A topological structure involving hesitant fuzzy sets

Abstract

This paper addresses an extension of the concept of topological spaces to that of hesitant fuzzy spaces. The topology is established using a conjunction and disjunction operator on hesitant fuzzy sets whose properties are discussed in the beginning. Some examples are discussed before introducing the concepts of interior and closure of a hesitant fuzzy set. The properties of these operators are verified to justify the definitions. The concepts of a hesitant fuzzy subspace topology and continuous mappings on hesitant fuzzy sets are studied. The concepts of connectedness and compactness of hesitant fuzzy topological spaces are studied and various results regarding them are proved.

Keywords

Hesitant Fuzzy Topological Spaces,continuous hesitant fuzzy mappings,connected hesitant fuzzy set,compact hesitant fuzzy topological spaces

1. Introduction

Fuzzy set theory has given a new tool to deal with linguistic concepts which are inherently vague. Novel ways for determining a suitable membership function for the given scenario are still being probed even though a lot of work has already been done in this area. among various extensions of fuzzy sets, that have been introduced, the important once include Atanassov’s intuitionistic fuzzy sets [1], L-fuzzy sets [2], type-2 fuzzy sets [3] and interval valued fuzzy sets [4]. One of the recent extensions of Fuzzy sets [5] in this direction is the notion of Hesitant fuzzy sets introduced by Vicent Torra [6]. The motivation for this extension is the hesitancy arising in the determination of the membership value of an element. Hesitancy does not arise just because of an error margin or a possibility distribution, but because there are some possible values of which there is a hesitation about which one would be the right one. These situations mainly arise in decision-making problems where there are a group of decision-makers to consider the evaluation of a scenario. In a hesitant fuzzy set, the membership function takes values from the power set of [0, 1]. This allows the use of all the values simultaneously which helps in dealing with the situation effectively. The main difference in hesitant fuzzy set theory is conceptual when compared to other extensions of fuzzy sets. Hesitant fuzzy set theory is having much applications in various fields like multi-criteria decision making, group decision making, decision support systems, evaluation processes, clustering algorithms, etc.

Topological spaces form a general framework for the study of various notions in analysis where the fundamental structure becomes the notion of an open set rather than distance function or a metric. The advantage of this is, thinking directly in terms of open sets can account for more clarity and greater generality. Fuzzy set theory provides a framework which is wider than that of classical set theory. On this theory many mathematical concepts rely on the effects of ordered structure can be embedded and fuzzy topology is one such branch which combines order structure and topology. This has led to the reformulation, and generalization of many techniques in various fields of real applications.

The main trademark in fuzzy topology [7] is its point like structure. In [8], it is shown that, the notion of fuzzy topology may be relevant to quantum particle physics in connection with string theory and ϵ^∞ theory. In [9], it is shown that, the concept of fuzzy spheres and fuzzy spaces topology can be used to explain the origin of the black hole entropy. Hesitant fuzzy topology being a more flexible variation of fuzzy topology which replaces a point like structure by a set like structure, such applications can be carried out more effectively in the context of hesitant fuzzy topology. This gives some motivation as well as a practical value for the concepts to be introduced.

The concept of a hesitant fuzzy topology uses the notions of hesitant fuzzy join and hesitant fuzzy meet. This notion of a hesitant fuzzy topology does not form a topology when it is restricted to the crisp case. So it is not a natural generalization of the concepts of general topology. Nevertheless, it uses all the concepts of general topology whose equivalent notions are discussed in the context of this hesitant fuzzy topology. Hesitant fuzzy topology simulates the topology in this scenario and thus it is different from many other generalized versions of topological structures introduced in the context of different types of fuzzy sets.

A series of aggregation operators for hesitant fuzzy information is developed in [10] and their applications in solving decision-making problems is also given. Wu et al. [11] introduced probabilistic interval-valued hesitant fuzzy sets and its aggregation operators, for developing a novel multi-attribute group decision making model to address the genuine loss of data in a reluctant fuzzy data condition. Another strategy to manage multi-criteria collective choice making (MCGDM) issues with unbalanced hesitant fuzzy linguistic term sets by thinking about the mental conduct of decision makers can be seen in [12]. Different types of distance and correlation measures for hesitant fuzzy information system are discussed in [13, 14]. In [15] a variety of distance measures for hesitant interval-valued fuzzy sets are proposed. In [16, 17], some shortcomings of existing distance measures of hesitant fuzzy sets are addressed and some new distance measures are introduced with applications. Some correlation coefficient formulas for hesitant fuzzy sets are derived in [18] and they are applied to clustering analysis under hesitant fuzzy environments.

The hesitant fuzzy preference relation (HFPR) which is defined in [19] is a valuable tool for decision makers to draw out their preference information over a set of alternatives. Different types of consistency indices are defined in [20] to measure the consistency level of an HFPR. In [21], an automatic algorithm is developed to improve the consistency of an incomplete HFPR. Multiplicative consistency of the hesitant fuzzy preference relation is discussed in [22] and a new multiplicative consistency concept for the HFPR is proposed in [23]. A new hesitant consistency measure, called interval consistency index is defined in [24] to estimate the consistency range of a hesitant fuzzy linguistic preference relation.

The hesitant fuzzy soft sets introduced in [25] is an important tool for solving decision making problems. Recent development this can be found in [26 –29]. The multiple attribute decision making (MADM) problems in an interval-valued dual hesitant fuzzy set are studied in [30]. A multi-criteria decision-making method for probabilistic hesitant fuzzy information is discussed in [31] and problems of multi-criteria group decision making with Probabilistic linguistic term sets is explored in [32].

