New Convergence Definitions for Sequences of Sets

Abstract

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 Wijsman  𝐼^∗-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 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 𝛿(𝐾)∶=lim_𝑛→∞1𝑛|{𝑘≤𝑛∶𝑘∈𝐾}|, (1) 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 sequence 𝑥=(𝑥_𝑘) is said to be statistically convergent to the number 𝜉 if, for every 𝜀>0, lim_𝑛→∞1𝑛𝑘≤𝑛∶𝑥_𝑘−𝜉≥𝜀=0. (2) In this case, we write 𝑠𝑡−lim𝑥_𝑘=𝜉. Statistical convergence is a natural generalization of ordinary convergence. If lim𝑥_𝑘=𝜉, then 𝑠𝑡−lim𝑥_𝑘=𝜉. The converse does not hold in general.

Definition 1 (see [13]). A family of sets 𝐼⊆2^ℕ is called an ideal on ℕ if and only if(i)∅∈𝐼;(ii)for each 𝐴,𝐵∈𝐼 one has 𝐴∪𝐵∈𝐼;(iii)for each 𝐴∈𝐼 and each 𝐵⊆𝐴 one has 𝐵∈𝐼.
An ideal is called nontrivial if ℕ∉𝐼, and nontrivial ideal is called admissible if {𝑛}∈𝐼 for each 𝑛∈ℕ.

Definition 2 (see [14]). A family of sets 𝐹⊆2^ℕ is a filter in ℕ if and only if(i)∅∉𝐹;(ii)for each 𝐴,𝐵∈𝐹 one has 𝐴∩𝐵∈𝐹;(iii)for each 𝐴∈𝐹 and each 𝐵⊇𝐴 one has 𝐵∈𝐹.

Proposition 3 (see [14]). 𝐼 is a nontrivial ideal in ℕ if and only if 𝐹=𝐹(𝐼)={𝑀=ℕ⧵𝐴∶𝐴∈𝐼} (3) is a filter in ℕ.

Definition 4 (see [14]). Let 𝐼 be a nontrivial ideal of subsets of ℕ. A number sequence (𝑥_𝑛)_𝑛∈ℕ is said to be 𝐼-convergent to 𝜉 (𝜉=𝐼−lim_𝑛→∞𝑥_𝑛) if and only if for each 𝜀>0 the set 𝑘∈ℕ∶𝑥_𝑘−𝜉≥𝜀 (4) belongs to 𝐼. The element 𝜉 is called the 𝐼 limit of the number sequence 𝑥=(𝑥_𝑛)_𝑛∈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. [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 ideal 𝐼⊆2^ℕ 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 𝐴_𝑖Δ𝐵_𝑖 (𝑖=1,2,…) is finite and ⋃^∞_𝑖=1𝐵_𝑖∈𝐼.

Definition 5 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 sequence 𝑥=(𝑥_𝑛) of elements of 𝑋 is said to be 𝐼^∗-convergence to 𝜉 if and only if there exists a set 𝑀∈𝐹(𝐼), 𝑀=𝑚=𝑚_𝑖∶𝑚_𝑖<𝑚_𝑖+1,𝑖∈ℕ⊂ℕ (5) such that lim_𝑘→∞𝑥_𝑚𝑘=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 that 𝐼-convergent sequence coincides with 𝐼^∗-convergent sequence for admissible ideals with property (AP), we need the following lemma.

Lemma 7 (see [13]). Let {𝑃_𝑖}^∞_𝑖=1 be a countable collection of subsets of ℕ such that 𝑃_𝑖∈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⧵𝑃_𝑖 is finite for all 𝑖.

Theorem 8 (see [13]). Let 𝐼⊆2^ℕ be an admissible ideals with property (AP) and 𝑥=(𝑥_𝑛) be a number sequence. Then 𝐼−lim_𝑛→∞𝑥_𝑛=𝜉 if and only if there exists a set 𝑃∈𝐹(𝐼), 𝑃={𝑝=(𝑝_𝑖)∶𝑝_𝑖<𝑝_𝑖+1,𝑖∈ℕ} such that lim_𝑘→∞𝑥_𝑝𝑘=𝜉.

Definition 9 (see [9]). Let (𝑋,𝑑) be a metric space. For any nonempty closed subsets 𝐴,𝐴_𝑘⊆𝑋, one says that the sequence {𝐴_𝑘} is Wijsman convergent to 𝐴: lim_𝑘→∞𝑑𝑥,𝐴_𝑘=𝑑(𝑥,𝐴) (6) for each 𝑥∈𝑋. In this case one writes 𝑊−lim_𝑘→∞𝐴_𝑘=𝐴.

