Abstract
In this paper, we introduce the concept of mapping on hesitant fuzzy soft multisets and present some results for this type of mappings. The notions of inverse image and identity mapping are defined, and their basic properties are investigated. Hence, kinds of mappings and the composition of two hesitant fuzzy soft multimapping with the same dimension are presented. The concept of hesitant fuzzy soft multitopology is defined, and certain types of hesitant fuzzy soft multimapping such as continuity, open, closed, and homeomorphism are presented in detail. Also, their properties and results are studied. In addition, the concept of hesitant fuzzy soft multiconnected spaces is introduced.
1. Introduction
Since the introduction of fuzzy sets by Zadeh [1], several extensions of this concept have been introduced. The most agreed one may be Atanassov’s intuitionistic fuzzy set (briefly, IFS or A-IFS) [2]. IFSs have the benefit that allows the user to model some uncertainties on the membership function of the elements. That is, fuzzy sets require a membership degree for each element in the universe set, whereas an IFS permits us to include some hesitation on this value. This is modeled with two functions that define an interval. Type 2 fuzzy sets [3, 4] are a generalization of the former, where the membership of a given element is presented as a fuzzy set. Other generalizations, such as type n fuzzy sets exist (see [3] for details about type n fuzzy sets). Dubois and Prade [3] report that Mizumoto and Tanaka [4] were the first to study type 2 fuzzy sets. Fuzzy multisets are another generalization of fuzzy sets. They are based on multisets (elements can be repeated in a multiset, for short, mset). In fuzzy multisets, the membership can be partial (instead of Boolean as for standard multisets). Tokat and Osmanoglu [5] introduced the concept of a soft mset as , where is a set of parameters and is a power set of an mset . In this paper, we adopt the notion of a soft mset in [5], since this way is better than the other [6, 7]. In 2013, Tokat et al. [7] introduced the concept of soft msets as a combination between soft sets and msets. Furthermore, in [7], the soft multitopology and its basic properties were given. Moreover, the soft multiconnectedness was studied in [5]. Additionally, the soft multicompactness on soft multitopological spaces was presented in [8]. In 2015, El-Sheikh et al. [9] introduced the concept of semicompact soft multispaces and the concept of soft multi-Lindelf spaces. Some other results and properties about soft multisets are presented in [10–12]. The concept of a generalized open soft mset is introduced in soft multitopological spaces, and its properties are presented in [10]. Several authors [13–15] discussed the concept of multisets, its generalizations, and its applications. In 2020, Hashmi et al. [16] introduced the notion of an m-polar neutrosophic set and m-polar neutrosophic topology and their applications to multicriteria decision-making (MCDM) in medical diagnosis and clustering analysis. They introduced a novel approach to census process by using Pythagorean m-polar fuzzy Dombis aggregation operators. Riaz and Hashmi [17] introduced the notion of linear Diophantine fuzzy set (LDFS) and its applications towards MCDM problem. Linear Diophantine fuzzy set (LDFS) is superior to IFSs, PFSs, and q-ROFSs. Riaz and Tehrim [18] introduced the concept of bipolar fuzzy soft mappings with application to bipolar disorders. Tehrim and Riaz [19] presented a novel extension of the TOPSIS method with bipolar neutrosophic soft topology and its applications to multicriteria group decision-making (MCGDM). Riaz et al. [20] presented the multiattribute group decision-making (MAGDM) methods to a hesitant fuzzy soft set. Moreover, Riaz et al. [21] developed the topological structure on a soft rough set by using pairwise soft rough approximations. The multicriteria group decision-making methods are introduced by using N-soft set and N-soft topology to deal with uncertainties in the real-world problems [22].
Recently, the concept of hesitant fuzzy sets was introduced firstly in 2010 by Torra [23] which permits the membership to have a set of possible values and presents some of its basic operations in expressing uncertainty and vagueness. Torra et al. [24] established the similarities and differences with the hesitant fuzzy sets and the previous generalization of fuzzy sets such as intuitionistic fuzzy sets, type 2 fuzzy sets, and type n fuzzy sets. Therefore, other authors [25, 26] introduced the concept of hesitant fuzzy soft sets, and they presented some of the applications in decision-making problems. In 2015, Dey and Pal [27] presented the concept of hesitant multifuzzy soft topological space. In 2019, Kandil et al. [28] introduced some important and basic issues of hesitant fuzzy soft multisets and studied some of its structural properties such as the neighborhood hesitant fuzzy soft multisets, interior hesitant fuzzy soft multisets, hesitant fuzzy soft multitopological spaces, and hesitant fuzzy soft multibasis. Finally, they showed how to apply the concept of hesitant fuzzy soft multisets in decision-making problems.
