912Wangshi6.pdf INTERNATIONAL JOURNAL OF COMPUTERS COMMUNICATIONS & CONTROL Special Issue on Fuzzy Sets and Applications (Celebration of the 50th Anniversary of Fuzzy Sets) ISSN 1841-9836, 10(6):889-903, December, 2015. Homeomorphism Problems of Fuzzy Real Number Space and The Space of Bounded Functions with Same Monotonicity on [-1,1] H.-D. Wang, S.-C. Guo, S.M. Hosseini Bamakan, Y. Shi Hua-Dong Wang 1. Key Laboratory of Big Data Mining and Knowledge Management, Chinese Academy of Sciences, Beijing 100190, China 2. Research Center on Fictitious Economy & Data Science, Chinese Academy of Sciences, Beijing 100190, China 3. College of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, China huadw2012@163.com Si-Cong Guo College of Sciences, Liaoning Technical University, Fuxin 123000, China Seyed Mojtaba Hosseini Bamakan 1. Key Laboratory of Big Data Mining and Knowledge Management, Chinese Academy of Sciences, Beijing 100190, China 2. Research Center on Fictitious Economy & Data Science, Chinese Academy of Sciences, Beijing 100190, China 3. School of Management, University of Chinese Academy of Sciences, Beijing, 100190, China s_mojtabahossini@yahoo.com Yong Shi* 1. Key Laboratory of Big Data Mining and Knowledge Management, Chinese Academy of Sciences, Beijing 100190, China 2. Research Center on Fictitious Economy & Data Science, Chinese Academy of Sciences, Beijing 100190, China *Corresponding author: yshi@ucas.ac.cn Abstract: In this paper, based on the fuzzy structured element, we prove that there is a bijection function between the fuzzy number space ε1 and the space B[−1, 1], which defined as a set of standard monotonic bounded functions with monotonicity on interval [−1, 1]. Furthermore, a new approach based upon the monotonic bounded functions has been proposed to create fuzzy numbers and represent them by suing fuzzy structured element. In order to make two different metrics based space in B[−1, 1], Hausdorff metric and Lp metric, which both are classical functional metrics, are adopted and their topological properties are discussed. In addition, by the means of introducing fuzzy functional to space B[−1, 1], we present two new fuzzy number’s metrics. Finally, according to the proof of homeomorphism between fuzzy number space ε1 and the space B[−1, 1], it’s argued that not only does it give a new way to study the fuzzy analysis theory, but also makes the study of fuzzy number space easier. Keywords: fuzzy numbers; fuzzy structured element, standard monotonic bounded functions, fuzzy functional, homeomorphism Copyright © 2006-2015 by CCC Publications 890 H.-D. Wang, S.-C. Guo, S.M. Hosseini Bamakan, Y. Shi 1 Introduction Fuzzy numbers, which are a generalization of real numbers, have been perfectly applied to model and show the fuzzy data. Recently, application of fuzzy numbers in data mining algorithms has been an interesting topic to the researchers in this domain, for instance, clustering [1, 2], classification [3] and regression [4, 5]. Generally, the efforts have been done in study of fuzzy mathematical analysis and its application falls into two main categories: First, studies on constructing fuzzy number metrics based on the fuzzy numbers and their topological properties. Many researchers proposed different metrics and many discussions on them have been proposed. For example, Hausdorff metric [6], Lp metric [7] and sendograph metric [8], were proposed as some of the most well-known widely used metrics. The second category consists of those studies which addresses the relationship between the fuzzy number space and other topological spaces, study the properties of the fuzzy number space and develop some new methods in the proposed spaces. Among these studies, Goetschel and Voxman(2003) introduced a homeomorphic mapping from θ-crisp fuzzy number space to Hilbert space ℓ2, which ranges in a convex cone (see [9]). Later, Gerg [10] generalized this mapping by extending the θ-crisp fuzzy number space to a more general one. In order to apply the functional analysis to the fuzzy-valued functions studies, in which variables are real numbers and function values are fuzzy numbers, Puri and Ralescu [11], proposed an embedding