A New Characterization of Compact Sets in Fuzzy Number Spaces

Abstract

We give a new characterization of compact subsets of the fuzzy number space equipped with the level convergence topology. Based on this, it is shown that compactness is equivalent to sequential compactness on the fuzzy number space endowed with the level convergence topology. Our results imply that some previous compactness criteria are wrong. A counterexample also is given to validate this judgment.

1. Introduction

The convergences on fuzzy number spaces and their applications have been extensively discussed by various authors [1–13]. One of the most important problems is the characterizations of compact subsets.

Fang and Huang [6] presented a characterization of compact subsets of fuzzy number space endowed with level convergence topology. In this paper, we further give a new characterization of compact subsets of the fuzzy number space equipped with level convergence topology. Based on this, we show that compactness is equivalent to sequential compactness on this type of fuzzy number space.

Diamond and Kloeden [2] presented a characterization of compact sets in fuzzy number spaces equipped with the supremum metric. Fang and Xue [14] also gave a characterization of compact subsets of one-dimensional fuzzy number spaces equipped with the supremum metric. We point out that the compactness criteria given by Fang and Xue are just a special 𝑚=1 case of the compactness criteria given by Diamond and Kloeden. It is found that there exist contradictions between the characterizations of compact sets given by us and the characterizations given in [2, 14]. Then it is shown that the characterizations in [2, 14] are incorrect by a counterexample.

2. Fuzzy Number Space

Let ℕ be the set of all natural numbers, let ℝ^𝑚 be 𝑚-dimensional Euclidean space, and let 𝐹(ℝ^𝑚) represent all fuzzy subsets on ℝ^𝑚, that is, functions from ℝ^𝑚 to [0,1]. For details, we refer the readers to [2, 12].

For 𝑢∈𝐹(ℝ^𝑚), let [𝑢]_𝛼 denote the 𝛼-cut of 𝑢: that is [𝑢]_𝛼=𝑥∈ℝ^𝑚∶𝑢(𝑥)≥𝛼,𝛼∈(0,1],supp𝑢={𝑥∈ℝ^𝑚∶𝑢(𝑥)>0},𝛼=0. (1) We call 𝑢∈𝐹(ℝ^𝑚) a fuzzy number if 𝑢 has the following properties: (1)𝑢is normal: there exists at least one 𝑥₀∈ℝ^𝑚 with 𝑢(𝑥₀)=1; (2)𝑢 is convex: 𝑢(𝜆𝑥+(1−𝜆)𝑦)≥min{𝑢(𝑥),𝑢(𝑦)} for 𝑥,𝑦∈ℝ^𝑚 and 𝜆∈[0,1];(3)𝑢 is upper semicontinuous; (4)[𝑢]₀ is a bounded set in ℝ^𝑚.

The set of all fuzzy numbers is denoted by 𝐸^𝑚.

Suppose that 𝐾(ℝ^𝑚) is the set of all nonempty compact sets of ℝ^𝑚 and that 𝐾_𝑐(ℝ^𝑚) is the set of all nonempty compact convex set of ℝ^𝑚. The following representation theorem is used widely in the theory of fuzzy numbers.

Proposition 1 (see [15]). Given 𝑢∈𝐸^𝑚, then (1)[𝑢]₁≠∅ and[𝑢]_𝜆∈𝐾_𝑐(ℝ^𝑚) for all 𝜆∈[0,1];(2)[𝑢]_𝜆=⋂_𝛾<𝜆[𝑢]_𝛾 for all 𝜆∈(0,1];(3)[𝑢]₀=⋃_𝛾>0[𝑢]_𝛾.
Moreover, if the family of sets {𝑣_𝛼∶𝛼∈[0,1]} satisfies conditions (1) through (3), then there exists a unique 𝑢∈𝐸^𝑚 such that [𝑢]_𝜆=𝑣_𝜆 for each 𝜆∈[0,1].