Chang [7] introduced the concept of a fuzzy topological space by generalizing many of the concepts of general topology. Lowen [33] came up with another definition of fuzzy topological space. His notion of fuzzy compactness preserves the Tychonoff theorem [34]. In [35] Lowen gives a comparison of the different kinds of compactness notions. Some of the limitations of earlier notions of compactness have been rectified to an extent in [36]. A different type of fuzzy compactness using fuzzy nets has been defined in [37].

In this paper, the basic concepts of topology are extended to the hesitant fuzzy domain. Section 2 discusses the basic notions regarding hesitant fuzzy sets. It gives the notions of hesitant fuzzy subset and discusses some of its properties. Conjunction and disjunction operators are developed to serve as hesitant intersection and union when it comes to the hesitant fuzzy topology. The notion of a hesitant fuzzy topology is introduced in Section 3. Some examples are studied before moving on to the closure and interior of a hesitant fuzzy set. Some of the properties of the closure and interior operators are verified. This section concludes with a study on the subspace topology. Section 4 studies the connectedness and compactness in hesitant fuzzy topological spaces. It begins by giving an extension principle to generalize crisp functions to functions on hesitant fuzzy subsets. The concept of a continuous hesitant fuzzy mapping and their equivalent forms are discussed. Finally, connectedness and compactness in hesitant fuzzy topological spaces are introduced. It is proved that continuous images of compact hesitant fuzzy spaces are compact. Similar result for connected spaces is also provided in this section. The section concludes by proving hereditary property of hesitant fuzzy compact spaces.

2. Preliminaries

This section introduces the basic concepts in hesitant fuzzy set theory.

Definition 2.1. [[6]] Let X be a reference set then a Hesitant fuzzy set(HFS) on X is defined in terms of a function h that when applied to X returns a subset of [0, 1] h : X → P [0, 1] where P [0, 1] denotes power set of [0, 1].

The empty hesitant set, the full hesitant set, the set to represent complete ignorance for x and the nonsense set are defined as follows:

empty set: h₀ (x) = {0} ∀ x ∈ X denoted by O^*

full set: h_X (x) = {1} ∀ x ∈ X denoted by I^*

complete ignorance h (x) = [0, 1] ∀ x ∈ X

set for a nonsense x : h (x) = φ ∀ x ∈ X

Given a hesitant fuzzy set h, its lower and upper bound are defined as follows: h^- (x) = min h (x) h⁺ (x) = max h (x)

For convenience we call h (x) a hesitant fuzzy element (HFE) [10]. Let l (h (x)) be the number of values in h (x). The set of all Hesitant fuzzy sets on X is denoted by HF (X)

Definition 2.2. ([10]) Score for a HFE, $s (h) = \frac{1}{l (h)} \sum_{γ \in h} γ$ is called the score function of h.

Note: If the HFE is of the form $⋃_{i = 1}^{n} [a_{i}, b_{i}]$ then $s (h (x)) = \sum_{i = 1}^{n} (b_{i} - a_{i}) / 2$ also if the HFE is infinite, i.e., h (x) = (a₁, a₂, a₃, ⋯) then $s (h (x)) = \frac{1}{2} (inf (h (x)) + sup (h (x)))$ .

Definition 2.3. Let h ∈ HF (X). Then the set $\underset{x \in X}{\cup} h (x)$ is called the image of h and is denoted by h (X). The set {x|x∈ X, s (h (x)) >0 }, is called the support of h and is denoted by h^*. h is called finite hesitant fuzzy set if h^* is a finite set, and an infinite hesitant fuzzy set otherwise.

Definition 2.4. Let Y ⊆ X and A ⊆ [0, 1]. We define ${\tilde{A}}_{Y} \in HF (X)$ as follows: ${\tilde{A}}_{Y} (x) = {\begin{matrix} A, & for x \in Y \\ {0}, & for x \in X \ Y \end{matrix}$ If Y is a singleton, say {y}, then ${\tilde{A}}_{{y}}$ is called a hesitant fuzzy point (or hesitant fuzzy singleton), and is denoted by ${\tilde{y}}_{A}$ . ${\tilde{y}}_{A} (x) = {\begin{matrix} A, & for x = y \\ {0}, & for x \in X \ {y} \end{matrix}$ A hesitant fuzzy point ${\tilde{x}}_{A}$ is said to belong to a hesitant fuzzy set h (denoted by ${\tilde{x}}_{A} \in h$ ) if A ⊆ h (x) If S is a set of hesitant fuzzy singletons, then we let foot (S) ={ y ∈ X|y_A ∈ S }.

Definition 2.5. ([38]) Let h₁, h₂ ∈ HF (X). Then we say that h₁ is a subset of h₂ denoted by, h₁ ⊆ h₂, ⇔ h₁ (x) ⊆ h₂ (x) ∀ x ∈ X.

h₁ = h₂ iff h₁ ⊆ h₂ and h₂ ⊆ h₁.

Definition 2.6. ([38]) Hesitant Equality:

Let h₁ and h₂ be two hesitant fuzzy sets on X, then we say that h₁ is equal to h₂ (denoted h₁ = h₂) iff h₁ (x) = h₂ (x) and h₁ is hesitantly equal to h₂ (denoted h₁ ≈ h₂) iff s (h₁ (x)) = s (h₂ (x) ∀ x ∈ X.