As an example, consider the following sequence of circles in the (𝑥,𝑦)-plane: 𝐴_𝑘={(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,𝐴_𝑘⊆𝑋, one says that the sequence {𝐴_𝑘} is Wijsman statistical convergent to 𝐴 if for 𝜀>0 and for each 𝑥∈𝑋, lim_𝑛→∞1𝑛𝑘≤𝑛∶𝑑𝑥,𝐴_𝑘−𝑑(𝑥,𝐴)≥𝜀=0. (7) In this case one writes 𝑠𝑡−lim_𝑊𝐴_𝑘=𝐴 or 𝐴_𝑘→𝐴(WS). Consider WS∶=𝐴_𝑘∶𝑠𝑡−lim_𝑊𝐴_𝑘=𝐴, (8) 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 𝐴_𝑘 of 𝑋, one says that the sequence {𝐴_𝑘} is bounded if sup_𝑘𝑑(𝑥,𝐴_𝑘)<∞ for each 𝑥∈𝑋.

2. Wijsman 𝐼-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 𝐼⊆2^ℕ be a proper ideal in ℕ. For any nonempty closed subsets 𝐴,𝐴_𝑘⊂𝑋, one says that the sequence {𝐴_𝑘} is Wijsman 𝐼-convergent to 𝐴, if, for each 𝜀>0 and for each 𝑥∈𝑋, the set 𝐴(𝑥,𝜀)=𝑘∈ℕ∶𝑑𝑥,𝐴_𝑘−𝑑(𝑥,𝐴)≥𝜀 (9) belongs to 𝐼. In this case, one writes 𝐼_𝑊−lim𝐴_𝑘=𝐴 or 𝐴_𝑘→𝐴(𝐼_𝑊), and the set of Wijsman 𝐼-convergent sequences of sets will be denoted by 𝐼_𝑊=𝐴_𝑘∶𝑘∈ℕ∶𝑑𝑥,𝐴_𝑘−𝑑(𝑥,𝐴)≥𝜀∈𝐼. (10)

Example 12. 𝐼⊆2^ℕ be a proper ideal in ℕ,  (𝑋,𝑑) a metric space, and 𝐴,𝐴_𝑘⊂𝑋 nonempty closed subsets. Let 𝑋=ℝ², {𝐴_𝑘} be following sequence: 𝐴_𝑘=𝑥,𝑦∈ℝ²∶𝑥²+𝑦²−2𝑘𝑦=0if,𝑘≠𝑛²𝑥,𝑦∈ℝ²∶𝑦=−1if,𝑘=𝑛²,𝐴=𝑥,𝑦∈ℝ²∶𝑦=0. (11)
For 𝑘=𝑛², 𝑑((𝑥,𝑦),𝐴_𝑛2)=|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^∗),𝐴_𝑘)→𝑑((𝑥^∗,𝑦^∗),𝐴)=|𝑦^∗|. Since the line equation is 𝑥−0𝑥^∗=𝑦−𝑘𝑦^∗,−𝑘 (12) 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 |𝑘|𝑥=⋅𝑥^∗(𝑥^∗)²+𝑦^∗−𝑘². (13) 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≠𝑛²𝑑𝑥^∗,𝑦^∗,𝐴_𝑘=(𝑥−𝑥^∗)²+𝑦−𝑦^∗2⟶𝑦^∗. (14) So we get 𝑑((𝑥^∗,𝑦^∗),𝐴_𝑘)→𝑑((𝑥^∗,𝑦^∗),𝐴)=|𝑦^∗|, for 𝑘≠𝑛².
For 𝑘=𝑛² and 𝑘≠𝑛², the set sequence {𝐴_𝑘} has two different limits. Thus {𝐴_𝑘} is not Wijsman convergent to set 𝐴, but 𝑘∈ℕ∶𝑑𝑥,𝑦,𝐴_𝑘=−𝑑𝑥,𝑦,𝐴≥𝜀𝑘∈ℕ∶𝑘=𝑛²⊂𝐼_𝑑. (15) Thus, suppose that 𝐴𝑥,𝑦,𝜀=𝑘∈ℕ∶𝑑𝑥,𝑦,𝐴_𝑘−𝑑𝑥,𝑦,𝐴≥𝜀 (16) for 𝜀>0 and for each (𝑥,𝑦)∈ℝ².
Since lim_𝑘→∞[|𝑑((𝑥,𝑦),𝐴_𝑘)−𝑑((𝑥,𝑦),𝐴)|]=0, for 𝑘≠𝑛², for each 𝜀>0, ∃𝑘_𝜀∈ℕ∶∀𝑘>𝑘_𝜀∶𝑑𝑥,𝑦,𝐴_𝑘−𝑑𝑥,𝑦,𝐴<𝜀. (17) Define the set 𝐴_𝑘𝜀(𝑥,𝑦) as 𝐴_𝑘𝜀𝑥,𝑦∶=𝑘∈ℕ∶𝑑𝑥,𝑦,𝐴_𝑘−𝑑𝑥,𝑦,𝐴>𝜀. (18)
Thus, since 𝐴(𝑥,𝑦,𝜀)=𝐴_𝑘𝜀(𝑥,𝑦)∪{𝑘∈ℕ∶𝑘=𝑛²} and 𝐴_𝑘𝜀(𝑥,𝑦)∈𝐼_𝑑 and {𝑘∈ℕ∶𝑘=𝑛²}∈𝐼_𝑑, we can write 𝐴𝑑𝑥,𝑦,𝜀∶=𝑘∈ℕ∶𝑥,𝑦,𝐴_𝑘−𝑑𝑥,𝑦,𝐴>𝜀∈𝐼_𝑑, (19) where 𝐼_𝑑={𝐴∶𝛿(𝐴)=0}. So the set sequence {𝐴_𝑛} is Wijsman 𝐼-convergent to set 𝐴.