Many metrics and topologies on 𝐸^𝑚 are based on the well-known Hausdorff metric. The Hausdorff metric 𝐻 on 𝐾(ℝ^𝑚) is defined by 𝐻(𝑈,𝑉)=max𝐻^∗(𝑈,𝑉),𝐻^∗(𝑉,𝑈) (2) for arbitrary 𝑈,𝑉∈𝐾(ℝ^𝑚), where 𝐻^∗(𝑈,𝑉)=sup_𝑢∈𝑈𝑑(u,𝑉)=sup_𝑢∈𝑈inf_𝑣∈𝑉𝑑(𝑢,𝑣). (3) Obviously, if [𝑥₁,𝑥₂] and [𝑦₁,𝑦₂] are bounded closed intervals of ℝ, then 𝐻𝑥₁,𝑥₂,𝑦₁,𝑦₂=max𝑥₁−𝑦₁,𝑥₂−𝑦₂. (4)

Throughout this paper, we suppose that the metric on ℝ^𝑚 is the Euclidean metric, and the metric on 𝐾(ℝ^𝑚) is the Hausdorff metric 𝐻. The Hausdorff metric has the following properties.

Proposition 2 (see [16, 17]). (𝑋,𝑑) is a metric space, and 𝐾(X) is the set of all compact set of 𝑋. Then (1)(𝑋,𝑑) complete ⇔(𝐾(𝑋),𝐻) complete;(2)(𝑋,𝑑) separable⇔(𝐾(𝑋),𝐻) separable;(3)(𝑋,𝑑) compact⇔(𝐾(𝑋),𝐻) compact.

In this paper, we consider two types of convergences on fuzzy number spaces. (i)Let 𝑢,𝑢_𝑛∈𝐸,𝑛=1,2,…. If lim_𝑛→∞𝑑_∞(𝑢_𝑛,𝑢)=0; then we say {𝑢_𝑛} supremum converges to 𝑢, denoted by 𝑢−−→_𝑛^𝑑∞𝑢, where the supremum metric 𝑑_∞ is defined by 𝑑_∞(𝑢,𝑣)=sup_[]𝛼∈0,1𝐻[𝑢]_𝛼,[𝑣]_𝛼 (5) for all 𝑢,𝑣∈𝐸^𝑚. (ii)Let 𝑢∈𝐸^𝑚 and let {𝑢_𝜉∶𝜉∈𝐷} be a net in 𝐸^𝑚, where 𝐷 is a direct set. If lim_𝜉∈D𝐻([𝑢_𝜉]_𝛼,[𝑢]_𝛼)=0 for each 𝛼∈[0,1], then we say {𝑢_𝜉} level converges to 𝑢, denoted by lim_𝜉∈𝐷𝑢_𝜉=𝑢(𝑙) or 𝑢→−_𝜉^𝑙𝑢.

Obviously, the supremum metric convergence is stronger than the level convergence on 𝐸^𝑚; that is, if {𝑢_𝑛} supremum metric converges to 𝑢, then it also level converges to 𝑢.

The symbol 𝜏(𝑙) is used to denote the topology induced by level convergence on 𝐸^𝑚; that is, 𝜏(𝑙) is the topology with ℬ_𝑢={𝑈_𝑢(𝛼,𝜀)∶𝜀>0,𝛼∈[0,1]} being a local subbase of 𝑢∈𝐸^𝑚, where 𝑈_𝑢(𝛼,𝜀)={𝑣∈𝐸^𝑚∶𝐻([𝑢]_𝛼,[𝑣]_𝛼)<𝜀} (see also [6]).

We use (𝐸^𝑚,𝑑_∞) or (𝐸^𝑚,𝜏(𝑙)) to denote the fuzzy number space 𝐸^𝑚 equipped with the supremum metric 𝑑_∞ or equipped with the level convergence topology 𝜏(𝑙), respectively.

3. Characterizations of Compact Sets and Sequentially Compact Sets in (𝐸^𝑚,𝜏(𝑙))

In this section, we give characterizations of compact sets and sequentially compact sets, respectively, in (𝐸^𝑚,𝜏(𝑙)). Then it is found that compactness is equivalent to sequential compactness on (𝐸^𝑚,𝜏(𝑙)). Some propositions and lemmas are needed at first.

Proposition 3 (see [6]). (𝐸^𝑚,𝜏(𝑙)) is a Hausdorff space and satisfies the first countability axiom.

