Completeness, metrizability and compactness in spaces of fuzzy-number-valued functions

COMPLETENESS, METRIZABILITY AND COMPACTNESS IN SPACES OF FUZZY-NUMBER-VALUED FUNCTIONS J.J. FONT, D. SANCHIS, AND M. SANCHIS Abstract. Fuzzy-number-valued functions, that is, functions defined on a topological space taking values in the space of fuzzy numbers, play a central role in the development of Fuzzy Analysis. In this paper we study completeness, metrizability and compactness of spaces of continuous fuzzy-number-valued functions. 1. Introduction Fuzzy Analysis is based on the notion of fuzzy number in the same way as Classical Analysis is based on the concept of real number. From 1986, the so-called representation theorem of real fuzzy numbers (see [19]) eased considerably the development of the theory concerning fuzzy-number-valued functions, that is, functions defined on a topological space taking values in E1, the space of fuzzy numbers. Such functions, as real-valued functions do in the classical setting, play a central role in Fuzzy Analysis. Namely, fuzzy-number-valued functions have become the main tool in several fuzzy contexts, such as fuzzy differential equations ([6]), fuzzy integrals ([37],[41]), fixed point theory ([25, 30, 40]) and fuzzy optimization ([20],[38], [39]). In this paper we address three topological aspects of the spaces of continuous fuzzy-number-valued functions when endowed with the most usual topologies. Namely we will study completeness, metrizability and compactness in this context. Only the latter concept, which is clearly related to Ascoli theorem, seems to have received certain attention in fuzzy literature (see, e.g., [33], [14]). The prototype of such result in Classical Analysis was proved by Ascoli in [5] and, independently, by Arzel`a, who acknowledged Ascoli’s priority in [4]. Nowadays, Arzel`a-Ascoli type theorems encompass the study of the (relative) compacity of a family of functions endowed with several topologies and their literature is extense. The applications of these results are numerous in different settings; namely, in the context of differential equations, in finding extremal curves, in most criteria for the consistency of systems involving inequalities, etc. In [14, Theorem 4.2], Fang and Xue set up a fuzzy version of Ascoli theorem which characterized compact subsets of the space C(K, (E1, d∞)) of all fuzzy-valued continuous functions on a compact metric space Kendowed with the topology of the uniform convergence. Unfortunately this version is not correct since, as pointed out in [15], it is based on a wrong characterization ([14, Theorem 2.4]) of the compact subsets of (E1, d∞). In the last section of this paper we fix [14, Theorem 4.2] by Key words and phrases. Continuous fuzzy number-valued functions; Metrizability; Completeness; bounded sets; Ascoli theorem. This research is supported by Spanish Government (grant MTM2016-77143-P). 1 2 J.J. FONT, D. SANCHIS, AND M. SANCHIS extending the fuzzy Ascoli theorem to a broader framework. The key concepts in our approach are bounded subsets of a topological space and αf-spaces. Thus, previously, in Section 3 we obtain a fuzzy characterization of bounded subsets and αf-spaces, and point out how such spaces appear in a natural way when considering the completeness of the spaces C(X, (E1, d∞)), Xa topological space, equipped with the topology ταof uniform convergence on the (bounded) members of a cover αof X. In Section 4, we establish an explicit criterion for Cτα(X, (E1, d∞)) to be metrizable. Finally, as mentioned above, in Section 5 we address the fuzzy Ascoli theorem (and also the weak fuzzy Ascoli theorem) for spaces C(X, (E1, d∞)). As a consequence of our results we show that C(X, (E1, d∞)) endowed with the topology of the uniform convergence satisfies the fuzzy Ascoli theorem if and only if Xis pseudocompact. 