Fuzzy Metric Space

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J. Appl. Math. & Computing Vol. 16(2004), No. 1 - 2, pp. 371 - 381

FUZZY METRIC SPACES

ZUN-QUAN XIA AND FANG-FANG GUO

Abstract. In this paper, fuzzy metric spaces are redefined, different fromthe previous ones in the way that fuzzy scalars instead of fuzzy numbersor real numbers are used to define fuzzy metric. It is proved that every

ordinary metric space can induce a fuzzy metric space that is completewhenever the original one does. We also prove that the fuzzy topologyinduced by fuzzy metric spaces defined in this paper is consistent with thegiven one. The results provide some foundations for the research on fuzzyoptimization and pattern recognition.

AMS Mathematics Subject Classification: 03E72, 90C70, 15A03.Key words and phrases : Fuzzy metric space, completeness of fuzzy metricspace, fuzzy topology, fuzzy closed set.

1. Introduction

How to define fuzzy metric is one of the fundamental problems in fuzzy mathe-matics which is wildly used in fuzzy optimization and pattern recognition. Thereare two approaches in this field till now. One is using fuzzy numbers to definemetric in ordinary spaces, firstly proposed by Kaleva (1984)[12], following whichfuzzy normed spaces, fuzzy topology induced by fuzzy metric spaces, fixed pointtheorem and other properties of fuzzy metric spaces are studied by a few re-searchers, see for instance, Felbin (1992)[7], George (1994)[8], George (1997)[9],Gregori (2000)[10], Hadzic (2002)[11] etc. The other one is using real numbersto measure the distances between fuzzy sets. The references of this approach canbe referred to, for instance, Dia (1990) [5], Chaudhuri (1996)[4], Boxer (1997)[2],

Received July 27, 2003. Revised October 20, 2003. Corresponding author.

This paper was supported by the National Foundations of Ph. D Units from the Ministry of

Education of China No. 20020141013, the Scientific Research Foundation of DUT No. 3004888.

c 2004 Korean Society For Computational & Applied Mathematics and Korean SIGCAM .

371


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372 Zun-Quan Xia and Fang-Fang Guo

Fan (1998)[6], Voxmam (1998)[16], Przemyslaw, (1998)[14], Brass (2002)[3]. Re-sults of these researches have been applied to many practical problems in fuzzyenvironment. While, usually, different measures are used in different problemsin other words, there does not exist a uniform measure that can be used in allkinds of fuzzy environments.

Therefore, it is still interesting to find some kind of new fuzzy measure suchthat it may be useful for solving some problems in fuzzy environment. Theattempt of the present paper is using fuzzy scalars (fuzzy points defined onthe real-valued space R) to measure the distances between fuzzy points, whichis consistent with the theory of fuzzy linear spaces in the sense of Xia andGuo (2003) [17] and hence more similar to the classical metric spaces. Thenew definitions in this paper are different from the previous ones because fuzzyscalars are used instead of fuzzy numbers or real numbers to measure the distancebetween two fuzzy points. It is the first time that fuzzy scalars are introduced

in measuring the distances between fuzzy points. Some other properties of fuzzymetric spaces, for instance, completeness and induced fuzzy topology are alsogiven in this paper.

For the convenience of reading, some basic concepts of fuzzy points and de-notations are presented below.

Fuzzy points are the fuzzy sets being of the following form in the sense of Pu(1980) [15],

x(y) =

, y = x,0, y=x,

y X,

whereXis a nonempty set and [0, 1].In this paper, fuzzy points are usually denoted by (x, ) and the set of all

the fuzzy points defined on Xis denoted byPF(X). Particularly, whenX=R,

fuzzy points are also called fuzzy scalars and the set of all the fuzzy scalarsis denoted by SF(R). A fuzzy set A can be regarded as a set of fuzzy pointsbelonging to it, i.e.,

A= {(x, )| A(x) }

or a set of fuzzy points on it,

A= {(x, )|A(x) =}.

This paper is organized as follows: In Section 2, fuzzy metric spaces, strongfuzzy metric spaces and fuzzy linear normed spaces are defined and some ex-amples are given to show the existence of these kinds of spaces; In Section 3,the convergence of sequences of fuzzy points and the completeness of inducedfuzzy metric spaces are considered; In the last section, it is proved that the fuzzy

topology induced by fuzzy metric spaces is consistent with the given one, see forinstance, Pu (1980), [15], which implies in another way the usefulness of thefuzzy metric spaces defined in Section 2.