Lemma 4. Each compact set of (𝐸^𝑚,𝜏(𝑙)) is sequentially compact.

Proof. By Proposition 3, (𝐸^𝑚,𝜏(𝑙)) satisfies the first countability axiom, and from the basic topology, every countable compact set of (𝐸^𝑚,𝜏(𝑙)) is sequentially compact. Since a compact set is obviously countable compact thus each compact set of (𝐸^𝑚,𝜏(𝑙)) is sequentially compact.

A set 𝑆 is called relatively compact if it has compact closure. A set 𝑈⊂𝐸^𝑚 is said to be uniformly support-bounded if there is a compact set 𝐾⊂ℝ^𝑚 such that [𝑢]₀⊂𝐾 for all 𝑢∈𝑈. Let ℱ be a family of functions from 𝑆⊂ℝ to (𝐾_𝑐(ℝ^𝑚),𝐻). Then(i)ℱ is said to be equi-left-continuous at 𝛼 if for each 𝜀>0 there exists 𝛿(𝛼,𝜀)>0 such that 𝐻(𝑓(𝛼),𝑓(𝛼^′))<𝜀 whenever 𝑓∈ℱ and 𝛼^′∈[𝛼−𝛿,𝛼].(ii)ℱ is said to be equi-right-continuous at 𝛼 if for each 𝜀>0 there exists 𝛿(𝛼,𝜀)>0 such that 𝐻(𝑓(𝛼),𝑓(𝛼^′))<𝜀 whenever 𝑓∈ℱ and 𝛼^′∈[𝛼,𝛼+𝛿].

It is said that ℱ is equi-left (right)-continuous on 𝑆 if it is equi-left (right)-continuous at each point of 𝑆.

Note that [𝑢]_• (where the • may stand for any subscript) can be seen as functions from [0,1] to 𝐾_𝑐(ℝ^𝑚). So we can consider whether {[𝑢]_•∶𝑢∈𝒰} is equi-left (right)-continuous or not for a set 𝒰 in 𝐸^𝑚.

Lemma 5. A subset 𝑈 of (𝐸^𝑚,𝜏(𝑙)) is relatively compact if and only if the following conditions are satisfied: (1)𝑈 is uniformly support-bounded.(2){[𝑢]_•∶𝑢∈𝑈} is equi-left-continuous on (0,1] and equi-right-continuous at 0.

