AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 852796 10.1155/2013/852796 852796 Research Article New Convergence Definitions for Sequences of Sets Kişi Ömer 1 http://orcid.org/0000-0003-0160-4001 Nuray Fatih 1, 2 Staněk Svatoslav 1 Faculty of Education Mathematics Education Department Cumhuriyet University Sivas Turkey cumhuriyet.edu.tr 2 Department of Mathematics Faculty of Science and Literature Afyon Kocatepe University Afyonkarahisar Turkey aku.edu.tr 2013 5 11 2013 2013 14 05 2013 26 09 2013 2013 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.

Several notions of convergence for subsets of metric space appear in the literature. In this paper, we define Wijsman  I-convergence and Wijsman  I*-convergence for sequences of sets and establish some basic theorems. Furthermore, we introduce the concepts of Wijsman  I-Cauchy sequence and WijsmanI*-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 ) 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 I-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 K of positive integers is defined by (1)δ(K):=limn1n|{kn:kK}|, where |kn:kK| denotes the number of elements of K not exceeding n ().

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

A number sequence x=(xk) is said to be statistically convergent to the number ξ if, for every ε>0, (2)limn1n|{kn:|xk-ξ|ε}|=0. In this case, we write st-limxk=ξ. Statistical convergence is a natural generalization of ordinary convergence. If limxk=ξ, then st-limxk=ξ. The converse does not hold in general.

Definition 1 (see [<xref ref-type="bibr" rid="B12">13</xref>]).

A family of sets I2 is called an ideal on if and only if

I;

for each A,BI one has ABI;

for each AI and each BA one has BI.

An ideal is called nontrivial if I, and nontrivial ideal is called admissible if {n}I for each n.

Definition 2 (see [<xref ref-type="bibr" rid="B11">14</xref>]).

A family of sets F2 is a filter in if and only if

F;

for each A,BF one has ABF;

for each AF and each BA one has BF.

Proposition 3 (see [<xref ref-type="bibr" rid="B11">14</xref>]).

I is a nontrivial ideal in if and only if (3)F=F(I)={M=A:AI} is a filter in .

Definition 4 (see [<xref ref-type="bibr" rid="B11">14</xref>]).

Let I be a nontrivial ideal of subsets of . A number sequence (xn)n is said to be I-convergent to ξ (ξ=I-limnxn) if and only if for each ε>0 the set (4){k:|xk-ξ|ε} belongs to I. The element ξ is called the I limit of the number sequence x=(xn)n.

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

Definition 5 (see [<xref ref-type="bibr" rid="B11">14</xref>]).

An admissible ideal I2 is said to have the property (AP) if for any sequence {A1,A2,} of mutually disjoint sets of I, there is sequence {B1,B2,} of sets such that each symmetric difference AiΔBi (i=1,2,) is finite and i=1BiI.

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

In , the concept of I*-convergence which is closely related to I-convergence has been introduced.

Definition 6 (see [<xref ref-type="bibr" rid="B11">14</xref>]).

A sequence x=(xn) of elements of X is said to be I*-convergence to ξ if and only if there exists a set MF(I), (5)M={m=(mi):mi<mi+1,i} such that limkxmk=ξ.

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

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

Lemma 7 (see [<xref ref-type="bibr" rid="B12">13</xref>]).

Let {Pi}i=1 be a countable collection of subsets of such that PiF(I) is a filter which associates with an admissible ideal I with property (AP). Then there exists a set P such that PF(I) and the set PPi is finite for all i.

Theorem 8 (see [<xref ref-type="bibr" rid="B12">13</xref>]).

Let I2 be an admissible ideals with property (AP) and x=(xn) be a number sequence. Then I-limnxn=ξ if and only if there exists a set PF(I), P={p=(pi):pi<pi+1,i} such that limkxpk=ξ.

Definition 9 (see [<xref ref-type="bibr" rid="B18">9</xref>]).

Let (X,d) be a metric space. For any nonempty closed subsets A,AkX, one says that the sequence {Ak} is Wijsman convergent to A: (6)limkd(x,Ak)=d(x,A) for each xX. In this case one writes W-limkAk=A.

As an example, consider the following sequence of circles in the (x,y)-plane: Ak={(x,y):x2+y2+2kx=0}. As k the sequence is Wijsman convergent to the y-axis A={(x,y):x=0}.

Definition 10 (see [<xref ref-type="bibr" rid="B14">16</xref>]).