Example 13. Let 𝐼⊆2^ℕ be a proper ideal in ℕ, (𝑋,𝑑) a metric space, and 𝐴,𝐴_𝑛⊂𝑋 nonempty closed subsets. Let 𝑋=ℝ², {𝐴_𝑛} be following sequence: 𝐴_𝑛=𝑥,𝑦∈ℝ²1∶0≤𝑥≤𝑛,0≤𝑦≤𝑛⋅𝑥,if,𝑛≠𝑘²𝑥,𝑦∈ℝ²∶𝑥≥0,𝑦=1,if,𝑛=𝑘²,𝐴=𝑥,𝑦∈ℝ²∶𝑥≥0,𝑦=0. (20) Since lim_𝑛→∞1𝑛𝑘≤𝑛∶𝑑𝑥,𝑦,𝐴_𝑛−𝑑𝑥,𝑦,𝐴≥𝜀=0, (21) the set sequence {𝐴_𝑛} is Wijsman statistical convergent to set 𝐴. Thus we can write 𝑠𝑡−lim_𝑊𝐴_𝑛=𝐴, but this sequence is not Wijsman convergent to set 𝐴. Because for 𝑛≠𝑘², lim_𝑛→∞𝑑((𝑥,𝑦),𝐴_𝑛)=𝑑((𝑥,𝑦),𝐴), but for 𝑛=𝑘², lim_𝑛→∞𝑑((𝑥,𝑦),𝐴_𝑛)≠𝑑((𝑥,𝑦),𝐴). Let 𝐼_𝑑⊂2^ℕ be proper ideal. Define set 𝐾 as 𝐾=𝐾(𝜀)=𝑛∈ℕ∶𝑑𝑥,𝑦,𝐴_𝑛−𝑑𝑥,𝑦,𝐴≥𝜀. (22)
If we take 𝐼_𝑑 for 𝐼, Wijsman ideal convergent coincides with Wijsman statistical convergent. Really, one has 𝑛∈ℕ∶𝑑𝑥,𝑦,𝐴_𝑛=−𝑑𝑥,𝑦,𝐴≥𝜀𝑛∈ℕ∶𝑛=𝑘²⊂𝐼_𝑑. (23)
Since the Wijsman topology is not first countable in general, if {𝐴_𝑘} is convergent to the set 𝐴 Wijsman sense, every subsequence of {𝐴_𝑘} 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 𝐼⊆2^ℕ be a proper ideal in ℕ and (𝑋,𝑑) be a separable metric space. For any nonempty closed subsets 𝐴,𝐴_𝑘⊂𝑋, one says that the sequence {𝐴_𝑘} is Wijsman 𝐼^∗-convergent to 𝐴, if and only if there exists a set 𝑀∈𝐹(𝐼),  𝑀={𝑚=(𝑚_𝑖)∶𝑚_𝑖<𝑚_𝑖+1,𝑖∈ℕ}⊂ℕ such that for each 𝑥∈𝑋lim_𝑘→∞𝑑𝑥,𝐴_𝑚𝑘=𝑑(𝑥,𝐴). (24) In this case, one writes 𝐼^∗_𝑊−lim𝐴_𝑘=𝐴.

