# New Convergence Definitions for Sequences of Sets

Authors: Ömer Kişi; Fatih Nuray
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

Several notions ofconvergence 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 Wijsman   I*-Cauchy sequence and then study their certain properties.


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## Body

## 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 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 setK of positive integers is defined by
(1)δ(K):=limn→∞1n|{k≤n:k∈K}|,
where |k≤n:k∈K| denotes the number of elements of K not exceeding n ([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 sequencex=(xk) is said to be statistically convergent to the number ξ if, for every ε>0,
(2)limn→∞1n|{k≤n:|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 [13]).
A family of setsI⊆2ℕ is called an ideal on ℕ if and only if(i)
∅∈I;(ii)
for eachA,B∈I one has A∪B∈I;(iii)
for eachA∈I and each B⊆A one has B∈I.
An ideal is called nontrivial ifℕ∉I, and nontrivial ideal is called admissible if {n}∈I for each n∈ℕ.Definition 2 (see [14]).
A family of setsF⊆2ℕ is a filter in ℕ if and only if(i)
∅∉F;(ii)
for eachA,B∈F one has A∩B∈F;(iii)
for eachA∈F and each B⊇A one has B∈F.Proposition 3 (see [14]).
I is a nontrivial ideal in ℕ if and only if
(3)F=F(I)={M=ℕ∖A:A∈I}
is a filter in ℕ.Definition 4 (see [14]).
LetI be a nontrivial ideal of subsets of ℕ. A number sequence (xn)n∈ℕ is said to be I-convergent to ξ (ξ=I-limn→∞xn) if and only if for each ε>0 the set
(4){k∈ℕ:|xk-ξ|≥ε}
belongs to I. The element ξ is called the I limit of the number sequence x=(xn)n∈ℕ.The concept ofI-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. [14] 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 ℕ and with the statistical convergence if I is the ideal of subsets of ℕ of natural density zero.Definition 5 (see [14]).
An admissible idealI⊆2ℕ 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=1∞Bi∈I.Definition5 is similar to the condition (APO) used in [15].In [14], the concept of I*-convergence which is closely related to I-convergence has been introduced.Definition 6 (see [14]).
A sequencex=(xn) of elements of X is said to be I*-convergence to ξ if and only if there exists a set M∈F(I),
(5)M={m=(mi):mi<mi+1,i∈ℕ}⊂ℕ
such that limk→∞xmk=ξ.In [14], it is proved that I-convergence and I*-convergence are equivalent for admissible ideals with property (AP).Also, in order to prove thatI-convergent sequence coincides with I*-convergent sequence for admissible ideals with property (AP), we need the following lemma.Lemma 7 (see [13]).
Let{Pi}i=1∞ be a countable collection of subsets of ℕ such that Pi∈F(I) is a filter which associates with an admissible ideal I with property (AP). Then there exists a set P⊂ℕ such that P∈F(I) and the set P∖Pi is finite for all i.Theorem 8 (see [13]).
LetI⊆2ℕ be an admissible ideals with property (AP) and x=(xn) be a number sequence. Then I-limn→∞xn=ξ if and only if there exists a set P∈F(I), P={p=(pi):pi<pi+1,i∈ℕ} such that limk→∞xpk=ξ.Definition 9 (see [9]).
Let(X,d) be a metric space. For any nonempty closed subsets A,Ak⊆X, one says that the sequence {Ak} is Wijsman convergent to A:
(6)limk→∞d(x,Ak)=d(x,A)
for each x∈X. In this case one writes W-limk→∞Ak=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 [16]).
Let(X,d) be a metric space. For any nonempty closed subsets A,Ak⊆X, one says that the sequence {Ak} is Wijsman statistical convergent to A if for ε>0 and for each x∈X,
(7)limn→∞1n|{k≤n:|d(x,Ak)-d(x,A)|≥ε}|=0.
In this case one writes st-limWAk=A or Ak→A(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 [16] 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 x∈X.