Let (X,d) be a metric space. For any nonempty closed subsets A,AkX, one says that the sequence {Ak} is Wijsman statistical convergent to A if for ε>0 and for each xX, (7)limn1n|{kn:|d(x,Ak)-d(x,A)|ε}|=0. In this case one writes st-limWAk=A or AkA(WS). Consider (8)WS{{Ak}:st-limWAk=A}, where WS denotes the set of Wijsman statistical convergence sequences.

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

Let (X,ρ) be a metric space. For any nonempty closed subsets Ak of X, one says that the sequence {Ak} is bounded if supkd(x,Ak)< for each xX.

2. Wijsman <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M123"><mml:mrow><mml:mi>I</mml:mi></mml:mrow></mml:math></inline-formula>-Convergence

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

Definition 11.

Let (X,d) be a metric space and I2 be a proper ideal in . For any nonempty closed subsets A,AkX, one says that the sequence {Ak} is Wijsman I-convergent to A, if, for each ε>0 and for each xX, the set (9)A(x,ε)={k:|d(x,Ak)-d(x,A)|ε} belongs to I. In this case, one writes IW-limAk=A or AkA(IW), and the set of Wijsman I-convergent sequences of sets will be denoted by (10)IW={{Ak}:{k:|d(x,Ak)-d(x,A)|ε}I}.

Example 12.

I 2 be a proper ideal in ,  (X,d) a metric space, and A,AkX nonempty closed subsets. Let X=2, {Ak} be following sequence: (11)Ak={{(x,y)2:x2+y2-2ky=0}if,kn2{(x,y)2:y=-1}if,k=n2,A={(x,y)2:y=0}.

For k=n2, d((x,y),An2)=|y+1|d((x,y);A)=|y|. Let us take a point (x*,y*) outside x2+y2-2ky=0. For kn2, we write d((x*,y*),Ak)d((x*,y*),A)=|y*|. Since the line equation is (12)x-0x*=y-ky*-k, where the line is passing from (0,k) the center point of the circle and (x*,y*) the outside of the circle, we write y=k+((y*-k)/x*)·x. If we write this y=k+((y*-k)/x*)·x value on the circle equation x2+y2-2ky=0, we can get (13)x=|k|·x*(x*)2+(y*-k)2. For k, if we take limit, it will be xx*. If we write x=(|k|·x*)/(x*)2+(y*-k)2 on the y=k+((y*-k)/x*)·x, we get y0(k). Thus, for kn2(14)d((x*,y*),Ak)=(x-x*)2+(y-y*)2|y*|. So we get d((x*,y*),Ak)d((x*,y*),A)=|y*|, for kn2.

For k=n2 and kn2, the set sequence {Ak} has two different limits. Thus {Ak} is not Wijsman convergent to set A, but (15){k:|d((x,y),Ak)-d((x,y),A)|ε}={k:k=n2}Id. Thus, suppose that (16)A(x,y,ε)={k:|d((x,y),Ak)-d((x,y),A)|ε} for ε>0 and for each (x,y)2.

Since limk[|d((x,y),Ak)-d((x,y),A)|]=0, for kn2, for each ε>0, (17)kε:k>kε:|d((x,y),Ak)-d((x,y),A)|<ε. Define the set Akε(x,y) as (18)Akε(x,y):={k:|d((x,y),Ak)-d((x,y),A)|>ε}.

Thus, since A(x,y,ε)=Akε(x,y){k:k=n2} and Akε(x,y)Id and {k:k=n2}Id, we can write (19)A(x,y,ε):={k:|d((x,y),Ak)-d((x,y),A)|>ε}Id, where Id={A:δ(A)=0}. So the set sequence {An} is Wijsman I-convergent to set A.

Example 13.

Let I2 be a proper ideal in , (X,d) a metric space, and A,AnX nonempty closed subsets. Let X=2, {An} be following sequence: (20)An={{(x,y)2:0xn,0y1n·x},if,nk2{(x,y)2:x0,y=1},if,n=k2,A={(x,y)2:x0,y=0}. Since (21)limn1n|{kn:|d((x,y),An)-d((x,y),A)|ε}|=0, the set sequence {An} is Wijsman statistical convergent to set A. Thus we can write st-limWAn=A, but this sequence is not Wijsman convergent to set A. Because for nk2, limnd((x,y),An)=d((x,y),A), but for n=k2, limnd((x,y),An)d((x,y),A). Let Id2 be proper ideal. Define set K as (22)K=K(ε)={n:|d((x,y),An)-d((x,y),A)|ε}.

