Abstract and Applied Analysis

Volume 2013 (2013), Article ID 852796, 6 pages

http://dx.doi.org/10.1155/2013/852796

## New Convergence Definitions for Sequences of Sets

^{1}Faculty of Education, Mathematics Education Department, Cumhuriyet University, Sivas, Turkey^{2}Department of Mathematics, Faculty of Science and Literature, Afyon Kocatepe University, Afyonkarahisar, Turkey

Received 14 May 2013; Accepted 26 September 2013

Academic Editor: Svatoslav Staněk

Copyright © 2013 Ömer Kişi and Fatih Nuray. 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.

#### Abstract

Several notions of *convergence* for subsets of metric space appear in the literature. In this paper, we define *Wijsman* -*convergence* and *Wijsman* -*convergence* for sequences of sets and establish some basic theorems. Furthermore, we introduce the concepts of *Wijsman* *I*-*Cauchy* sequence and *Wijsman* -*Cauchy* sequence and then study their certain properties.

#### 1. Introduction and Background

The concept of convergence of sequences of points has been extended by several authors (see [1–9]) to the concept of convergence of sequences of sets. The one of these such extensions that we will consider in this paper is Wijsman convergence. We will define -convergence for sequences of sets and establish some basic results regarding these notions.

Let us start with fundamental definitions from the literature. The natural density of a set of positive integers is defined by where denotes the number of elements of not exceeding ([10]).

Statistical convergence of sequences of points was introduced by Fast [11]. In [12], Schoenberg established some basic properties of statistical convergence and also studied the concept as a summability method.

A number sequence is said to be statistically convergent to the number if, for every , In this case, we write . Statistical convergence is a natural generalization of ordinary convergence. If , then . The converse does not hold in general.

*Definition 1 (see [13]). * A family of sets is called an ideal on if and only if(i);(ii)for each one has ;(iii)for each and each one has .

An ideal is called nontrivial if , and nontrivial ideal is called admissible if for each .

*Definition 2 (see [14]). *A family of sets is a filter in if and only if(i);(ii)for each one has ;(iii)for each and each one has .

Proposition 3 (see [14]). * is a nontrivial ideal in if and only if
**
is a filter in . *

*Definition 4 (see [14]). *Let be a nontrivial ideal of subsets of . A number sequence is said to be -convergent to () if and only if for each the set
belongs to . The element is called the limit of the number sequence .

The concept of -convergence of real sequences is a generalization of statistical convergence which is based on the structure of the ideal of subsets of the set of natural numbers. Kostyrko et al. [14] introduced the concept of -convergence of sequences in a metric space and studied some properties of this convergence. -convergence of real sequences coincides with the ordinary convergence if is the ideal of all finite subsets of and with the statistical convergence if is the ideal of subsets of of natural density zero.

*Definition 5 (see [14]). *An admissible ideal is said to have the property (AP) if for any sequence of mutually disjoint sets of , there is sequence of sets such that each symmetric difference () is finite and .

Definition 5 is similar to the condition (APO) used in [15].

In [14], the concept of -convergence which is closely related to -convergence has been introduced.

*Definition 6 (see [14]). *A sequence of elements of is said to be -convergence to if and only if there exists a set ,
such that .

In [14], it is proved that -convergence and -convergence are equivalent for admissible ideals with property (AP).

Also, in order to prove that -convergent sequence coincides with -convergent sequence for admissible ideals with property (AP), we need the following lemma.

Lemma 7 (see [13]). *Let be a countable collection of subsets of such that is a filter which associates with an admissible ideal with property (AP). Then there exists a set such that and the set is finite for all .*

Theorem 8 (see [13]). *Let be an admissible ideals with property (AP) and be a number sequence. Then if and only if there exists a set , such that .*

*Definition 9 (see [9]). *Let be a metric space. For any nonempty closed subsets , one says that the sequence is Wijsman convergent to :
for each . In this case one writes .

As an example, consider the following sequence of circles in the -plane: . As the sequence is Wijsman convergent to the -axis .

*Definition 10 (see [16]). *Let be a metric space. For any nonempty closed subsets , one says that the sequence is Wijsman statistical convergent to if for and for each ,
In this case one writes or . Consider
where denotes the set of Wijsman statistical convergence sequences.

Also the concept of bounded sequence for sequences of sets was given by Nuray and Rhoades [16] as follows.

Let be a metric space. For any nonempty closed subsets of , one says that the sequence is bounded if for each .

#### 2. Wijsman -Convergence

In this section, we will define Wijsman -convergence and Wijsman -convergence of sequences of sets, give the relationship between them, and establish some basic theorems.

*Definition 11. *Let be a metric space and be a proper ideal in . For any nonempty closed subsets , one says that the sequence is Wijsman -convergent to , if, for each and for each , the set
belongs to . In this case, one writes or , and the set of Wijsman -convergent sequences of sets will be denoted by

*Example 12. * be a proper ideal in , a metric space, and nonempty closed subsets. Let , be following sequence:

For , . Let us take a point outside . For , we write . Since the line equation is
where the line is passing from the center point of the circle and the outside of the circle, we write . If we write this value on the circle equation , we can get
For , if we take limit, it will be . If we write on the , we get . Thus, for
So we get , for .