theorem that the fuzzy number space ε1 can be embedded into a Banach space X, with the help of the Radstrom embedding theorem of compact convex set. This theorem establishes the theoretical link between the fuzzy number space and the Banach space. However, because of do not considering any specific structure of Banach space, it is not easy to implement (it is not applicable anymore). Thus, by adopting the mapping of Goetschel and Voxman, Wu and Ma [12, 13] embedded fuzzy number space into the concrete Banach space C[0, 1] × C[0, 1] (C[0, 1] = {f : f is a bounded left-continuous function on (0, 1], and f has right limit on (0, 1], especially f is right-continuous at 0}), and present a specific isometrically isomorphic operator. Although the proposed embedding operator is proved to be as same as the embedding operator given by Puri and Ralescu [11] in the sense of isometrical isomorphism, the embedding operator has a specific form. This paper is organized as follows: Section 2 and Section 3 introduced definitions and no- tations employed throughout the paper. In section 4, we introduced a specific fuzzy number, namely; fuzzy structured element. Then two important theorems which are the local mapping theorem and the structured element representation theorem of fuzzy number, are proved. As a result, we obtain a conclusion that there exist an one-one mapping from B[−1, 1] with the same order standard monotonic bounded function family on [0, 1] to the fuzzy real number space ε1. In section 5, we introduced Lp metric and Hausdorff metric into B[−1, 1] and some of its topo- logical properties, such as completeness and separability are discussed. In section 6, by means of a fuzzy functional induced by fuzzy structured element, two fuzzy number metrics induced in the given metrics of B[−1, 1]. This section discussed the homeomorphism Problems between B[−1, 1] and the space ε1. Finally, we conclude and provide future works in Section 7. 2 Notion of the fuzzy numbers Fuzzy numbers are the natural generalization of real and crisp numbers. A fuzzy number is a set of the real line with the upper semi-continuous and quasi-concave membership function. The definition implies that α-cut (Aα) of a fuzzy subset A is a closed interval in [A α l , A α r ] for any α ∈ (0, 1]. The support of a fuzzy number A is a crisp set so that suppA = cl({x : A(x) > 0}) = [A0l , A 0 r](the closure of the support of A). Thus, by supposing suppA to be a bounded closed Homeomorphism Problems of Fuzzy Real Number Space and The Space of Bounded Functions with Same Monotonicity on [-1,1] 891 interval, A defined as a bounded fuzzy number. Denote all bounded fuzzy numbers on real line R as Ñc(R)(or ε 1). Theorem 1. [12] If u ∈ Ñc(R), then let u(α) = inf{x : x ∈ uα}, u(α) = sup{x : x ∈ uα}, here u(α) and u(α) are two functions that satisfy the following conditions (1)–(4) on [0, 1]: (1) u(α) is a bounded left continuous nondecreasing function on (0, 1]; (2) u(α) is a bounded left continuous nonincreasing function on (0, 1]; (3) u(α) and u(α) are right continuous at α = 0; (4) u(α) ≤ u(α). Conversely, if functions u(α) and u(α) satisfy the conditions (1)–(4) on [0, 1], then there exists an unique u ∈ Ñc(R) such that uα = [u(α), u(α)] for each α ∈ [0, 1]. The theorem says that for any fuzzy number A, it can be uniquely determined by two mono- tonic functions u(α), u(α) on interval [0, 1]. 3 Notions of the Extended set-valued function and general in- verse function Let f be a monotonic and bounded function on [a, b] and x0 ∈ (a, b) be a discontinuous point in f. By considering f as a monotone increasing function, f can be a surjective function from [a, b] to (−∞, +∞) by the following formula: f(x0) = [f(x0−), f(x0+)], f(a) = (−∞, f(a+)], f(b) = [f(b−), +∞), Here, we denote a new function f̂, which f̂ is a monotonic set-valued function extended by f and it also called extensional set-valued function of f. Furthermore, we denote all the family of function f which are bounded and have the same monotonicity on [a, b], by D[a, b]. 