Proof. Necessity. If 𝑈 is relatively compact in (𝐸^𝑚,𝜏(𝑙)), then, by Lemma 4, 𝑈 is sequentially compact in (𝐸^𝑚,𝜏(𝑙)), and thus {[𝑢]₀∶𝑢∈𝑈} is compact in 𝐾_𝑐(ℝ^𝑚). So {[𝑢]₀∶𝑢∈𝑈} is bounded in 𝐾_𝑐(ℝ^𝑚); then obviously 𝑈 is uniformly support-bounded; that is, condition (1) holds.
Now we prove condition (2). In the opposing case where {[𝑢]_𝛼∶𝑢∈𝑈} is not equi-left-continuous at 𝛼₀∈(0,1]. Then there exist 𝜀₀>0 and two sequences {𝑢_𝑛}⊆𝑈 and {𝛼_𝑛}⊆(0,1] with 𝛼_𝑛→𝛼₀−,𝑛=1,2,… such that 𝐻𝑢_𝑛𝛼𝑛,𝑢_𝑛𝛼0>𝜀₀. (6) Since 𝑈 is compact, by Lemma 4, 𝑈 is sequentially compact. We may assume without loss of generality that 𝑢→−_𝑛^𝑙𝑢₀∈𝑈. Note that for a given 𝛽<𝛼₀, there is an 𝑁 such that 𝛼₀>𝛼_𝑛>𝛽 for all 𝑛>𝑁; hence [𝑢_𝑛]_𝛼0⊆[𝑢_𝑛]_𝛼𝑛⊆[𝑢_𝑛]_𝛽 for all 𝑛>𝑁, and thus by (1) 𝐻𝑢_0𝛽,𝑢_0𝛼0=lim_𝑚𝐻𝑢_𝑛𝛽,𝑢_𝑛𝛼0≥lim_𝑚𝐻𝑢_𝑛𝛼𝑛,𝑢_𝑛𝛼0≥𝜀₀ (7) for all 𝛽<𝛼₀; this contradicts with [𝑢₀]_𝛼0=⋂_𝛽<𝛼0[𝑢₀]_𝛽. Hence {[𝑢]_•∶𝑢∈𝑈} is equi-left-continuous on (0,1]. Similarly, we can prove that {[𝑢]_•∶𝑢∈𝑈} is equi-right-continuous at 0.
Sufficiency. Notice that (𝐸^𝑚,𝜏(𝑙)) can be seen as a subset of the product space ∏_𝛼∈[0,1](𝐾_𝑐(ℝ^𝑚),𝐻). Let 𝑈 be the closure of 𝑈 in ∏_𝛼∈[0,1](𝐾_𝑐(ℝ^𝑚),𝐻). Given 𝑣∈𝑈, there is a net {𝑢_𝜉∶𝜉∈𝐷} of 𝑈 such that 𝑣=lim_𝜉∈𝐷𝑢_𝜉. Then obviously [𝑣]_𝛼∈𝐾_𝑐ℝ^𝑚,[𝑣]_𝜇⊆[𝑣]_𝜈 (8) for all 𝛼∈[0,1] and 𝜇≥𝜈. Given 𝛾∈(0,1] and 𝜀>0, from the equi-left-continuity of {[𝑢]_•∶𝑢∈𝑈} at 𝛾, there is a 𝛿>0 such that 𝐻𝑢_𝜉𝛾,𝑢_{𝜉𝛾−𝛿}<𝜀3 (9) for all 𝜉∈𝐷. Since 𝑣=lim_𝜉∈𝐷𝑢_𝜉, there exists 𝑘∈𝐷 such that 𝐻[𝑣]_𝛾,𝑢_𝑘𝛾<𝜀3,𝐻[𝑣]_𝛾−𝛿,𝑢_{𝑘𝛾−𝛿}<𝜀3. (10) Thus 𝐻[𝑣]_𝛾,[𝑣]_𝛾−𝛿≤𝐻[𝑣]_𝛾,𝑢_𝑘𝛾+𝐻𝑢_𝑘𝛾,𝑢_{𝑘𝛾−𝛿}+𝐻𝑢_{𝑘𝛾−𝛿},[𝑣]_𝛾−𝛿<𝜀, (11) and so lim_𝛿→0𝐻[𝑣]_𝛾,[𝑣]_𝛾−𝛿=0 (12) for all 𝛾∈(0,1]. Combined with (2) and (3), we know that [𝑣]_𝛼=_𝛽<𝛼[𝑣]_𝛽 (13) for all 𝛼∈(0,1]. Similarly, we can prove that [𝑣]₀=_𝛽>0[𝑣]_𝛽. (14) Then 𝑣∈𝐸^𝑚 from Proposition 1, (2), (13), and (14). So 𝑈⊂𝐸^𝑚 from the arbitrariness of 𝑣∈𝑈. This means that the closure of 𝑈 in ∏_𝛼∈[0,1](𝐾_𝑐(ℝ^𝑚),𝐻) is just the closure of 𝑈 in (𝐸^𝑚,𝜏(𝑙)).
Since 𝑈 is uniformly support-bounded, then {[𝑢]_𝛼∶𝑢∈𝑈} is bounded in (𝐾_𝑐(ℝ^𝑚),𝐻) for each 𝛼∈[0,1]. By Proposition 2, {[𝑢]_𝛼∶𝑢∈𝑈} is compact in (𝐾_c(ℝ^𝑚),𝐻) for each 𝛼∈[0,1]; then from the Tychonoff product theorem ∏_𝛼∈[0,1]{[𝑢]_𝛼∶𝑢∈𝑈} is compact in ∏_𝛼∈[0,1](𝐾_𝑐(ℝ^𝑚),𝐻). So 𝑈⊂∏_𝛼∈[0,1]{[𝑢]_𝛼∶𝑢∈𝑈} is compact in ∏_𝛼∈[0,1](𝐾_𝑐(ℝ^𝑚),𝐻). Since 𝑈⊂𝐸^𝑚, 𝑈 is also a compact set in (𝐸^𝑚,𝜏(𝑙)).