Definition 2.7. ([38]) Hesitant subset:

Let h₁ and h₂ be two hesitant fuzzy sets on X, then we say that h₁ is a hesitant subset of h₂ (denoted by h₁ ⪯ h₂) iff s (h₁ (x)) ≤ s (h₂ (x)) ∀ x ∈ X.

Definition 2.8. ([38]) Hesitant proper subset:

h₁ ≺ h₂ if s (h₁ (x)) ≤ s (h₂ (x) ∀ x ∈ X and s (h₁ (x)) < s (h₂ (x) for atleast one x ∈ X.

Definition 2.9. ([6]) Given a hesitant fuzzy set represented by its membership function h its compliment is defined as follows h^c (x) = ⋃ _γ∈h(x) {1 - γ}

Definition 2.10. Given two hesitant fuzzy sets represented by their membership functions h₁ and h₂, we define a score based intersection of h₁ and h₂ (denoted by $h_{1} \tilde{\land} h_{2}$ )as $(h_{1} \tilde{\land} h_{2}) (x) = {\begin{matrix} h_{1} (x) & if h_{1} (x) ≺ h_{2} (x) \\ h_{2} (x) & if h_{2} (x) ≺ h_{1} (x) \\ h_{1} (x) ⋃ h_{2} (x) & if h_{1} (x) \approx h_{2} (x) \end{matrix}$ and a score based union of h₁ and h₂ (denoted by $h_{1} \tilde{\lor} h_{2}$ )as $(h_{1} \tilde{\lor} h_{2}) (x) = {\begin{matrix} h_{1} (x) & if h_{1} (x) ≻ h_{2} (x) \\ h_{2} (x) & if h_{2} (x) ≻ h_{1} (x) \\ h_{1} (x) ⋃ h_{2} (x) & if h_{1} (x) \approx h_{2} (x) \end{matrix}$ For a collection, {h_i|i∈ I } of hesitant fuzzy subsets of X, where I is a non empty index set, we have ∀x ∈ X $\begin{array}{l} ({\tilde{\lor}}_{i \in I} h_{i}) (x) = {\tilde{\lor}}_{i \in I} h_{i} (x) \\ ({\tilde{\land}}_{i \in I} h_{i}) (x) = {\tilde{\land}}_{i \in I} h_{i} (x) \end{array}$

Proposition 2.11. Let h₁, h₂ ∈ HF (X) then

$(h_{1} \tilde{\land} h_{2})^{c} (x) = (h_{1})^{c} (x) \tilde{\lor} (h_{2})^{c} (x)$

$(h_{1} \tilde{\lor} h_{2})^{c} (x) = (h_{1})^{c} (x) \tilde{\land} (h_{2})^{c} (x)$

Proof.

$(h_{1} \tilde{\land} h_{2}) (x) = {\begin{matrix} h_{1} (x) & if h_{1} (x) ≺ h_{2} (x) \\ h_{2} (x) & if h_{2} (x) ≺ h_{1} (x) \\ h_{1} (x) ⋃ h_{2} (x) & if h_{1} (x) \approx h_{2} (x) \end{matrix}$ $(h_{1} \tilde{\land} h_{2})^{c} (x) = {\begin{matrix} ⋃_{γ \in h_{1} (x)} {1 - γ} & if h_{1} (x) ≺ h_{2} (x) \\ ⋃_{γ \in h_{2} (x)} {1 - γ} & if h_{2} (x) ≺ h_{1} (x) \\ ⋃_{γ \in h_{1} (x) ⋃ h_{2} (x)} {1 - γ} & if h_{1} (x) \approx h_{2} (x) \end{matrix}$ Now, $h_{1} ≺ h_{2} \Rightarrow h_{1}^{c} ≻ h_{2}^{c}$ , therefore $(h_{1} \tilde{\land} h_{2}) (x) = {\begin{matrix} h_{1}^{c} (x) & if h_{1}^{c} (x) ≻ h_{2}^{c} (x) \\ h_{2}^{c} (x) & if h_{2}^{c} (x) ≻ h_{1}^{c} (x) \\ h_{1}^{c} (x) ⋃ h_{2}^{c} (x) & if h_{1}^{c} (x) \approx h_{2}^{c} (x) \end{matrix}$ now, $h_{1}^{c} (x) \tilde{\lor} h_{2}^{c} (x) = {\begin{matrix} h_{1}^{c} (x) & if h_{1}^{c} (x) ≻ h_{2}^{c} (x) \\ h_{2}^{c} (x) & if h_{2}^{c} (x) ≻ h_{1}^{c} (x) \\ h_{1}^{c} (x) ⋃ h_{2}^{c} (x) & if h_{1}^{c} (x) \approx h_{2}^{c} (x) \end{matrix}$

proof similar to 1.

□

Proposition 2.12. Let h₁, h₂ ∈ HF (X) then $h_{1} ⪯ h_{2} \Rightarrow h_{1}^{c} ≽ h_{2}^{c}$ i.e., the compliment operetor is order reversing.