For and , the set sequence has two different limits. Thus is not Wijsman convergent to set , but
Thus, suppose that
for and for each .

Since , for , for each ,
Define the set as

Thus, since and and , we can write
where . So the set sequence is Wijsman -convergent to set .

*Example 13. *Let be a proper ideal in , a metric space, and nonempty closed subsets. Let , be following sequence:
Since
the set sequence is Wijsman statistical convergent to set . Thus we can write , but this sequence is not Wijsman convergent to set . Because for , , but for , . Let be proper ideal. Define set as

If we take for , Wijsman ideal convergent coincides with Wijsman statistical convergent. Really, one has

Since the Wijsman topology is not first countable in general, if is convergent to the set Wijsman sense, every subsequence of may not be convergent to . But if is separable, then every subsequence of a convergent set sequence is convergent to the same limit.

*Definition 14. *Let be a proper ideal in and be a separable metric space. For any nonempty closed subsets , one says that the sequence is Wijsman -convergent to , if and only if there exists a set , such that for each
In this case, one writes .

*Definition 15. *Let be an admissible ideal in and be a separable metric space. For any nonempty closed subset in , one says that the sequence is Wijsman -Cauchy sequence if for each and for each , there exists a number such that
belongs to .

*Definition 16. *Let be an admissible ideal in and be a separable metric space. For any nonempty closed subsets , one says that the sequence is Wijsman -Cauchy sequences if there exists a set , such that the subsequence is Wijsman Cauchy in ; that is,

Now we will prove that Wijsman -convergence implies the Wijsman -Cauchy condition.

Theorem 17. *Let be an arbitrary admissible ideal and let be a separable metric space. Then implies that is Wijsman -Cauchy sequence. *

*Proof. *Let be an arbitrary admissible ideal and . Then for each and for each , we have
that belongs to . Since is an admissible ideal, there exists an such that .

Let . Taking into account the inequality
we observe that if , then
On the other hand, since , we have . Here we conclude that ; hence . Observe that for each and for each . This gives that ; that is is Wijsman -Cauchy sequence.

Theorem 18. *Let be an admissible ideal and let be a separable metric space. If is Wijsman -Cauchy sequence, then it is Wijsman -Cauchy sequence. *

*Proof. *Let be Wijsman -Cauchy sequence; then by the definition, there exists a set , such that
for each , for each , and for all .

Let . Then for every , we have
Now let . It is clear that and that
belongs to . Therefore, for every , we can find a such that ; that is, is Wijsman -Cauchy sequence. Hence the proof is complete.

In order to prove that Wijsman -convergent sequence coincides with Wijsman -convergent sequence for admissible ideals with property (AP), we need the following lemma.

Lemma 19. *Let be an admissible ideal with property (AP) and a separable metric space. If , then there exists a set such that . *

Theorem 20. *Let be an admissible ideal with property (AP), let be an arbitrary separable metric space and . Then, , if and only if there exists a set , such that . *

Now we prove that, a Wijsman -Cauchy sequence coincides with a Wijsman -Cauchy sequence for admissible ideals with property (AP).

Theorem 21. *If is an admissible ideal with property (AP) and if is a separable metric space, then the concepts Wijsman -Cauchy sequence and Wijsman -Cauchy sequence coincide. *

*Proof. *If a sequence is Wijsman -Cauchy, then it is Wijsman -Cauchy by Theorem 18 where does not need to have the (AP) property. Now it is sufficient to prove that is Wijsman -Cauchy sequence in under assumption that is a Wijsman -Cauchy sequence. Let be a Wijsman -Cauchy sequence. Then by definition, there exists a such that
for each and for each .

Let , where . It is clear that for . Since has (AP) property, then by Lemma 7 there exists a set such that and is finite for all . Now we show that
To prove this, let , , and such that . If then is finite set, therefore there exists such that
for all . Hence it follows that
for .

Thus, for any , there exists and :
This shows that the sequences is a Wijsman -Cauchy sequence.

Theorem 22. *Let be an admissible ideal and a separable metric space. Then implies that is a Wijsman -Cauchy sequence. *

*Proof. *Let . Then by definition there exists a set , such that
for each and for each , and ,
Therefore,
Hence, is a Wijsman -Cauchy sequence.

Theorem 23. *Let be an admissible ideal and a separable metric space. If the ideal has property (AP) and if is an arbitrary metric space, then for arbitrary sequence of elements of implies .*

*Proof. *Suppose that satisfies condition (AP). Let . Then
for each and for each . Put
for , and . Obviously for . By condition (AP) there exists a sequence of sets such that are finite sets for and . It is sufficient to prove that for , , we have .

Let . Choose such that . Then
Since , are finite sets, there exists such that
If and , so and by (44) . But then for each , so we have .