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Fuzzy metric spaces 373

2. Fuzzy metric spaces

The purpose of this section mainly consists in defining fuzzy metric spaces,strong fuzzy metric spaces and fuzzy normed linear spaces. To do so, we firstgive some definitions related to fuzzy scalars.

Definition 1. Suppose (x, ) and (y, ) are two fuzzy scalars. A series of defi-nitions contains the following ones:

(1) we say (a, ) (b, ) ifa > bor (a, ) = (b, );(2) (a, ) is said to be no less than(b, ) ifa b, denoted by (a, ) (b, )

or (b, ) (a, );(3) (a, ) is said to be nonnegative if a 0. The set of all the nonnegative

fuzzy scalars is denoted by S+F(R).

Obviously, the orders defined in Definition 1(1) and Definition 1(2) are bothpartial orders. Note that whenR is considered as a subset ofSF(R), (R, ) and(R, ) are the same as (R, ). Thus both and can be viewed as some kindof generalization of the ordinary complete order . It is obvious that the orderdefined in Definition 1(1) is stronger than the one in Definition 1(2).

We now present the definition of fuzzy metric spaces. It will be seen that it isvery similar to the definition of ordinary metric spaces except that is replacedby in the triangle inequality. This is because that there exist no reasonablecomplete order in SF(R)

+.

Definition 2. Suppose Xis a nonempty set and

dF :

PF(

X)

PF(

X)

S+F(

R)

is a mapping. (PF(X), dF) is said to be a fuzzy metric space if for any {(x, ),(y, ), (z, )} PF(X), dF satisfies the following three conditions,

(1) Nonnegative: dF((x, ), (y, )) = 0 iffx= y and = = 1;(2) Symmetric:

dF((x, ), (y, )) =dF((y, ), (x, ));(3) Triangle inequality:

dF((x, ), (z, )) dF((x, ), (y, )) + dF((y, ), (z, )).dF is called a fuzzy metric defined in PF(X) and dF((x, ), (y, )) is called afuzzy distancebetween the two fuzzy points.

Note that fuzzy metric spaces have fuzzy points as their elements, i.e., they

are sets of fuzzy points. There are many fuzzy metric spaces in the sense ofDefinition 2. To show this, some examples are presented below.


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Example 1. Suppose (X, d) is an ordinary metric space. The distance of anytwo fuzzy points (x, ), (y, ) in PF(X) is defined by

dF((x, ), (y, )) = (d(x, y), min{, }),

where d(x, y) is the distance between x and ydefined in (X, d).Then (PF(X), dF)is a fuzzy metric space.

Proof. It suffices to prove that dFsatisfies the three conditions in Definition 2.Nonnegative: Suppose (x, ) and (y, ) are two fuzzy points in PF(X). Since

d(x, y) is a distance between x and y, one has d(x, y) 0. It follows fromDefinition 1 that dF((x, ), (y, )) = (d(x, y), min{, }) is a nonnegative fuzzyscalar. It is obvious that dF((x, ), (y, )) = 0 iffd(x, y) = 0 and min{, }= 1which is equal to that x= y and = = 1.

Symmetric: For any{(x, ), (y, )} PF(X), one has

dF((x, ), (y, )) = (d(x, y), min{, })= (d(y, x), min{, })= dF((y, ), (x, )).

Triangle inequality: For any{(x, ), (y, ),(z, )} PF(X), we have

dF((x, ), (z, )) = (d(x, z), min{, }) (d(x, y) + d(y, z), min{,,})= (d(x, y), min{, }) + (d(y, z), min{, })= d((x, ), (y, )) + d((y, ), (z, )).

Example 2. We denote Rn the usual n-dimensional Euclidean space. Suppose

Lis a fuzzy linear space defined in Rn. The distance between arbitrary two fuzzypoints (x, ), (y, )belonging to L, denoted bydFE((x, ), (y, )), is defined by

dFE((x, ), (y, )) = (dE(x, y), min{, }),

wheredE is the usual Euclidean distance. Then (L, dEF) is also a fuzzy metricspace, where L is also viewed as the set of fuzzy points belonging to the fuzzyset L.

Proof. Since Rn is a metric space in the ordinary sense and L can be regardedas a subset ofPF(R

n), dFE is a fuzzy metric from Example 1.

The two examples given above show that a fuzzy (linear) metric space can

be constructed by a (linear) metric space in the usual sense, called an induced(linear) metric spaceof it and the metric of the space is called an induced metricof the original one.