Proof. h₁ ⪯ h₂⇒s (h₁ (x)) ≤ s (h₂ (x)) ∀ x ∈ X

$\Rightarrow \frac{\sum_{γ \in h_{1} (x)} γ}{n_{1}} \leq \frac{\sum_{α \in h_{2} (x)} α}{n_{2}}$ where l (h₁ (x)) = n₁ and l (h₂ (x)) = n₂

$\Rightarrow - \frac{\sum_{γ \in h_{1} (x)} γ}{n_{1}} \geq - \frac{\sum_{α \in h_{2} (x)} α}{n_{2}}$

$\Rightarrow 1 - \frac{\sum_{γ \in h_{1} (x)} γ}{n_{1}} \geq 1 - \frac{\sum_{α \in h_{2} (x)} α}{n_{2}}$

$\Rightarrow \frac{\sum_{γ \in h_{1} (x)} 1 - γ}{n_{1}} \geq \frac{\sum_{α \in h_{2} (x)} 1 - α}{n_{2}}$

$\Rightarrow \frac{\sum_{γ^{'} \in h_{1}^{c} (x)} γ^{'}}{n_{1}} \geq \frac{\sum_{α^{'} \in h_{2}^{c} (x)} α^{'}}{n_{2}}$ where γ′ = 1 - γ and α′ = 1 - α

$\Rightarrow h_{1}^{c} ≽ h_{2}^{c}$ □

Definition 2.13. Let h₁, h₂ ∈ HF (X). Then the difference of the two hesitant fuzzy sets h₁ and h₂ is defined as (h₁ \ h₂) (x) = h₁ (x) \ h₂ (x).

Note: Let h ∈ HF (X) and Y ⊆ X then consider h′ : Y ⟶ P [0, 1] where h′ = h|_Y. h′ is the restriction of h to Y.

3. Hesitant fuzzy topology

Here the concept of a hesitant fuzzy topology is proposed and interior and closure of hesitant fuzzy sets using this notion of hesitant fuzzy topology is explored. The subspace topology is discussed briefly in the end of this section.

Definition 3.1. A hesitant fuzzy topology on a set X is a collection τ of hesitant fuzzy sets in X, (τ ⊆ HF (X)) satisfying the following:

O^* ∈ τ and I^* ∈ τ i . e . , h₀ ∈ τ and h_X ∈ τ

If h₁, h₂ ∈ τ then $h_{1} \tilde{\land} h_{2} \in τ$

If h_j ∈ τ for each j ∈ J then ${\tilde{V}}_{j \in J} h_{j} \in τ$ where J is the index set.

Then (X, τ) is called a hesitant fuzzy topological space over X.

Definition 3.2. Let (X, τ) be a hesitant fuzzy topological space over X then the members of τ are called the hesitant open sets in X. h ∈ HF (X) is called a hesitant closed set in X if h^c ∈ τ. (for the sake of underatanding let us denote the set of closed sets in X as τ^c).

Proposition 3.3. Let (X, τ) be a hesitant fuzzy topological space over X then

O^* ∈ τ and I^* are closed sets in X.

If h_j is a closed set in X for each j ∈ J then ${\tilde{\land}}_{j \in J} h_{j}$ is a closed set in X where J is the index set.

If h₁, h₂ be closed sets in X then $h_{1} \tilde{\lor} h_{2}$ is a closed set in X.

Proof.

$h_{0}^{c} = h_{X}$ and $h_{X}^{c} = h_{0},$ hence they are closed in X.

If h_j is a closed set in X for each j ∈ J then $h_{j}^{c} \in τ$ .

Then we have ! ${({\tilde{\land}}_{j \in J} h_{j})}^{c} = {\tilde{\lor}}_{j \in J} h_{j}^{c} \in τ$ .

If h₁, h₂ be closed sets in X then $h_{1}^{c}, h_{2}^{c} \in τ$ .

Then we have $(h_{1} \tilde{\lor} h_{2})^{c} = h_{1}^{c} \tilde{\land} h_{2}^{c} \in τ$ □

Example 3.4. Consider a set X. If τ = {h₀, h_X} then (X, τ) is called the hesitant fuzzy indiscrete topology on X. (X, HF (X)) is called the hesitant fuzzy discrete topology on X where HF (X) is the set of all hesitant fuzzy subsets of X.

Example 3.5. Let X = {x, y, z} be the universal set and h₁, h₂, h₃, h₄ be hesitant fuzzy sets on X as given below:

h₁ (x) = {.4} ,	h₁ (y) = {.6} ,	h₁ (z) = {.1, . 3}
h₂ (x) = {.2, . 3} ,	h₂ (y) = {.7, . 8, . 9} ,	h₂ (z) = {.4, . 5}
h₃ (x) = {.2, . 3} ,	h₃ (y) = {.6} ,	h₃ (z) = {.1, . 3}
h₄ (x) = {.4} ,	h₄ (y) = {.7, . 8, . 9} ,	h₄ (z) = {.4, . 5}

Let τ₁ = {h₀, h_X, h₁, h₂, h₃, h₄}

Then (X, τ₁) is a hesitant fuzzy topological space.

Example 3.6. Let X = [0, 1] ; h_n : X ⟶ P [0, 1] for $n \in ℕ \ {1}$ . $\begin{matrix} h_{n} (x) & = & (x - \frac{δ_{x}}{n}, x + \frac{δ_{x}}{n}); n \in ℕ \ {1} \\ where δ_{x} = min (x, 1 - x) \end{matrix}$ Then (X, τ) is a hesitant fuzzy topological space for $τ = {h_{0}, h_{X}} \cup {h_{n} | n \in ℕ \ {1}}$ .