2. Preliminaries and notation Given a fuzzy subset uon the real numbers R, the λ-level set of uis defined by [u]λ={x∈R:u(x)≥λ}for λ∈(0,1] and [u]0= clR{x∈R:u(x)>0}for λ= 0. Now, the fuzzy number space E1is the set of such usatisfying the following properties: (1) uis normal, i.e., there exists an x0∈Rwith u(x0) = 1. (2) uis convex, i.e., u(λx + (1 −λ)y)≥min {u(x), u(y)}for all x, y ∈R, λ ∈[0,1]. (3) u(x) is upper-semicontinuous. (4) [u]0is a compact set in R. Notice that, if u∈E1, then the λ-level set [u]λof uis a compact interval for each λ∈[0,1]. We denote [u]λby [u−(λ), u+(λ)]. Every real number rcan be consider a fuzzy number: indeed, if rcan be identified with the fuzzy number ˜rdefined by ˜r=(1 if t=r, 0 if t6=r. From now on, we do not distinguish between rand ˜r. The following two results are useful in the theory of fuzzy numbers. THEOREM 2.1. [19] Let u∈E1and [u]λ= [u−(λ), u+(λ)],λ∈[0,1]. Then the pair of functions u−(λ)and u+(λ)has the following properties: (i) u−(λ)is a bounded left continuous nondecreasing function on (0,1]. (ii) u+(λ)is a bounded left continuous nonincreasing function on (0,1]. (iii) u−(λ)and u+(λ)are right continuous at λ= 0. (iv) u−(1) ≤u+(1). Conversely, if the pair of functions α(λ)and β(λ)satisfies the above conditions (i)-(iv), then there exists a unique u∈E1such that [u]λ= [α(λ), β(λ)] for each λ∈[0,1]. THEOREM 2.2. [19, 12] For u, v ∈E1, define d∞(u, v) = sup λ∈[0,1] max |u−(λ)−v−(λ)|,|u+(λ)−v+(λ)|. COMPLETENESS, METRIZABILITY AND COMPACTNESS IN Cτα(X, E1) 3 Then d∞is a metric on E1called the supremum metric on E1, and (E1, d∞)is a complete metric space. Notice that, by the definition of d∞, the reals Rendowed with the euclidean topology can be topologically identified with the closed subspace ˜ R={˜x:x∈R} of (E1, d∞) where ˜x+(λ) = ˜x−(λ) = xfor all λ∈[0,1]. As a metric space, we will always consider E1equipped with the metric d∞. Throughout this paper, Xwill stand for a Tychonoff space, that is, a completely regular Hausdorff space. A subset Bof a space Xis said to be bounded (in X) if every real-valued continuous function on Xis bounded on Bor, equivalently, every locally finite sequence {Un:n∈B}of pairwise disjoint open sets meeting Bis finite. Spaces which are bounded in themselves are called pseudocompact. Given a space X, the family of all bounded subsets of Xis denoted by β. If αis a cover of a space X, we say that a function ffrom a space Xinto a space Yis αf-continuous if the restriction of fto each member of αcan be extended to a continuous function on the whole X. A space Xis called an αf-space if every real-valued αf-continuous function on Xis continuous. For α⊆β, locally pseudocompact spaces and kr-spaces (spaces Xwhere a realvalued function is continuous whenever its restriction to each compact subset of Xis continuous) are examples of αf-spaces. Thus, locally compact spaces, first countable spaces (in particular, metrizable spaces) are αf-spaces too. The theory of z-closed projections [27], the distribution of the functor of the Dieudonn´e completion [9, 31], compactness of functions spaces in the topology of the pointwise convergence [2], and locally pseudocompact groups [34] are some of the frameworks where αf-spaces arise in a natural way. We encourage the reader unfamiliar with the techniques of the theory of bounded subsets to consult [35]. Let F(X, E1) denote the set of all functions from a set Xinto E1. For a cover, say α, of X, we denote by ταthe