3.1 Discontinue monotonic function with set-valued extensional at disconti- nuity For discontinue monotonic increasing function f, x0 is a discontinuous point in the range of [−1, 1]. Here, f(x0 − 0) = m1 and f(x0 + 0) = m2, by considering our default suppose that f is an increasing function, then m1 < m2 and f(x0) is an interval number between [m1, m2]. If functional values of all discontinuities redefined as closed interval with left-hand and right- hand limited values, then this new function is called monotonic bounded set-valued function extensional from f that we denote it by f̂. Obviously, inverse function f̂−1 of f̂ exist. 3.2 Continuous non-strictly monotonic function Suppose f is a non-strictly increasing function, then there exists at least one pair points {x1, x2} on [−1.1] such that value of f is equal to constant c = f(x1) = f(x2) on interval [x1, x2]. And suppose x1, x2 are two endpoints so that increasing function f is equal to constant, that is, when x < x1, f(x) < c and when x > x2, f(x) > c. Here, we define inverse function f −1(x) which is continue close to 0 at discontinuity, i.e. when x2 ≤ 0, f̂ −1(c) = limx→c+0 f −1(x) = y+; 892 H.-D. Wang, S.-C. Guo, S.M. Hosseini Bamakan, Y. Shi when x1 ≥ 0, f̂ −1(c) = limx→c−0 f −1(x) = y−; when 0 ∈ [x1, x2], define f −1(c) as set of two points {y−, y+}, denote by f̂ −1(c) = {y−, y+}. It is quite straightforward to verify that, if f is an increasing and bounded function on [−1, 1] and f̂ is the extensional set-valued function of f, then the inverse function of f̂ can be equivalently defined as: f̂−1(x) =      sup{t : f̂(t) = x,−1 6 t < 0}, −∞ < x 6 f(0−) 0, f(0−) 6 x 6 f(0+) inf{t : f̂(t) = x, 0 < t 6 1}, f(0+) 6 x < +∞ . (1) Example 2. To make the above concept more understandable, let consider f as a monotonic bounded function on [0, 2], f(x) = { x, 0 6 x 6 1 1 + x, 1 < x 6 2 . Then its extensional set-valued function f̂(x) and its inverse function f̂−1(x) are defined as the following: f̂(x) =      x, 0 6 x < 1 [1, 2], x = 1 1 + x, 1 < x 6 2 , f̂−1(x) =      x, 0 6 x < 1 1, 1 6 x < 2 x − 1, 2 6 x 6 3 . The f(x) and f̂(x) can be illustrated by Figure 1 y x21 0 1 2 y x21 0 1 2 ( )f x � ( )f x Figure 1: Set-valued function f̂ extended by f 4 Fuzzy structured element and transformation In order to establish the relationship between the fuzzy real number space ε1 and the mono- tone function space on interval [−1, 1], we introduce a method, namely the Fuzzy Structured Element, which was proposed by Sicong Guo in [14]. Definition 3. Let E be a fuzzy set on real line R and E(x) is membership function of E. Then, E is called a fuzzy structured element, if E(x) satisfies the following properties: 1) E(0) = 1; 2) E(x) is monotonic increasing and right-continuous on [−1, 0), monotonic decreasing and left- continuous on (0, 1]; Homeomorphism Problems of Fuzzy Real Number Space and The Space of Bounded Functions with Same Monotonicity on [-1,1] 893 3) For any x ∈ (−∞,−1) ∪ (1, +∞), E(x) = 0. Further, E is called a normal fuzzy structured element if the fuzzy structured element E satisfies: (1) E(x) > 0 for all x ∈ (−1, 1); (2) E(x) is continuous, strictly monotonic increasing on [−1, 0) and also continuous, strictly monotonic decreasing on (0, 1]. According to Definition 3, it is easy to know that the fuzzy structured element is a special fuzzy number on real line R, which can be used to express the concept of fuzzy zero 0̃. Let E be a fuzzy number. E is called a triangular structured element if it has membership function µE(x), where µE(x) =      1 − x, x ∈ [0, 1] 1 + x, x ∈ [−1, 0] 0, otherwise (2) As it is shown in Figure 2. Obviously, E is a special fuzzy structured element. 1 11! o x ( )E x Figure 2: Triangular structured element E Based on the fuzzy structured element, we can give the following two theorems: Theorem 4 (Local Mapping Theorem). Suppose E is a fuzzy structured element on R with membership function E(x). f(x) is monotonically bounded on [−1, 1] and f̂(x) is extensional set- valued function of f(x). Then f̂(E) is a bounded closed fuzzy number and membership function of f̂(E) is E(f̂−1(x)), where f̂−1(x) is the inverse function of f̂(x) (If f(x) is strictly increasing and continuous on [−1, 1], then f̂−1(x) is a ordinary inverse function of f(x)). Proof: Let A = f̂(E), A(y) defined as a membership function of f̂(E). Suppose that