Example 3.7. Let f be a bijection from X to [0, 1], R be an equivalence relation on X. Let [x] _i ; i ∈ {1, 2, . . . , n} be equivalence classes w.r.t R of X. Then define $h_{i} (y) = {\begin{matrix} f ([x]_{i}) & if y \in [x]_{i} \\ {0}; & otherwise \end{matrix}$ if i = j ; y ∈ [x_i] then $h_{i} (y) \tilde{\land} h_{j} (y) = h_{i} (y) = h_{i} (y) \tilde{\lor} h_{j} (y)$

if i = j ; y ∉ [x_i] then $h_{i} (y) \tilde{\land} h_{j} (y) = h_{0} = h_{i} (y) \tilde{\lor} h_{j} (y)$

if i ≠ j then $h_{i} (y) \tilde{\land} h_{j} (y) = h_{0}$ and $h_{i} (y) \tilde{\lor} h_{j} (y) = {\begin{matrix} h_{i} (y); if y \in [x]_{i} \\ h_{j} (y); if y \in [x]_{j} \\ h_{0}; if y \notin [x]_{i} and y \notin [x]_{j} \end{matrix}$ Then (X, τ_R) is a hesitant fuzzy topological space for τ_R = {h₀, h_X} ∪ {h_i|i ∈ {1, 2, . . . , n}}.

Definition 3.8. Let X₁, X₂ ⊆ HF (X). X₁ is said to be finer then X₂ denoted by $X_{1} \tilde{\subseteq} X_{2}$ if for any h_i ∈ X₁, there exists h_j ∈ X₂ such that h_i ⊆ h_j.

Note that h₁ ⊆ h₂, ⇔ h₁ (x) ⊆ h₂ (x) ∀ x ∈ X

Definition 3.9. Let (X, τ₁) and (X, τ₂) be hesitant fuzzy topological spaces, then τ₁ is said to be finer than τ₂ if $τ_{1} \tilde{\subseteq} τ_{2}$ .

Definition 3.10. A base for a hesitant fuzzy topological space (X, τ) is a sub collection $B$ of τ such that each hesitant fuzzy set h of τ can be written as a hesitant fuzzy union of members of $B .$ i.e., $\forall h \in τ; h = {\tilde{\lor}}_{h_{i} \in B^{'}} h_{i}$ for some $B^{'} \subseteq B$

Let $S \subset HF (X)$ , then ${h_{S_{1}} \tilde{\land} h_{S_{2}} \tilde{\land} . . . \tilde{\land} h_{S_{n}}$ and $h_{S_{1}}, h_{S_{2}}, . . ., h_{S_{n}} \in S} ⋃ {h_{X}, h_{0}}$ forms a basis for a unique hesitant fuzzy topology on X, and we call that topology, $τ_{S}$ , the topology generated by $S$ .

Note: If $B$ is a basis for a topology τ and $B^{'} \subseteq B$ then

1. $B^{'} = φ \Rightarrow h_{0} \in τ$

2. ${\tilde{\lor}}_{h_{i} \in B} h_{i} = h_{X}$

3. for any $h_{1}, h_{2} \in B; h_{1} \tilde{\land} h_{2} \in B$

If any of these properties are violated then that subcollection does not form a basis for the hesitant fuzzy topology under consideration.

Proposition 3.11. Let (X, τ₁) and (X, τ₂) be two hesitant fuzzy topological spaces over X. Then (X, τ₁ ∩ τ₂) is a hesitant fuzzy topological space over X.

Proof.

h₀, h_X ∈ τ₁ and τ₂; hence h₀, h_X ∈ τ₁ ∩ τ₂.

$h_{1}, h_{2} \in τ_{1} \Rightarrow h_{1} \tilde{\land} h_{2} \in τ_{1}$ and similarly $h_{1}, h_{2} \in τ_{2} \Rightarrow h_{1} \tilde{\land} h_{2} \in τ_{2}$ hence $h_{1}, h_{2} \in τ_{1} \cap τ_{2} \Rightarrow h_{1} \tilde{\land} h_{2} \in τ_{1} \cap τ_{2}$

Let h_i ∈ τ₁ ∩ τ₂ ∀ i ∈ I then ${\tilde{\lor}}_{i \in I} h_{i} \in τ_{1}$ and ${\tilde{\lor}}_{i \in I} h_{i} \in τ_{2}$ $\Rightarrow {\tilde{\lor}}_{i \in I} h_{i} \in τ_{1} \cup τ_{2}$ □

Example 3.12. The union of two hesitant fuzzy topologies need not be a hesitant fuzzy topology as illustrated in the following example.

Let X = {x, y, z} and τ₁ be the topology given in Example 3.

h₅ (x) = {.3} ,	h₅ (y) = {.6, . 7} ,	h₅ (z) = {.3, . 7}
h₆ (x) = {.5, . 8, . 9} ,	h₆ (y) = {.2, . 5, . 6} ,	h₆ (z) = {.9}
h₇ (x) = {.3} ,	h₇ (y) = {.2, . 5, . 6} ,	h₇ (z) = {.3, . 7}
h₈ (x) = {.5, . 8, . 9} ,	h₈ (y) = {.6, . 7} ,	h₈ (z) = {.9}

τ₂ = {h₀, h_X, h₅, h₆, h₇, h₈}

Now (X, τ₁) and (X, τ₂) are hesitant fuzzy topological spaces. But τ₁ ∪ τ₂ is not a hesitant fuzzy topology since $h_{1} \tilde{\land} h_{5} \notin τ_{1} \cup τ_{2}$

Definition 3.13. Let (X, τ) be a hesitant fuzzy topological space over X. Let h ∈ HF (X), the hesitant fuzzy closure of h, denoted by $\bar{h}$ is given by $\bar{h} \approx \tilde{\land} h_{i}$ such that h ⪯ h_i and $h_{i}^{c} \in τ$ . i.e., closure of h is the intersection of all hesitant fuzzy closed sets containing h (h ⪯ h_i).