The main goal of this paper is to introduce the definition of mapping on hesitant fuzzy soft multisets and present some results for this form of mappings. The notions of inverse image and identity mapping are introduced, and their basic properties are investigated in detail. The types of mappings are also given on hesitant fuzzy soft multisets, and their properties are established. Therefore, the composition of two hesitant fuzzy soft multimapping with the same dimension is presented. In addition, the concepts of hesitant fuzzy soft multitopologies and hesitant fuzzy soft multi-subspaces are introduced. Some types of hesitant fuzzy soft multimapping such as continuity, open, closed, and homeomorphism are presented in detail. Also, their properties and results are investigated. Finally, the concept of hesitant fuzzy soft multiconnected space is introduced.
2. Preliminaries
The aim of this section is to present the basic concepts and properties of multisets, soft multisets, hesitant fuzzy sets, and hesitant fuzzy soft multisets.
Definition 1. (see [29]). An mset drawn from the set is represented by a count function defined as : , where represents the set of nonnegative integers.
Here, is the number of occurrences of the element in the mset . The mset is drawn from the set , and it is written as , where is the number of occurrences of the element , in the mset .
Definition 2. (see [29]). A domain is defined as a set of elements from which msets are constructed. The mset space is the set of all msets whose elements are in such that no element in the mset occurs more than times.
The mset space is the set of all msets over a domain such that there is no limit on the number of occurrences of an element in an mset. If , then .
Definition 3. (see [29]). Let be an mset drawn from the set . If , then is called an empty mset and denoted by , i.e., .
Definition 4. (see [5]). Let be a universal multiset, be a set of parameters, and . Then, an ordered pair is called a soft mset, where is a mapping given by ; is the power set of an mset . For all , mset is represented by count function , where represents the set of nonnegative integers and represents the support set of .
Definition 5. (see [5]). Let and be two soft msets over . Then,(1) is said to be a sub-soft mset of and denoted by if(i)(ii), for all , (2)Two soft msets and over are equal if is a sub-soft mset of and is a sub-soft mset of .(3)The union of two soft msets and over is the soft mset , where and , for all , . It is denoted by .(4)The intersection of two soft msets and over is the soft mset , where and, for all , . It is denoted by .(5)A soft mset over is said to be a null soft mset and denoted by if for all , .(6)A soft mset over is said to be an absolute soft mset and denoted by if for all , .
Definition 6. (see [5]). The complement of a soft mset is denoted by and is defined by , where is a mapping given by for all , where , for all .
Definition 7. (see [5]). Let be a universal mset and be a set of parameters. Then, the collection of all soft msets over with parameters from is called a soft multiclass and is denoted by .
Definition 8. (see [23]). Let be a reference set, then a hesitant fuzzy (briefly, an ) set is defined in terms of a function from into the power set of .
Definition 9. (see [23]). Let , and be hesitant fuzzy sets over a set . Then, the following operations are defined:(1)Full set : for all (2)Null set : for all (3)Lower bound: (4)Upper bound: (5)-Upper bound: (6)-Lower bound: (7)Complement: (8)Union: or, equivalently, for (9)Intersection: or, equivalently, for
Definition 10. (see [28]). A hesitant fuzzy multiset of dimension (briefly, set) on a nonempty mset is denoted by and is defined in terms of when applied to , and is a set of some distinct values in sorting into increasing order, indicating the possible membership degrees of the elements to the multiset .
Definition 11. (see [28]). Let and be two sets on a nonempty mset . is called a hesitant fuzzy submset of if for each , and denoted by .
Definition 12. (see [28]). A pair is a hesitant fuzzy soft mset of dimension if is a mapping from to , where is the set of all hesitant fuzzy msets of dimension defined over an mset and , i.e., for all , and is the membership function of .
Definition 13. (see [28]). An set over is said to be(1)A relative null set and is denoted by , if for all , (2)A relative absolute set and is denoted by , if for all ,
Definition 14. (see [28]). Let and be two hesitant fuzzy soft multisets of dimension , then is called a hesitant fuzzy soft multi-subset (briefly, ) of of dimension if(1)(2) is a hesitant fuzzy submset of , for every , i.e., for all , , Hence, this relationship is denoted by , and is called an superset of .