#### 3. Wijsman -Limit Points and Wijsman -Cluster Points Sequences of Sets

In this section, we introduce Wijsman -limit points of sequences of sets and Wijsman -cluster points of sequences of sets, prove some basic properties of these concepts, and establish some basic theorems.

*Definition 24. *Let a proper ideal in and a separable metric space. For any nonempty closed subsets , , one says that the sequences and are almost equal with respect to if
and we write -a.a.n .

*Definition 25. *Let be a proper ideal in and let be a separable metric space; is nonempty closed subset of . If is subsequence of and , then we abbreviate by . If , then subsequence is called thin subsequence of . If , then subsequence is called nonthin subsequence of .

*Definition 26. *Let be a proper ideal in and let be a separable metric space, for any nonempty closed subsets . One has the following.(i) is said to be a Wijsman -limit point of provided that there is a set such that and for each .(ii) is said to be a Wijsman -cluster point of if and only if for each , for each , we have

Denote by , , and the set of all Wijsman -limit, Wijsman -cluster, and Wijsman limit points of , respectively.

For the sequences , . Let . Then for each sequence , we have which means that .

Theorem 27. *Let be a proper ideal in and let be a separable metric space. Then for each sequence one has .*

*Proof. *Let . Then, there exists such that and
According to (47), there exists such that for each , for each and , . Hence,
Then, the set on the right hand side of (48) does not belong to ; therefore
which means that .

Theorem 28. *Let be a proper ideal in and let be a separable metric space. Then for each sequence one has .*

*Proof. *Let . Then for each and for each , we have
Let
for . is decreasing sequence of infinite subsets of . Hence such that which means that .

Theorem 29. *Let a proper ideal in , a separable metric space, and nonempty subsets of . If -a.a.k for , then and . *

*Proof. *If a.a.k for , then
Let . For each and for each we have
. If -a.a.k, then which means that ; hence . Similarly we can also prove that . So we have .

Now, we show that . Let . Then there exists a set such that and
, and hence . Then there exists
such that
which means that . Similarly we can also prove that . Therefore we have .

#### References

- J.-P. Aubin and H. Frankowska,
*Set-Valued Analysis*, vol. 2, Birkhäuser, Boston, Mass, USA, 1990. View at MathSciNet - M. Baronti and P. L. Papini, “Convergence of sequences of sets,” in
*Methods of Functional Analysis in Approximation Theory*, vol. 76, pp. 135–155, Birkhäuser, Basel, Switzerland, 1986. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Beer, “Convergence of continuous linear functionals and their level sets,”
*Archiv der Mathematik*, vol. 52, no. 5, pp. 482–491, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Beer, “On convergence of closed sets in a metric space and distance functions,”
*Bulletin of the Australian Mathematical Society*, vol. 31, no. 3, pp. 421–432, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Borwein and J. Vanderwerff, “Dual Kadec-Klee norms and the relationships between Wijsman, slice, and Mosco convergence,”
*The Michigan Mathematical Journal*, vol. 41, no. 2, pp. 371–387, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - O. Kisi and F. Nuray, “On ${S}_{\lambda}^{L}$ (I)-asymptotically statistical equivalence of sequences of sets,”
*Mathematical Analysis*, vol. 2013, Article ID 602963, 6 pages, 2013. View at Publisher · View at Google Scholar - Y. Sonntag and C. Zălinescu, “Set convergences: an attempt of classification,”
*Transactions of the American Mathematical Society*, vol. 340, no. 1, pp. 199–226, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. A. Wijsman, “Convergence of sequences of convex sets, cones and functions,”
*Bulletin of the American Mathematical Society*, vol. 70, pp. 186–188, 1964. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. A. Wijsman, “Convergence of sequences of convex sets, cones and functions. II,”
*Transactions of the American Mathematical Society*, vol. 123, pp. 32–45, 1966. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. C. Buck, “Generalized asymptotic density,”
*The American Journal of Mathematics*, vol. 75, pp. 335–346, 1953. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Fast, “Sur la convergence statistique,”
*Colloquium Mathematicae*, vol. 2, pp. 241–244, 1951. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - I. J. Schoenberg, “The integrability of certain functions and related summability methods,”
*The American Mathematical Monthly*, vol. 66, pp. 361–375, 1959. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P. Kostyrko, M. Mačaj, T. Šalát, and M. Sleziak, “$I$-convergence and extremal $I$-limit points,”
*Mathematica Slovaca*, vol. 55, no. 4, pp. 443–464, 2005. View at Google Scholar · View at MathSciNet - P. Kostyrko, T. Šalát, and W. Wilczyński, “$I$-convergence,”
*Real Analysis Exchange*, vol. 26, no. 2, pp. 669–685, 2000. View at Google Scholar · View at MathSciNet - A. R. Freedman and J. J. Sember, “Densities and summability,”
*Pacific Journal of Mathematics*, vol. 95, no. 2, pp. 293–305, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Nuray and B. E. Rhoades, “Statistical convergence of sequences of sets,”
*Fasciculi Mathematici*, no. 49, pp. 87–99, 2012. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet