Some Results on Wijsman Ideal Convergence in Intuitionistic Fuzzy Metric Spaces

In 1951, Fast [1] initiated the theory of statistical convergence. It is an extremely effective tool to study the convergence of numerical problems in sequence spaces by the idea of density. Statistical convergence of the sequence of sets was examined by Nuray and Rhoades [2]. Ulusu and Nuray [3] studied the Wijsman lacunary statistical convergence sequence of sets and connected with the Wijsman statistical convergence. Esi et al. [4] introduced the Wijsman λ-statistical convergence of interval numbers. Kostyrko et al. [5] generalized the statistical convergence and introduced the notion of ideal J–convergence. Salát et al. [6, 7] investigated it from the sequence space viewpoint and associated with the summability theory. Further, it was analyzed by Khan et al. [8] with the help of a bounded operator. In 2008, Das et al. [9] analyzed J and J ∗–convergence for double sequences. Kisi and Nuray [10] initiated new convergence definitions for the sequence of sets. Furthermore, Gümüş [11] studied the Wijsman ideal convergent sequence of sets using the Orlicz function. In 1965, Zadeh [12] started the fuzzy sets theory. This theory has proved its usefulness and ability to solve many problems that classical logic was unable to handle. Karmosil et al. [13] introduced the fuzzy metric space, which has the most significant applications in quantum particle physics. Afterward, numerous researchers have studied the concept of fuzzy metric spaces in different ways. George et al. [14, 15] modified the notion of fuzzy metric space and determined a Hausdorff topology for fuzzy metric spaces. Atanassov [16] generalized the fuzzy sets and introduced the notion of intuitionistic fuzzy sets in 1986. Park [17] examined the notion of IFMS, and Saadati and Park [18] further analyzed the intuitionistic fuzzy topological spaces. Moreover, statistical convergence, ideal convergence, and different properties of sequences in intuitionistic fuzzy normed spaces were examined by Mursaleen et al. [19–21]. Also one can refer to Sengül and Et [22], Sengül et al. [23], Et and Yilmazer [24], Mohiuddine and Alamri [25], andMohiuddine et al. [26, 27].


Introduction
In 1951, Fast [1] initiated the theory of statistical convergence. It is an extremely effective tool to study the convergence of numerical problems in sequence spaces by the idea of density. Statistical convergence of the sequence of sets was examined by Nuray and Rhoades [2]. Ulusu and Nuray [3] studied the Wijsman lacunary statistical convergence sequence of sets and connected with the Wijsman statistical convergence. Esi et al. [4] introduced the Wijsman λ-statistical convergence of interval numbers. Kostyrko et al. [5] generalized the statistical convergence and introduced the notion of ideal J -convergence. Salát et al. [6,7] investigated it from the sequence space viewpoint and associated with the summability theory. Further, it was analyzed by Khan et al. [8] with the help of a bounded operator. In 2008, Das et al. [9] analyzed J and J * -convergence for double sequences. Kisi and Nuray [10] initiated new convergence definitions for the sequence of sets. Furthermore, Gümüş [11] studied the Wijsman ideal convergent sequence of sets using the Orlicz function.
In 1965, Zadeh [12] started the fuzzy sets theory. This theory has proved its usefulness and ability to solve many problems that classical logic was unable to handle. Karmosil et al. [13] introduced the fuzzy metric space, which has the most significant applications in quantum particle physics.
Afterward, numerous researchers have studied the concept of fuzzy metric spaces in different ways. George et al. [14,15] modified the notion of fuzzy metric space and determined a Hausdorff topology for fuzzy metric spaces. Atanassov [16] generalized the fuzzy sets and introduced the notion of intuitionistic fuzzy sets in 1986. Park [17] examined the notion of IFMS, and Saadati and Park [18] further analyzed the intuitionistic fuzzy topological spaces. Moreover, statistical convergence, ideal convergence, and different properties of sequences in intuitionistic fuzzy normed spaces were examined by Mursaleen et al. [19][20][21]. Also one can refer to Sengül and Et [22], Sengül et al. [23], Et and Yilmazer [24], Mohiuddine and Alamri [25], and Mohiuddine et al. [26,27].

Preliminaries
We recall some concepts and results which are needed in sequel.
Definition 1 [5]. A family of subsets J ⊆ 2 ℕ is known as an ideal in a non-empty set ℕ , if for any C, D ∈ J ⇒ C ∪ D ∈ J , (3) for any C ∈ J and D ⊆ C, ⇒D ∈ J .
Remark 2 [5]. An ideal J is known as non-trivial if ℕ ∉ J . A nontrivial ideal J is known as admissible if ffng: n ∈ ℕg ∈ J .
Definition 3 [5]. A nonempty subset F ⊆ 2 ℕ is known as filter in ℕ if (1) for every ∅∉ F, for every C ∈ F with C ⊆ D, one obtain D ∈ F: Proposition 4 [5]. For every ideal J , there is a filter FðJ Þ associated with J defined as follows: Definition 5 [5]. Let fC 1 , C 1 , ⋯g be a mutually disjoint sequence of sets of J . Then, there is sequence of sets fD 1 , D 1 , ⋯g so that ∪ ∞ j=1 D j ∈ J and each symmetric difference C j △D j ðj = 1, 2, ⋯Þ is finite. In this case, admissible ideal J is known as property ðAPÞ.
Lemma 6 [28]. Suppose J be an admissible ideal alongside property ðAPÞ. Let a countable collection of subsets fC k g ∞ k=1 of positive integer ℕ in such a way that C k ∈ FðJ Þ. Then, there exists a set C ⊂ ℕ such that C \ C k is finite for all k ∈ ℕ and C ∈ FðJ Þ. Definition 7 [29]. Let ðM, dÞ be a metric space and fC k g be a sequence of nonempty closed subsets of M which is said to be Wijsman convergent to the closed C of M, if In other words, W − lim In 2012, Nuray and Rhoades [2] initiated the theory of Wijsman statistical convergence for a sequence of sets. Furthermore, Kisi and Nuray [10] extended it into J -convergence.
Definition 8 [10]. Suppose ðM, dÞ is a metric space. A nonempty closed subset fC k g of M is known as Wijsman Jconvergent to a closed set C, if for every x ∈ M, one has Hence, one writes J W − lim k→∞ C k = C.
Definition 9 [10]. Suppose ðM, dÞ is a metric space. A nonempty closed subset fC k g of M is known as Wijsman J -Cauchy if for each x ∈ M , there exists a positive integer m = mðϵÞ so that the set Definition 10 [10]. Suppose ðM, dÞ is a separable metric space and fC k g, C is nonempty closed subsets of M. A sequence fC k g is known as Wijsman J * -convergent to C if and only if ∃P ∈ FðJ Þ and P = fp = ðp j < p j+1 , j ∈ ℕÞg ⊂ ℕ in such a manner that One writes J * W − lim k→∞ C k = C.
Definition 11 [10]. Suppose ðM, dÞ is a separable metric space and J is an admissible ideal. A sequence fC k g of nonempty closed subsets of M is known as the Wijsman Remark 12 [10]. In general, the Wijsman topology is not first-countable, if sequence of nonempty sets fC k g is Wijsman convergent to set C, then every subsequence of fC k g may not be convergent to C. Every subsequence of the convergent sequence fC k g converges to the same limit provided that M is a separable metric space.
Definition 15 [18]. Let ðM, η, φ, * ,⋄Þ be an IFMS and C be a nonempty subset of M. For all s > 0 and x ∈ M, we define and where ηðx, C, sÞ and φðx, C, sÞ are the degree of nearness and nonnearness of x to C at s, respectively.
Saadati and Park [18] studied the notion of convergence sequence with respect to IFMS which are defined as follows: Definition 16 [18]. Let ðM, η, φ, * ,⋄Þ be an IFMS. A sequence Definition 17 [20]. An IFMS ðM, η, φ, * ,⋄Þ is known as separable if it contains a countable dense subset, i.e., there is a countable set fx k g along with subsequent property: for any s > 0 and for all ξ ∈ M, there is at least one x n in order that 3. Wijsman J and J * -convergence in IFMS Throughout this section, we denote J to be the admissible ideal in ℕ. We begin with the following definitions as follows.
Definition 18. Let ðM, η, φ, * ,⋄Þ be an IFMS. A sequence of sets fC k g is said be Wijsman convergent to C if for every ϵ > 0 and s > 0 there exists k 0 ∈ ℕ such that The set of all Wijsman limit point of the sequence fC k g is denoted by L fC k g .
Definition 19. Let ðM, η, φ, * ,⋄Þ be an IFMS and J be a proper ideal in ℕ. A sequence fC k g of nonempty closed subsets of M is known as Wijsman J -convergent to C with respect to IFM ðη, φÞ, if for every 0 < ϵ < 1, for each x ∈ M and for all s > 0 such that We write ðη, φÞ − J W − lim Example 20. Suppose ðM, η, φ, * ,⋄Þ is an IFMS and C, fC k g is nonempty closed subsets of M. Assume M = ℝ 2 and fC k g are sequence defined by Since Therefore, the sequence of sets fC k g is Wijsman statistical convergent to the set C. Now, define the set S as If we assume J = J d , then the Wijsman statistical convergence coincides with the Wijsman ideal convergence. Therefore, Definition 21. Let ðM, η, φ, * ,⋄Þ be a separable IFMS and J be an admissible ideal in ℕ. A sequence fC k g of nonempty closed subsets of M is known as Wijsman J -Cauchy with respect to IFM ðη, φÞ, if for each 0 < ϵ < 1, for each x ∈ M and for all s > 0, ∃ l = lðϵÞ such that Definition 22. Let ðM, η, φ, * ,⋄Þ be a separable IFMS and fC k g be any nonempty closed subset of M. The sequence fC k g is known as Wijsman J * -Cauchy with respect to IFM ðη, φÞ, if there exists P = fp = ðp j Þ: p j < p j+1 , j ∈ ℕg ⊂ ℕ and P ∈ FðJ Þ with the result that the subsequence C P = fC p k g is Wijsman Cauchy in M, i.e.