Definition 15. (see [28]). Let be a hesitant fuzzy soft multitopological space and , be two sets over a hesitant fuzzy soft mset (for short, ). A hesitant fuzzy soft mset is called neighborhood of if there exists an open hesitant fuzzy soft mset such that .
Definition 16. (see [28]). Let be a hesitant fuzzy soft multitopological space and and be two sets over such that . Then, is called an interior hesitant fuzzy soft mset of if is a neighborhood of . Additionally, the union of all interior hesitant fuzzy soft mset of is called the interior of , and it is denoted by .
Theorem 1 (see [28]). Let be a hesitant fuzzy soft multitopological space and , be two sets over , then(1) is the largest open hesitant fuzzy soft mset contained in (2) is an open hesitant fuzzy soft mset if and only if (3)(4)If , then
3. Mappings in Hesitant Fuzzy Soft Multisets
The purpose of this section is to present a concept of mapping in hesitant fuzzy soft multisets. The main properties of the current branch are studied, and some results of this type of sets are established. Also, the concept of inverse mapping in hesitant fuzzy soft multisets is defined. Therefore, the composition of two hesitant fuzzy soft multimappings is introduced. Finally, some examples are used to explain the current definitions in a friendly way.
It should be noted that, in this section, let be a universal set, be a set of parameters, and be a multiset over . The union and intersection of hesitant fuzzy sets are defined by Torra [23], but these definitions did not preserve the dimension, so we introduce the following definitions.
Definition 17. Union of two sets and over is the set , where , for all ,where . It is written as .
Example 1. Let , , and . The hesitant fuzzy soft msets of dimension , , are defined as , , . , , . Hence, such that , , .
Definition 18. Intersection of two sets and over is the set , where , for all ,It is written as .
Example 2. Let , and . The hesitant fuzzy soft msets of dimension , , are defined as , , . , , . Hence, such that , , .
Theorem 2. Let , and be three elements in . Then,(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13) if and only if (14) if and only if (15)If , then (16) if and only if
Proof. Straightforward.
Example 3. From Example 2, the complements of are defined as , , . , , . Then, such that , , .
Also, such that , , . Then, the complement of is , , . Hence, .
Definition 19. Let and be two families of hesitant fuzzy soft msets over msets and with dimension and sets of parameters and , respectively. Let and be two mappings. Now, a mapping is defined as follows: for a hesitant fuzzy soft mset in , is a hesitant fuzzy soft mset in obtained as follows: for and ,Hence, is called an image hesitant fuzzy soft mset with dimension of a hesitant fuzzy soft mset .
Example 4. Let and be two msets, and . Also, let and be two mappings defined as , , , and and , , , and . Choose a hesitant fuzzy soft mset in such as . Then, the hesitant fuzzy soft mset image of under is obtained as as . Similarly, .Then, .
Definition 20. Let be a mapping such that and be two mappings. If is a hesitant fuzzy soft mset in , then the inverse image of is a hesitant fuzzy soft mset in , denoted by , defined as follows: for and ,
Example 5. From Example 4, let . Since , then and . Since , thenHence, the inverse image of is , , .
Definition 21. Let be a mapping such that and be two mappings. Let and be two hesitant fuzzy soft msets in . For , : the union and intersection of two images and in are defined as
Definition 22. Let be a mapping such that and be two mappings. Let and be two hesitant fuzzy soft msets in . For , : the union and intersection of two inverse images and in are defined as
Theorem 3. Let be a mapping such that and be two mappings. If , are two hesitant fuzzy soft msets in and is a family of hesitant fuzzy soft msets in , then(1).(2).(3). In general, .(4). In general, .(5)If , then .
Proof. The proof of parts and are obvious.
(3) For , , let , where , thenandConsidering only the nontrivial case, we haveBy Definition 21, we haveHence, the proof is complete.
(4) Let , where . For , , and by using Definition 19 and considering only the nontrivial case, we haveHence, .
(5) If for nontrivial case in Definition 19, , , then
Remark 3.1. The inclusion in Theorem 3, parts and , cannot be replaced by equality relation. Moreover, the converse of part is not necessarily true as shown in the following example.