Now, we arrive at one of the main results of this section.

Theorem 6. A subset 𝑈 of (𝐸^𝑚,𝜏(𝑙)) is compact if and only if the following conditions are satisfied: (1)𝑈 is closed in (𝐸^𝑚,𝜏(𝑙)).(2)𝑈 is uniformly support-bounded.(3){[𝑢]_•∶𝑢∈𝑈} is equi-left-continuous on (0,1] and equi-right-continuous at 0.

Proof. Note that (𝐸^𝑚,𝜏(𝑙)) is a Hausdorff space, so 𝑈 is compact if and only if 𝑈 is closed and relatively compact. The remainder part of proof follows from Lemma 5 immediately.

Fang and Huang [6] proposed a characterization of compact set in (𝐸^𝑚,𝜏(𝑙)). They used concepts “eventually equi-left-continuous” and “eventually equi-right-continuous.” (i)A net {𝑢_𝑘}_𝑘∈𝐷 in (𝐸^𝑚,𝜏(𝑙)) is said to be eventually equi-left-continuous at 𝛼∈(0,1], if for each 𝜀>0, there exist a 𝑘₀∈𝐷 and a 𝛿>0 such that 𝐻([𝑢_𝑘]_𝛼−𝛿,[𝑢_𝑘]_𝛼)<𝜀 for all 𝑘≥𝑘₀. (ii)A net {𝑢_𝑘}_𝑘∈𝐷 in (𝐸^𝑚,𝜏(𝑙)) is eventually equi-right-continuity at 𝛼∈[0,1), if for each 𝜀>0, there exist a 𝑘₀∈𝐷 and a 𝛿>0 such that 𝐻([𝑢_𝑘]_𝛼+𝛿,[𝑢_𝑘]_𝛼)<𝜀 for all 𝑘≥𝑘₀.

They [6] gave the following compact characterization on (𝐸^𝑚,𝜏(𝑙)).

Proposition 7. A closed subset 𝑈 of (𝐸^𝑚,𝜏(𝑙)) is compact if and only if the following conditions are satisfied. (1)𝑈 is uniformly support-bounded.(2) Each net in 𝑈 has a subnet which is eventually equi-left-continuous on (0,1] and eventually equi-right-continuous at 0.

The readers may compare the condition (3) in Theorem 6 with the condition (2) in Proposition 7. In fact, it can be checked that these two conditions are equivalent. However, since the former is a stronger statement in formal, based on it, we can obtain many interesting results. The following Corollary 8 and Theorem 10 are such examples.

Corollary 8. Suppose that 𝑓 is a continuous function from [𝑎,𝑏] to (𝐸^𝑚,𝜏(𝑙)); then {[𝑓(𝑥)]_•∶𝑥∈[𝑎,𝑏]} is equi-left-continuous on (0,1] and equi-right-continuous at 0.

Proof. Since [𝑎,𝑏] is a compact subset of ℝ, we have that 𝑓[𝑎,𝑏] is a compact set in (𝐸^𝑚,𝜏(𝑙)). The desired result follows immediately from Theorem 6.

A set 𝑆 in a topological space is said to be sequentially compact if every sequence in 𝑆 has a subsequence that converges to a point of 𝑆 (see also [18]).

Theorem 9. A subset 𝑈 of (𝐸^𝑚,𝜏(𝑙)) is sequentially compact if and only if the following statements are true. (1)𝑈 is closed in (𝐸^𝑚,𝜏(𝑙)).(2)𝑈 is uniformly support-bounded.(3){[𝑢]_•∶𝑢∈𝑈} is equi-left-continuous on (0,1] and equi-right-continuous at 0.