f(x) is increasing and bounded on [−1, 1]. By extension principle, we have A(y) = ∨ y∈f̂(x) E(x), f̂(E) = ⋃ x∈R E(x) ∗ f̂(x), where E(x) ∗ f̂(x)(y) = { E(x), y ∈ f̂(x) 0, otherwise . From the former equation, the membership function of f̂(E) is f̂(E)(y). When y ∈ f̂(x), counterpart membership degree defined as E(x). Denote α–cut of E by Eα = [e − α , e + α ]. It follows from the concept of fuzzy structured element that E0 = [e − 0 , e + 0 ] ⊆ [−1, 1]. Since f(x) is increasing bounded on interval [−1, 1] and f̂(x) is surjection on R, it follows that for α ∈ (0, 1], [f̂(E)]α = f̂(Eα) = f̂[e − α , e + α ] = [f(e − α ), f(e + α )], 894 H.-D. Wang, S.-C. Guo, S.M. Hosseini Bamakan, Y. Shi For α = 0, [f̂(E)]0 = suppf̂(E) = ∪α∈(0,1][f̂(E)]α = ∪α∈(0,1][f̂(Eα)] = [f(e − 0 +), f(e + 0 −)] (3) Thus Eα, α ∈ [0, 1] are bounded closed sets. For all α1, α2 ∈ (0, 1], if α1 ≤ α2, then Eα1 ⊆ Eα2, that is e − α2 ≤ e−α1, e + α1 ≤ e+α2 . Since f(x) is monotone increasing, we have f(e−α1−) ≤ f(e − α2 −), f(e+α2+) ≤ f(e + α1 +) Therefore, [f̂(E)]α1 ⊆ [f̂(E)]α2 . Furthermore, it follows from Eq.(3) that [f̂(E)]0 = ∪α∈(0,1][f̂(E)]α ⊇ [f̂(E)]α1, It means that f̂(E) is a convex set on real line R. Since E is a fuzzy number and 1-cut set E1 of E is nonempty and [f̂(E)]1 = f̂(E1) also is nonempty, hence we can say that f̂(E) is a normal fuzzy number. From definition of bounded closed fuzzy number, we know f̂(E) ∈ Ñc(R). Since f̂(E)(y) = ∨ y∈f̂(x) E(x) = E(xy) as y ∈ f̂(x), where xy = f̂ −1(y). It follows that f̂(E)(y) = E(f̂−1(y)), or f̂(E)(x) = E(f̂−1(x)). If f(x) is monotonic decreasing on [−1, 1], the proof can be shown in a similar manner. ✷ Theorem 5 (Theorem of Structured Element Expression of Fuzzy Number ). For a given regular fuzzy structured element E and any bounded fuzzy number A, there exists a monotonic bounded function f on [−1, 1] such that A = f(E) (strictly, exists a extended set-valued function f̂ such that A = f̂(E)). We called it fuzzy number A generated by the fuzzy structured element. Proof: From fuzzy number expression theorem, fuzzy number u can be expressed by a family set {uα : uα = [u(α), u(α)], α ∈ [0, 1]}. Therefore, we just need to prove that there exists a monotone bounded function f(x) on [−1, 1] such that f(E) = u, that is, for all α ∈ [0, 1], [f(E)]α = [u(α), u(α)]. Let f(x) = { u(E(x)), x ∈ [−1, 0] u(E(x)), x ∈ (0, 1] . It follows from Theorem 1 that f(x) is a monotone increasing bounded function on [−1, 1]. From the local mapping principle, f(E) is a bounded closed fuzzy number. It follows from the extension principle that [f(E)]α = f(Eα). Denote E(x) on [−1, 0] as lE(x) and E(x) on [0, 1] as rE(x). Since E is strictly increasing on [−1, 0] and is also a bijection from [−1, 0] to [0, 1] , E is strictly decreasing on [0, 1] and is also a bijection from [0, 1] to [0, 1], so lE(x), rE(x) inverse and are denoted by l−1E (α), r −1 E (α), then Eα = [E(α), E(α)] = [l −1 E (α), r −1 E (α)]. Since u(α), u(α) are left-continuous on (−1, 0] and are right-continuous at α = 0, also E(x) is continuous, we know that u(E(x)) = u(lE(x)) is left-continuous on (−1, 0] and is right-continuous at x = −1, u(E(x)) = u(rE(x)) is right-continuous on [0, 1) and is left-continuous at x = 1. Since Homeomorphism Problems of Fuzzy Real Number Space and The Space of Bounded Functions with Same Monotonicity on [-1,1] 895 f(x) is increasing, it follows that for all α ∈ (0, 1], f(Eα) = f[l −1 E (α), r −1 E (α)] = [f(l −1 E (α)−), f(r −1 E (α)+)] = [u(E(l−1E (α)+)), u(E(r −1 E (α)−))] = [u(E(l−1E (α))), u(E(r −1 E (α)))] = [u(α), u(α)] for α = 0, f(E0) = f[l −1 E (0), r −1 E (0)] = [f(−1+), f(1−)] = [u(E(−1+)), u(E(1−))] = [u(E(−1)), u(E(1))] = [u(0), u(0)] Therefore, we conclude that [f(E)]α = [u(α), u(α)] for all α ∈ [0, 1]. That’s to say that f(E) = u. We complete the proof of this theorem. ✷ When no confusion can arise in the following discussions, we will use f(x) to denote the extended function f̂(x) and use f(E) to instead of f̂(E), respectively. Theorem 6. Let f be a monotonic bounded function and E be a fuzzy structured element on R and fuzzy number u = f(E). For all α ∈ [0, 1], Eα = [e − α , e + α ]. Then (1) If f(x) is increasing on [−1, 1], then α-cut