Example 6. Let and be two msets, and . Also, let be a mapping such that and be two mappings defined as , , and and , , , and . Then,(1).(2)Let and be two hesitant fuzzy soft msets in . Assume that , where , then . Therefore, In similar way, and . Also, as , but , . Hence, .(3)Let and be two hesitant fuzzy soft msets in . Then, and . Hence, but .
Theorem 4. Let be a mapping such that and be two mappings. If and are two hesitant fuzzy soft msets in and is a family of hesitant fuzzy soft msets in , then(1)(2).(3). In general, .(4). In general, .(5)If , then .
Proof. The proof of parts and is obvious.(3)Let , where . For , , and for nontrivial case, we have By using Definition 22, we haveHence, .(4)Let , where . For , , and for nontrivial case, we have Hence, .(5)If , then for , we haveHence, .
Remark 3.2. The converse in Theorem 4 part is not necessarily true as shown in the following example.
Example 7. Let and be two msets, and . Also, let and be two mappings defined as , , and and , , , and . Choose two hesitant fuzzy soft msets and in such as and . Then, the inverse image of under is obtained as . Also, . Hence, , but .
Definition 23. Let and be two mappings. An mapping is called(1)One-one (or injection) if and are one-one (or injection)(2)Onto (or surjection) if and are onto (or surjection)(3)Bijection if and are bijection
Theorem 5. Let and be two mappings of dimension . Then, and are equal if and only if and .
Proof. Immediate.
Definition 24. Let and be two mappings of dimension . Their composition is also a hesitant fuzzy soft multimapping with dimension from into such that, for every in ,This composition is defined as, for and ,
Theorem 6. Let and be two mappings of dimension . Then,(1) is injection if and are injection or, equivalently, if and are injection(2) is surjection if and are surjection or, equivalently, if and are surjection(3) is bijection if and are bijection or, equivalently, if and are bijection
Proof. (1)Let such that . Therefore, . Since is injection, then . Also, is injection, so . Hence, is injection.(2)Let , then there exists such that as is surjection. Since is also surjection, then there exists such that . Thus, which completes the proof.(3)Immediately by part 1 and 2.
Definition 25. A bijection hesitant fuzzy soft multimapping is called invertable. Also, the inverse of , denoted by , is defined as , for each in , in .
Theorem 7. Let and be two bijection mappings of dimension . Then, .
Proof. If and are bijection mappings with dimension , then there exist and defined as , whenever , , , and , whenever and . Hence, . Since are bijection, then is also bijection. Therefore, exists such that . Also, . Then, .
Theorem 8. Let and be two mappings of dimension . Then,(1), where (2), where
Proof. (1)For , and , we have Hence, .(2)The proof is similar to that of part .
Remark 3.3. The inclusion in Theorem 9 parts and cannot be replaced by equality relation as shown in the following example.
Example 8. (1)From Example 7, .(2)From Example 6 part , but . Hence, .
Corollary 1. Let be an mapping of dimension . Then,(1), where if is surjection(2), where if is injection
Proof. Immediate by using Theorem 8.
Definition 26. A hesitant fuzzy soft multimapping with dimension , where is said to be identity if are identity mappings.
Theorem 9. Let and be two mappings of dimension . Then, for the identity mapping , we have(1)(2)
Proof. Immediate.
4. Continuous Mappings on Hesitant Fuzzy Soft Multispaces
The aim of this section is to introduce the concept of hesitant fuzzy soft multitopology. Therefore, some types of hesitant fuzzy soft multimapping are presented in detail such as continuity, open, closed, and homeomorphism. Also, their properties and results are obtained.
Definition 27. The subcollection of members of is called a hesitant fuzzy soft multitopology of dimension on , if the following conditions are satisfied:(1)(2)If , then (3)If , , then The pair is called a hesitant fuzzy soft multitopological space. Each member of is called an open hesitant fuzzy soft mset. Also, the complement of an open hesitant fuzzy soft mset is called closed. The family of all closed hesitant fuzzy soft msets is denoted by .
Definition 28. Let be a hesitant fuzzy soft multitopological space. A subfamily is called a hesitant fuzzy soft multibasis for if every member of can be written as arbitrary hesitant fuzzy soft multiunion of some elements of .
Definition 29. Let be a hesitant fuzzy soft multitopological space and be an set over . The closure of is denoted by and defined as
Theorem 10. Let be a hesitant fuzzy soft multitopological space and and be two sets over , then(1) is the smallest closed hesitant fuzzy soft mset containing (2) is a closed hesitant fuzzy soft mset if and only if (3)(4)If , then
Proof. The proof is omitted.