Theorem 3.14. Let (X, τ) be a hesitant fuzzy topological space and h, h₁, h₂ ∈ HF (X). Then

$\bar{h_{0}} = h_{0}$ and $\bar{h_{X}} = h_{X}$ .

$h ⪯ \bar{h}$

h is a hesitant fuzzy closed set iff $h \approx \bar{h}$

$\bar{\bar{h}} \approx \bar{h}$

$h_{1} ⪯ h_{2} \Rightarrow \bar{h_{1}} ⪯ \bar{h_{2}}$

$\bar{h_{1} \tilde{\lor} h_{2}} \approx \bar{h_{1}} \tilde{\lor} \bar{h_{2}}$

$\bar{h_{1} \tilde{\land} h_{2}} ⪯ \bar{h_{1}} \tilde{\land} \bar{h_{2}}$

Proof.

obvious

$\bar{h} \approx \tilde{\land} h_{i}$ such that h ⪯ h_i and $h_{i}^{c} \in τ$ .

Now h ⪯ h_i $\Rightarrow h ⪯ \tilde{\land} h_{i}$ $\Rightarrow h ⪯ \bar{h}$ .

Let h be a hesitant fuzzy closed set, then $\bar{h} \approx \tilde{\land} h_{i}$ such that h ⪯ h_i and $h_{i}^{c} \in τ$ .

h^c ∈ τ⇒h ∈ {h_i|h ⪯ h_i}

Now h ∈ {h_i} and h ⪯ h_i ∀ i (h is a closed set containing h_i)

$\Rightarrow \tilde{\land} h_{i} \approx h$ $∴ \bar{h} \approx h$ conversely $h \approx \bar{h} \Rightarrow \tilde{\land} h_{i} \approx h \Rightarrow$ h is closed as $h^{c} = (\tilde{\land} h_{i})^{c} = \tilde{\lor} h_{i}^{c} \in τ$

$\bar{h}$ is a hesitant fuzzy closed set, hence using (3) above we get $\bar{\bar{h}} = \bar{h}$ .

Let $H_{1} = {h_{i} : h_{1} ⪯ h_{i} and h_{i}^{c} \in τ}$ and $H_{2} = {h_{j} : h_{2} ⪯ h_{j} and h_{j}^{c} \in τ}$

Since h₁ ⪯ h₂; for any h_i ∈ H₁ we have an h_j ∈ H₂ such that h_i ⪯ h_j.

$\Rightarrow \bar{h_{1}} = \tilde{\land} h_{i} ⪯ \tilde{\land} h_{j} \approx \bar{h_{2}}$

$\Rightarrow \bar{h_{1}} ⪯ \bar{h_{2}}$ .

from (5) above we have $h_{1} ⪯ h_{1} \tilde{\lor} h_{2} \Rightarrow \bar{h_{1}} ⪯ \bar{h_{1} \tilde{\lor} h_{2}}$ and

$h_{2} ⪯ h_{1} \tilde{\lor} h_{2} \Rightarrow \bar{h_{2}} ⪯ \bar{h_{1} \tilde{\lor} h_{2}}$

$\Rightarrow \bar{h_{1}} \tilde{\lor} \bar{h_{2}} ⪯ \bar{h_{1} \tilde{\lor} h_{2}}$

conversely, we have $h_{1} ⪯ \bar{h_{1}}$ and $h_{2} ⪯ \bar{h_{2}}$

$\Rightarrow h_{1} \tilde{\lor} h_{2} ⪯ \bar{h_{1}} \tilde{\lor} \bar{h_{2}}$

$\Rightarrow \bar{h_{1} \tilde{\lor} h_{2}} ⪯ \bar{\bar{h_{1}} \tilde{\lor} \bar{h_{2}}} \approx \bar{h_{1}} \tilde{\lor} \bar{h_{2}}$ (since $\bar{h_{1}} \tilde{\lor} \bar{h_{2}}$ ) is closed.

Hence $\bar{h_{1} \tilde{\lor} h_{2}} \approx \bar{h_{1}} \tilde{\lor} \bar{h_{2}}$

$h_{1} ⪯ \bar{h_{1}}$ and $h_{2} ⪯ \bar{h_{2}}$

$\Rightarrow h_{1} \tilde{\land} h_{2} ⪯ \bar{h_{1}} \tilde{\land} \bar{h_{2}}$

$\Rightarrow \bar{h_{1} \tilde{\land} h_{2}} ⪯ \bar{\bar{h_{1}} \tilde{\land} \bar{h_{2}}} \approx \bar{h_{1}} \tilde{\land} \bar{h_{2}}$ (since $\bar{h_{1}} \tilde{\land} \bar{h_{2}}$ ) is closed.

Hence $\bar{h_{1} \tilde{\land} h_{2}} ⪯ \bar{h_{1}} \tilde{\land} \bar{h_{2}}$ □

Example 3.15. Let X = {x, y, z} and τ₁ be the topology given in Example 3.

