FUZZY METRIC SPACES ZUN-QUAN XIA AND FANG-FANG GUO

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 from the previous ones in the way that fuzzy scalars instead of fuzzy numbers or real numbers are used to define fuzzy metric. It is proved that every ordinary metric space can induce a fuzzy metric space that is complete whenever the original one does. We also prove that the fuzzy topology induced by fuzzy metric spaces defined in this paper is consistent with the given one. The results provide some foundations for the research on fuzzy optimization and pattern recognition. AMS Mathematics Subject Classification: 03E72, 90C70, 15A03. Key words and phrases : Fuzzy metric space, completeness of fuzzy metric space, fuzzy topology, fuzzy closed set. 1. Introduction How to define fuzzy metric is one of the fundamental problems in fuzzy mathematics which is wildly used in fuzzy optimization and pattern recognition. There are two approaches in this field till now. One is using fuzzy numbers to define metric in ordinary spaces, firstly proposed by Kaleva (1984)[12], following which fuzzy normed spaces, fuzzy topology induced by fuzzy metric spaces, fixed point theorem and other properties of fuzzy metric spaces are studied by a few researchers, 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 numbers to measure the distances between fuzzy sets. The references of this approach can be 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

372 Zun-Quan Xia and Fang-Fang Guo Fan (1998)[6], Voxmam (1998)[16], Przemyslaw, (1998)[14], Brass (2002)[3]. Results of these researches have been applied to many practical problems in fuzzy environment. While, usually, different measures are used in different problems in other words, there does not exist a uniform measure that can be used in all kinds of fuzzy environments. Therefore, it is still interesting to find some kind of new fuzzy measure such that it may be useful for solving some problems in fuzzy environment. The attempt of the present paper is using fuzzy scalars (fuzzy points defined on the real-valued space R) to measure the distances between fuzzy points, which is consistent with the theory of fuzzy linear spaces in the sense of Xia and Guo (2003) [17] and hence more similar to the classical metric spaces. The new definitions in this paper are different from the previous ones because fuzzy scalars are used instead of fuzzy numbers or real numbers to measure the distance between two fuzzy points. It is the first time that fuzzy scalars are introduced in measuring the distances between fuzzy points. Some other properties of fuzzy metric spaces, for instance, completeness and induced fuzzy topology are also given in this paper. For the convenience of reading, some basic concepts of fuzzy points and denotations are presented below. Fuzzy points are the fuzzy sets being of the following form in the sense of Pu (1980) [15], { λ, y = x, x λ (y) = y X, 0, y x, where X is a nonempty set and λ [0, 1]. In this paper, fuzzy points are usually denoted by (x, λ) and the set of all the fuzzy points defined on X is denoted by P F (X). Particularly, when X = R, fuzzy points are also called fuzzy scalars and the set of all the fuzzy scalars is denoted by S F (R). A fuzzy set A can be regarded as a set of fuzzy points belonging to it, i.e., A = {(x, λ) A(x) λ } or a set of fuzzy points on it, A = {(x, λ) A(x) =λ}. This paper is organized as follows: In Section 2, fuzzy metric spaces, strong fuzzy metric spaces and fuzzy linear normed spaces are defined and some examples 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 induced fuzzy 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 for instance, Pu (1980), [15], which implies in another way the usefulness of the fuzzy metric spaces defined in Section 2.