Definition 30. Let be a mapping such that and be two mappings. Let and be two hesitant fuzzy soft multitopologies of dimension over and respectively. A function is said to be(1)Continuous if for all (2)Open if for all (3)Closed if for all (4)Homeomorphism if it is bijection, continuous, and its inverse is also continuous
Theorem 11. Let be a mapping such that and be two mappings. Let and are two hesitant fuzzy soft multitopologies of dimension over and respectively. Then, the following conditions are equivalent:(1) is continuous(2) for all , where is a base for (3) is closed for all closed (4) for all (5), where (6), where .
Proof. The proof is omitted for parts 1, 2, and 3, and these statements are equivalent.(1)Since , by using Theorem 4, we get . Therefore, by using Theorem 1, , but is continuous, so is open. Hence, .(2)By using part , is closed, but . Then, . Now, by using Theorems 3 and 8, we have .(3)Since , by using Theorem 4, we get as is continuous. Therefore, .
Remark 4. The inclusion in Theorem 11 parts , and cannot be replaced by equality relation as shown in the following example.
Example 9. From Example 4, let be a hesitant fuzzy soft multitopological space with dimension , where , (for short, ) and be a hesitant fuzzy soft multitopological space with dimension , where , then(1)Choose is an element in , therefore . Then, . Now, we need to estimate ; so, By the similar way, we get . Hence, . Then, .(2)Choose is an element in . Then, .By the similar way, we get , but . Hence, . Also, one may extend an example for part in Theorem 11 by the same technique.
Theorem 12. Let and be two mappings of dimension and be three topologies over respectively. If are continuous, then is also continuous.
Proof. Immediate.
Theorem 13. Let be an mapping of dimension and be two topologies over , respectively. If for every , where is a base for , then is an open hesitant fuzzy soft multimapping.
Proof. Let .Then, . Therefore, . According to the given hypothesis, ; hence, which completes the proof.
Theorem 14. Let be a mapping such that and be two mappings. Let and are two hesitant fuzzy soft multitopologies of dimension over and , respectively. If , then(1) is open if and only if (2) is closed if and only if
Proof. (1)Let be an open hesitant fuzzy soft multimapping and since . Then, by using Theorem 3, . By taking the interior for both sides, but is open, then . Hence, . Conversely, let and by using the given hypothesis, we have . Then, , but we know that . Hence, is an open hesitant fuzzy soft multimapping.(2)By the similar way of part .
Theorem 15. Let be a mapping such that and be two mappings. Let and are two hesitant fuzzy soft multitopologies of dimension over and , respectively. Then, the following conditions are equivalent:(1) is a homeomorphism hesitant fuzzy soft multimapping(2) is a bijection, open, and continuous hesitant fuzzy soft multimapping(3) is a bijection, closed, and continuous hesitant fuzzy soft multi mapping Proof. Straightforward.
5. Connectedness on Hesitant Fuzzy Soft Multitopological Spaces
The aim of this section is to introduce the concept of hesitant fuzzy soft multiconnected space and present their results and properties in detail. Moreover, the concept of hesitant fuzzy soft multi-subspace is introduced.
Definition 31. Let be a hesitant fuzzy soft multitopological space with dimension . A hesitant fuzzy soft multiseparation of is a pair of no-null open hesitant fuzzy soft msets over such that
Definition 32. Let be a hesitant fuzzy soft multitopological space with dimension . It is said to be hesitant fuzzy soft multiconnected if there does not exist a hesitant fuzzy soft multiseparation of . Otherwise, is said to be a hesitant fuzzy soft multi-disconnected.
Example 10. Let be an mset, be a set of parameters, and be a hesitant fuzzy soft multitopology with dimension over , where , and . Since, , and , then a hesitant fuzzy soft multitopological space is connected.
Theorem 16. Let be a hesitant fuzzy soft multitopological space with dimension . If the only hesitant fuzzy soft msets over that are both open and closed in are and , then a hesitant fuzzy soft multitopological space is connected.
Proof. Let and be a hesitant fuzzy soft multiseparation of . If , then which is a contradiction. Hence, . Since and , then . Therefore, is both open and closed hesitant fuzzy soft mset different from and which is a contradiction. Hence, a hesitant fuzzy soft multitopological space is connected.