Proof. Necessity. Given a limit point of 𝑢 of  𝑈, since (𝐸^𝑚,𝜏(𝑙)) is first countable, there is a sequence {𝑢_𝑛,𝑛=1,2,…} of 𝑈 such that 𝑢=lim_𝑛→∞𝑢_𝑛, and then 𝑢∈𝑈 according to the sequential compactness of 𝑈. Thus 𝑈 is a closed set from the arbitrariness of 𝑢. So statement (1) holds. Statements (2) and (3) can be proved similarly as in Lemma 5.
Sufficiency. By Theorem 6, if statements (1), (2), and (3) hold, then 𝑈 is compact, and thus 𝑈 is sequentially compact from Lemma 4.

The following statement is another main result of this section.

Theorem 10. A subset 𝑈 of (𝐸^𝑚,𝜏(𝑙)) is compact if and only if it is sequentially compact.

Proof. The desired result follows immediately from Theorems 6 and 9.

4. Applications: To See Some Characterizations of Compact Sets in (𝐸^𝑚,𝑑_∞)

Many authors discussed the characterizations of compact sets in (𝐸^𝑚,𝑑_∞) and obtained many interesting conclusions. However, by using the results in Section 3, it is found that some of those results are incorrect.

Diamond and Kloeden [2, Proposition 8.2.1] have presented the following compactness criteria of sets in (𝐸^𝑚,𝑑_∞).

Theorem 11. A closed set 𝑈 of (𝐸^𝑚,𝑑_∞) is compact if and only if (1)𝑈 is uniformly support-bounded,(2)𝑈^∗={𝑢^∗∶𝑢∈𝑈} is equi-left-continuous on [0,1] uniformly in 𝑝∈𝑆^𝑛−1: that is, given 𝛼∈[0,1], for each 𝜀>0, there is a 𝛿>0 such that 𝑢^∗(𝛼,𝑝)≤𝑢^∗(𝛽,𝑝)≤𝑢^∗(𝛼,𝑝)+𝜀 for all 𝛽∈[𝛼−𝛿,𝛼], 𝑝∈𝑆^𝑛−1, and 𝑢∈𝑈, where the support function 𝑢^∗∶[0,1]×𝑆^𝑛−1→ℝ of 𝑢∈𝐸^𝑚 is defined by 𝑢^∗𝛼,𝑝=sup𝑝,𝑥∶𝑥∈[𝑢]_𝛼. (15)

Remark 12. Note that 𝐻[𝑢]_𝛼,[𝑣]_𝛼=sup𝑢^∗𝛼,𝑝−𝑣^∗𝛼,𝑝∶𝑝∈𝑆^𝑛−1, (16) so condition (2) in Theorem 11 is equivalent to {[𝑢]_•∶𝑢∈𝑈} is equi-left-continuous on (0,1].

Fang and Xue [14, Theorem 2.3] gave the following characterization of compact subsets in (𝐸¹,𝑑_∞).

Theorem 13. A subset 𝑈 in (𝐸¹,𝑑_∞) is compact if and only if the following three conditions are satisfied: (1)𝑈 is uniformly support-bounded; (2)𝑈 is a closed subset in (𝐸¹,𝑑_∞);(3){𝑢⁺(⋅)∶𝑢∈𝑈} and {𝑢⁻(⋅)∶𝑢∈𝑈} are equi-left-continuous on (0,1].

Remark 14. Notice that [𝑢]_𝛼 is a bounded interval [𝑢⁻(𝛼),𝑢⁺(𝛼)] for all 𝑢∈𝐸¹, so condition (3) holds if and only if {[𝑢]_•∶𝑢∈𝑈} is equi-left-continuous on (0,1]. Thus Theorem 13 is just the 𝑚=1 case of Theorem 11.

Comparing Theorem 6 with Theorem 11, we can see that there exist conflicts: the supremum metric convergence is stronger than the level convergence on 𝐸^𝑚; however the compactness characterization of (𝐸^𝑚,𝑑_∞) given in Theorem 11 is weaker than that of (𝐸^𝑚,𝜏(𝑙)) given in Theorem 6.

We find that Theorems 11 and 13 are incorrect; the following is a counterexample.