Definition 15. Let 𝐼⊆2^ℕ be an admissible ideal in ℕ and (𝑋,𝑑) be a separable metric space. For any nonempty closed subset 𝐴_𝑛 in 𝑋, one says that the sequence {𝐴_𝑛} is Wijsman 𝐼-Cauchy sequence if for each 𝜀>0 and for each 𝑥∈𝑋, there exists a number 𝑁=𝑁(𝜀) such that 𝑛∈ℕ∶𝑑𝑥,𝐴_𝑛−𝑑𝑥,𝐴_𝑁≥𝜀 (25) belongs to 𝐼.

Definition 16. Let 𝐼⊆2^ℕ be an admissible ideal in ℕ and (𝑋,𝑑) be a separable metric space. For any nonempty closed subsets 𝐴_𝑘⊂𝑋, one says that the sequence {𝐴_𝑘} is Wijsman 𝐼^∗-Cauchy sequences if there exists a set 𝑀={𝑚=(𝑚_𝑖)∶𝑚_𝑖<𝑚_𝑖+1,𝑖∈ℕ}⊂ℕ, 𝑀∈𝐹(𝐼) such that the subsequence 𝐴_𝑀={𝐴_𝑚𝑘} is Wijsman Cauchy in 𝑋; that is, lim_{𝑘,𝑝→∞}𝑑𝑥,𝐴_𝑚𝑘−𝑑𝑥,𝐴_𝑚𝑝=0. (26)

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 𝐼_𝑊−lim𝐴_𝑛=𝐴 implies that {𝐴_𝑛} is Wijsman 𝐼-Cauchy sequence.

Proof. Let 𝐼 be an arbitrary admissible ideal and 𝐼_𝑊−lim𝐴_𝑛=𝐴. Then for each 𝜀>0 and for each 𝑥∈𝑋, we have 𝐴(𝑥,𝜀)=𝑛∈ℕ∶𝑑𝑥,𝐴_𝑛−𝑑(𝑥,𝐴)≥𝜀 (27) that belongs to 𝐼. Since 𝐼 is an admissible ideal, there exists an 𝑛₀∈ℕ such that 𝑛₀∉𝐴(𝑥,𝜀).
Let 𝐵(𝑥,𝜀)={𝑛∈ℕ∶|𝑑(𝑥,𝐴_𝑛)−𝑑(𝑥,𝐴_𝑛0)|≥2𝜀}. Taking into account the inequality 𝑑𝑥,𝐴_𝑛−𝑑𝑥,𝐴_𝑛0≤𝑑𝑥,𝐴_𝑛−𝑑(𝑥,𝐴)+𝑑𝑥,𝐴_𝑛0−𝑑(𝑥,𝐴), (28) we observe that if 𝑛∈𝐵(𝑥,𝜀), then 𝑑𝑥,𝐴_𝑛−𝑑(𝑥,𝐴)+𝑑𝑥,𝐴_𝑛0−𝑑(𝑥,𝐴)≥2𝜀. (29) On the other hand, since 𝑛₀∉𝐴(𝑥,𝜀), we have |𝑑(𝑥,𝐴_𝑛0)−𝑑(𝑥,𝐴)|<𝜀. Here we conclude that |𝑑(𝑥,𝐴_𝑛)−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 {𝐴_𝑛} is Wijsman 𝐼-Cauchy sequence.

Theorem 18. Let 𝐼 be an admissible ideal and let 𝑋 be a separable metric space. If {𝐴_𝑛} is Wijsman 𝐼^∗-Cauchy sequence, then it is Wijsman 𝐼-Cauchy sequence.

Proof. Let {𝐴_𝑛} be Wijsman 𝐼^∗-Cauchy sequence; then by the definition, there exists a set 𝑀={𝑚=(𝑚_𝑖)∶𝑚_𝑖<𝑚_𝑖+1,𝑖∈ℕ}⊂ℕ, 𝑀∈𝐹(𝐼) such that 𝑑𝑥,𝐴_𝑚𝑘−𝑑𝑥,𝐴_𝑚𝑝<𝜀 (30) for each 𝜀>0, for each 𝑥∈𝑋, and for all 𝑘,𝑝>𝑘₀=𝑘₀(𝜀).
Let 𝑁=𝑁(𝜀)=𝑚_𝑘0+1. Then for every 𝜀>0, we have 𝑑𝑥,𝐴_𝑚𝑘−𝑑𝑥,𝐴_𝑁<𝜀,𝑘>𝑘₀. (31) Now let 𝐻=ℕ⧵𝑀. It is clear that 𝐻∈𝐼 and that 𝐴(𝑥,𝜀)=𝑛∈ℕ∶𝑑𝑥,𝐴_𝑛−𝑑𝑥,𝐴_𝑁𝑚≥𝜀⊂𝐻∪₁,𝑚₂,…,𝑚_𝑘0 (32) belongs to 𝐼. Therefore, for every 𝜀>0, we can find a 𝑁=𝑁(𝜀) such that 𝐴(𝑥,𝜀)∈𝐼; that is, {𝐴_𝑛} 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 𝐼⊆2^ℕ be an admissible ideal with property (AP) and (𝑋,𝑑) a separable metric space. If 𝐼_𝑊−lim_𝑛→∞𝑑(𝑥,𝐴_𝑛)=𝑑(𝑥,𝐴), then there exists a set 𝑃∈𝐹(𝐼)𝑃={𝑝=(𝑝_𝑖)∶𝑝_𝑖<𝑝_𝑖+1,𝑖∈ℕ} such that 𝐼_𝑊−lim_𝑘→∞𝑑(𝑥,𝐴_𝑝𝑘)=𝑑(𝑥,𝐴).

