A New Characterization of Compact Sets in Fuzzy Number Spaces

and Applied Analysis 3 Proof. Necessity. If U is relatively compact in (Em, τ(l)), then, by Lemma 4,U is sequentially compact in (Em, τ(l)), and thus {[u] 0 : u ∈ U} is compact in K c (R). So {[u] 0 : u ∈ U} is bounded in K c (R); then obviously U is uniformly supportbounded; 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 α 0 ∈ (0, 1]. Then there exist ε 0 > 0 and two sequences {u n } ⊆ U and {α n } ⊆ (0, 1] with α n → α 0 −, n = 1, 2, . . . such that H([u n ] α n , [u n ] α 0 ) > ε 0 . (6) Since U is compact, by Lemma 4, U is sequentially compact. We may assume without loss of generality that u n l 󳨀 → u 0 ∈ U. Note that for a given β < α 0 , there is anN such that α 0 > α n > β for all n > N; hence [u n ] α 0 ⊆ [u n ] α n ⊆ [u n ] β for all n > N, and thus by (1) H([u 0 ] β , [u 0 ] α 0 ) = lim m H([u n ] β , [u n ] α 0 ) ≥ lim m H([u n ] α n , [u n ] α 0 ) ≥ ε 0 (7) for all β < α 0 ; this contradicts with [u 0 ] α 0 = ⋂ β<α 0 [u 0 ] β . Hence {[u] ∙ : u ∈ U} is equi-left-continuous on (0, 1]. Similarly, we can prove that {[u] ∙ : u ∈ U} is equi-rightcontinuous at 0. Sufficiency. Notice that (Em, τ(l)) can be seen as a subset of the product space∏ α∈[0,1] (K c (R),H). Let U be the closure of U in∏ α∈[0,1] (K c (R),H). Given V ∈ U, there is a net {u ξ : ξ ∈ D} of U such that V = lim ξ∈D u ξ . Then obviously [V] α ∈ K c (R m ) , [V] μ ⊆ [V]] (8) for all α ∈ [0, 1] and μ ≥ ]. 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 H([u ξ ] γ , [u ξ ] γ−δ ) < ε

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  = 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.

Fuzzy Number Space
Let N be the set of all natural numbers, let R  be dimensional Euclidean space, and let (R  ) represent all fuzzy subsets on R  , that is, functions from R  to [0, 1].For details, we refer the readers to [2,12].
For  ∈ (R  ), let []  denote the -cut of : that is We call  ∈ (R  ) a fuzzy number if  has the following properties: (1)  is normal: there exists at least one  0 ∈ R  with ( 0 ) = 1; (2)  is convex: ( + (1 − )) ≥ min{(), ()} for ,  ∈ R  and  ∈ [0, 1]; (3)  is upper semicontinuous; The set of all fuzzy numbers is denoted by   .Suppose that (R  ) is the set of all nonempty compact sets of R  and that   (R  ) is the set of all nonempty compact convex set of R  .The following representation theorem is used widely in the theory of fuzzy numbers.
In this paper, we consider two types of convergences on fuzzy number spaces.
then we say {  } supremum converges to , denoted by    ∞   → , where the supremum metric  ∞ is defined by for all , V ∈   .
We use (  ,  ∞ ) or (  , ()) to denote the fuzzy number space   equipped with the supremum metric  ∞ or equipped with the level convergence topology (), respectively.

Characterizations of Compact Sets and
Sequentially Compact Sets in (  ,()) In this section, we give characterizations of compact sets and sequentially compact sets, respectively, in (  , ()).
Then it is found that compactness is equivalent to sequential compactness on (  , ()).Some propositions and lemmas are needed at first.
Proof.By Proposition 3, (  , ()) satisfies the first countability axiom, and from the basic topology, every countable compact set of (  , ()) is sequentially compact.Since a compact set is obviously countable compact thus each compact set of (  , ()) is sequentially compact.
A set  is called relatively compact if it has compact closure.A set  ⊂   is said to be uniformly supportbounded if there is a compact set  ⊂ R  such that [] 0 ⊂  for all  ∈ .Let F be a family of functions from  ⊂ R to (  (R  ), ).Then (i) F is said to be equi-left-continuous at  if for each  > 0 there exists (, ) > 0 such that ((), (  )) <  whenever  ∈ F and   ∈ [ − , ].(ii) F is said to be equi-right-continuous at  if for each  > 0 there exists (, ) > 0 such that ((), (  )) <  whenever  ∈ F and It is said that F is equi-left (right)-continuous on  if it is equi-left (right)-continuous at each point of .
Note that [] • (where the • may stand for any subscript) can be seen as functions from [0, 1] to   (R  ).So we can consider whether {[] • :  ∈ U} is equi-left (right)-continuous or not for a set U in   .Lemma 5. A subset  of (  , ()) is relatively compact if and only if the following conditions are satisfied: (1)  is uniformly support-bounded.
Since  is uniformly support-bounded, then Now, we arrive at one of the main results of this section.Theorem 6.A subset  of (  , ()) is compact if and only if the following conditions are satisfied: (1)  is closed in (  , ()).
(3) {[] • :  ∈ } is equi-left-continuous on (0, 1] and equi-right-continuous at 0. Proof.Note that (  , ()) is a Hausdorff space, so  is compact if and only if  is closed and relatively compact.The remainder part of proof follows from Lemma 5 immediately.
They [6] gave the following compact characterization on (  , ()).Proposition 7. A closed subset  of (  , ()) is compact if and only if the following conditions are satisfied.
(2) Each net in  has a subnet which is eventually equileft-continuous on (0, 1] and eventually equi-rightcontinuous 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,