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A countable set derived by fuzzy set Huan Huang 5 Department of Mathematics, Jimei University, Xiamen 361021, China. 1 0 2 t c Abstract O 9 In this paper, it shows that for each fuzzy set u on Rm, the set D(u) is at most 1 countable. Based on this, it modifies the proof of assertion (I) in step 2 of the sufficiency part of Theorem 4.1 in paper:Characterizations of compact sets in fuzzy ] M sets spaces with L metric, http://arxiv.org/abs/1509.00447. p G . h Key words: Fuzzy sets; Level cut sets; Countable; t a m [ 1 v 1 Introduction 3 5 9 6 In this paper, we show that for each fuzzy set u on Rm, the set D(u) = {α ∈ 0 . (0,1) : [u] * {u > α} } is at most countable. 0 α 1 5 1 : 2 Preliminaries v i X r a We introduced some basic concepts about fuzzy sets. For details, we refer to [1,2]. We use F(Rm) to represent all fuzzy subsets on Rm, i.e. functions from Rm to [0,1]. For u ∈ F(Rm), let [u] denote the α-cut of u, i.e. α {x ∈ Rm : u(x) ≥ α}, α ∈ (0,1], [u] = α suppu = {x ∈ Rm : u(x) > 0}, α = 0.  Let {u > α} denotethe strong α-cut of u, i.e., {u > α} = {x ∈ Rm : u(x) > α}. Email address: [email protected] (H. Huang). Preprint submitted to Elsevier 3 Main results Let u ∈ F(Rm), t ∈ Rm and r be a positive number in R. Define a function S (·,·) : Sm−1 ×[0,1] → {−∞}∪R by u,t,r −∞, if [u] ∩B(t,r) = ∅, α S (e,α) = u,t,r  sup{he,x−ti : x ∈ [u]α ∩B(t,r)}, if [u]α ∩B(t,r) 6= ∅,  where B(t,r) denote the closed ball {x ∈ Rm : d(t,x) ≤ r}. We say α ∈ (0,1) is a discontinuous point of S (e,·) if u,t,r (i) S (e,α) ∈ R, and u,t,r (ii)S (e,β) = −∞forallβ > αor−∞ < lim S (e,β) < lim S (e,β). u,t,r β→α+ u,t,r β→α− u,t,r Denote the set of all discontinuous points of S (e,·) by D . Then D u,t,r u,t,r,e u,t,r,e is at most countable because S (e,·) is a monotone function on [0,1]. u,t,r Theorem 3.1 Let u be a fuzzy set on Rm, t be a point in Rm, and r be a positive real number. Then D := D is at most countable. u,t,r e∈Sm−1 u,t,r,e S Proof Let ̟ be a countable dense subset of Sm−1. Then D = D (1) u,t,r u,t,r,e e∈̟ [ In fact, suppose that α ∈ D , then there exists e ∈ Sm−1 such that α ∈ u,t,r D . Hence S(u,t,r)(e,α) > −∞. This is equivalent to [u] ∩B(t,r) 6= ∅. u,t,r,e α Therefore S(u,t,r)(f,α) > −∞ for all f ∈ Sm−1. (2) To show α ∈ D , we divide the proof into two cases. e∈̟ u,t,r,e S Case 1. S(u,t,r)(e,β) = −∞ for all β > α. In this case, [u] ∩ B(t,r) = ∅ for any β > α, and so S(u,t,r)(f,β) = −∞ β when f ∈ Sm−1 and β > α. Combined with (2), we know α ∈ D for each u,t,r,f f ∈ Sm−1. Thus α ∈ D . e∈̟ u,t,r,e S Case 2. −∞ < lim S (e,β) < lim S (e,β). β→α+ u,t,r β→α− u,t,r In this case, there is an α > α such that [u] ∩B(t,r) 6= ∅ when λ ∈ [0,α ]. 0 λ 0 Set ξ := lim S (e,β)− lim S (e,β) > 0. (3) u,t,r u,t,r β→α− β→α+ Notice that, for all β ∈ [0,1] with [u] ∩B(t,r) 6= ∅, β |S (e,β)−S (f,β)| u,t,r u,t,r 2 = |sup{he,x−ti : x ∈ [u] ∩B(t,r)}−sup{hf,x−ti : x ∈ [u] ∩B(t,r)}| β β ≤ sup{| he−f,x−ti |: x ∈ [u] ∩B(t,r)} β ≤ ke−fk·r, hence, for any λ ∈ [0,α ], 0 | S (e,λ)−S (f,λ) |≤ ke−fk·r, u,t,r u,t,r and so, combined with (3), we know there exists δ > 0 such that, for all f ∈ Sm−1 ∩B(e,δ), lim S (f,β)− lim S (f,β) > ξ/2, u,t,r u,t,r β→α− β→α+ this means that α ∈ D when f ∈ Sm−1∩B(e,δ). Thus there exists g ∈ ̟ u,t,r,f such that α ∈ D , i.e., α ∈ D . u,t,r,g e∈̟ u,t,r,e S Now we obtain (1). Since ̟ is countable and D is at most countable, we u,t,r,e know D is at most countable. ✷ u,t,r Remark 3.1 In the proof of Theorem 3.1, in order to show D is at most u,t,r countable, it proves that D = D . This kind of trick was used in u,t,r e∈̟ u,t,r,e the proof of Lemma 4 in [3] to show a set is at most countable. S Theorem 3.2 Let u be a fuzzy set on Rm, then D(u) = {α ∈ (0,1) : [u] * α {u > α} } is at most countable. Proof Ifα ∈ D(u),thenthereisay ∈ Rm suchthaty ∈ [u] buty ∈/ {u > α}. α Thus, d(y,{u > α}) > ε > 0. (4) Choose a q ∈ Qm = {(z1,z2,...,zm) ∈ Rm : zi ∈ Q, i = 1,2,...,m} which satisfies that ky − qk > 0. We assure that α ∈ D for some r ∈ Q with u,q,r r ≥ ky −qk. In fact, let e = (y −q)/ky−qk. Then S (e,α) ≥ he,y −qi = ky −qk. (5) u,q,r If [u] ∩B(q,r) = ∅ for any β > α, then S (e,β) = −∞ for all β > α, and β u,q,r thus α ∈ D . u,q,r,e If there exists β > α such that [u] ∩ B(q,r) 6= ∅. Pick an arbitrary x ∈ β {u > α}∩B(q,r), then, by (4), kx−qk ≤ r, kx−yk > ε. 3 If x = q, then he,x−qi = 0. Suppose that x 6= q. Notice that hy −q,x−qi kx−qk2 +ky −qk2 −kx−yk2 = cosα = , ky −qk·kx−qk 2ky −qk·kx−qk where α is the angle between two vectors x−q and y −q. Thus 1 kx−qk2 kx−yk2 he,x−qi = +ky −qk− 2 ky −qk ky −qk ! 1 r2 ε2 ≤ +ky −qk− , 2 ky −qk ky −qk! and so there exists a δ > 0 such that for all r ∈ [ky −qk,ky−qk+δ), 1 ε2 he,x−qi ≤ ky −qk− . (6) 4ky −qk Combined with (5) and (6), it then follows from the arbitrariness of x ∈ {u > α}∩B(q,r) that lim S (e,β) < S (e,α) u,q,r u,q,r β→α+ when r ∈ [ky−qk,ky−qk+δ). This implies that there exists r ∈ Q such that α ∈ D ⊂ D . u,q,r,e u,q,r Now we know D(u) ⊆ D . u,q,r q∈Q[m,r∈Q It then follows from Theorem 3.1 that D(u) is at most countable. ✷ 4 An application In this section, we give an application of the main result. The following is assertion (I) in step 2 of the sufficiency part of the proof of Theorem 4.1 in paper: Characterizations of compact sets in fuzzy sets spaces with L metric, see http://arxiv.org/abs/1509.00447. p (I) P(v) is at most countable. We can show this assertion in the following way. In fact, given α ∈ P(v), it holds that {v > α} & [v] , and hence α ∈ D(v) := α {γ ∈ (0,1) : [v] * {v > γ} }. Thus P(v) ⊆ D(v). By Theorem 3.2, D(v) is γ at most countable. So P(v) is at most countable. 4 References [1] C. Wu, M. Ma, The Basic of Fuzzy Analysis (in Chinese), National Defence Industry press, Beijing, 1991. [2] P. Diamond, P. Kloeden, Metric Space of Fuzzy Sets–Theory and Application, World Scientific, Singapore, 1994. [3] W. Trutschnig, Characterization of the sendograph-convergence of fuzzy sets by means of their L - and lelewise convergence, Fuzzy Sets and Systems 161 p (2010) 1064-1077. 5