Then

$h_{1}^{c} (x) = {. 6},$	$h_{1}^{c} (y) = {. 4},$	$h_{1}^{c} (z) = {. 7, . 9}$
$h_{2}^{c} (x) = {. 7, . 8},$	$h_{2}^{c} (y) = {. 1, . 2, . 3},$	$h_{2}^{c} (z) = {. 5, . 6}$
$h_{3}^{c} (x) = {. 7, . 8},$	$h_{3}^{c} (y) = {. 4},$	$h_{3}^{c} (z) = {. 7, . 9}$
$h_{4}^{c} (x) = {. 6},$	$h_{4}^{c} (y) = {. 1, . 2, . 3},$	$h_{4}^{c} (z) = {. 5, . 6}$

Let us consider two hesitant fuzzy sets h₉, h₁₀ ∈ HF (X)

h₉ (x) = {.8} ,	h₉ (y) = {.4} ,	h₉ (z) = {.1, . 2}
h₁₀ (x) = {.4, . 8} ,	h₁₀ (y) = {.4} ,	h₁₀ (z) = {.9}
$(h_{9} \tilde{\land} h_{10}) (x) = {. 4, . 8},$	$(h_{9} \tilde{\land} h_{10}) (y) = {. 4},$	$(h_{9} \tilde{\land} h_{10}) (z) = {. 1, . 2}$
$(\bar{h_{9} \tilde{\land} h_{10}}) (x) = {. 6},$	$(\bar{h_{9} \tilde{\land} h_{10}}) (y) = {. 4},$	$(\bar{h_{9} \tilde{\land} h_{10}}) (z) = {. 7, . 9}$
$\bar{h_{9}} (x) = {1},$	$\bar{h_{9}} (y) = {1},$	$\bar{h_{9}} (z) = {1}$
$\bar{h_{10}} (x) = {1},$	$\bar{h_{10}} (y) = {1},$	$\bar{h_{10}} (z) = {1}$
$(\bar{h_{9}} \tilde{\land} \bar{h_{10}}) (x) = {1},$	$(\bar{h_{9}} \tilde{\land} \bar{h_{10}}) (x) = {1},$	$(\bar{h_{9}} \tilde{\land} \bar{h_{10}}) (x) = {1}$

Therefore $\bar{h_{9} \tilde{\land} h_{10}} ⪯ \bar{h_{9}} \tilde{\land} \bar{h_{10}}$ and $\bar{h_{9} \tilde{\land} h_{10}} \bar{h_{9}} \tilde{\land} \bar{h_{10}}$ .

Definition 3.16. Let (X, τ) be a hesitant fuzzy topological space over X. Let h ∈ HF (X), the hesitant fuzzy interior of h, denoted by h^o is given by $h^{o} \approx \tilde{\lor} h_{i}$ such that h_i ⪯ h and h_i ∈ τ. i.e., interior of h is the union of all hesitant fuzzy open sets contained in h (h_i ⪯ h).

Theorem 3.17. Let (X, τ) be a hesitant fuzzy topological space and h, h₁, h₂ ∈ HF (X). Then

$h_{0}^{o} = h_{0}$ and $h_{X}^{o} = h_{X}$ .

h^o ⪯ h

h is a hesitant fuzzy open set iff h ≈ h^o

(h^o) ^o ≈ h^o

$h_{1} ⪯ h_{2} \Rightarrow h_{1}^{o} ⪯ h_{2}^{o}$

$(h_{1} \tilde{\lor} h_{2})^{o} ≽ h_{1}^{o} \tilde{\lor} h_{2}^{o}$

$(h_{1} \tilde{\land} h_{2})^{o} \approx h_{1}^{o} \tilde{\land} h_{2}^{o}$

Theorem 3.18. Let (X . τ) be a hesitant fuzzy topological space then ∀h ∈ HF (X) we have

$(h^{o})^{c} \approx \bar{(h^{c})}$

$(h^{c})^{o} \approx (\bar{h})^{c}$

Proof. 1. h^o ⪯ h⇒h^c ⪯ (h^o) ^c by Proposition 2

By Theorem 3 5 we have $\bar{(h^{c})} ⪯ \bar{(h^{o})^{c}} \approx (h^{o})^{c}$ as (h^o) ^c is closed.

Conversely by Theorem 3 2 we have $h^{c} ⪯ \bar{(h^{c})}$

$\Rightarrow (\bar{(h^{c})})^{c} ⪯ (h^{c})^{c} = h$ .

$\bar{(h^{c})}$ is closed, so $(\bar{(h^{c})})^{c}$ is open.

Therefore $(\bar{(h^{c})})^{c} ⪯ h^{o}$

Now taking compliment on both sides we have $(h^{o})^{c} ⪯ ((\bar{(h^{c})})^{c})^{c} \approx \bar{(h^{c})}$

2. From 1. above we have $(h^{o})^{c} \approx \bar{(h^{c})}$ . Taking compliment on both sides we get $((h^{o})^{c})^{c} \approx (\bar{(h^{c})})^{c}$ $\Rightarrow h^{o} \approx (\bar{(h^{c})})^{c}$ .

Replace h by h^c to get $(h^{c})^{o} \approx (\bar{((h^{c})^{c})})^{c} \approx (\bar{h})^{c}$ □

Definition 3.19. Let (X, τ) be a hesitant fuzzy topological space over X and h ∈ HF (X). Let ${\tilde{y}}_{A}$ be a hesitant fuzzy point. Then ${\tilde{y}}_{A}$ is said to be a hesitant fuzzy interior point of h if there exist a hesitant fuzzy open set h₁ ∈ τ such that ${\tilde{y}}_{A} \in h_{1} ⪯ h$ . If ${\tilde{y}}_{A}$ is an interior point of a hesitant fuzzy set h ; then h is called the hesitant fuzzy neighbourhood of ${\tilde{y}}_{A}$ . $(i . e ., \exists \overset{´}{h} \in τ such that {\tilde{y}}_{A} \in \overset{´}{h} ⪯ h)$

Proposition 3.20. Let (X, τ) be a hesitant fuzzy topological space over X and ${\tilde{y}}_{A} \in HF (X)$ be a hesitant fuzzy point then

each ${\tilde{y}}_{A} \in HF (X)$ has a hesitant fuzzy neighbourhood.

