Mappings and Connectedness on Hesitant Fuzzy Soft Multispaces

In this paper, we introduce the concept of mapping on hesitant fuzzy soft multisets and present some results for this type of mappings. )e 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. )e 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.


Introduction
Since the introduction of fuzzy sets by Zadeh [1], several extensions of this concept have been introduced. e 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.
at 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. is 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. ey 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 (F, E) as F: E ⟶ P * (U), where E is a set of parameters and P * (U) is a power set of an mset U. 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-Lindel€ of spaces. Some other results and properties about soft multisets are presented in [10][11][12]. e concept of a generalized open soft mset is introduced in soft multitopological spaces, and its properties are presented in [10]. Several authors [13][14][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. ey 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. e multicriteria group decision-making methods are introduced by using N-soft set and N-soft topology to deal with uncertainties in the realworld 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. erefore, other authors [25,26] introduced the concept of hesitant fuzzy soft sets, and they presented some of the applications in decisionmaking 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.
e 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. e notions of inverse image and identity mapping are introduced, and their basic properties are investigated in detail. e types of mappings are also given on hesitant fuzzy soft multisets, and their properties are established. erefore, 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 multisubspaces 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.

Preliminaries
e 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 X drawn from the set U is represented by a count function C X defined as C X : U ⟶ N, where N represents the set of nonnegative integers.
Here, C X (x) is the number of occurrences of the element x in the mset X.
e mset X is drawn from the set U � x 1 , x 2 , . . . , x n , and it is written as where m i is the number of occurrences of the element x i , i � 1, 2, 3, ..., n in the mset X.
Definition 2 (see [29]). A domain U is defined as a set of elements from which msets are constructed. e mset space [U] w is the set of all msets whose elements are in U such that no element in the mset occurs more than w times. e mset space [U] ∞ is the set of all msets over a domain U such that there is no limit on the number of occurrences of an element in an mset.
Definition 3 (see [29]). Let X be an mset drawn from the set U. If C X (x) � 0 for all x ∈ U, then X is called an empty mset and denoted by ϕ, i.e., ϕ(x) � 0 for all x ∈ U.
Definition 4 (see [5]). Let X be a universal multiset, E be a set of parameters, and A ⊆ E. en, an ordered pair (F, A) is called a soft mset, where F is a mapping given by F: A ⟶ P * (X); P * (X) is the power set of an mset X. For all e ∈ A, F(e) mset is represented by count function C F(e) : X * ⟶ N, where N represents the set of nonnegative integers and X * represents the support set of X.
Definition 5 (see [5]). Let (F, A) and (G, B) be two soft msets over X. en, (1) (F, A) is said to be a sub-soft mset of (G, B) and denoted by (F, (2) Two soft msets (F, A) and (G, B) over X are equal if (F, A) is a sub-soft mset of (G, B) and (G, B) is a subsoft mset of (F, A). (3) e union of two soft msets (F, A) and (G, Definition 6 (see [5]). e complement of a soft mset (F, A) is denoted by (F, A) c and is defined by where F c : A ⟶ P * (X) is a mapping given by F c (e) � (X/F(e)) for all

