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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 820141, 6 pages
A New Characterization of Compact Sets in Fuzzy Number Spaces
1Department of Mathematics, Jimei University, Xiamen 361021, China
2Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
Received 14 August 2013; Accepted 18 October 2013
Academic Editor: Juan Carlos Cortés López
Copyright © 2013 Huan Huang and Congxin Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 [1–13]. One of the most important problems is the characterizations of compact subsets.
Fang and Huang  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  presented a characterization of compact sets in fuzzy number spaces equipped with the supremum metric. Fang and Xue  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 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
For , let denote the -cut of : that is We call a fuzzy number if has the following properties: (1)is normal: there exists at least one with ; (2) is convex: for and ;(3) is upper semicontinuous; (4) is a bounded set in .
The set of all fuzzy numbers is denoted by .
Suppose that is the set of all nonempty compact sets of and that is the set of all nonempty compact convex set of . The following representation theorem is used widely in the theory of fuzzy numbers.
Many metrics and topologies on are based on the well-known Hausdorff metric. The Hausdorff metric on is defined by for arbitrary , where Obviously, if and are bounded closed intervals of , then
Throughout this paper, we suppose that the metric on is the Euclidean metric, and the metric on is the Hausdorff metric . The Hausdorff metric has the following properties.
In this paper, we consider two types of convergences on fuzzy number spaces. (i)Let . If ; then we say supremum converges to , denoted by , where the supremum metric is defined by for all . (ii)Let and let be a net in , where is a direct set. If for each , then we say level converges to , denoted by or .
Obviously, the supremum metric convergence is stronger than the level convergence on ; that is, if supremum metric converges to , then it also level converges to .
The symbol is used to denote the topology induced by level convergence on ; that is, is the topology with being a local subbase of , where (see also ).
We use or to denote the fuzzy number space equipped with the supremum metric or equipped with the level convergence topology , respectively.
3. 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.
Proposition 3 (see ). is a Hausdorff space and satisfies the first countability axiom.
Lemma 4. Each compact set of is sequentially compact.
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 support-bounded if there is a compact set such that for all . Let be a family of functions from to . Then(i) is said to be equi-left-continuous at if for each there exists such that whenever and .(ii) is said to be equi-right-continuous at if for each there exists such that whenever and .
It is said that 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 to . So we can consider whether is equi-left (right)-continuous or not for a set in .
Lemma 5. A subset of is relatively compact if and only if the following conditions are satisfied: (1) is uniformly support-bounded.(2) is equi-left-continuous on and equi-right-continuous at 0.
Proof. Necessity. If is relatively compact in , then, by Lemma 4, is sequentially compact in , and thus is compact in . So is bounded in ; then obviously is uniformly support-bounded; that is, condition (1) holds.
Now we prove condition (2). In the opposing case where is not equi-left-continuous at . Then there exist and two sequences and with such that Since is compact, by Lemma 4, is sequentially compact. We may assume without loss of generality that . Note that for a given , there is an such that for all ; hence for all , and thus by (1) for all ; this contradicts with . Hence is equi-left-continuous on . Similarly, we can prove that is equi-right-continuous at 0.
Sufficiency. Notice that can be seen as a subset of the product space . Let be the closure of in . Given , there is a net of such that . Then obviously for all and . Given and , from the equi-left-continuity of at , there is a such that for all . Since , there exists such that Thus and so for all . Combined with (2) and (3), we know that for all . Similarly, we can prove that Then from Proposition 1, (2), (13), and (14). So from the arbitrariness of . This means that the closure of in is just the closure of in .
Since is uniformly support-bounded, then is bounded in for each . By Proposition 2, is compact in for each ; then from the Tychonoff product theorem is compact in . So is compact in . Since , is also a compact set in .
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 .(2) is uniformly support-bounded.(3) is equi-left-continuous on 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.
Fang and Huang  proposed a characterization of compact set in . They used concepts “eventually equi-left-continuous” and “eventually equi-right-continuous.” (i)A net in is said to be eventually equi-left-continuous at , if for each , there exist a and a such that for all . (ii)A net in is eventually equi-right-continuity at , if for each , there exist a and a such that for all .
They  gave the following compact characterization on .
Proposition 7. A closed subset of is compact if and only if the following conditions are satisfied. (1) is uniformly support-bounded.(2) Each net in has a subnet which is eventually equi-left-continuous on 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 is a continuous function from to ; then is equi-left-continuous on and equi-right-continuous at .
Proof. Since is a compact subset of , we have that is a compact set in . The desired result follows immediately from Theorem 6.
A set in a topological space is said to be sequentially compact if every sequence in has a subsequence that converges to a point of (see also ).
Theorem 9. A subset of is sequentially compact if and only if the following statements are true. (1) is closed in .(2) is uniformly support-bounded.(3) is equi-left-continuous on and equi-right-continuous at 0.
Proof. Necessity. Given a limit point of of , since is first countable, there is a sequence of such that , and then according to the sequential compactness of . Thus is a closed set from the arbitrariness of . 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 is compact, and thus is sequentially compact from Lemma 4.
The following statement is another main result of this section.
Theorem 10. A subset of is compact if and only if it is sequentially compact.
4. Applications: To See Some Characterizations of Compact Sets in
Many authors discussed the characterizations of compact sets in 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 ] have presented the following compactness criteria of sets in .
Theorem 11. A closed set of is compact if and only if (1) is uniformly support-bounded,(2) is equi-left-continuous on uniformly in : that is, given , for each , there is a such that for all , , and , where the support function of is defined by
Fang and Xue [14, Theorem 2.3] gave the following characterization of compact subsets in .
Theorem 13. A subset in is compact if and only if the following three conditions are satisfied: (1) is uniformly support-bounded; (2) is a closed subset in ;(3) and are equi-left-continuous on .
Comparing Theorem 6 with Theorem 11, we can see that there exist conflicts: the supremum metric convergence is stronger than the level convergence on ; however the compactness characterization of given in Theorem 11 is weaker than that of given in Theorem 6.
Example 15. Consider a fuzzy number sequence defined by Then for all .
Now we show that is equi-left-continuous on . In fact, given , the argument is divided into two cases:
(A) . Then for all , So fixed an , given , then for all and .
Let be a fuzzy number in which is defined by Then for all . So and therefore for all ; thus we know that .
Notice that , ; then is uniformly support-bounded. Combined with above discussion, we know satisfies conditions (1)–(3) of Theorem 13, and that it is not a compact set in . 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 of a metric space to . However, since it is based on the above theorem, it is wrong too.
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.
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.
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