## 2. WijsmanI-Convergence

In this section, we will define WijsmanI-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 I⊆2ℕ be a proper ideal in ℕ. For any nonempty closed subsets A,Ak⊂X, one says that the sequence {Ak} is Wijsman I-convergent to A, if, for each ε>0 and for each x∈X, the set
(9)A(x,ε)={k∈ℕ:|d(x,Ak)-d(x,A)|≥ε}
belongs to I. In this case, one writes IW-limAk=A or Ak→A(IW), and the set of Wijsman I-convergent sequences of sets will be denoted by
(10)IW={{Ak}:{k∈ℕ:|d(x,Ak)-d(x,A)|≥ε}∈I}.Example 12.
I
⊆
2
ℕ be a proper ideal in ℕ,  (X,d) a metric space, and A,Ak⊂X nonempty closed subsets. Let X=ℝ2, {Ak} be following sequence:
(11)Ak={{(x,y)∈ℝ2:x2+y2-2ky=0}if,k≠n2{(x,y)∈ℝ2:y=-1}if,k=n2,A={(x,y)∈ℝ2:y=0}.
Fork=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 k≠n2, 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 x→x*. If we write x=(|k|·x*)/(x*)2+(y*-k)2 on the y=k+((y*-k)/x*)·x, we get y→0(k→∞). Thus, for k≠n2(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 k≠n2.
Fork=n2 and k≠n2, the set sequence {Ak} has two different limits. Thus {Ak} is not Wijsman convergent to set A, but
(15){k∈ℕ:|d((x,y),Ak)-d((x,y),A)|≥ε}={k∈ℕ:k=n2}⊂Id.
Thus, suppose that
(16)A(x,y,ε)={k∈ℕ:|d((x,y),Ak)-d((x,y),A)|≥ε}
for ε>0 and for each (x,y)∈ℝ2.
Sincelimk→∞[|d((x,y),Ak)-d((x,y),A)|]=0, for k≠n2, for each ε>0,
(17)∃kε∈ℕ:∀k>kε:|d((x,y),Ak)-d((x,y),A)|<ε.
Define the set Akε(x,y) as
(18)Akε(x,y):={k∈ℕ:|d((x,y),Ak)-d((x,y),A)|>ε}.
Thus, sinceA(x,y,ε)=Akε(x,y)∪{k∈ℕ:k=n2} and Akε(x,y)∈Id and {k∈ℕ:k=n2}∈Id, we can write
(19)A(x,y,ε):={k∈ℕ:|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.
LetI⊆2ℕ be a proper ideal in ℕ, (X,d) a metric space, and A,An⊂X nonempty closed subsets. Let X=ℝ2, {An} be following sequence:
(20)An={{(x,y)∈ℝ2:0≤x≤n,0≤y≤1n·x},if,n≠k2{(x,y)∈ℝ2:x≥0,y=1},if,n=k2,A={(x,y)∈ℝ2:x≥0,y=0}.
Since
(21)limn→∞1n|{k≤n:|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 n≠k2, limn→∞d((x,y),An)=d((x,y),A), but for n=k2, limn→∞d((x,y),An)≠d((x,y),A). Let Id⊂2ℕ be proper ideal. Define set K as
(22)K=K(ε)={n∈ℕ:|d((x,y),An)-d((x,y),A)|≥ε}.
If we takeId for I, Wijsman ideal convergent coincides with Wijsman statistical convergent. Really, one has
(23){n∈ℕ:|d((x,y),An)-d((x,y),A)|≥ε}={n∈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.
LetI⊆2ℕ be a proper ideal in ℕ and (X,d) be a separable metric space. For any nonempty closed subsets A,Ak⊂X, one says that the sequence {Ak} is Wijsman I*-convergent to A, if and only if there exists a set M∈F(I),  M={m=(mi):mi<mi+1,i∈ℕ}⊂ℕ such that for each x∈X(24)limk→∞d(x,Amk)=d(x,A).
In this case, one writes IW*-limAk=A.Definition 15.
LetI⊆2ℕ be an admissible ideal in ℕ 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 x∈X, there exists a number N=N(ε) such that
(25){n∈ℕ:|d(x,An)-d(x,AN)|≥ε}
belongs to I.Definition 16.
LetI⊆2ℕ be an admissible ideal in ℕ and (X,d) be a separable metric space. For any nonempty closed subsets Ak⊂X, one says that the sequence {Ak} is Wijsman I*-Cauchy sequences if there exists a set M={m=(mi):mi<mi+1,i∈ℕ}⊂ℕ, M∈F(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 WijsmanI-convergence implies the Wijsman I-Cauchy condition.Theorem 17.
LetI 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.
LetI be an arbitrary admissible ideal and IW-limAn=A. Then for each ε>0 and for each x∈X, we have
(27)A(x,ε)={n∈ℕ:|d(x,An)-d(x,A)|≥ε}