of fuzzy number u is closed interval uα = { [f(e−α−), f(e + α +)], α ∈ (0, 1], [f(e−α +), f(e + α−)], α = 0, . (4) (2) If f(x) is monotonic decreasing function on [−1, 1], then α-cut of u is closed interval uα = { [f(e+α−), f(e − α +)], α ∈ (0, 1], [f(e+α +), f(e − α−)], α = 0, (5) Proof: Based on Theorem 4 and function f satisfies the monotone condition of Local Mapping Theorem, it follows that for any α ∈ (0, 1], we have [f(E)]α = f(Eα). Since f is monotone on closed interval Eα = [e − α , e + α ] ⊆ [−1, 1], it follows that: If f is increasing, for α ∈ (0, 1], we have uα = f(Eα) = f[e − α , e + α ] = [inf{y : y ∈ f[e−α , e + α ]}, sup{y : y ∈ f[e − α , e + α ]}] = [f(e−α−), f(e + α +)], and for α = 0, it holds that u0 = suppf(E) = f(E)0̇ = f(E0̇) = lim α→0 [f(e−α−), f(e + α +)] = [f(e − 0 +), f(e + 0 −)]. 896 H.-D. Wang, S.-C. Guo, S.M. Hosseini Bamakan, Y. Shi If f is decreasing, we have uα = f(Eα) = f[e − α , e + α ] = [inf{y : y ∈ f[e−α , e + α ]}, sup{y : y ∈ f[e − α , e + α ]}] = [f(e+α +), f(e − α−)], ✷ From the Local Mapping Theorem 4, we know that given a fuzzy structured element E, it will be transformed into an fuzzy number A = f(E) with any a monotonic function f on [−1, 1]. When f is not a monotonic function, the fuzzy set f(A) can not be guaranteed to be a fuzzy number. Theorem 5 show us that for any bounded fuzzy number A, we always can find a monotonic bounded function f on [−1, 1] such that f(E) = A. Therefore, the two theorems reveal to us that there exists a deep relationship between the family of bounded monotonic function on [−1, 1] and the fuzzy number space. 5 The same order standard monotonic bounded function classes B[−1, 1] Let f be monotonic bounded function on [−1, 1]. If for any discontinuity x in [−1, 1], we have f(x) = 1 2 [f(x+) + f(x−)], (6) where f(x+)(f(x−)) is the right-limit(left-limit) of f(x) at the point x, then f(x) is called a standard monotonic bounded function on [−1, 1]. All same order standard monotonic bounded function on [−1, 1] is denoted by B[−1, 1]. It is obvious that a continuous monotonic bounded function on D[−1, 1] is also a standard monotonic bounded function. Definition 7. Suppose that f ∈ D[−1, 1], we define f̌(x) =        f(−1+), x = −1 [f(x−) + f(x+)]/2, x ∈ (−1, 1) f(1−), x = 1 (7) where f̌(x) is as a standardized form of f(x). Obviously, f̌ ∈ B[−1, 1]. If f is a standard monotonic bounded function, then f̌ = f. In the following we introduced two distance formulas: dp(f, g) = [ ∫ 1 −1 |f(x) − g(x)|pdx ]1/p , for all f, g ∈ B[−1, 1], (8) dH(f, g) = sup x∈[−1,1] |f(x) − g(x)|, for all f, g ∈ B[−1, 1]. (9) where 1 ≤ p < +∞. Theorem 8. Let E be a normal fuzzy structured element, K is a bounded closed interval on R. Denote Bf(K) = {f : f ∈ B[−1, 1] and [f(−1), f(1)] ⊆ K}, (10) Metric spaces (B[−1, 1], dH) and (Bf(K), dp) both are complete. Homeomorphism Problems of Fuzzy Real Number Space and The Space of Bounded Functions with Same Monotonicity on [-1,1] 897 Proof: First, we prove completeness of space (B[−1, 1], dp). Suppose each elements in (B[−1, 1], dp) are increasing, for decreasing situation has similarity conclusions. Suppose given sequence {xn(t)}, where xn(t) ∈ (B[−1, 1], dp), n = 1, 2, · · · . Let dp(xn, xm) → 0, (as n, m → ∞), that is, sequence {xn(t)} satisfies Cauchy uniformly convergence conditions. Suppose x0(t) is limit of sequence xn(t),i.e. lim n→∞ xn(t) = x0(t) for all t ∈ [−1, 1]. Since all xn(t1) is monotonic increasing function, then xn(t1) ≤ xn(t2) for all t1, t2 ∈ [−1, 1]. So lim n→∞ xn(t1) ≤ lim n→∞ xn(t2), and x0(t1) ≤ x0(t2). Thus x0(t) is monotonic increasing function. Now we prove g(t) is standard function on [−1, 1]. Suppose t0 is a discontinuous point of g(t). We might as well suppose that g(t) is’t standard, that is g(t0) 6= 1 2 [g(t0+) + g(t0−)]. Letδ = ∣ ∣g(t0) − 1 2 [g(t0+) + g(t0−)] ∣ ∣ . Furthermore, since fn(t), n = 1, 2, · · · , are standard, we have E(t0+) |fn(t0+) − g(t0+)| < ε, E(t0−) |fn(t0−) − g(t0−)| < ε, E(t0) |fn(t0) − g(t0)| < ε. Take ε = ε0 < δ/2. When n ≥ n0(ε0), we have |g(t0) − 1 2 [g(t0+) + g(t0−)]| ≤ |fn(t0) − g(t0)| + 1 2 |fn(t0+) − g(t0+)| + 1 2 |fn(t0−) − g(t0−)| ≤ ε0 + ε0 = 2ε0 < δ. a contradiction. Thus, g(t) is standard on [−1, 1]. 