Journal of Function Spaces
and lim k,l→∞ Definition 23. Let ðM, η, φ, * ,⋄Þ be a separable IFMS and J be an proper ideal in ℕ. Let fC k g be nonempty closed subsets of M. The sequence fC k g is known as Wijsman J * -convergent to C with respect to ðη, φÞ, if there exists P ∈ FðJ Þ, where P = fp = ðp j Þ: p j < p j+1 , j ∈ ℕg ⊂ ℕ such that for each s > 0, we have and In such case, we write ðη, φÞ − J * W − limC k = C.
In the following theorem, we prove that every Wijsman J -convergent implies the Wijsman J -Cauchy condition in IFMS: Theorem 24. Let ðM, η, φ, * ,⋄Þ be a separable IFMS and let J be an arbitrary admissible ideal. Then, every Wijsman J -convergent sequence of closet sets fC k g is Wijsman J -Cauchy with respect to IFM ðη, φÞ.
Proof. Suppose ðη, φÞ − J W − lim k→∞ C k = C. Then, for every 0 < ϵ < 1, for all s > 0 and x ∈ X, the set belongs to J . Since J is an admissible ideal, then there exists k 0 ∈ ℕ with the result that k 0 ∉ Uðϵ, sÞ. Now, suppose that Considering the inequality and Observe that if k ∈ Vðϵ, sÞ, therefore and From another point of view, since k 0 ∉ Uðϵ, sÞ, we obtain We achieve that Hence, k ∈ Uðϵ, sÞ. This implies that Uðϵ, sÞ ⊂ Vðϵ, sÞ ∈ J for every 0 < ϵ < 1 and for all s > 0 and x ∈ M. Therefore, Vðϵ, sÞ ∈ J , so the sequence is fC k g which is Wijsman J -Cauchy. Theorem 25. Let ðM, η, φ, * ,⋄Þ be a separable IFMS and let J be an admissible ideal. Then, every Wijsman J * -Cauchy sequence of closed sets is Wijsman J -Cauchy.
Proof. Suppose that sequence fC k g is Wijsman J * -Cauchy with respect to IFM ðη, φÞ. Then, for each x ∈ M and for each 0 < ϵ < 1, there exists P ∈ FðJ Þ, where P = fðp j Þ: p j < p j+1 , j ∈ ℕg in such a way that and Suppose N = NðϵÞ = p k 0 +1 . Therefore, for each ϵ > 0, one obtains Journal of Function Spaces Now, suppose that K = ℕ \ P. Clearly, K ∈ J and Hence, for all s > 0 and for each 0 < ϵ < 1, one can determine N = NðϵÞ so that Qðϵ, sÞ ∈ J , that is, sequence fC k g is Wijsman J -Cauchy.