Theorem 20. Let 𝐼⊆2^ℕ be an admissible ideal with property (AP), let (𝑋,𝑑) be an arbitrary separable metric space and 𝑥=(𝑥_𝑛)∈𝑋. Then, 𝐼_𝑊−lim_𝑛→∞𝑑(𝑥,𝐴_𝑛)=𝑑(𝑥,𝐴), if and only if there exists a set 𝑃∈𝐹(𝐼), 𝑃={𝑝=(𝑝_𝑖)∶𝑝_𝑖<𝑝_𝑖+1,𝑖∈ℕ} such that 𝐼_𝑊−lim_𝑘→∞𝑑(𝑥,𝐴_𝑝𝑘)=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 𝐼⊆2^ℕ 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 {𝐴_𝑛} is Wijsman 𝐼^∗-Cauchy sequence in 𝑋 under assumption that {𝐴_𝑛} is a Wijsman 𝐼-Cauchy sequence. Let {𝐴_𝑛} be a Wijsman 𝐼-Cauchy sequence. Then by definition, there exists a 𝑁=𝑁(𝜀) such that 𝐴(𝑥,𝜀)=𝑛∈ℕ∶𝑑𝑥,𝐴_𝑛−𝑑𝑥,𝐴_𝑁≥𝜀∈𝐼 (33) for each 𝜀>0 and for each 𝑥∈𝑋.
Let 𝑃_𝑖={𝑛∈ℕ∶|𝑑(𝑥,𝐴_𝑛)−𝑑(𝑥,𝐴_𝑚𝑖)|<1/𝑖}, 𝑖=1,2,… where 𝑚_𝑖=𝑁(1/𝑖). It is clear that 𝑃_𝑖∈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⧵𝑃_𝑖 is finite for all 𝑖. Now we show that lim_{𝑛,𝑚→∞}𝑑𝑥,𝐴_𝑛−𝑑𝑥,𝐴_𝑚=0. (34) To prove this, let 𝜀>0,  𝑥∈𝑋, and 𝑗∈ℕ such that 𝑗>2/𝜀. If 𝑚,𝑛∈𝑃 then 𝑃⧵𝑃_𝑖 is finite set, therefore there exists 𝑘=𝑘(𝑗) such that 𝑑𝑥,𝐴_𝑛−𝑑𝑥,𝐴_𝑚𝑗<1𝑗,𝑑𝑥,𝐴_𝑚−𝑑𝑥,𝐴_𝑚𝑗<1𝑗 (35) for all 𝑚,𝑛>𝑘(𝑗). Hence it follows that 𝑑𝑥,𝐴_𝑛−𝑑𝑥,𝐴_𝑚<𝑑𝑥,𝐴_𝑛−𝑑𝑥,𝐴_𝑚𝑗+𝑑𝑥,𝐴_𝑚−𝑑𝑥,𝐴_𝑚𝑗<𝜀 (36) for 𝑚,𝑛>𝑘(𝑗).
Thus, for any 𝜀>0, there exists 𝑘=𝑘(𝜀) and 𝑛,𝑚∈𝑃∈𝐹(𝐼): 𝑑𝑥,𝐴_𝑛−𝑑𝑥,𝐴_𝑚<𝜀. (37) This shows that the sequences {𝐴_𝑛} is a Wijsman 𝐼^∗-Cauchy sequence.

Theorem 22. Let 𝐼 be an admissible ideal and (𝑋,𝑑) a separable metric space. Then 𝐼^∗_𝑊−lim𝐴_𝑘=𝐴 implies that {𝐴_𝑛} is a Wijsman 𝐼-Cauchy sequence.