Example 11. By using Theorem 16, the hesitant fuzzy soft multi-indiscrete topological space with dimension is connected.
Remark 5. The converse of Theorem 16 is not necessarily true as shown in the following example.
Example 12. Let be an mset, be a set of parameters, and be a hesitant fuzzy soft multitopology with dimension over , where . Then, a hesitant fuzzy soft multitopological space is connected, but there exists an open and closed hesitant fuzzy soft mset different from and .
Example 13. Let be an mset, be a set of parameters, and be a hesitant fuzzy soft multitopology with dimension over , where and . Since, and , a hesitant fuzzy soft multitopological space is disconnected.
Definition 33. Let be a hesitant fuzzy soft multitopological space with dimension and be a nonempty hesitant fuzzy soft multi-subset of . The family is said to be a hesitant fuzzy soft multitopology with dimension on , and is called a hesitant fuzzy soft multi-subspace of .
Theorem 17. If the hesitant fuzzy soft msets with dimension k, and , form a hesitant fuzzy soft multiseparation of and is a hesitant fuzzy soft multiconnected subspace of , then lies entirely within either or .
Proof. Since , then , and , are open. Suppose that does not lie entirely within neither nor . Then, and . Also, . Hence, and are two hesitant fuzzy soft multiseparation of , i.e., is disconnected which is a contradiction. Then, lies entirely within either or .
6. Conclusions
The fuzzy set theory, which was originally introduced by Zadeh [1] in 1965, is one of the most efficient decision aid techniques providing the ability to deal with imprecise and vague information. Nonetheless, to cope with imperfect or imprecise information that two or more sources of vagueness appear simultaneously, the traditional fuzzy set shows some limitations. Hence, it has been extended into several different forms, such as the type fuzzy set, the type fuzzy set, the interval-valued fuzzy set, and the fuzzy multisets. All these extensions are based on the same rationale that it is not clear to assign the membership degree of an element to a fixed set. Recently, the concept of hesitant fuzzy sets is introduced firstly in 2010 by Torra [23] which permits the membership to have a set of possible values and presents some of its basic operations in expressing uncertainty and vagueness. Torra and Narukawa [24] established the similarities and differences with the hesitant fuzzy sets and the previous generalization of fuzzy sets such as intuitionistic fuzzy sets, type 2 fuzzy sets, and type n fuzzy sets. Therefore, other authors [25, 26] introduced the concept of hesitant fuzzy soft sets, and they presented some of the applications in decision-making problems. In 2015, Dey and Pal [27] presented the concept of a hesitant multifuzzy soft topological space. In 2019, Kandil et al. [28] introduced some important and basic issues of hesitant fuzzy soft multisets and studied some of its structural properties such as the neighborhood hesitant fuzzy soft multisets, interior hesitant fuzzy soft multisets, hesitant fuzzy soft multitopological spaces, and hesitant fuzzy soft multibasis. Finally, they showed how to apply the concept of hesitant fuzzy soft multisets in decision-making problems.
In this paper, we introduced some important and basic issues of hesitant fuzzy soft multisets. The main properties of the current branch are studied, and some operations of this type of sets are established. Also, the concept of hesitant fuzzy soft multitopological spaces is defined. It should be mentioned that the concept of hesitant fuzzy soft multisets is a generalization of the previous concepts such as hesitant fuzzy soft sets, hesitant fuzzy multisets, hesitant fuzzy sets, and fuzzy sets. The concept of mapping on hesitant fuzzy soft multisets is introduced, and some results for this type of mappings are presented. The notions of inverse image and identity mapping are introduced, and their basic properties are investigated in detail. Also, the types of mappings on hesitant fuzzy soft multisets are given, and their properties are established. Therefore, the composition of two hesitant fuzzy soft multi mapping with the same dimension is presented. Moreover, we introduce the concepts of hesitant fuzzy soft multitopologies and hesitant fuzzy soft multi-subspaces. Some types of hesitant fuzzy soft multimapping such as continuity, open, closed, and homeomorphism are presented in detail. Also, their properties and results are investigated. Finally, the concept of hesitant fuzzy soft multiconnected space is introduced. The future work in this approach is introducing the near continuous hesitant fuzzy soft multimappings. Also, we will investigate the concepts of locally connected, hyperconnected in hesitant fuzzy soft multispaces and their applications in real-life problems.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.