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 first give some definitions related to fuzzy scalars. Definition 1. Suppose (x, λ) and (y, γ) are two fuzzy scalars. A series of definitions contains the following ones: (1) we say (a, λ) (b, γ) ifa>bor (a, λ) =(b, γ); (2) (a, λ) is said to be no less than (b, γ) ifa b, denoted by (a, λ) (b, γ) or (b, γ) (a, λ); (3) (a, λ) 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 both partial orders. Note that when R is considered as a subset of S F (R), (R, ) and (R, ) are the same as (R, ). Thus both and can be viewed as some kind of generalization of the ordinary complete order. It is obvious that the order defined 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 is very similar to the definition of ordinary metric spaces except that is replaced by in the triangle inequality. This is because that there exist no reasonable complete order in S F (R) +. Definition 2. Suppose X is a nonempty set and d F : P F (X) P F (X) S + F (R) is a mapping. (P F (X),d F ) is said to be a fuzzy metric space if for any {(x, λ), (y, γ), (z,ρ)} P F (X), d F satisfies the following three conditions, (1) Nonnegative: d F ((x, λ), (y, γ)) = 0 iff x = y and λ = γ =1; (2) Symmetric: d F ((x, λ), (y, γ)) = d F ((y, γ), (x, λ)); (3) Triangle inequality: d F ((x, λ), (z,ρ)) d F ((x, λ), (y, γ)) + d F ((y, γ), (z,ρ)). d F is called a fuzzy metric defined in P F (X) and d F ((x, λ), (y, γ)) is called a fuzzy distance between 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 of Definition 2. To show this, some examples are presented below.

374 Zun-Quan Xia and Fang-Fang Guo Example 1. Suppose (X, d) is an ordinary metric space. The distance of any two fuzzy points (x, λ), (y, γ) inp F (X) is defined by d F ((x, λ), (y, γ)) = (d(x, y), min{λ, γ}), where d(x, y) is the distance between x and y defined in (X, d).then (P F (X),d F ) is a fuzzy metric space. Proof. It suffices to prove that d F satisfies the three conditions in Definition 2. Nonnegative: Suppose (x, λ) and (y, γ) are two fuzzy points in P F (X). Since d(x, y) is a distance between x and y, one has d(x, y) 0. It follows from Definition 1 that d F ((x, λ), (y, γ)) = (d(x, y), min{λ, γ}) is a nonnegative fuzzy scalar. It is obvious that d F ((x, λ), (y, γ)) = 0 iff d(x, y) = 0 and min{λ, γ} =1 which is equal to that x = y and λ = γ =1. Symmetric: For any {(x, λ), (y, γ)} P F (X), one has d F ((x, λ), (y, γ)) = (d(x, y), min{λ, γ}) = (d(y, x), min{γ,λ}) = d F ((y, γ), (x, λ)). Triangle inequality: For any {(x, λ), (y, γ), (z,ρ)} P F (X), we have d F ((x, λ), (z,ρ)) = (d(x, z), min{λ, ρ}) (d(x, y)+d(y, z), min{λ, ρ, γ}) = (d(x, y), min{λ, γ})+(d(y, z), min{γ,ρ}) = d((x, λ), (y, γ)) + d((y, γ), (z,ρ)). Example 2. We denote R n the usual n-dimensional Euclidean space. Suppose L is a fuzzy linear space defined in R n. The distance between arbitrary two fuzzy points (x, λ), (y, γ) belonging to L, denoted by d FE ((x, λ), (y, γ)), is defined by d FE ((x, λ), (y, γ)) = (d E (x, y), min{λ, γ}), where d E is the usual Euclidean distance. Then (L, d EF ) is also a fuzzy metric space, where L is also viewed as the set of fuzzy points belonging to the fuzzy set L. Proof. Since R n is a metric space in the ordinary sense and L can be regarded as a subset of P F (R n ), d FE 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 space of it and the metric of the space is called an induced metric of the original one.