Proof. Let 𝐼^∗_𝑊−lim𝐴_𝑘=𝐴. Then by definition there exists a set 𝑀∈𝐹(𝐼),  𝑀={𝑚=(𝑚_𝑖)∶𝑚_𝑖<𝑚_𝑖+1,𝑖∈ℕ}⊂ℕ such that lim_𝑘→∞𝑑𝑥,𝐴_𝑚𝑘=𝑑(𝑥,𝐴) (38) for each 𝜀>0 and for each 𝑥∈𝑋, and 𝑘,𝑝>𝑘₀, 𝑑𝑥,𝐴_𝑚𝑘−𝑑𝑥,𝐴_𝑚𝑝<𝑑𝑥,𝐴_𝑚𝑘−𝑑(𝑥,𝐴)+𝑑𝑥,𝐴_𝑚𝑝(<𝜀−𝑑𝑥,𝐴)2+𝜀2=𝜀. (39) Therefore, lim_{𝑘,𝑝→∞}𝑑𝑥,𝐴_𝑚𝑘−𝑑𝑥,𝐴_𝑚𝑝=0. (40) Hence, {𝐴_𝑛} 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 {𝐴_𝑛}_𝑛∈ℕ of elements of 𝑋𝐼_𝑊−lim𝐴_𝑛=𝐴 implies 𝐼^∗_𝑊−lim𝐴_𝑛=𝐴.

Proof. Suppose that 𝐼 satisfies condition (AP). Let 𝐼_𝑊−lim𝐴_𝑛=𝐴. Then 𝑇(𝜀,𝑥)=𝑛∈ℕ∶𝑑𝑥,𝐴_𝑛−𝑑(𝑥,𝐴)≥𝜀∈𝐼 (41) for each 𝜀>0 and for each 𝑥∈𝑋. Put 𝑇₁=𝑛∈ℕ∶𝑑𝑥,𝐴_𝑛(,𝑇−𝑑𝑥,𝐴)≥1_𝑛=1𝑛∈ℕ∶𝑛≤𝑑𝑥,𝐴_𝑛(<1−𝑑𝑥,𝐴)𝑛−1 (42) for 𝑛≥2, and 𝑛∈ℕ. Obviously 𝑇_𝑖∩𝑇_𝑗=∅ for 𝑖≠𝑗. By condition (AP) there exists a sequence of sets {𝑉_𝑛}_𝑛∈ℕ such that 𝑇_𝑗Δ𝑉_𝑗 are finite sets for 𝑗∈ℕ and 𝑉=⋃^∞_𝑗=1𝑉_𝑗∈𝐼. It is sufficient to prove that for 𝑀=ℕ⧵𝑉, 𝑀={𝑚=(𝑚_𝑖)∶𝑚_𝑖<𝑚_𝑖+1,𝑖∈ℕ}∈𝐹(𝐼), we have lim_𝑘→∞𝑑(𝑥,𝐴_𝑚𝑘)=𝑑(𝑥,𝐴).
Let 𝛾>0. Choose 𝑘∈ℕ such that 1/(𝑘+1)<𝛾. Then 𝑛∈ℕ∶𝑑𝑥,𝐴_𝑛−𝑑(𝑥,𝐴)≥𝛾⊂^𝑘+1_𝑗=1𝑇_𝑗. (43) Since 𝑇_𝑗Δ𝑉_𝑗, 𝑗=1,2,… are finite sets, there exists 𝑛₀∈ℕ such that ^𝑘+1_𝑗=1𝑉_𝑗∩𝑛∈ℕ∶𝑛>𝑛₀=^𝑘+1_𝑗=1𝑇_𝑗∩𝑛∈ℕ∶𝑛>𝑛₀. (44) If 𝑛>𝑛₀ and 𝑛∉𝑉, so 𝑛∉⋃^𝑘+1_𝑗=1𝑉_𝑗 and by (44) 𝑛∉⋃^𝑘+1_𝑗=1𝑇_𝑗. But then |𝑑(𝑥,𝐴_𝑛)−𝑑(𝑥,𝐴)|<1/(𝑛+1)<𝛾 for each 𝑥∈𝑋, so we have lim_𝑘→∞𝑑(𝑥,𝐴_𝑚𝑘)=𝑑(𝑥,𝐴).

3. Wijsman 𝐼-Limit Points and Wijsman 𝐼-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 𝐼⊆2^ℕ a proper ideal in ℕ and (𝑋,𝑑) a separable metric space. For any nonempty closed subsets 𝐴_𝑛, 𝐵_𝑛⊂𝑋, one says that the sequences {𝐴_𝑛} and {𝐵_𝑛} are almost equal with respect to 𝐼 if 𝑛∈ℕ∶𝐴_𝑛≠𝐵_𝑛∈𝐼, (45) and we write 𝐼-a.a.n 𝐴_𝑛=𝐵_𝑛.