If we take Id for I, Wijsman ideal convergent coincides with Wijsman statistical convergent. Really, one has (23){n:|d((x,y),An)-d((x,y),A)|ε}={n:n=k2}Id.

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

Definition 14.

Let I2 be a proper ideal in and (X,d) be a separable metric space. For any nonempty closed subsets A,AkX, one says that the sequence {Ak} is Wijsman I*-convergent to A, if and only if there exists a set MF(I),  M={m=(mi):mi<mi+1,i} such that for each xX(24)limkd(x,Amk)=d(x,A). In this case, one writes IW*-limAk=A.

Definition 15.

Let I2 be an admissible ideal in and (X,d) be a separable metric space. For any nonempty closed subset An in X, one says that the sequence {An} is Wijsman I-Cauchy sequence if for each ε>0 and for each xX, there exists a number N=N(ε) such that (25){n:|d(x,An)-d(x,AN)|ε} belongs to I.

Definition 16.

Let I2 be an admissible ideal in and (X,d) be a separable metric space. For any nonempty closed subsets AkX, one says that the sequence {Ak} is Wijsman I*-Cauchy sequences if there exists a set M={m=(mi):mi<mi+1,i}, MF(I) such that the subsequence AM={Amk} is Wijsman Cauchy in X; that is, (26)limk,p|d(x,Amk)-d(x,Amp)|=0.

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

Theorem 17.

Let I be an arbitrary admissible ideal and let X be a separable metric space. Then IW-limAn=A implies that {An} is Wijsman I-Cauchy sequence.

Proof.

Let I be an arbitrary admissible ideal and IW-limAn=A. Then for each ε>0 and for each xX, we have (27)A(x,ε)={n:|d(x,An)-d(x,A)|ε} that belongs to I. Since I is an admissible ideal, there exists an n0 such that n0A(x,ε).

Let B(x,ε)={n:|d(x,An)-d(x,An0)|2ε}. Taking into account the inequality (28)|d(x,An)-d(x,An0)||d(x,An)-d(x,A)|+|d(x,An0)-d(x,A)|, we observe that if nB(x,ε), then (29)|d(x,An)-d(x,A)|+|d(x,An0)-d(x,A)|2ε. On the other hand, since n0A(x,ε), we have |d(x,An0)-d(x,A)|<ε. Here we conclude that |d(x,An)-d(x,A)|ε; hence nA(x,ε). Observe that B(x,ε)A(x,ε)I for each ε>0 and for each xX. This gives that B(x,ε)I; that is {An} is Wijsman I-Cauchy sequence.

Theorem 18.

Let I be an admissible ideal and let X be a separable metric space. If {An} is Wijsman I*-Cauchy sequence, then it is Wijsman I-Cauchy sequence.

Proof.

Let {An} be Wijsman I*-Cauchy sequence; then by the definition, there exists a set M={m=(mi):mi<mi+1,i}, MF(I) such that (30)|d(x,Amk)-d(x,Amp)|<ε for each ε>0, for each xX, and for all k,p>k0=k0(ε).

Let N=N(ε)=mk0+1. Then for every ε>0, we have (31)|d(x,Amk)-d(x,AN)|<ε,k>k0. Now let H=M. It is clear that HI and that (32)A(x,ε)={n:|d(x,An)-d(x,AN)|ε}H{m1,m2,,mk0} belongs to I. Therefore, for every ε>0, we can find a N=N(ε) such that A(x,ε)I; that is, {An} is Wijsman I-Cauchy sequence. Hence the proof is complete.

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

Lemma 19.

Let I2 be an admissible ideal with property (AP) and (X,d) a separable metric space. If IW-limnd(x,An)=d(x,A), then there exists a set PF(I)P={p=(pi):pi<pi+1,i} such that IW-limkd(x,Apk)=d(x,A).

Theorem 20.

Let I2 be an admissible ideal with property (AP), let (X,d) be an arbitrary separable metric space and x=(xn)X. Then, IW-limnd(x,An)=d(x,A), if and only if there exists a set PF(I), P={p=(pi):pi<pi+1,i} such that IW-limkd(x,Apk)=d(x,A).

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

Theorem 21.