topology of uniform convergence on members of α. It is a well-known fact that ((F(X, E1), τα) is a Tychonoff space. Indeed, the family of all subsets of F(X, E1)×F(X, E1) of the form U(A, ε) = (f, g)∈F(X, E1)×(F(X, E1) : sup a∈A d∞(f(a), g(a)) < ε , for all A∈αand all ε > 0, is a subbase for a (Hausdorff) uniformity Uαon F(X, E1) which induces the topology τα. In the sequel we will use the well-known fact that every uniformity with a countable base is metrizable (see, e.g., [13, Theorem 8.1.21]). Our terminology and notation are standard. For instance, Nstands for the set of natural numbers and f|Ameans the restriction of a function fto a subset A. For a subset F(X, E1) of F(X, E1), the space (F(X, E1), τα) is denoted by Fτα(X, E1). The symbol C(X, E1) (respectively, C(X, R)) stands for the set of all continuous functions from Xinto (E1, d∞), that is, the set of all fuzzy-number-valued continuous functions on X(respectively, the set of all real-valued continuous functions on X). Notice that we obtain τp, the topology of the pointwise convergence, by taking αthe cover of Xconsisting of its points (equivalently, of all its finite subsets). The paper 4 J.J. FONT, D. SANCHIS, AND M. SANCHIS [23] covers a wide variety of topics about C(X, E1) endowed with the topology τp. If α={X}, then we obtain the topology, τu, of uniform convergence on X. The cover kof all compact subsets of a topological space Xinduces the so-called compactopen topology on C(X, E1) denoted by τco. It is worth noting that the pointwise convergence topology on F(X, E1) coincides with the product topology on (E1)X. This equivalent to consider τpon C(X, E1) when Xis equipped with the discrete topology. For notions which are not explicitly defined here, the reader might consult [13]. 3. Completeness of Cτα(X, E1) In the first part of this section we shall show that the notions of bounded subset and of αf-space (α⊆β) can be characterized by means of fuzzy-number-valued continuous functions. In the second part we shall show that αf-spaces occur naturally when we study completeness of Cτα(X, E1). A similar claim can be made about the metrizability of Cτα(X, E1) as we will see in the next section. A subset Aof a metric space (X, d) is precompact if for every ε > 0, there is a finite subset {x1, x2, . . . , xn}such that A⊆Sn i=1 Bε(xi) where, as usual, Bε(xi) stands for the ball of center xiand radius ε. Recall that a metric space Xis compact if and only if Xis precompact and complete. Since boundedness is equivalent to precompactness for subsets of R, we can replace bounded by precompact in the definition of bounded subset. In this direction, we have THEOREM 3.1. For a subset Bof a space X, the following are equivalent: (1) Bis bounded in X. (2) For each continuous function f:X→E1,f(B)is precompact. Proof. (2)=⇒(1) follows from the fact that precompact subsets of Rare precompact in E1. In order to prove (1)=⇒(2), notice that, being E1a complete metric space of non-measurable cardinality, it is realcompact. Thus, by [18, Theorem 11.8], E1 is homeomorphic to a closed subset Tof a product, RS, of real lines. Now, if f:X→E1is a continuous function, we can consider the bounded subset (πs◦ f)(B) for all s∈S. Then clE1f(B) = clTf(B) is a closed subset of the compact set Qs∈SclR(πs◦f)) (B) and, consequently, clE1f(B) is compact. Thus, f(B) is precompact.  