that belongs to I. Since I is an admissible ideal, there exists an n0∈ℕ such that n0∉A(x,ε).
LetB(x,ε)={n∈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 n∈B(x,ε), then
(29)|d(x,An)-d(x,A)|+|d(x,An0)-d(x,A)|≥2ε.
On the other hand, since n0∉A(x,ε), we have |d(x,An0)-d(x,A)|<ε. Here we conclude that |d(x,An)-d(x,A)|≥ε; hence n∈A(x,ε). Observe that B(x,ε)⊂A(x,ε)∈I for each ε>0 and for each x∈X. This gives that B(x,ε)∈I; that is {An} is Wijsman I-Cauchy sequence.Theorem 18.
LetI 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∈ℕ}⊂ℕ, M∈F(I) such that
(30)|d(x,Amk)-d(x,Amp)|<ε
for each ε>0, for each x∈X, and for all k,p>k0=k0(ε).
LetN=N(ε)=mk0+1. Then for every ε>0, we have
(31)|d(x,Amk)-d(x,AN)|<ε,k>k0.
Now let H=ℕ∖M. It is clear that H∈I and that
(32)A(x,ε)={n∈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 WijsmanI-convergent sequence coincides with Wijsman I*-convergent sequence for admissible ideals with property (AP), we need the following lemma.Lemma 19.
LetI⊆2ℕ be an admissible ideal with property (AP) and (X,d) a separable metric space. If IW-limn→∞d(x,An)=d(x,A), then there exists a set P∈F(I)P={p=(pi):pi<pi+1,i∈ℕ} such that IW-limk→∞d(x,Apk)=d(x,A).Theorem 20.
LetI⊆2ℕ be an admissible ideal with property (AP), let (X,d) be an arbitrary separable metric space and x=(xn)∈X. Then, IW-limn→∞d(x,An)=d(x,A), if and only if there exists a set P∈F(I), P={p=(pi):pi<pi+1,i∈ℕ} such that IW-limk→∞d(x,Apk)=d(x,A).Now we prove that, a WijsmanI-Cauchy sequence coincides with a Wijsman I*-Cauchy sequence for admissible ideals with property (AP).Theorem 21.
IfI⊆2ℕ 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 WijsmanI*-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∈ℕ:|d(x,An)-d(x,AN)|≥ε}∈I
for each ε>0 and for each x∈X.
LetPi={n∈ℕ:|d(x,An)-d(x,Ami)|<1/i}, i=1,2,… where mi=N(1/i). It is clear that Pi∈F(I) for i=1,2,…. Since I has (AP) property, then by Lemma 7 there exists a set P⊂ℕ such that P∈F(I) and P∖Pi 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,  x∈X, and j∈ℕ such that j>2/ε. If m,n∈P then P∖Pi 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 anyε>0, there exists k=k(ε) and n,m∈P∈F(I):
(37)|d(x,An)-d(x,Am)|<ε.
This shows that the sequences {An} is a Wijsman I*-Cauchy sequence.Theorem 22.
LetI 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.
LetIW*-limAk=A. Then by definition there exists a set M∈F(I),  M={m=(mi):mi<mi+1,i∈ℕ}⊂ℕ such that
(38)limk→∞d(x,Amk)=d(x,A)
for each ε>0 and for each x∈X, 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.
LetI 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∈ℕ of elements of XIW-limAn=A implies IW*-limAn=A.Proof.
Suppose thatI satisfies condition (AP). Let IW-limAn=A. Then
(41)T(ε,x)={n∈ℕ:|d(x,An)-d(x,A)|≥ε}∈I
for each ε>0 and for each x∈X. Put
(42)T1={n∈ℕ:|d(x,An)-d(x,A)|≥1},Tn={n∈ℕ:1n≤|d(x,An)-d(x,A)|<1n-1}
for n≥2, and n∈ℕ. Obviously Ti∩Tj=∅ for i≠j. By condition (AP) there exists a sequence of sets {Vn}n∈ℕ such that TjΔVj are finite sets for j∈ℕ and V=⋃j=1∞Vj∈I. It is sufficient to prove that for M=ℕ∖V, M={m=(mi):mi<mi+1,i∈ℕ}∈F(I), we have limk→∞d(x,Amk)=d(x,A).
Letγ>0. Choose k∈ℕ such that 1/(k+1)<γ. Then
(43){n∈ℕ:|d(x,An)-d(x,A)|≥γ}⊂⋃j=1k+1Tj.
Since TjΔVj, j=1,2,… are finite sets, there exists n0∈ℕ such that
(44)(⋃j=1k+1Vj)∩{n∈ℕ:n>n0}=(⋃j=1k+1Tj)∩{n∈ℕ:n>n0}.
If n>n0 and n∉V, so n∉⋃j=1k+1Vj and by (44) n∉⋃j=1k+1Tj. But then |d(x,An)-d(x,A)|<1/(n+1)<γ for each x∈X, so we have limk→∞d(x,Amk)=d(x,A).


