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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.

The convergences on fuzzy number spaces and their applications have been extensively discussed by various authors [

Fang and Huang [

Diamond and Kloeden [

Let

For

The set of all fuzzy numbers is denoted by

Suppose that

Given

Moreover, if the family of sets

Many metrics and topologies on

Throughout this paper, we suppose that the metric on

In this paper, we consider two types of convergences on fuzzy number spaces.

Let

Let

Obviously, the supremum metric convergence is stronger than the level convergence on

The symbol

We use

In this section, we give characterizations of compact sets and sequentially compact sets, respectively, in

Each compact set of

By Proposition

A set

It is said that

Note that

A subset

Now we prove condition (

Since

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

A subset

Note that

Fang and Huang [

A net

A net

They [

A closed subset

The readers may compare the condition (

Suppose that

Since

A set

A subset

The following statement is another main result of this section.

A subset

The desired result follows immediately from Theorems

Many authors discussed the characterizations of compact sets in

Diamond and Kloeden [

A closed set

Note that

Fang and Xue [

A subset

Notice that

Comparing Theorem

We find that Theorems

Consider a fuzzy number sequence

Now we show that

Let

On the other hand, given

Notice that

Fang and Xue [

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.

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

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.