THEOREM 3.2. If α⊆β, then the following are equivalent: (1) Xis an αf-space. (2) Every αf-continuous function from Xinto E1is continuous. Proof. Since E1is a Tychonoff space, (1)=⇒(2) is a consequence of Lemma 8 in [7]. Thus, we only need to prove (2)=⇒(1). To this end, it suffices to notice that if f:X→Ris an αf-continuous function, then fcan be regarded as an αf-continuous function from Xinto E1because Ris a (closed) subspace of E1. Let us now move on to the study of completeness of Cτα(X, E1). We say that a space Xis topologically complete if Xis homeomorphic to a closed subspace of a product of metrizable spaces. It is known that, for every space X, there exists a unique, up to homeomorphisms that leave Xpointwise fixed, topologically complete COMPLETENESS, METRIZABILITY AND COMPACTNESS IN Cτα(X, E1) 5 space γX, in which Xis dense and every continuous function ffrom Xinto a topologically complete space Mcan be extended to a continuous function fγon γX. Such γX is called the Dieudonn´e topological completion of X. For these and related results, the reader might consult [18]. Bounded subsets can be characterized as subsets whose closure in γX is compact (see [18, Problem 8E.] or [3, Proposition 5.1]). The following lemma is straightforward. We write τ1≥τ2, if the topology τ1 is finer than the topology τ2. LEMMA 3.3. Let Xbe a space. If τ≥τp, then Cτ(X, R)is topologically embedded as a closed subspace of Cτ(X, E1). It is a well-known fact that, for an arbitrary space X,Cτu(X, (M, d)) is complete whenever the metric space (M, d) is (see, for example, [13, Exercise 8.3.C.(a)]). Thus, we have THEOREM 3.4. (Compare with [14, Theorem 3.5]) For an arbitrary space X, Cτu(X, E1)is a complete metric space. Proof. The space (E1, d∞) being complete [19], completeness follows from [13, Exercise 8.3.C.(a)]. On the other hand, notice that the entourage of the diagonal (f, g)∈C(X, E1)×(C(X, E1) : sup x∈X d∞(f(x), g(x)) <1/n  for all n∈N, forms a countable subbase for the uniformity of Cτu(X, E1). Thus, Cτu(X, E1) is metrizable by [13, Theorem 8.1.21].  Now we take up a result which plays a pivotal role in the characterization of the completeness of Cτα(X, E1). Recall that there exists an isometric embedding jof (E1, d∞) into a Banach space (E, k.k) which preserves convex combinations, that is, jsatisfies the following two properties: (1) d∞(u, v) = kj(u)−j(v)kfor all u, v ∈E1, and (2) j(λu + (1 −λ)v) = λj(u) + (1 −λ)j(v) for all u, v ∈E1and all λ∈[0,1] (see [11] for the details). Moreover, j(E1) is a closed cone of (E, k.k) with vertex 0. Recall also that a subset Aof a space Xis called C-embedded in Xif every real-valued continuous function on Aextends continuously to the whole X. THEOREM 3.5. For a subset Aof a space X, the following assertions are equivalent: (1) Ais C-embedded in X. (2) Every continuous fuzzy-number-valued function fon Awith separable range has a continuous extension Fto the whole X; furthermore F(X)is included in the closed convex hull of f(A). Proof. (2)=⇒(1) can be easily verified since Ris a closed convex subset of E1. In order to see (1)=⇒(2), let jbe the above isometric embedding of (E1, d∞) into a Banach space (E, k.k) . From now on, we identify E1with j(E1). Thus, we can consider fas a function from Ato the closed (so complete) convex hull conv(f(A)) of f(A) in (E, k.k). Being E1a (complete) cone, we have that conv(f(A)) ⊂ E1. Since Ais C-embedded in X, by Theorem 2.4 of [17] and Theorem 4.7 of [36], every continuous pseudometric don Awhich induces a separable topology extends continuously to a continuous pseudometric on X. Hence we can apply 6 J.J. FONT, D. SANCHIS, AND M. SANCHIS Theorem 2.3 of [1] to conclude that fhas a continuous extension Fto Xwith F(X)⊂conv(f(A)).  REMARK 3.6. An argument similar to the one used in the previous theorem enables us to characterize C?-embedded subsets A(that is, every bounded real-valued continuous function on Aextends to a continuous function on X). Indeed, we can apply Theorem 2.9 of [1] to obtain: For a subset Aof a space X, the following assertions are equivalent: (1) Ais C?