Fuzzy metric spaces 375 Since S + F (R) is not a complete ordered set, in the triangle inequality of Definition 2, is replaced by which is much weaker than it. A natural question is that whether there exist some kind of fuzzy metric spaces satisfying the triangle inequality with some partial order stronger than, for example,. The answer is positive and they are called strong fuzzy metric spaces. Definition 3. Suppose X is a nonempty set and d F : P F (X) P F (X) S + F (R) is a mapping. (P F (X), d F ) is said to be a strong fuzzy metric space if it satisfies the first two conditions in Definition 2 and for any (x, λ), (y, γ), (z,ρ) inp F (X), one has (3 ) d F ((x, λ), (z,ρ)) d F ((x, λ), (y, γ)) + d F ((y, γ), (z,ρ)). It is obvious from Definition 2 and Definition 3 that every strong fuzzy metric space 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 R n. The distance between arbitrary two fuzzy points (x, λ) and (y, γ) on L is defined by d FE ((x, λ), (y, γ)) = (d E (x, y), min{λ, γ}), (1) where d E is the Euclidean distance. Then (L, d FE ) is a strong fuzzy metric space where L denote the set of fuzzy points on the fuzzy set L. Proof. The first two conditions can be proved just as Example 1. Here we only prove the third one. Given arbitrary three fuzzy points on L,(x, λ), (y, γ)and(z,ρ). Since (R n,d E ) is a metric space, one has d E (x, z) d E (y, z)+d E (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 λ F such that y =(1 λ)x+λz. Let α = min{λ, ρ}. We have that {x, z} L α. Since L is a fuzzy linear space, L α is a linear subspace of R n (see the Representation Theorem of fuzzy linear spaces due to Lowen (1980), [13]). It follows that y L α, i.e., γ = L(y) α = min{λ, ρ}. This implies that min{λ, ρ, γ} = min{λ, ρ}. Thus, one has d FE ((x, λ), (z,ρ)) = (d E (x, z), min{λ, ρ}) =(d E (x, y)+d E (y, z), min{λ, ρ, γ}) = d FE ((x, λ), (y, γ)) + d FE ((y, γ), (z,ρ)). Consequently, condition (3 ) is satisfied.

376 Zun-Quan Xia and Fang-Fang Guo Note that the strong fuzzy metric space given above is a set of fuzzy points on some fuzzy linear space. Different from it, the fuzzy metric space in Example 2 comprises fuzzy points belonging to a fuzzy linear space. The difference is caused by that is replaced by the partial order which is much stronger than it. Definition 4. Suppose that L is a fuzzy linear space. (L, ) is said to be a fuzzy linear normed space if the mapping : L S + F (R) satisfies: (a) (x, λ) = 0 if and only if x = 0 and λ =1; (b) For any k R and (x, λ) L, one has k(x, λ) = k (x, λ) ; (c) For any {(x, λ), (y, γ)} L, one has (x, λ)+(y, γ) (x, λ) + (y, γ). The mapping : x x is called the fuzzy norm of (L, ). Note that a fuzzy linear normed space L has fuzzy points belonging to the fuzzy set L as its elements. Example 4. Let (G, G ) be a linear normed space defined on R. L is a fuzzy linear space defined in G and FG is a mapping from L to S F (R) + defined by (x, λ) FG := ( x G,λ), (x, λ) L. Then (L, FG ) is a fuzzy linear normed space which can be verified similar to Example 2. The following proposition given without proof shows the relationship between fuzzy linear normed spaces and fuzzy metric spaces. Proposition 1. Suppose (L, FG ) is a fuzzy linear normed space. (L, d FG ) is a fuzzy metric space, where d FG is defined by d FG ((x, λ), (y, γ)) := (x, λ) (y, γ) FG. Proof. It is omitted. Then Taking G = R n in Example 4, we have the following proposition, which shows the relationship between fuzzy norm and inner product of fuzzy points. Proposition 2. Suppose (L, FE ) is a fuzzy linear normed space defined in R n. For any (x, λ) L, one has < (x, λ), (x, λ) >= (x, λ) 2 FE, where the inner product is defined in the sense of Xia and Guo (2003) [17], i.e., < (x, λ), (y, γ) >= (< x, y >, min{λ, γ}).