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Definition 7 (see [5]). Let X be a universal mset and E be a set of parameters. en, the collection of all soft msets over X with parameters from E is called a soft multiclass and is denoted by SMS(X) E .
Definition 8 (see [23]). Let U be a reference set, then a hesitant fuzzy (briefly, an HF) set is defined in terms of a function h from U into the power set of [0, 1].
Definition 9 (see [23]). Let h, h 1 , and h 2 be hesitant fuzzy sets over a set U. en, the following operations are defined: Definition 10 (see [28]). A hesitant fuzzy multiset of dimension k (briefly, HF k M set) on a nonempty mset X is denoted by A � < (m/x), h A (x) > : x∈ m X and is defined in terms of h A (x) when applied to X, and h A (x) is a set of some distinct values in [0, 1] sorting into increasing order, indicating the possible membership degrees of the elements x∈ m X to the multiset A.
Definition 11 (see [28] Definition 12 (see [28]). A pair (F, E) is a hesitant fuzzy soft mset of dimension k if F is a mapping from E to HF k M(X), where HF k M(X) is the set of all hesitant fuzzy msets of dimension k defined over an mset X and F(e) ∈ HF k M(X) ∀e ∈ E, i.e., F(e) � < (m/x), h F(e) (x) > |x∈ m X} for all e ∈ E, and h F(e) is the membership function of F(e).
Definition 13 (see [28]). An HF k SM set (F, E) over (X, E) is said to be (1) A relative null HF k SM set and is denoted by 0 X E , if h F(e) (x) � 0, 0, k− times , . . . , 0 for all x∈ m X, e ∈ E (2) A relative absolute HF k SM set and is denoted by Definition 14 (see [28]). Let (F, A) and (G, B) be two hesitant fuzzy soft multisets of dimension k, then (F, A) is called a hesitant fuzzy soft multi-subset (briefly, HF k SM) of Hence, this relationship is denoted by (F, A)⊑(G, B), and (G, B) is called an HF k SM superset of (F, A).
Definition 15 (see [28]). Let (X E , τ E ) be a hesitant fuzzy soft multitopological space and (F, A), (G, B) be two HF k SM sets over a hesitant fuzzy soft mset (X, E) (for short, Definition 16 (see [28]). Let (X E , τ E ) be a hesitant fuzzy soft multitopological space and (F, A) and (G, B) be two Additionally, the union of all interior hesitant fuzzy soft mset of (F, A) is called the interior of (F, A), and it is denoted by (F, A) o .
Theorem 1 (see [28]). Let (X E , τ E ) be a hesitant fuzzy soft multitopological space and (F, A), (G, B) be two HF k SM sets over (X, E), then

Mappings in Hesitant Fuzzy Soft Multisets
e purpose of this section is to present a concept of mapping in hesitant fuzzy soft multisets. e 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. erefore, 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 U be a universal set, E be a set of parameters, and X be a multiset over U.
e 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.

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It is written as (F,A)⊔(G,B) � (H,C).
It is written as Definition 19. Let HF k SM(X E ) and HF k SM(Y E′ ) be two families of hesitant fuzzy soft msets over msets X and Y with dimension k and sets of parameters E and E ′ , respectively. Let u: X * ⟶ Y * and p: E ⟶ E ′ be two mappings. Now, a mapping f � (u, p): HF k SM(X E ) ⟶ HF k SM(Y E′ ) is defined as follows: for a hesitant fuzzy soft mset (F, A) in HF k SM(X E ), f((F, A)) is a hesitant fuzzy soft mset in HF k SM(Y E′ ) obtained as follows: for e ′ ∈ p(E) ⊆ E ′ and y ∈ Y * , Hence, f((F, A)) is called an image hesitant fuzzy soft mset with dimension k of a hesitant fuzzy soft mset (F, A).
Advances in Fuzzy Systems en, (f (((F, tA)), p(E)) � (e 1 ′ , < (3/x), 0, 0, 0 Definition 20. Let f � (u, p): HF k SM(X E ) ⟶ HF k SM (Y E′ ) be a mapping such that u: X * ⟶ Y * and p: E ⟶ E ′ be two mappings. If (H, B) is a hesitant fuzzy soft mset in HF k SM(Y E′ ), then the inverse image of (H, B) is a hesitant fuzzy soft mset in HF k SM(X E ), denoted by f − 1 ((H, B)), defined as follows: for e ∈ p − 1 (B) ⊆ E and x ∈ X * , Hence, the inverse image of (H, B) is f − 1 ((H, B)) � (e 1 , < (2/a), 0, 0, 0 Definition 21. Let f � (u, p): HF k SM(X E ) ⟶ HF k SM(Y E′ ) be a mapping such that u: X * ⟶ Y * and p: E ⟶ E ′ be two mappings. Let (F, A) and (G, B) be two hesitant fuzzy soft msets in HF k SM(X E ). For e ′ ∈ E ′ , y ∈ Y * : the union and intersection of two images f((F, A)) and f ((G, B)) in HF k SM(Y E′ ) are defined as Definition 22. Let f � (u, p): HF k SM(X E ) ⟶ HF k SM (Y E′ ) be a mapping such that u: X * ⟶ Y * and p: E ⟶ E ′ be two mappings. Let (F, A) and (G, B) be two hesitant fuzzy soft msets in HF k SM(Y E′ ). For e ∈ E, x ∈ X * : the union and intersection of two inverse images f − 1 ((F, A)) and f − 1 ((G, B)) in HF k SM(X E ) are defined as