Example 15. Consider a fuzzy number sequence {𝑢_𝑛,𝑛=1,2,…}⊂𝐸¹ defined by 𝑢_𝑛(𝑥)=11,𝑥=0,3+23(1−𝑥)^𝑛,0<𝑥≤1,0,otherwise.𝑛=1,2,…, (17) Then 𝑢_𝑛𝛼=30,1−21𝛼−2^1/𝑛,13[]1<𝛼≤1,0,1,0≤𝛼≤3,𝑛=1,2,…, (18) for all 𝛼∈[0,1].

Now we show that {[𝑢_𝑛]_•,𝑛=1,2,…} is equi-left-continuous on (0,1]. In fact, given 𝛼∈(0,1], the argument is divided into two cases:

(A) 𝛼∈(1/3,1]. Then for all 𝛽∈[(1/2)(𝛼+1/3),𝛼], 𝐻𝑢_𝑛𝛼,𝑢_𝑛𝛽=321𝛼−2^1/𝑛−321𝛽−2^1/𝑛≤1𝑛32⋅121𝛼+3−12^1/𝑛−1≤3𝛼−𝛽41𝛼−4⁻¹𝛼−𝛽. (19) So fixed an 𝛼∈(1/3,1], given 𝜀>0, then 𝐻𝑢_𝑛𝛼,𝑢_𝑛𝛽≤𝜀 (20) for all 𝑛∈ℕ and 𝛽∈[𝛼−𝜀((3/4)𝛼−1/4),𝛼]∩[(1/2)(𝛼+1/3),𝛼].

(B) 𝛼∈[0,1/3]. Then for all 𝛽∈[0,𝛼], 𝐻𝑢_𝑛𝛼,𝑢_𝑛𝛽=0. (21) Combined with (20) and (21), we know that {[𝑢_𝑛]_•,𝑛=1,2,…} is equi-left-continuous on (0,1].

Let 𝑢 be a fuzzy number in 𝐸¹ which is defined by 𝑢(𝑥)=11,𝑥=0,3,0<𝑥≤1,0,otherwise. (22) Then [𝑢]_𝛼={0},13[]1<𝛼≤1,0,1,0≤𝛼≤3, (23) for all 𝛼∈[0,1]. So 𝐻[𝑢]_𝛼,𝑢_𝑛𝛼=31−21𝛼−2^1/𝑛,131<𝛼≤1,0,0≤𝛼≤3, (24) and therefore 𝑑_∞𝑢_𝑛3,𝑢=sup1−21𝛼−2^1/𝑛∶13<𝛼≤1=1 (25) for all 𝑛=1,2,…; thus we know that 𝑑_∞(𝑢_𝑛,𝑢)↛0.

On the other hand, given 𝛼∈[0,1], 𝐻(𝑢_𝛼,[𝑢_𝑛]_𝛼)→0(𝑛→∞); hence 𝑢⟶_𝑛^𝑙𝑢. (26) It follows from (25) and (26) that {𝑢_𝑛,𝑛=1,2,…} has no limit point in (𝐸¹,𝑑_∞). So it is a closed set and is not a compact set in (𝐸¹,𝑑_∞).

Notice that [𝑢_𝑛]₀⊆[0,1], 𝑛=1,2,…; then {𝑢_𝑛,𝑛=1,2,…} is uniformly support-bounded. Combined with above discussion, we know {𝑢_𝑛,𝑛=1,2,…} satisfies conditions (1)–(3) of Theorem 13, and that it is not a compact set in (𝐸¹,𝑑_∞). This shows that Theorem 13 is incorrect.

Remark 16. Fang and Xue [14, Theorem  4.1] give a characterization of compact subsets of all continuous functions from a compact subset 𝐾 of a metric space 𝑋 to (𝐸¹,𝑑_∞). However, since it is based on the above theorem, it is wrong too.

5. Conclusion

In this paper, we give a characterization of compact sets in fuzzy number space. The result can be used to discuss the analysis properties of fuzzy numbers and fuzzy-number-valued functions. It can also be used to the applied areas including fuzzy neural networks, fuzzy systems, and so forth.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was supported by the National Science Foundation of China (Grant no. 61103052). The authors would like to thank the reviewers for their invaluable comments and suggestions.