Definition 25. Let 𝐼⊆2^ℕ be a proper ideal in ℕ and let (𝑋,𝑑) be a separable metric space; 𝐴_𝑛 is nonempty closed subset of 𝑋. If {𝐴_𝑛}_𝐾 is subsequence of {𝐴_𝑛} and 𝐾∶={𝑛(𝑗)∶𝑗∈ℕ}, then we abbreviate {𝐴_𝑛𝑗} by {𝐴_𝑛}_𝐾. If 𝐾∈𝐼, then {𝐴_𝑛}_𝐾 subsequence is called thin subsequence of {𝐴_𝑛}. If 𝐾∉𝐼, then {𝐴_𝑛}_𝐾 subsequence is called nonthin subsequence of {𝐴_𝑛}.

Definition 26. Let 𝐼⊆2^ℕ be a proper ideal in ℕ and let (𝑋,𝑑) be a separable metric space, for any nonempty closed subsets 𝐴_𝑘⊂𝑋. One has the following.(i)𝐴∈𝑋 is said to be a Wijsman 𝐼-limit point of {𝐴_𝑛} provided that there is a set 𝑀={𝑚=(𝑚_𝑖)∶𝑚_𝑖<𝑚_𝑖+1,𝑖∈ℕ}⊂ℕ such that 𝑀∉𝐼 and for each 𝑥∈𝑋  lim_𝑘→∞𝑑(𝑥,𝐴_𝑚𝑘)=𝑑(𝑥,𝐴).(ii)𝐴∈𝑋 is said to be a Wijsman 𝐼-cluster point of {𝐴_𝑛} if and only if for each 𝜀>0, for each 𝑥∈𝑋, we have 𝑛∈ℕ∶𝑑𝑥,𝐴_𝑛−𝑑(𝑥,𝐴)<𝜀∉𝐼. (46)

Denote by 𝐼_𝑊(Λ_{𝐴𝑛}), 𝐼_𝑊(Γ_{𝐴𝑛}), and 𝐿_{𝐴𝑛} the set of all Wijsman 𝐼-limit, Wijsman 𝐼-cluster, and Wijsman limit points of {𝐴_𝑛}, respectively.

For the sequences {𝐴_𝑛},  𝐼_𝑊(Γ_{𝐴𝑛})⊆𝐼_𝑊(𝐿_{𝐴𝑛}). Let 𝐴∈𝐼_𝑊(Γ_{𝐴𝑛}). Then for each sequence {𝐴_𝑛}⊂𝑋, we have lim_𝑘→∞𝑑(𝑥,𝐴_𝑚𝑘)=𝑑(𝑥,𝐴) which means that 𝐴∈𝐿_{𝐴𝑛}.

Theorem 27. Let 𝐼⊆2^ℕ be a proper ideal in ℕ and let (𝑋,𝑑) be a separable metric space. Then for each sequence {𝐴_𝑛}⊂𝑋 one has 𝐼_𝑊(Λ_{𝐴𝑛})⊂𝐼_𝑊(Γ_{𝐴𝑛}).

Proof. Let 𝐴∈𝐼_𝑊(Λ_{𝐴𝑛}). Then, there exists 𝑀={𝑚₁<𝑚₂<⋯}⊂ℕ such that 𝑀={𝑚=(𝑚_𝑖)∶𝑚_𝑖<𝑚_𝑖+1,𝑖∈ℕ}∉𝐼 and lim_𝑘→∞𝑑𝑥,𝐴_𝑚𝑘=𝑑(𝑥,𝐴). (47) According to (47), there exists 𝑘₀∈ℕ such that for each 𝜀>0, for each 𝑥∈𝑋 and 𝑘>𝑘₀, |𝑑(𝑥,𝐴_𝑚𝑘)−𝑑(𝑥,𝐴)|<𝜀. Hence, 𝑘∈ℕ∶𝑑𝑥,𝐴_𝑚𝑘(𝑚−𝑑𝑥,𝐴)<𝜀⊇𝑀⧵₁,𝑚₂,…,𝑚_𝑘𝑜. (48) Then, the set on the right hand side of (48) does not belong to 𝐼; therefore 𝑘∈ℕ∶𝑑𝑥,𝐴_𝑚𝑘−𝑑(𝑥,𝐴)<𝜀∉𝐼 (49) which means that 𝐴∈𝐼_𝑊(Γ_{𝐴𝑛}).

Theorem 28. Let 𝐼⊆2^ℕ be a proper ideal in ℕ and let (𝑋,𝑑) be a separable metric space. Then for each sequence {𝐴_𝑛}⊂𝑋 one has 𝐼_𝑊(Γ_{𝐴𝑛})⊆𝐿_{𝐴𝑛}.