Theorem 26.
Let J be an admissible ideal including property (AP) and ðM, η, φ, * ,⋄Þ be a separable IFMS. Then, the notion of Wijsman J * -Cauchy sequence of sets coincides with Wijsman J -Cauchy with respect to ðη, φÞ and vice-versa.
Proof. The direct part is already proven in Theorem 25. Now, suppose that sequence fC k g is Wijsman J -Cauchy sequence with respect to IFM ðη, φÞ. Then by definition, if for every 0 < ϵ < 1, for each x ∈ X and for all s > 0, there exists a m = mðϵÞ such that Now, suppose that where m j = mð1/jÞ, j = 1, 2, 3, ⋯. Obviously, for j = 1, 2, 3 ⋯ , P j ðϵ, sÞ ∈ FðJ Þ. Using Lemma 6, there exists P ⊂ ℕ so that P ∈ FðJ Þ and P \ P j are finite for all j. Now, we prove that and lim k,l→∞ To show the above equations, let ϵ > 0, and r ∈ ℕ such that r > 2/ϵ. If k, l ∈ P, then P \ P j is a finite set; therefore, there exists w = wðrÞ in order that and for all k, l > wðrÞ. Then, the above inequalities follow that for k, l > wðrÞ and Therefore, for each ϵ > 0, ∃w = wðϵÞ and k, l ∈ P ∈ FðIÞ, we achieve This proves that the sequence fC k g is a Wijsman J * -Cauchy.
Theorem 27. Let ðM, η, φ, * ,⋄Þ be a separable IFMS and let J be an admissible ideal. Then implies that sequence fC k g is a Wijsman J -Cauchy sequence with respect to IFM ðη, φÞ.
Proof. Suppose that ðη, φÞ − J * W − lim k→∞ C k = C. Then, there exists P = fp = ðp j Þ: p j < p j+1 , j ∈ ℕg ⊂ ℕ with P ∈ F ðJ Þ so that C P = fC p k g for any ϵ > 0 and k, l > k 0 . Suppose r ∈ ℕ and ϵ > 0 in such a way that r > 2/ϵ. If k, l ∈ P, then P \ P j is a finite set; therefore, there exists kðrÞ = k so that and Therefore, and lim k,l→∞ Hence, sequence fC k g is Wijsman J -Cauchy with respect to IFM ðη, φÞ.

Wijsman J -cluster points and Wijsman Jlimit points in IFMS
Throughout this section, we denote J to be the proper ideal in ℕ and define Wijsman J -cluster and J -limit points of the sequence of sets in intuitionistic fuzzy metric space and obtain some results.
Definition 28. Let ðM, η, φ, * ,⋄Þ be a separable IFMS. An element C ∈ M is known as the Wijsman J -cluster point of fC k g if and only if for any x ∈ M and for all ϵ, s > 0, one has We denote J ðη,φÞ W ðΓ fC k g Þ as the collection of all Wijsman J -cluster points.
Definition 29. Let ðM, η, φ, * ,⋄Þ be a separable IFMS. An element C ∈ M is known as Wijsman J -limit point of sequence fC k g of nonempty closed subsets of M provided that P = fp = ðp j Þ: p j < p j+1 , j ∈ ℕg ⊂ ℕ in such a way that P ∉ J , and for any x ∈ M and s > 0, we obtain We denote J ðη,φÞ W ðΛ fC k g Þ as the collection of all Wijsman J -limit points.
Proof. Suppose C ∈ J ðη,φÞ W ðΛ fC k g Þ. Then, there exists P = fp 1 < p 2 <⋯g ⊂ ℕ such that P = fp = ðp j Þ: p j < p j+1 , j ∈ ℕg ∉ J and for all s > 0 and x ∈ M, we have and According to Equations (54) and (55), there exists k 0 ∈ ℕ so that for each ϵ > 0 and for any x ∈ X and k > k 0 and φ x, Hence, Then, the right-hand side of (58) does not belong to J , and then which means that C ∈ J ðη,φÞ W ðΓ fC k g Þ.

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