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SinceS+F(R) is not a complete ordered set, in the triangle inequality of Defi-nition 2, is replaced by which is much weaker than it. A natural question isthat whether there exist some kind of fuzzy metric spaces satisfying the triangleinequality with some partial order stronger than , for example, . The answeris positive and they are called strong fuzzy metric spaces.

Definition 3. SupposeXis a nonempty set anddF :PF(X)PF(X) S+F(R)

is a mapping. (PF(X), dF) is said to be a strong fuzzy metric spaceif it satisfiesthe first two conditions in Definition 2 and for any (x, ), (y, ),(z, ) inPF(X),one has

(3) dF((x, ), (z, )) dF((x, ), (y, )) + dF((y, ), (z, )).

It is obvious from Definition 2 and Definition 3 that every strong fuzzy metricspace is a fuzzy metric space. The following example shows the existence of

strong fuzzy metric spaces and the difference between these two kinds of spaces.

Example 3. L is a fuzzy linear space defined in Rn. The distance betweenarbitrary two fuzzy points (x, ) and (y, ) onL is defined by

dFE((x, ), (y, )) = (dE(x, y), min{, }), (1)

where dE is the Euclidean distance. Then (L, dFE) is a strong fuzzy metricspace where L denote the set of fuzzy points onthe fuzzy set L.

Proof. The first two conditions can be proved just as Example 1. Here we onlyprove the third one.

Given arbitrary three fuzzy points on L, (x, ), (y, )and(z, ). Since (Rn, dE)is a metric space, one has

dE(x, z) dE(y, z) + dE(x, y). (2)

In the case of that inequality (2) holds strictly, it is obvious from Definition 1(1)that condition (3) is satisfied. In the other case, there must exists some Fsuch thaty = (1)x+z. Let = min{, }. We have that {x, z} L. SinceLis a fuzzy linear space, L is a linear subspace ofR

n (see the RepresentationTheorem of fuzzy linear spaces due to Lowen (1980), [13]). It follows thaty L,i.e., = L(y) = min{, }. This implies that min{,,} = min{, }.Thus, one has

dFE((x, ), (z, )) = (dE(x, z), min{, })= (dE(x, y) + dE(y, z), min{,,})= dFE((x, ), (y, )) + dFE((y, ), (z, )).

Consequently, condition (3) is satisfied.


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Note that the strong fuzzy metric space given above is a set of fuzzy pointsonsome fuzzy linear space. Different from it, the fuzzy metric space in Example2 comprises fuzzy points belonging to a fuzzy linear space. The difference iscaused by that is replaced by the partial order which is much stronger thanit.

Definition 4. Suppose that L is a fuzzy linear space. (L, ) is said to be afuzzy linear normed spaceif the mapping :L S+F(R) satisfies:

(a) (x, )= 0 if and only ifx= 0 and = 1;(b) For any k R and (x, ) L, one has k(x, )= |k| (x, );(c) For any{(x, ), (y, )} L, one has(x, ) + (y, ) (x, ) + (y, ).

The mapping : x xis called the fuzzy normof (L, ). Note that afuzzy linear normed space L has fuzzy points belonging to the fuzzy set Las its

elements.

Example 4. Let (G, ||||G) be a linear normed space defined on R. Lis a fuzzylinear space defined in G and FGis a mapping from L to SF(R)

+ defined by

(x, )FG:= (xG, ), (x, ) L.

Then (L, FG) is a fuzzy linear normed space which can be verified similar toExample 2.

The following proposition given without proof shows the relationship betweenfuzzy linear normed spaces and fuzzy metric spaces.

Proposition 1. Suppose (L, FG) is a fuzzy linear normed space. Then

(L, dFG) is a fuzzy metric space, wheredFG is defined by

dFG((x, ), (y, )) :=(x, ) (y, )FG.

Proof. It is omitted.

TakingG = Rn in Example 4, we have the following proposition, which showsthe relationship between fuzzy norm and inner product of fuzzy points.

Proposition 2. Suppose (L, FE) is a fuzzy linear normed space defined inRn. For any(x, ) L, one has

=(x, )2FE,

where the inner product is defined in the sense of Xia and Guo (2003) [17], i.e.,

= (< x,y >, min{, }).


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Proof. From the definition of inner product of fuzzy points, one has

= (< x,x >, )= (x2E, )= (xE, ) (xE, )= (x, )2FE ,

where E is the Euclidean norm.