Proof. Let 𝐴∈𝐼_𝑊(Γ_{𝐴𝑛}). Then for each 𝜀>0 and for each 𝑥∈𝑋, we have 𝑛∈ℕ∶𝑑𝑥,𝐴_𝑛−𝑑(𝑥,𝐴)<𝜀∉𝐼. (50) Let 𝐾_𝑛∶=𝑛∈ℕ∶𝑑𝑥,𝐴_𝑛(<1−𝑑𝑥,𝐴)𝑛 (51) for 𝑛∈ℕ. {𝐾_𝑛}^∞_𝑛=1 is decreasing sequence of infinite subsets of ℕ. Hence 𝐾={𝑛=(𝑛_𝑖)∶𝑛_𝑖<𝑛_𝑖+1,𝑖∈ℕ}∉𝐼 such that lim_𝑛→∞𝑑(𝑥,𝐴_𝑛𝑖)=𝑑(𝑥,𝐴) which means that 𝐴∈𝐿_{𝐴𝑛}.

Theorem 29. Let 𝐼⊆2^ℕ a proper ideal in ℕ, (𝑋,𝑑) a separable metric space, and 𝐴_𝑘,𝐵_𝑘 nonempty subsets of 𝑋. If {𝐴_𝑘}={𝐵_𝑘}  𝐼-a.a.k for 𝑘∈ℕ, then 𝐼_𝑊(Γ_{𝐴𝑘})=𝐼_𝑊(Γ_{𝐵𝑘}) and 𝐼_𝑊(Λ_{𝐴𝑘})=𝐼_𝑊(Λ_{𝐵𝑘}).

Proof. If {𝐴_𝑘}={𝐵_𝑘} a.a.k for 𝑘∈ℕ, then 𝐾∶=𝑘∈ℕ∶𝐴_𝑘≠𝐵_𝑘∈𝐼 (52) Let 𝐴∈𝐼_𝑊(Γ_{𝐴𝑘}). For each 𝜀>0 and for each 𝑥∈𝑋 we have 𝑘∈ℕ∶𝑑𝑥,𝐴_𝑘−𝑑(𝑥,𝐴)<𝜀∉𝐼, (53)∀𝜀>0. If {𝐴_𝑘}={𝐵_𝑘}  𝐼-a.a.k, then {𝑘∈ℕ∶|𝑑(𝑥,𝐵_𝑘)−𝑑(𝑥,𝐴)|<𝜀}∉𝐼 which means that 𝐴∈𝐼_𝑊(Γ_{𝐵𝑘}); hence 𝐼_𝑊(Γ_{𝐴𝑘}⊂𝐼_𝑊(Γ_{𝐵𝑘}). Similarly we can also prove that 𝐼_𝑊(Γ_{𝐵𝑘})⊂𝐼_𝑊(Γ_{𝐴𝑘}. So we have 𝐼_𝑊(Γ_{𝐴𝑘}=𝐼_𝑊(Γ_{𝐵𝑘}).
Now, we show that 𝐼_𝑊(Λ_{𝐴𝑘})=𝐼_𝑊(Λ_{𝐵𝑘}). Let 𝐴∈𝐼_𝑊(Λ_{𝐴𝑘}). Then there exists a set 𝑀={𝑚=(𝑚_𝑖)∶𝑚_𝑖<𝑚_𝑖+1,𝑖∈ℕ}⊂ℕ such that 𝑀∉𝐼 and lim_𝑘→∞𝑑𝑥,𝐴_𝑚𝑘=𝑑(𝑥,𝐴),𝑀=𝑘∶𝑘∈𝑀and𝐴_𝑘≠𝐵_𝑘∪𝑘∶𝑘∈𝑀and𝐴_𝑘=𝐵_𝑘, (54)𝑀∉𝐼, and hence {𝑘∶𝑘∈𝑀and𝐴_𝑘=𝐵_𝑘}∉𝐼. Then there exists 𝑃=𝑝=𝑝_𝑖∶𝑝_𝑖<𝑝_𝑖+1,𝑖∈ℕ∉𝐼 (55) such that lim_𝑘→∞𝑑𝑥,𝐵_𝑝𝑘=𝑑(𝑥,𝐴) (56) which means that 𝐴∈𝐼_𝑊(Λ_{𝐵𝑘}). Similarly we can also prove that 𝐼_𝑊(Λ_{𝐵𝑘})⊂𝐼_𝑊(Λ_{𝐴𝑘}). Therefore we have 𝐼_𝑊(Λ_{𝐴𝑘})=𝐼_𝑊(Λ_{𝐵𝑘}).