ) = 1 . Thus the sequence Proof: Suppose that A is a closed subset of X and n →∞ let {xn } be a Cauchy sequence in A . Clearly, {x } in the interval-valued fuzzy metric n

space ( , M ,*I ) . Additionally, if

is interval-valued

A ⊂ X , {xn } is also a Cauchy sequence in X . Since

42


( X , M ,*I ) is complete, there exists a x ∈ X such fuzzy metric space and let A be a dense subset of X . If every Cauchy sequence of A is convergent in X , that xn → x . Then x ∈ A = A and thus {xn } is then ( X , M ,*I ) is complete. convergent in A . Hence ( A, M A ,*I ) is complete. Theorem 12: Let ( X , M ,*I ) be an interval-valued Conversely, suppose that ( A, M A ,*I ) is complete fuzzy metric space. For any t > 0 and r , s ∈ ( I ) , if and A is not closed. Let x ∈ A and x ∉ A . Then t there is a sequence {xn } of points in A that con- (1 − r )*I (1 − r ) ≥ (1 − s ) , then B( x, r , ) ⊂ B( x, s , t ) . 2 verges to x . It is obvious that {xn } is a Cauchy set Proof: Let B ( y, r , ) be an open ball with center y 2 quence. Thus, for each 0 < ε < 1 and each t > 0 , t there exists a n0 ∈ such that M ( xn , xm , t ) > 1 − ε and interval number radius r . For all y ∈ B ( x, r , ) 2 for all m, n ≥ n0 . Since {xn } ⊂ A , we know that M ( xn , xm , t ) = M A ( xn , xm , t ) . Therefore we know that B ( y, r , t ) ∩ B ( x, r , t ) ≠ ∅ . Assume 2 2 M A ( xn , xm , t ) > 1 − ε for all m, n ≥ n0 . Thus {xn } t t is a Cauchy sequence in A . Since ( A, M A ,*I ) is that z ∈ B ( y, r , ) ∩ B ( x, r , ) , then we have 2 2 complete, there exists a y ∈ A such that xn → y . That t t t M ( x, y, t ) ≥ M ( x, z, ) *I M ( x, z, ) M ( y, z , ) 2 2 2 is, for each 0 < ε < 1 and each t > 0 , there exists a n1 ∈ such that M A ( y, xn , t ) > 1 − ε for all n ≥ n1 . > (1 − r ) *I (1 − r ) Since {xn } ⊂ A and y ∈ A , M ( y, xn , t ) = ≥ 1−s M A ( y, xn , t ) . Hence {xn } converges to both x and Hence y ∈ B( x, s , t ) and thus y in X . But we know that x ∉ A and y ∈ A , i.e., t B ( x , r , ) ⊂ B ( x, s , t ) . x ≠ y . This is a contradiction. 2 Remark 12: Theorem 10 shows that every closed subspace of a complete interval-valued fuzzy metric space is complete. Theorem 11: Let ( X , M ,*I ) be an interval-valued

fuzzy metric space and let A be a subset of X . If every Cauchy sequence of A is convergent in X , then every Cauchy sequence of A is convergent in X . Proof: Let {xn } be a Cauchy sequence of A . For any

0 < ε < 1 and t > 0 . Choose r , s ∈ ( I ) such that (1 − r ) *I (1 − r ) *I (1 − s ) ≥ 1 − ε . For each xn ∈ A ,

Theorem 13: A subset of an interval-valued fuzzy metric space ( X , M ,*I ) is nowhere dense if and only if every nonempty open set in X contains an open ball whose closure is disjoint from A . Proof: Let U be a nonempty open subset of X . Since A is a nowhere dense subset, there exists a nonempty open set V ⊂ U such that V ∩ A = ∅ . Let x ∈ V . Then there exist s ∈ ( I ) and t > 0 such that B( x, s , t ) ⊂ V . Choose r ∈ (I ) such that

(1 − r ) *I (1 − r ) ≥ 1 − s

.

By

Theorem

12,

0 < r < 1 and each t > 0 , there exists a yn ∈ A such

t t B ( x, r , ) ⊂ B ( x, s , t ) . Thus B ( x, r , ) ⊂ V and 2 2 that M ( xn , yn , t ) > 1 − r . Since {xn } is a Cauchy t sequence, for any 0 < s < 1 and t > 0 , there exist a B ( x, r , ) ∩ A = ∅ . Conversely, if A is not nowhere 2 n0 ∈ such that M ( xn , xm , t ) > 1 − s for all dense, i.e., Int ( A) ≠ ∅ , then there exists a nonempty m, n ≥ n0 . Then open set U such that U ⊂ A ⊂ A . Let B( x, r , t ) be M ( yn , ym ,3t ) ≥ M ( yn , xn , t )*I M ( xn , x m , t )*I M ( xm , y m , t ) an open ball such that B ( x, r , t ) ⊂ U . Then > (1 − r )*I (1 − s )