2) Now we prove the metric space (Bf(K), dp) is complete. Suppose that fn is a Cauchy sequence in Bf(K), dp, then for any ε > 0, there exists a positive integer N such that for any m, n > N, we have dH(fm, fn) = [ ∫ 1 −1 |fm(t) − fn(t)| p dx ]1/p < ε. This indicates that {fn} is a Cauchy sequence of Lp[−1, 1]. We know that Lp[−1, 1] is complete space, so {fn} is converse in Lp[−1, 1]. Suppose h is a limit of sequence {fn}. Similar to the proof in 1) that h(x) is increasing and bounded in interval [−1, 1]. Therefore, h(−1+) and h(1−) exist. Let f(x) =      h(−1+), x = −1 [h(x+) + h(x−)] /2, x ∈ (−1, 1) h(1−), x = 1 . It is obvious that f(x) ∈ B[−1, 1] and also is a limit of Cauchy sequence fn. f(x) and h(x) have different values which happened only on discontinuity, so we have dp(fn, f) = dp(fn, g). Thus, f is a limit of Cauchy sequence in (Bf(K), dp), that is, {fn} converges in (Bf(K), dp). Hereby, the completeness of (Bf(K), dp) have been proved. ✷ 898 H.-D. Wang, S.-C. Guo, S.M. Hosseini Bamakan, Y. Shi Theorem 9. Let E be a normal fuzzy structured element and K ∈ I(R) be a nonempty set, denoted by: Bf(K) = {f : f ∈ B[−1, 1] and [f(−1), f(1)] ⊆ K}, then the metric space (Bf(K), dp) is complete. Example 10. The metric space (B[−1, 1], dp) is not complete. For example, fn ∈ B[−1, 1] is defined by fn+1(x) = { fn(x), x ∈ [−1, 1 − 1/n 2] n, x ∈ (1 − 1/n2, 1] , (n ≥ 1), where f1(x) = 0, x ∈ [−1, 1]. It is obvious that fn(x)(n ≥ 1) are bounded functions. Suppose m ≤ n, we have dp(fm, fn) = [ ∫ 1 −1 |fm(t) − fn(t)| p dt ]1/p < [ 1 (n + 1)2 + 1 (n + 2)2 + · · · + 1 m2 ]1/p < [ 1 n − 1 m ]1/p < 1 n → 0(m, n → ∞) Thus, {fn} is a Cauchy Sequence, their standard function sequence {f̌n(x)} is a Cauchy Sequence in B[−1, 1]. It is easy to understand that {f̌n(x)} do not convergence to any upper bounded function. Theorem 11. Metric space (B[−1, 1], dH) is not a separable space and metric space (B[−1, 1], dp) is a separable space. Proof: 1) It is sufficient to construct an uncountable set Bf of B[−1, 1] with the property that if f, g ∈ Bf , then dH(f, g) = 1. For each t ∈ (−1, 1), define ft ∈ B[−1, 1] by ft(x) =      0 x ∈ [−1, t) 0.5 x = t 1 x ∈ (t, 1] . Denote Bf = {ft : t ∈ (−1, 1)}. Consequently, if t1 6= t2, then dH(ft1, ft2) = 1. 2) Since Lp[−1, 1] is a separable space with respect to the metric dp and B[−1, 1] ⊂ Lp[−1, 1], then B[−1, 1] is separable with respect to metric dp (For any separable metric space X, any nonempty subset of this space is also separable). ✷ In general, f ∈ B[−1, 1],−f ∈ B[−1, 1] unless f is a constant-valued function. Because, if f isn’t constant-valued function, despite −f is also monotonic function, but it is not same order with f. Hence, B[−1, 1] cannot form group with respect to operation of addition,just can form a semigroup. It should be noted that each element in B[−1, 1] is not a closed form with respect to ordinary subtraction operator. We can take an example, function obtain by two monotonic function subtracted may be non-monotonic. Therefore, B[−1, 1] can’t form linear space with respect to addition and number multiply operation. Theorem 12. B[−1, 1] is a convex cone with 0 as its vertex. The theorem is obvious, so the proof is omitted. Homeomorphism Problems of Fuzzy Real Number Space and The Space of Bounded Functions with Same Monotonicity on [-1,1] 899 6 Topological relationship between B[−1, 1] and Ñc(R) 6.1 Two types of fuzzy number metric spaces induced by the fuzzy structured element Let E be a symmetrical regular fuzzy structured element on real line R and Ñc(R) be the set of all bounded closed fuzzy numbers. For given function f ∈ B[−1, 1], there exists corresponding unique fuzzy number such that Af = f(E). In other words, fuzzy structured element determines a mapping from B[−1, 1] to Ñc(R). Denote HE : B[−1, 1] → Ñc(R), f → HE(f) = f(E) ∈ Ñc(R). Then HE is called fuzzy functional induced by fuzzy structured element E. Using metrics dp and dH on B[−1, 1], mapping HE induces distances dNp(A, B) = dp(H −1 E (A), H −1 E (B)), (11) dNH(A, B) = dH(H −1 E (A), H −1 E (B)), (12) on Ñc(R), where H −1 E (A), H −1 E (B) are preimage of mapping HE at A and B, respectively. Sup- pose A = f(E), B = f(E), where f, g ∈ B[−1, 1], then Eq.