-embedded in X, and (2) every continuous fuzzy-numbervalued function fon Awith precompact range has a continuous extension Fto the whole X; furthermore F(X) is included in the closed convex hull of f(A). The relationship between αf-spaces and the completeness of Cτα(X, E1) is given by the next result. Let us first recall that, given a space X, a family α⊆βis a bornology if it satisfies the following two conditions: (i) αis a cover of X; and (ii) if Aand Bbelong to α, then there exists C∈αsuch that both Aand Bare included in C. Indeed, any cover αof Xgeneretes a bornology bαby taking finite unions of elements of αand both uniformities coincide. THEOREM 3.7. Assume α⊆βis a bornology in a space X. Then Cτα(X, E1)is complete if and only if Xis an αf-space. Proof. Assume that Xis an αf-space and consider a Cauchy net {fi}i∈Iin Cτα(X, E1). By [19], (E1, d∞) is a complete metric space and consequently each element fihas a continuous extension, say fγ i, to γX. By Theorem 3.4, for each subset Bin α, the net {fγ i|clγX B}i∈Iconverges uniformly to a function fB∈C(clγX B, E1). Let g be the function from Xinto (E1, d∞) defined by the rule g|B=fBwhenever Bis a bounded subset of Xin α. Since αis a bornology, a standard argument shows that gis well defined. Moreover, since each compact subset of a topological space is C-embedded, by Theorem 3.5 the restriction of gto each subset B∈αcan be continuously extended to γX. Thus, gis αf-continuous. Since Xis an αf-space, g is actually continuous. One can easily verify that gis the limit of the net {fi}i∈Iin Cτα(X, E1). Thus, the function space Cτα(X, E1) is complete. Suppose now that Cτα(X, E1) is complete. Let fbe a αf-continuous fuzzy-numbervalued function on Xand consider the set {B⊂X:B∈α}directed by inclusion. Define now a net {fB}B∈αwhere fBis the continuous extension of f|Bto X. It is clear that the net {fB}B∈αconverges to fin Fτα(X, E1). The completeness of Cτα(X, E1) implies that fis a continuous function. Thus, Xis an αf-space.  COROLLARY 3.8. (1) Cτβ(X, E1)is complete if and only if Xis a bf-space. (2) Cτco (X, E1)is complete if and only if Xis a kr-space. (3) Cτp(X, E1)is complete if and only if Xis discrete. 4. Metrizability of Cτα(X, E1) We now address the question of metrizability of the space Cτα(X, E1). The following is an explicit criterion for Cτα(X, E1) to be metrizable. Let α⊆βbe a COMPLETENESS, METRIZABILITY AND COMPACTNESS IN Cτα(X, E1) 7 cover of a space X. We say that Xis hemi-α-bounded if there is a countable family A={An:n∈N} ⊂ αsuch that X=Sn∈NAn, and any A∈αis a subset of some finite union An1∪An2∪ · · · Ankof elements of A. Hemi-β-bounded (respectively, hemi-k-bounded) spaces are usually called hemibounded (respectively, hemicompact) spaces. THEOREM 4.1. Let αbe a cover of a space X. If every element of αis a closed set, then the following conditions are equivalent: (1) Xis a hemi-α-bounded space. (2) Cτα(X, E1)is metrizable. (3) Cτα(X, E1)is first countable. (4) Cτα(X, E1)has a countable base at the zero function. Proof. (1)=⇒(2) Let {An:n∈N} ⊂ αbe a countably family which witnesses hemiα-boundedness of X. It is an easy matter to check that B={U(An,1/k) : n, k ∈N} is a countable subbase for the uniformity Uα. The result now follows from [13, Theorem 8.1.21]. (2)=⇒(3) and (3)=⇒(4) are trivial. To see (4)=⇒(1), set V(A, ε) := f∈C(X, E1) : sup x∈A d∞(0, f(x)) < ε with A∈αand ε > 0, and consider a countable family B={(V(An, εn) : n∈N} such that finite intersections of elements of Bform a base at 0. Fix now A∈α. Given a neighborhood