Fuzzy metric spaces 377 Proof. From the definition of inner product of fuzzy points, one has < (x, λ), (x, λ) > = (< x, x >, λ) = ( x 2 E,λ) = ( x E,λ) ( x E,λ) = (x, λ) 2 FE, 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 fuzzy points and the completeness of induced fuzzy metric spaces. Since fuzzy scalars are used to measure the distances between fuzzy points, the convergence of a sequence of fuzzy scalars is considered first. Definition 5. Let {(a n,λ n )} be a sequence of fuzzy scalars. It is said to be convergent to a fuzzy scalar (a, λ), λ 0, denoted by lim n (a n,λ n )=(a, λ) if lim n a n = a, {λ i λ i <λ,i N} is a finite set and there exists a subsequence of {λ i }, denoted by {λ l }, such that lim n λ l = λ. The requirement that almost all the λ i N satisfy λ i λ is natural since we hope that the degree of the convergence is not less than λ. A new definition of the convergence of a sequence of fuzzy points is presented below based on the fuzzy metric given in the last section. Definition 6. Suppose (P F (X), d F ) is the induced fuzzy metric space of (X, d) and {(x n,λ n )} is a sequence of fuzzy points in (P F (X), d F ). {(x n,λ n )} is said to be convergent to a fuzzy point (x, λ), if lim n d F ((x n,λ n ), (x, λ)) = 0 λ and for any γ (0, 1] such that lim n d F ((x n,λ n ), (x, γ)) = 0 γ, one has λ γ. (x, λ) is called the limit of the sequence, denoted by lim n (x n,λ n )=(x, λ). Proposition 3. Suppose {(x n,λ n )} is a sequence of fuzzy points in (P F (X), d F ) and (x, λ) (P F (X), d F ), λ 0. We have that lim n (x n,λ n )=(x, λ) if and only if lim n x n = x, {λ i λ i <λ,i N} is a finite set and there exists a subsequence of {λ i }, denoted by {λ l }, such that lim n λ l = λ. Proof. It is omitted.

378 Zun-Quan Xia and Fang-Fang Guo Definition 7. A sequence of fuzzy points (x n,λ n ) (P F (X), d F ) is said to be a Cauchy sequence if there exists some λ (0, 1] such that lim d F ((x m+n,λ m+n ), (x n,λ n )) = 0 λ, m N. n Note that every Cauchy sequence of fuzzy points defined above has a unique fuzzy point as its limit, which is very similar to the classical one. We now begin to consider the completeness of fuzzy metric spaces. Definition 8. An induced fuzzy metric space is said to be complete if any Cauchy sequence in it has a unique limit in the space. Theorem 1. Suppose (P F (X), d F ) is the induced fuzzy metric space of an ordinary metric space (X, d). Then it is complete iff (X, d) is complete. Proof. Necessity (Only if) : It is obvious. Sufficiency (If) : Suppose {(x n,λ n )} is an arbitrary Cauchy sequence of (P F (X), d F ). Since (X, d) is complete and lim n d(x m+n,x n ) = 0 for any m N, there must exists some x X such that lim n x n = x. For any m N, denote the index set {l λ l = min{λ m+n,λ n },n=1, 2 } by L m. From the definition of Cauchy sequences of fuzzy points, there exists some λ (0, 1] such that for any m N, the set {λ l λ l <λ,l L m } is finite and there exists a subsequence of {λ l } l Lm, denoted by {λ k }, which is also a subsequence of {λ n }, such that lim k λ k = λ. It is obvious that {l λ l <λ,l L m } {n λ n <λ,n=1, 2, }. Consequently, {λ n λ n <λ,n=1, 2, } is also a finite set. From the above arguments, we have lim n (x n,λ n ) = (x, λ). It implies that there exists a limit of {x n,λ n } in P F (X). In the following we prove the uniqueness. By contradiction, assume that there is another limit of the same Cauchy sequence {(x n,λ n )}. Since we know x is the unique limit of {x n }, we can denote by (x, γ) the limit different from (x, λ), γ λ, sayγ>λ. Then we have {λ n λ n <γ} is a finite set. From the above arguments, we know that lim k λ k = λ and {λ k λ k <λ} is a finite set. Thus, taking ρ = λ+γ 2, we have {λ k } [λ, ρ] {λ n λ n <γ} is an infinite set. This contradicts that {λ n λ n <γ} 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 can be seen through the counter-example given below.