Theorem 3. Let f � (u, p): HF k SM(X E ) ⟶ HF k SM (Y E′ ) be a mapping such that u: X * ⟶ Y * and p: E ⟶ E ′ be two mappings. If (F, A), (G, B) are two hesitant fuzzy soft msets in HF k S(X E ) and (F i , A i ) is a family of hesitant fuzzy soft msets in HF k SM(X E ), then
(

(14)
Advances in Fuzzy Systems e inclusion in eorem 3, parts 2 and 4, cannot be replaced by equality relation. Moreover, the converse of part 5 is not necessarily true as shown in the following example.

Proof.
e proof of parts 1and 2 is obvious.
x ∈ X * , and for nontrivial case, we have

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By using Definition 22, we have ((G, B)).

Remark 3.2.
e converse in eorem 4 part 5 is not necessarily true as shown in the following example.  ((G, B)), but (F, A)⊑(G, B). Proof. Immediate.

Definition
24. Let f � (u, p): HF k SM(X E ) ⟶ HF k SM(Y E′ ) and g � (r, t): HF k SM(Y E′ ) ⟶ HF k SM (Z E″ ) be two HFSMmappings of dimension k. eir composition g∘fis also a hesitant fuzzy soft multimapping with dimension k from HF k SM(X E ) into HF k SM(Z E″ ) such that, for every (F, A) in HF k SM(X E ), is composition is defined as, for e ″ ∈ t(E ′ ) ⊆ E ″ and  ((G, B)).

Remark 3.3.
e inclusion in eorem 9 parts 1 and 2 cannot be replaced by equality relation as shown in the following example.

Example 8
Advances in Fuzzy Systems Proof. Immediate by using eorem 8.

Continuous Mappings on Hesitant Fuzzy
Soft Multispaces e aim of this section is to introduce the concept of hesitant fuzzy soft multitopology. erefore, 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.
e subcollection τ E of members of HF k SM(X) E is called a hesitant fuzzy soft multitopology of dimension kon (X, E), if the following conditions are satisfied:   (F, A)⊑(G, B), then (F, A)⊑(G, B) Proof.
e proof is omitted.
Definition 30. Let (f � (u, p): HF k SM(X E ) ⟶ HF k SM (Y E′ )) be a mapping such that u: X * ⟶ Y * and p: E ⟶ E ′ be two mappings. Let τ E and η E ′ be two hesitant fuzzy soft multitopologies of dimension k over X E and Y E ′ respectively. A function f is said to be (1) Continuous if f − 1 ((G, B))∈ τ E for all (G, B)∈ η E′ (2) Open if f((F, A))∈ η E′ for all (F, A)∈ τ E (3) Closed if f((F, A))∈ η c E′ for all (F, A)∈ τ c limitations. Hence, it has been extended into several different forms, such as the type 2fuzzy set, the type nfuzzy 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. erefore, 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. e 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. e concept of mapping on hesitant fuzzy soft multisets is introduced, and some results for this type of mappings are presented. e 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. erefore, 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. e 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
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.