V(A, 1/2) of 0, there exist {V(An1, εn1), V (An2, εn2), . . . , V (Ank, εnk)} such that V(An1, εn1)∩V(An2, εn2)∩. . . ∩V(Ank)⊂V(A, 1/2). We claim that A⊆An1∪An2∪ · · · ∪ Ank. Indeed, suppose, contrary to what we claim, that there is x∈A\(An1∪An2∪ · · · ∪ Ank). The set An1∪An2∪Ank being closed, there is f∈C(X, E1) such that f(x) = 1 and f|An1∪An2∪Ank= 0 (see Proposition 3.5 of [23]). Then f∈V(An1, εn1)∩V(An2, εn2)∩. . . ∩V(Ank) but f /∈V(A, 1/2). This contradiction yields the claim. In addition, for any x∈X, there exists A∈αcontaining x. Hence the previous argument proves that there are An1, An2,· · · , Ankwith A⊂An1∪An2∪· · ·∪Ank. Thus, the family {An:n∈N}is a cover of X. We have just showed that {An:n∈N} witnesses the hemi-α-boundedness of X. REMARK 4.2. (1) The condition that the elements of the cover αare closed is not as restrictive as it may seem. Indeed, if α⊆βis a cover of X, then we can consider the cover bα⊆βdefined as bα={clXA:A∈α}. Then the function spaces Cτα(X, E1) and Cτbα(X, E1) are homeomorphic. An even more strong conclusion is possible: actually, they are uniformly isomorphic. The proof can be left to the reader as an easy exercise. 8 J.J. FONT, D. SANCHIS, AND M. SANCHIS (2) For the function space Cτα(X, R), the equivalence of (2) and (4) is a consequence of the well-known Birkhoff-Kakutani theorem which states that a topological group Gis metrizable if and only if it is Hausdorff and first countable. However, it is worth noting that Cτα(X, E1) fails to be a topological group so that Birkhoff-Kakutani theorem does not apply in this context. COROLLARY 4.3. (1) Cτβ(X, E1)is metrizable if and only if Xis hemibounded. (2) Cτco (X, E1)is metrizable if and only if Xis hemicompact. (3) Cτp(X, E1)is metrizable if and only if Xis countable. 5. αf-spaces and the Ascoli theorem Given a metric or a uniform space, Y, an Ascoli type theorem characterizes compactness in a function space C(X, Y ) in terms of equicontinuity plus natural conditions. For example, the basic Ascoli theorem deals with compactness in the function space C([0,1]) of all real-valued continuous functions on the unit interval endowed with the uniform topology. It states that a subset Fof C([0,1]) is compact if, and only if, Fis closed, bounded, and equicontinuous. Since the classical HeineBorel theorem states that a subset of Rnis compact if, and only if, it is closed and bounded, the basic Ascoli theorem can be viewed as fixing the problems of Heine–Borel theorem in C([0,1]). Recall that, in contrast to Heine–Borel theorem, in infinite-dimensional normed vector spaces, closed and bounded sets need not be compact and closed balls are never compact. In the realm of function spaces of fuzzy-valued functions, we study Ascoli theorem in the spirit commented in the previous paragraph, that is, by means of equicontinuity plus extra conditions. The key idea is to analyze the relationship between αf-spaces and compactness in Cτα(X, E1). As a fairly direct consequence of our results, we characterize when Cτu(X, E1) satisfies the fuzzy Ascoli theorem. Our approach follows the link between the exponential map and the Ascoli theorems as developed in [28]. First, some concepts are in order. Given a space Xand a metric space (Y, d), a family {fi}i∈I⊂F(X, Y ) is called equicontinuous if for each x∈Xand each ε > 0, there is a neighborhood Vof xsuch that d(fi(y), fi(x)) < ε for all y∈V and all i∈I. In the case when Xis also a metric space, say (X, d0), the family {fi}i∈I⊂F(X, Y ) is said to be uniformly equicontinuous if for every ε > 0, there is δ > 0 such that d(fi(y), fi(x)) < ε for all i∈Iwhenever d0(x, y)< δ. It is a well-known fact that, for a