Fuzzy metric spaces 379 Example 5. Consider the strong fuzzy linear metric space (L, d FE ), where { L = (x, λ) x R \{0}, λ= 2} 1 {(0, 1)} and d FE is induced by the ordinary Euclidean metric d E. The sequence {( 1 n, 1 2 )} in L is a Cauchy sequence in the sense of Definition 7. However, the limit of the sequence, (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 the ordinary metric spaces. Except the relationship between distances and inner products of fuzzy points mentioned in the Section 2, a conclusion similar to that every metric space can induce a topology will be proved in this section. Here we 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 ordinary metric space. To do this, the definition of fuzzy closed sets with respect to induced fuzzy metric spaces is given first. Suppose (X, d) is an ordinary metric space. Since a fuzzy set A in X can be viewed as a set of fuzzy points belonging to it, A can be regarded as a subset of P F (X), called a fuzzy set in the induced fuzzy metric space (P F (X), d F ) in the following definition. Definition 9. A fuzzy set A in (P F (X), d F ) is said to be closed if the limit of any Cauchy sequence in A belongs to it. A fuzzy set A in (P F (X), d F ) is said to be open if A 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 sets is reasonable. Proposition 4. A fuzzy set A in (P F (X), d F ) is closed if and only if every α-cut set of A, α [0, 1], is a closed set in (X, d) in the ordinary sense. Proof. It is omitted. 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 points is given.

380 Zun-Quan Xia and Fang-Fang Guo Lemma 1. Any subsequence of a Cauchy sequence of fuzzy points is also a Cauchy sequence and has the same limit as the original one. Proof. It is obvious from Definition 7. Theorem 2. Suppose (P F (X), d F ) is the induced fuzzy metric space of a metric space (X, d). Then (X, T F ) is a fuzzy topology space in the sense of Pu (1980) [15], called the fuzzy topology space induced by (P F (X), d F ), where T F is defined by T F = {A P F (X) A is a fuzzy closed set in (P F (X), d F )}. Proof. It suffices to prove that T F satisfies the three conditions in the definition of fuzzy topology due to Pu (1980) [15]. (1) It is obvious that X and are fuzzy closed sets. (2) For any {A, B} T F, we prove in the following that A B T F. For any Cauchy sequence of fuzzy points {(y n,γ n )} included in A B, A or B, say A, must contain a subsequence {(y m,γ m )} of {(y n,γ n )}. From Lemma 1, {(y m,γ m )} is also a Cauchy sequence and hence has a limit. Since A is a closed fuzzy set, the limit of {(y m,γ m )} which is also the limit of {(y n,γ n )} is included in A. In consequence, the limit of {(y n,γ n )} is included in A B, which implies that A B T F. (3) For any {A i } i I T F, where I is an arbitrary index set, it only need to be proved that i I A i T F. For any Cauchy sequence in i I A i, denoted by {(x n,λ n )}, we have that {(x n,λ n )} A i for any i I. Since every A i is a closed fuzzy set, the limit of {(x n,λ n )} is in A i for any i I. It follows that i I A i is a closed fuzzy set in the sense of Definition 9. Therefore one has i I A i T F. The proof is completed. From the theorem given above we know that every fuzzy metric space can induce a fuzzy topology space, which implies in another way that the fuzzy measure defined in this paper is not only reasonable but also significant. References 1. R. Biswas, Fuzzy inner product spaces and fuzzy norm functions, Inf. Sci. 53 (1991), 185-190. 2. L.Boxer, On Hausdirff-like metirc for fuzzy sets, Pattern Recognition Letters 18 (1997), 115-118. 3. P. Brass, On the noexistence of Hausdorff-like metrics for fuzzy sets, Pattern Recognition Letters 23 (2002), 39-43.

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