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

In this section, we introduce WijsmanI-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.
LetI⊆2ℕ a proper ideal in ℕ and (X,d) a separable metric space. For any nonempty closed subsets An, Bn⊂X, one says that the sequences {An} and {Bn} are almost equal with respect to I if
(45){n∈ℕ:An≠Bn}∈I,
and we write I-a.a.n An=Bn.Definition 25.
LetI⊆2ℕ be a proper ideal in ℕ 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∈ℕ}, then we abbreviate {Anj} by {An}K. If K∈I, then {An}K subsequence is called thin subsequence of {An}. If K∉I, then {An}K subsequence is called nonthin subsequence of {An}.Definition 26.
LetI⊆2ℕ be a proper ideal in ℕ and let (X,d) be a separable metric space, for any nonempty closed subsets Ak⊂X. One has the following.(i)
A∈X is said to be a Wijsman I-limit point of {An} provided that there is a set M={m=(mi):mi<mi+1,i∈ℕ}⊂ℕ such that M∉I and for each x∈X  limk→∞d(x,Amk)=d(x,A).(ii)
A∈X is said to be a Wijsman I-cluster point of {An} if and only if for each ε>0, for each x∈X, we have
(46){n∈ℕ:|d(x,An)-d(x,A)|<ε}∉I.Denote byIW(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 A∈IW(Γ{An}). Then for each sequence {An}⊂X, we have limk→∞d(x,Amk)=d(x,A) which means that A∈L{An}.Theorem 27.
LetI⊆2ℕ be a proper ideal in ℕ and let (X,d) be a separable metric space. Then for each sequence {An}⊂X one has IW(Λ{An})⊂IW(Γ{An}).Proof.
LetA∈IW(Λ{An}). Then, there exists M={m1<m2<⋯}⊂ℕ such that M={m=(mi):mi<mi+1,i∈ℕ}∉I and
(47)limk→∞d(x,Amk)=d(x,A).
According to (47), there exists k0∈ℕ such that for each ε>0, for each x∈X and k>k0, |d(x,Amk)-d(x,A)|<ε. Hence,
(48){k∈ℕ:|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∈ℕ:|d(x,Amk)-d(x,A)|<ε}∉I
which means that A∈IW(Γ{An}).Theorem 28.
LetI⊆2ℕ be a proper ideal in ℕ and let (X,d) be a separable metric space. Then for each sequence {An}⊂X one has IW(Γ{An})⊆L{An}.Proof.
LetA∈IW(Γ{An}). Then for each ε>0 and for each x∈X, we have
(50){n∈ℕ:|d(x,An)-d(x,A)|<ε}∉I.
Let
(51)Kn:={n∈ℕ:|d(x,An)-d(x,A)|<1n}
for n∈ℕ. {Kn}n=1∞ is decreasing sequence of infinite subsets of ℕ. Hence K={n=(ni):ni<ni+1,i∈ℕ}∉I such that limn→∞d(x,Ani)=d(x,A) which means that A∈L{An}.Theorem 29.
LetI⊆2ℕ a proper ideal in ℕ, (X,d) a separable metric space, and Ak,Bk nonempty subsets of X. If {Ak}={Bk}  I-a.a.k for k∈ℕ, then IW(Γ{Ak})=IW(Γ{Bk}) and IW(Λ{Ak})=IW(Λ{Bk}).Proof.
If{Ak}={Bk} a.a.k for k∈ℕ, then
(52)K:={k∈ℕ:Ak≠Bk}∈I
Let A∈IW(Γ{Ak}). For each ε>0 and for each x∈X we have
(53){k∈ℕ:|d(x,Ak)-d(x,A)|<ε}∉I,∀ε>0. If {Ak}={Bk}  I-a.a.k, then {k∈ℕ:|d(x,Bk)-d(x,A)|<ε}∉I which means that A∈IW(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 thatIW(Λ{Ak})=IW(Λ{Bk}). Let A∈IW(Λ{Ak}). Then there exists a set M={m=(mi):mi<mi+1,i∈ℕ}⊂ℕ such that M∉I and
(54)limk→∞d(x,Amk)=d(x,A),M={k:k∈MandAk≠Bk}∪{k:k∈MandAk=Bk},M∉I, and hence {k:k∈MandAk=Bk}∉I. Then there exists
(55)P={p=(pi):pi<pi+1,i∈ℕ}∉I
such that
(56)limk→∞d(x,Bpk)=d(x,A)
which means that A∈IW(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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*Source: 852796-2013-11-05.xml*