If I2 is an admissible ideal with property (AP) and if (X,d) is a separable metric space, then the concepts Wijsman I-Cauchy sequence and Wijsman I*-Cauchy sequence coincide.

Proof.

If a sequence is Wijsman I*-Cauchy, then it is Wijsman I-Cauchy by Theorem 18 where I does not need to have the (AP) property. Now it is sufficient to prove that {An} is Wijsman I*-Cauchy sequence in X under assumption that {An} is a Wijsman I-Cauchy sequence. Let {An} be a Wijsman I-Cauchy sequence. Then by definition, there exists a N=N(ε) such that (33)A(x,ε)={n:|d(x,An)-d(x,AN)|ε}I for each ε>0 and for each xX.

Let Pi={n:|d(x,An)-d(x,Ami)|<1/i}, i=1,2, where mi=N(1/i). It is clear that PiF(I) for i=1,2,. Since I has (AP) property, then by Lemma 7 there exists a set P such that PF(I) and PPi is finite for all i. Now we show that (34)limn,m|d(x,An)-d(x,Am)|=0. To prove this, let ε>0,  xX, and j such that j>2/ε. If m,nP then PPi is finite set, therefore there exists k=k(j) such that (35)|d(x,An)-d(x,Amj)|<1j,|d(x,Am)-d(x,Amj)|<1j for all m,n>k(j). Hence it follows that (36)|d(x,An)-d(x,Am)|<|d(x,An)-d(x,Amj)|+|d(x,Am)-d(x,Amj)|<ε for m,n>k(j).

Thus, for any ε>0, there exists k=k(ε) and n,mPF(I): (37)|d(x,An)-d(x,Am)|<ε. This shows that the sequences {An} is a Wijsman I*-Cauchy sequence.

Theorem 22.

Let I be an admissible ideal and (X,d) a separable metric space. Then IW*-limAk=A implies that {An} is a Wijsman I-Cauchy sequence.

Proof.

Let IW*-limAk=A. Then by definition there exists a set MF(I),  M={m=(mi):mi<mi+1,i} such that (38)limkd(x,Amk)=d(x,A) for each ε>0 and for each xX, and k,p>k0, (39)|d(x,Amk)-d(x,Amp)|<|d(x,Amk)-d(x,A)|+|d(x,Amp)-d(x,A)|<ε2+ε2=ε. Therefore, (40)limk,p|d(x,Amk)-d(x,Amp)|=0. Hence, {An} is a Wijsman I-Cauchy sequence.

Theorem 23.

Let I be an admissible ideal and (X,d) a separable metric space. If the ideal I has property (AP) and if (X,d) is an arbitrary metric space, then for arbitrary sequence {An}n of elements of XIW-limAn=A implies IW*-limAn=A.

Proof.

Suppose that I satisfies condition (AP). Let IW-limAn=A. Then (41)T(ε,x)={n:|d(x,An)-d(x,A)|ε}I for each ε>0 and for each xX. Put (42)T1={n:|d(x,An)-d(x,A)|1},Tn={n:1n|d(x,An)-d(x,A)|<1n-1} for n2, and n. Obviously TiTj= for ij. By condition (AP) there exists a sequence of sets {Vn}n such that TjΔVj are finite sets for j and V=j=1VjI. It is sufficient to prove that for M=V, M={m=(mi):mi<mi+1,i}F(I), we have limkd(x,Amk)=d(x,A).

Let γ>0. Choose k such that 1/(k+1)<γ. Then (43){n:|d(x,An)-d(x,A)|γ}j=1k+1Tj. Since TjΔVj, j=1,2, are finite sets, there exists n0 such that (44)(j=1k+1Vj){n:n>n0}=(j=1k+1Tj){n:n>n0}. If n>n0 and nV, so nj=1k+1Vj and by (44) nj=1k+1Tj. But then |d(x,An)-d(x,A)|<1/(n+1)<γ for each xX, so we have limkd(x,Amk)=d(x,A).

3. Wijsman <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M431"><mml:mrow><mml:mi>I</mml:mi></mml:mrow></mml:math></inline-formula>-Limit Points and Wijsman <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M432"><mml:mrow><mml:mi>I</mml:mi></mml:mrow></mml:math></inline-formula>-Cluster Points Sequences of Sets

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

Definition 24.