compact metric space (X, d0), equicontinuity is equivalent to uniform equicontinuity. A subset U⊂E1such that the families {u+(−) : u∈U}and {u−(−) : u∈U} are uniformly equicontinuous on ]0,1] is named ]0,1]-uniformly equicontinuous. The motivation for the results of this section is the following fuzzy version of the classical Ascoli theorem stated by Fang and Xue. THEOREM 5.1. ([14, Theorem 4.2]) If Kis a compact metric space, then a closed subset Fof Cτu(K, E1)is compact if and only if the following conditions are satisfied: (i) For each k∈K, the set {f(k) : f∈F}is d∞-bounded, that is, it is contained in a ball of center 0in the metric space (E1, d∞). COMPLETENESS, METRIZABILITY AND COMPACTNESS IN Cτα(X, E1) 9 (ii) Fis equicontinuous on K. (iii) For each k∈K, the set {f(k) : f∈F}is ]0,1]-uniformly equicontinuous, i.e., Fis pointwise ]0,1]-uniformly equicontinuous. Unfortunately the above theorem in based on a wrong characterization ([14, Theorem 2.4]) of the compact subsets of E1, as pointed out in [15]. To provide a right characterization we need to introduce several concepts. Namely, given a function f: [0,1] →R, let f(λ0+) denote the limit of fwhen λapproaches λ0from above (right). DEFINITION 5.2. Let {fi}i∈Ibe a family of functions defined from the unit interval [0,1] into the reals. Given λ0∈[0,1[ such that fi(λ0+) exists for all i∈I, the family {fi}i∈Iis said to be almost-right-equicontinuous at λ0if, for every ε > 0, there is δ > 0 such that |fi(λ)−fi(λ0+)|< ε for all i∈Iwhenever λ∈]λ0, λ0+δ[. Such a family of real-valued functions is said to be left-equicontinuous at a point λ0∈]0,1] if, for all ε > 0 and for all i∈I, there is δ > 0 such that |fi(λ)−fi(λ0)|< ε whenever λ∈]λ0−δ, λ0]. The family {fi}i∈Iis called left-equicontinuous (resp. almost-right-equicontinuous) if it is left-equicontinuous (resp. almost-right-equicontinuous) at every point of ]0,1] (respectively, at every point of [0,1[). We say that a subset U⊂E1is both-sided equicontinuous if both {u+(−) : u∈U} and {u−(−) : u∈U}are almost-right-equicontinuous and left-equicontinuous. DEFINITION 5.3. A subset Fof F(X, E1) is said to be pointwise both-sided equicontinuous if, for all x∈X,{f(x) : f∈F}is both-sided equicontinuous. DEFINITION 5.4. A subset Fof F(X, E1) is said to be pointwise d∞-bounded if, for all x∈X,{f(x) : f∈F}is d∞-bounded in E1. If τis a topology on C(X, E1), then (X, E1, τ) is said to satisfy the weak fuzzy Ascoli theorem if each τ-closed, pointwise d∞-bounded, equicontinuous, pointwise both-sided equicontinuous subset of C(X, E1) is τ-compact. If also each τ-compact subset is τ-closed, pointwise d∞-bounded, equicontinuous, and pointwise both-sided equicontinuous, then (X, E1, τ) is said to satisfy the fuzzy Ascoli theorem. The aim of this section is to establish fuzzy versions of the (weak) Ascoli theorem for topologies of the uniform convergence on the members of a cover of a space X, but first we need several previous results. PROPOSITION 5.5. If U⊂E1is both-sided equicontinuous, then so is clE1U. Proof. Fix λ0∈]0,1]. By hypothesis, given ε > 0, there is δ > 0 such that, for all u∈U,|u+(λ)−u+(λ0)|< ε/3 whenever λ∈]λ0−δ, λ0]. Now, if v∈clE1U, we can choose u∈Uwith sup λ∈]0,1] v+(λ)−u+(λ)< ε. 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Zhang, The convergence for a sequence of fuzzy integrals of fuzzy number-valued functions on the fuzzy set, Fuzzy Sets and Systems 59 (1993) no. 1, 43–57. Current address: Institut Universitari de Matem`atiques i Aplicacions de Castell´o (IMAC), Universitat Jaume I, Campus del Riu Sec. s/n, 12071 Castell´o (Spain) E-mail address:[email protected], [email protected], [email protected]