3. The completeness of fuzzy metric spaces

In this section, we mainly consider the convergence of a sequence of fuzzypoints and the completeness of induced fuzzy metric spaces. Since fuzzy scalarsare used to measure the distances between fuzzy points, the convergence of a

sequence of fuzzy scalars is considered first.

Definition 5. Let {(an, n)} be a sequence of fuzzy scalars. It is said to beconvergentto a fuzzy scalar (a, ), = 0, denoted by limn(an, n) = (a, ) iflimn an= a,{i|i < , i N}is a finite set and there exists a subsequenceof{i}, denoted by{l}, such that limn l = .

The requirement that almost all thei Nsatisfy i is natural since wehope that the degree of the convergence is not less than . A new definition ofthe convergence of a sequence of fuzzy points is presented below based on thefuzzy metric given in the last section.

Definition 6. Suppose (PF(X), dF) is the induced fuzzy metric space of (X, d)and {(xn, n)} is a sequence of fuzzy points in (PF(X), dF). {(xn, n)} is saidto beconvergentto a fuzzy point (x, ), if limn dF((xn, n), (x, )) = 0 andfor any (0, 1] such that limn dF((xn, n), (x, )) = 0, one has .(x, ) is called the limit of the sequence, denoted by limn(xn, n) = (x, ).

Proposition 3. Suppose{(xn, n)}is a sequence of fuzzy points in(PF(X), dF)and(x, ) (PF(X), dF), = 0. We have that limn(xn, n) = (x, ) if andonly if limn xn = x, {i|i < , i N} is a finite set and there exists asubsequence of{i}, denoted by{l}, such that limn l =.



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Definition 7. A sequence of fuzzy points (xn, n) (PF(X), dF) is said to beaCauchy sequenceif there exists some (0, 1] such that

limn

dF((xm+n, m+n), (xn, n)) = 0, m N.

Note that every Cauchy sequence of fuzzy points defined above has a uniquefuzzy point as its limit, which is very similar to the classical one. We now beginto consider the completeness of fuzzy metric spaces.

Definition 8. An induced fuzzy metric space is said to be complete if anyCauchy sequence in it has a unique limit in the space.

Theorem 1. Suppose(PF(X), dF) is the induced fuzzy metric space of an or-dinary metric space(X, d). Then it is complete iff(X, d) is complete.

Proof. Necessity (Only if) : It is obvious.Sufficiency (If) : Suppose {(xn, n)} is an arbitrary Cauchy sequence of

(PF(X), dF). Since (X, d) is complete and limn d(xm+n, xn) = 0 for anym N, there must exists some x X such that limn xn = x. For anym N, denote the index set {l|l = min{m+n, n}, n= 1, 2 } by Lm.

From the definition of Cauchy sequences of fuzzy points, there exists some (0, 1] such that for anym N, the set{l|l < , l Lm}is finite and thereexists a subsequence of{l}lLm, denoted by {k}, which is also a subsequenceof{n}, such that limk k =. It is obvious that

{l|l < , l Lm} {n|n < , n= 1, 2, }.

Consequently, {n|n < , n = 1, 2, } is also a finite set. From the above

arguments, we have limn(xn, n) = (x, ). It implies that there exists alimit of {xn, n} in PF(X). In the following we prove the uniqueness. Bycontradiction, assume that there is another limit of the same Cauchy sequence{(xn, n)}. Since we know x is the unique limit of{xn}, we can denote by (x, )the limit different from (x, ), = , say > . Then we have{n|n < }is a finite set. From the above arguments, we know that limk k = and{k|k < } is a finite set. Thus, taking =

+2

, we have

{k} [, ] {n|n < }

is an infinite set. This contradicts that {n|n < } is a finite set. Therefore,there is only one limit of the Cauchy sequence.

Note that a strong fuzzy linear metric space is generally not complete. It canbe seen through the counter-example given below.


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Example 5. Consider the strong fuzzy linear metric space (L, dFE), where

L=

(x, )|x R \ {0}, = 12

{(0, 1)}

anddFE is induced by the ordinary Euclidean metricdE . The sequence {(1n

, 12

)}inL is a Cauchy sequence in the sense of Definition 7. However, the limit of thesequence, (0, 1

2) is not on the space L.