Let I2 a proper ideal in and (X,d) a separable metric space. For any nonempty closed subsets An, BnX, one says that the sequences {An} and {Bn} are almost equal with respect to I if (45){n:AnBn}I, and we write I-a.a.n An=Bn.

Definition 25.

Let I2 be a proper ideal in and let (X,d) be a separable metric space; An is nonempty closed subset of X. If {An}K is subsequence of {An} and K:={n(j):j}, then we abbreviate {Anj} by {An}K. If KI, then {An}K subsequence is called thin subsequence of {An}. If KI, then {An}K subsequence is called nonthin subsequence of {An}.

Definition 26.

Let I2 be a proper ideal in and let (X,d) be a separable metric space, for any nonempty closed subsets AkX. One has the following.

AX is said to be a Wijsman I-limit point of {An} provided that there is a set M={m=(mi):mi<mi+1,i} such that MI and for each xX  limkd(x,Amk)=d(x,A).

AX is said to be a Wijsman I-cluster point of {An} if and only if for each ε>0, for each xX, we have (46){n:|d(x,An)-d(x,A)|<ε}I.

Denote by IW(Λ{An}), IW(Γ{An}), and L{An} the set of all Wijsman I-limit, Wijsman I-cluster, and Wijsman limit points of {An}, respectively.

For the sequences {An},  IW(Γ{An})IW(L{An}). Let AIW(Γ{An}). Then for each sequence {An}X, we have limkd(x,Amk)=d(x,A) which means that AL{An}.

Theorem 27.

Let I2 be a proper ideal in and let (X,d) be a separable metric space. Then for each sequence {An}X one has IW(Λ{An})IW(Γ{An}).

Proof.

Let AIW(Λ{An}). Then, there exists M={m1<m2<} such that M={m=(mi):mi<mi+1,i}I and (47)limkd(x,Amk)=d(x,A). According to (47), there exists k0 such that for each ε>0, for each xX and k>k0, |d(x,Amk)-d(x,A)|<ε. Hence, (48){k:|d(x,Amk)-d(x,A)|<ε}M{m1,m2,,mko}. Then, the set on the right hand side of (48) does not belong to I; therefore (49){k:|d(x,Amk)-d(x,A)|<ε}I which means that AIW(Γ{An}).

Theorem 28.

Let I2 be a proper ideal in and let (X,d) be a separable metric space. Then for each sequence {An}X one has IW(Γ{An})L{An}.

Proof.

Let AIW(Γ{An}). Then for each ε>0 and for each xX, we have (50){n:|d(x,An)-d(x,A)|<ε}I. Let (51)Kn:={n:|d(x,An)-d(x,A)|<1n} for n. {Kn}n=1 is decreasing sequence of infinite subsets of . Hence K={n=(ni):ni<ni+1,i}I such that limnd(x,Ani)=d(x,A) which means that AL{An}.

Theorem 29.

Let I2 a proper ideal in , (X,d) a separable metric space, and Ak,Bk nonempty subsets of X. If {Ak}={Bk}  I-a.a.k for k, then IW(Γ{Ak})=IW(Γ{Bk}) and IW(Λ{Ak})=IW(Λ{Bk}).

Proof.

If {Ak}={Bk} a.a.k for k, then (52)K:={k:AkBk}I Let AIW(Γ{Ak}). For each ε>0 and for each xX we have (53){k:|d(x,Ak)-d(x,A)|<ε}I,ε>0. If {Ak}={Bk}  I-a.a.k, then {k:|d(x,Bk)-d(x,A)|<ε}I which means that AIW(Γ{Bk}); hence IW(Γ{Ak}IW(Γ{Bk}). Similarly we can also prove that IW(Γ{Bk})IW(Γ{Ak}. So we have IW(Γ{Ak}=IW(Γ{Bk}).

Now, we show that IW(Λ{Ak})=IW(Λ{Bk}). Let AIW(Λ{Ak}). Then there exists a set M={m=(mi):mi<mi+1,i} such that MI and (54)limkd(x,Amk)=d(x,A),M={k:kMandAkBk}{k:kMandAk=Bk},MI, and hence {k:kMandAk=Bk}I. Then there exists (55)P={p=(pi):pi<pi+1,i}I such that (56)limkd(x,Bpk)=d(x,A) which means that AIW(Λ{Bk}). Similarly we can also prove that IW(Λ{Bk})IW(Λ{Ak}). Therefore we have IW(Λ{Ak})=IW(Λ{Bk}).

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