≥ 1− ε Hence {xn } is convergent in X . Corollary 4: Let ( X , M ,*I ) be an interval-valued

t B( x, r , ) ∩ A = ∅ . This leads to a contradiction. 2 Theorem 14: (Baire’s Theorem) Let U n (n = 1, 2,3,…) be a countable number of dense open sets in the com-

43


plete interval-valued fuzzy metric space ( X , M ,*I ) .

rem.

∞

Then ∩ U n is dense in X .

5. Conclusions

n =1

Proof: Let V0 be a nonempty open set of X . Since

As a natural generalization of the fuzzy metric space, the main contributions of this paper are: (1) A new conU1 is dense in X , we know that V0 ∩ U1 = ∅ . Let cept of the interval-valued t-norm was introduced, and x ∈ V0 ∩ U1 . Since V0 ∩ U1 is open, there exist the sequence of interval numbers was given. Meantime, 0 < r < 1 and t1 > 0 such that B ( x1 , r1 , t1 ) ⊂ V0 ∩ U1 . the continuity of an interval-valued t-norm was defined by means of the convergence of sequence of interval Choose r1′ < r1 and t1′ = min{t1 ,1} such that numbers. (2) The notion of interval-valued fuzzy metric space was presented on the basis of the continuous inB ( x1 , r1′ , t1′) ⊂ V0 ∩ U1 . Let V1 = B( x1 , r1′ , t1′) . Since terval-valued t-norm. Specially, it should be pointed out U 2 is dense in X , we have V1 ∩ U 2 ≠ ∅ . Let that the condition (C6) must be appended in this definition, and the main reason was explained in Remark 4. xx ∈ V1 ∩ U 2 . Since V1 ∩ U 2 is open, there exist That is to say, there are some differences between the notion of interval-valued fuzzy metric space and the one 1 and t2 > 0 such that of fuzzy metric space. (3) Some related properties of the 0 < r2 < 2 topology by an interval-valued fuzzy metric space were ′ B( x2 , r2 , t2 ) ⊂ V1 ∩ U 2 . Choose and examined. Moreover, it is worth mentioning that Baire's r2 < r2 Theorem has also been extended to interval-valued fuzzy 1 t2′ = min{t2 , } such that B ( x2 , r2′ , t2′ ) ⊂ V1 ∩ U 2 . Let metric spaces. 2 The interval-valued fuzzy metric space will be proved to be an important basis for the construction of interV2 = B( x2 , r2′ , t2′ ) . val-valued fuzzy topology. At the same time, it will enSimilarly continuing in this manner we can obtain a rich the contents of fuzzy mathematics. In our future resequence {xn } and a sequence {tn′ } such that search, we intend to establish some fixed point theorems for contraction type mappings in interval-valued fuzzy 1 and Vn = B ( xn , rn′ , tn′ ) ⊂ Vn −1 ∩ U n . Now metric spaces. 0 < tn′
0 and 0 < ε < 1 , choose n0 such

The authors would like to express their sincere thanks 1 1 1 1 < t and ( ) = [ , ] < ε . Then for all to Editor-in-Chief, Associate Editor and two anonymous n0 n0 n0 n0 referees for their valuable comments and suggestions which have helped immensely in improving the quality n ≥ n0 and m ≥ n , we have

that

1 1 M ( xn , xm , t ) ≥ M ( xn , xm , ) ≥ 1 − ( ) > 1 − ε . n n Therefore {xn } is a Cauchy sequence. Since

( X , M ,*I ) is complete, there exists a x ∈ X such that xn → x . Since xk ∈ B ( xn , rn′, tn′ ) for all k ≥ n , we

know

that

x ∈ Vn ⊂ Vn −1 ∩ U n ∞

x ∈ B( xn , rn′, tn′ ) = Vn . for

V0 ∩ (∩U n ) ≠ ∅ . Thus n =1

n∈

all

+

.

Hence

of the paper. This work was supported by “QingLan” Talent Engineering Funds by Tianshui Normal University, the Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality (No.PHR201007117) and the Beijing Municipal Education Commission Foundation of China (No. KM201210038001).

Therefore

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Remark 13: Since each metric induces an interval-valued metric and interval-valued fuzzy metric is a generalization of fuzzy metric, Baire’s theorem for complete metric space [26] and Baire’s theorem for complete fuzzy metric space [7] are two particular cases of the above theo-

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