4. The fuzzy topology spaces induced by fuzzy metric spaces

Fuzzy metric spaces given in this paper have many similar properties to theordinary metric spaces. Except the relationship between distances and innerproducts of fuzzy points mentioned in the Section 2, a conclusion similar to thatevery metric space can induce a topology will be proved in this section. Herewe introduce fuzzy topology in the sense of Pu (1980) [15] via fuzzy closed sets.It provides a convenient method to construct a fuzzy topology of any ordinarymetric space. To do this, the definition of fuzzy closed sets with respect toinduced fuzzy metric spaces is given first. Suppose (X, d) is an ordinary metricspace. Since a fuzzy set A in Xcan be viewed as a set of fuzzy points belongingto it, Acan be regarded as a subset ofPF(X), called a fuzzy set in the inducedfuzzy metric space (PF(X), dF) in the following definition.

Definition 9. A fuzzy set A in (PF(X), dF) is said to be closed if the limit ofany Cauchy sequence in A belongs to it. A fuzzy set A in (PF(X), dF) is saidto be open ifA is a fuzzy closed set, where A is defined by A(x) = 1 A(x),for any x X.

The following proposition shows that the new definition of fuzzy closed setsis reasonable.

Proposition 4. A fuzzy set A in (PF(X), dF) is closed if and only if every-cut set ofA, [0, 1], is a closed set in(X, d) in the ordinary sense.


In the following we will show that every induced fuzzy metric space can induce

a fuzzy topology. To prove it, a lemma about Cauchy sequences of fuzzy pointsis given.


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Lemma 1. Any subsequence of a Cauchy sequence of fuzzy points is also aCauchy sequence and has the same limit as the original one.

Proof. It is obvious from Definition 7.

Theorem 2. Suppose(PF(X), dF)is the induced fuzzy metric space of a metricspace (X, d). Then (X, TF) is a fuzzy topology space in the sense of Pu (1980)[15], called the fuzzy topology space induced by(PF(X), dF), whereTF is definedby

TF ={A PF(X)|A is a fuzzy closed set in (PF(X), dF)}.

Proof. It suffices to prove that TF satisfies the three conditions in the definitionof fuzzy topology due to Pu (1980) [15].

(1) It is obvious that X and are fuzzy closed sets.(2) For any {A, B} TF, we prove in the following that A B TF. For

any Cauchy sequence of fuzzy points {(yn, n)} included in A B, A or B,say A, must contain a subsequence {(ym, m)} of{(yn, n)}. From Lemma 1,{(ym, m)}is also a Cauchy sequence and hence has a limit. Since A is a closedfuzzy set, the limit of{(ym, m)}which is also the limit of{(yn, n)}is includedinA. In consequence, the limit of{(yn, n)}is included in A B, which impliesthat A B TF.

(3) For any {Ai}iI TF, where I is an arbitrary index set, it only need tobe proved that

iIAi TF. For any Cauchy sequence in

iIAi, denoted by

{(xn, n)}, we have that{(xn, n)} Ai for anyi I. Since everyAi is a closedfuzzy set, the limit of{(xn, n)}is inAi for any i I. It follows that

iIAi is

a closed fuzzy set in the sense of Definition 9. Therefore one has iI

Ai TF.The proof is completed.

From the theorem given above we know that every fuzzy metric space caninduce a fuzzy topology space, which implies in another way that the fuzzymeasure defined in this paper is not only reasonable but also significant.

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13. R. Lowen,Convex fuzzy sets, Fuzzy Sets and Systems 3 (1980), 291-310.14. G. Przemyslaw, Metrics and orders in space of fuzzy numbers, Fuzzy Sets and Systems

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100 (1998), 353-365.17. Z. Q. Xia and F. F. Guo, Fuzzy linear spaces, Int. J. Pure and Applied Mathematics, to

appear.

Fang-Fang Guois a student for Ph.D. under the supervision of Prof. Xia. She received hermasters degree from Liaoning Normal University in 2001. Her research interesting focus

on fuzzy convex analysis, fuzzy optimization and numerical method and fuzzy reasoning.Laboratory 2, CORA, Department of Applied Mathematics, Dalian University of Technol-ogy, Dalian 116024, P. R. Chinae-mail: [email protected]

Zun-Quan Xiais a professor in Department of Applied Mathematics, Dalian University ofTechnology, Dalian, China. He graduated from Institute of Mathematics, Fudan University,Shanghai, as a graduate student in 1968, His research areas are (smooth, nonsmooth,discrete and numerical) optimization and applications (in science and technology) ORmethods and applications.

Laboratory 2, CORA, Department of Applied Mathematics, Dalian University of Technol-ogy, Dalian 116024, P. R. Chinae-mail: [email protected]

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