(11) and Eq.(12) can also rewrite as dp(f, g) = dp(HE(f), HE(g)), (13) dH(f, g) = dH(HE(f), HE(g)), (14) (Ñc(R), dNp) and (Ñc(R), dNH) are said to be distance space induced by (B[−1, 1], dp) and (B[−1, 1], dH), respectively. It is easy to understand that HE is an isometric bijection of B[−1, 1] onto Ñc(R). Using isometric bijection HE, we can translate metric of elements in fuzzy number space to metric between the same order standard monotonic bounded functions in range of [−1, 1]. Then, what is the relationship between those metrics and the other metrics on fuzzy numbers? Before discussing the relationship, a Lemma need to be presented here: For u ∈ Ñc(R), E is a normal fuzzy structured element. If g ∈ B[−1, 1] such that u = g(E), as defined in the following: gu(x) = { u(E(x)) −1 ≤ x ≤ 0 u(E(x)) 0 < x ≤ 1 . Lemma 13. Suppose that E is a normal fuzzy structured element, fuzzy number u ∈ Ñc(R), uα = [u(α), u(α)](α ∈ [0, 1]). If u = f(E), f ∈ B[−1, 1], then f(x) = ǧu(x), x ∈ [−1, 1], where ǧu is the standard function of gu. Proof: The proof has been provided in the following two steps: 900 H.-D. Wang, S.-C. Guo, S.M. Hosseini Bamakan, Y. Shi 1) First, we prove ǧu = u. According to the decomposition theorem, we only need to prove that [ǧu(E)]α = uα for any α ∈ (0, 1]. Denote Eα = [e − α , e + α ]. From the extension principle and g(x) is a increasing function, it follows that [ǧu(E)]α = ǧu(Eα) = ǧu([e − α , e + α ]) = [ǧu(e − α−), ǧu(e + α +)]. Since u(α), u(α) are continuous on (0, 1], we have u(E(x)) is left-continuous on (−1, 0] and u(E(x)) is right-continuous on [0, 1). Therefore, [ǧu(e − α−), ǧu(e + α +)] = [gu(e − α−), gu(e + α +)] = [ u ( E(e−α ) ) , u ( E(e+α ) )] = [ u ( α ) , u ( α )] That is, [ǧu(E)]α = uα. Thus, ǧu = u. 2) Here we prove f is unique in B[−1, 1]. Suppose f1, f2 ∈ B[−1, 1] such that f̂1(E) = f̂2(E) = u, then f̂ −1 1 = f̂ −1 2 . Furthermore, we have f̂1 = f̂2, then f1 = f2. Thus, f is unique in B[−1, 1]. Therefore, f(x) = ǧu(x), x ∈ [−1, 1]. The proof is complete. ✷ The following theorem shows the relation between the induced fuzzy number metrics dNH, dNp and the previous metrics of fuzzy numbers. Theorem 14. Let E be regular structured element, u, v ∈ Ñc(R), there are f, g ∈ B[−1, 1] such that u = f(E), v = g(E). Denote uα = [u(α), u(α)], vα = [v(α), v(α)], then dNp(u, v) = [ ∫ 1 −1 |f(x) − g(x)|pdx ]1/p = [ ∫ 1 0 |u(α) − v(α)|p dE(α) + |u(α) − v(α)|p dE(α) ]1/p (15) dNH(u, v) = sup x∈[−1,1] |f(x) − g(x)| = sup x∈[−1,1] (|u(α) − v(α)|∨ |u(α) − v(α)|) (16) Proof: 1) We have from Lemma 13 that dNp(u, v) = dp(f, g) = dp(f̌u, ǧv), And ǧu(x) = gu(x) and ǧv(x) = gv(x) are bounded almost everywhere on [−1, 1] respectively, it follows that dp(f̌u, ǧv) = dp(fu, gv). Therefore, dNp(u, v) = dp(fu, gv) = [ ∫ 1 −1 |fu(x) − gv(x)| p dx ]1/p = [ ∫ 0 −1 |u(E(x)) − v(E(x))| p dx + ∫ 1 0 |u(E(x)) − v(E(x))| p dx ]1/p Denote E(x) = lE(x) for x ∈ [−1, 0] and rE(x) = E(x) for x ∈ [0, 1]. Since E is a regular fuzzy structured element, we know that lE is bijective from [−1, 0] to [0, 1], rE is bijective from [0, 1] to [0, 1]. Thus, we can say that l−1E , r −1 E exist and they are monotone bijections. It obvious that E = l−1E and E = r −1 E both are differentiable almost everywhere. Therefore, we have dNp(u, v) = [ ∫ 1 0 |u(α) − v(α)| p dl−1E (α) + ∫ 0 1 |ū(α) − v̄(α)| p dr−1E (α) ]1/p = [ ∫ 1 0 |u(α) − v(α)| p dE(α) − ∫ 1 0 |ū(α) − v̄(α)| p dĒ(α) ]1/p Homeomorphism Problems of Fuzzy Real Number Space and The Space of Bounded Functions with Same Monotonicity on [-1,1] 901 2) From Lemma 13, we obtain dNH(u, v) = dH(f, g) = dHf̌u, ǧv), Furthermore, fu(x), gu(x) both are left-continuous on (−1, 0] and right-continuous on [0, 1), and are right-continuous at x = −1 and left-continuous at x = 1, we have |fu(x) − gv(x)| ≤ |fu(x−) − gv(x−)|∨ |fu(x+) − gv(x+)|, x ∈ (−1, 1) |fu(x) − gv(x)| = |fu(x) − gv(x)|, x ∈ {−1, 1} Since f̌u(x), ǧu(x) are standard functions of fu(x), gv(x) for x ∈ (−1, 1), we have |f̌u(x−) − ǧv(x−)| = |fu(x−) − gv(x−)|; |f̌u(x+) − ǧv(x+)| = |fu(x+) − gv(x+)|. It follows that dH(f̌u, ǧv) = dH(fu, gv). Moreover, we have dH(fu, gv) = sup x∈[−1,1] {|fu(x) − gv(x)|} = sup x∈[−1,0] {|u(E(x)) − v(E(x))|}∨ sup x∈[0,1] {|ū(E(x)) − v̄(E(x))|} Let α = E(x) on [−1, 0] and [0, 1], respectively. Since E is a regular fuzzy structured element, it follows that sup x∈[−1,0] {|u(E(x)) − v(E(x))|} = sup α∈[0,1] {|u(α) − v(α)|} , sup x∈[0,1] {|ū(E(x)) − v̄(E(x))|} = sup α∈[0,1] {|ū(α) − v̄(α)|} , completing the proof of the theorem. ✷ Since integral variables E(α), Ē(α) in the Eq.(6.1) are two general functions, the definite integral is a Riemann-Stieltjes integral. When the E is a triangular structured element (2), we have dE(α) = d(α − 1) = dα, dĒ(α) = d(−α + 1) = −dα. Then the Eq.(2) becomes the following form: dNp(u, v) = [ ∫ 1 0 |u(α) − v(α)|p dα −|u(α) − v(α)|p dα ]1/p . From Eq.(16) in the Theorem 14, we also note that the induced fuzzy number metric dNH(u, v) is the same as the Hausdorff metric(Diamond,1989) [6]. 6.2 Homeomorphism between the fuzzy number space Ñc(R) and B[−1, 1] Proposition 15. Suppose that (X, dX), (Y, dY ) are two metric spaces. F is an isometric bijection from (X, dX) to (Y, dY ). Then F is continuous and inverse mapping F −1 of F exists and is also continuous. 902 H.-D. Wang, S.-C. Guo, S.M. Hosseini Bamakan, Y. Shi Proof: Since F is a bijection of (X, dX) into (Y, dY ), there exists inverse mapping F −1 which is also one-to-one mapping. By definition of continuous mapping, for all x0 ∈ X and any positive number ε, there always exists a positive number δ such that dY (F(x), F(x0)) < ε as dX(x, x0) < δ. Since dX(x, x0) = dY (F(x), F(x0)), given ε, it is sufficient by taking δ ≤ ε (For instance, take δ = ε/2).Hence, F is continuous. Similarly, we can also prove inverse mapping F−1 which is also continuous. ✷ Given a bounded closed interval K, let the uniformly bounded fuzzy number set as Ñc(K) = {u : u ∈ Ñc(R) and supp u ⊆ K}, Since there exists a bijection HE of B[−1, 1] into Ñc(R) and HE and inverse function H −1 E are continuous, thus we have conclusions as follows: Theorem 16. Metric spaces (B[−1, 1], dH) and (Ñc(R), dNH) are homeomorphic. Metric spaces (Bf(K), dp) and (Ñc(K)), dNp) are homeomorphic. Since space (Ñc(R), dNH) and (B[−1, 1], dH) are homeomorphic, that is, both metric spaces are topologically equivalent. So elements in both of them have consistent properties on met- rics. There are one-to-one relationship between fuzzy number sequence {un} of (Ñc(R), dNH) and function sequence {fn}, fuzzy number sequence on (Ñc(R), dNH) and function sequence on (B[−1, 1], dH) have completely same properties. Similarly, fuzzy number sequence on (Ñc(K), dNp) and function sequence on (Bf(K), dH) have completely same properties. Therefore, the proper- ties of convergence sequence of general metric spaces are also founded to the convergence fuzzy number sequence. Thus, they are trivial to the following corollaries. Corollary 17. Fuzzy number metric space (Ñc(R), dNH) is complete and (Ñc(R), dNp) is not complete. Corollary 18. For any nonempty closed interval K on R, (Ñc(K), dNp) is a complete metric space. Corollary 19. For any fuzzy number sequence {un} of fuzzy number space (Ñc(R), dN) only has a limit almost everywhere. That is, the limit of convergence sequence is unique.(Where dN represents dNH or dNp). Corollary 20. Suppose that {un} is a fuzzy number sequence of (Ñc(R), dNH), {fn} is a function sequence on (B[−1, 1], dH). For all n ≥ 1, un = fn(E). Then fuzzy number sequence {un} converge if and only if function sequence {fn} converge. Let lim n→∞ un = u0, lim n→∞ fn = f0, then u0 = f0(E). If using dNp , dp, Bf(K) (defined as 10) instead of dNH, dH and B[−1, 1] in Corollary 20, respectively. The Corollary is still founded. 7 Conclusion By using monotonic mapping of the fuzzy structured element, we have proved that the bounded fuzzy number space is homeomorphic to the space B[−1, 1] of monotonic bounded function with same monotonicity on [−1, 1]. Therefore, the problem of the fuzzy number space can be transformed to one’s of space B[−1, 1], such as the convergence of sequence of fuzzy numbers, the convergence of the fuzzy series , continuous of fuzzy-valued function and so on. To some extent, our study provides a new way for the study of fuzzy analysis. 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