If h₁ and h₂ are hesitant fuzzy neighbourhoods of some ${\tilde{y}}_{A} \in HF (X)$ then $h_{1} \tilde{\land} h_{2}$ is also a hesitant fuzzy neighbourhood of ${\tilde{y}}_{A}$ .

If h₁ is a hesitant fuzzy neighbourhood of ${\tilde{y}}_{A} \in HF (X)$ and h₁ ⪯ h₂ then h₂ is also a hesitant fuzzy neighbourhood of ${\tilde{y}}_{A}$ .

Proof.

for any ${\tilde{y}}_{A} \in HF (X)$ , we have ${\tilde{y}}_{A} ⪯ h_{X} \in τ$ . ⇒h_X is a hesitant fuzzy neighbourhood of ${\tilde{y}}_{A}$ .

if h₁ and h₂ are hesitant fuzzy neighbourhoods of some ${\tilde{y}}_{A} \in HF (X)$ then ∃h₃, h₄ ∈ τ such that ${\tilde{y}}_{A} \in h_{3} ⪯ h_{1}$ and ${\tilde{y}}_{A} \in h_{4} ⪯ h_{2}$ .

Now h₃ ⪯ h₁ and $h_{4} ⪯ h_{2} \Rightarrow h_{3} \tilde{\land} h_{4} ⪯ h_{1} \tilde{\land} h_{2}$ .

${\tilde{y}}_{A} \in h_{3}$ and ${\tilde{y}}_{A} \in h_{4} \Rightarrow {\tilde{y}}_{A} \in h_{3} \tilde{\land} h_{4} \in τ$ as (h₃, h₄ ∈ τ)

$\Rightarrow {\tilde{y}}_{A} \in h_{3} \tilde{\land} h_{4} ⪯ h_{1} \tilde{\land} h_{2}$

if h₁ is a hesitant fuzzy neighbourhood of ${\tilde{y}}_{A}$ then $\exists h \in τ such that {\tilde{y}}_{A} \in h ⪯ h_{1}$ . Now $h_{1} ⪯ h_{2} \Rightarrow {\tilde{y}}_{A} \in h ⪯ h_{2}$

Hence h₂ is a hesitant fuzzy neighbourhood of ${\tilde{y}}_{A}$ □

Definition 3.21. Let (X, τ₁) and (Y, τ₂) be two hesitant fuzzy topological spaces, $\tilde{f} : HF (X) \to HF (Y)$ be a hesitant fuzzy mapping. $\tilde{f} : (X, τ_{1}) \to (Y, τ_{2})$ is called open if it maps every open subset in (X, τ₁) to an open subset in (Y, τ₂). i.e., ∀h₁ ∈ τ₁ ; f (h₁) ∈ τ₂.

Definition 3.22. Let (X, τ) be a hesitant fuzzy topological space over X and Y be a nonempty subset of X then τ_Y = {h|_Y : h ∈ τ} is said to be the hesitant fuzzy subspace topology on Y.

The following proposition follows from the definition.

Proposition 3.23. Let (X, τ) be a hesitant fuzzy topological space and h₁ ∈ HF (X). τ_Y be the subspace topology on Y then

h₁ is a hesitant fuzzy open set in Y iff h₁ = h₂|_Y for some h₂ ∈ τ .

h₁ is a hesitant fuzzy closed set in Y iff h₁ = h₂|_Y for some closed set h₂ in (X, τ).

Proof.

If h₁ is a hesitant fuzzy open set in Y then by definition 3, ∃h₂ ∈ τ . such that h₁ = h₂|_Y.

Conversely if h₁ = h₂|_Y for some h₂ ∈ τ ., then h₁ ∈ τ_Y.∴ h₁ is a hesitant fuzzy open set in Y.

If h₁ is a hesitant fuzzy closed set in Y, then $h_{1}^{c}$ is a hesitant fuzzy open set in Y. Then by part (1), $h_{1}^{c} = h |_{Y}$ for some h ∈ τ . then $(h_{1}^{c})^{c} = h^{c} |_{Y} \Rightarrow h_{1} = h_{2} |_{Y}$ where h₂ = h^c which is closed in X.

Conversely if h₁ = h₂|_Y for some closed set h₂ in (X, τ), then $h_{1}^{c} = h_{2}^{c} |_{Y}$ , since $h_{2}^{c}$ is open in X, by part(1), $h_{1}^{c}$ is open in Y. Hence h₁ is closed in Y.□

Proposition 3.24. Let (x, τ) be a hesitant fuzzy topological space, Y ⊂ X and {h_t : t ∈ T} ⊂ τ, h ∈ τ. Then

${({\tilde{\lor}}_{t \in T} h_{t}) |}_{Y} \approx {\tilde{\lor}}_{t \in T} {(h_{t}) |}_{Y}$

${({\tilde{\land}}_{t \in T} h_{t}) |}_{Y} \approx {\tilde{\land}}_{t \in T} {(h_{t}) |}_{Y}$

h^c|_Y ≈(h|_Y) ^c

Proof.

let $\begin{matrix} h \in {({\tilde{\lor}}_{t \in T} h_{t}) |}_{Y} \Leftrightarrow & h \in {\tilde{\lor}}_{t \in T} h_{t} & h \in Y \\ \Leftrightarrow & h = h_{t} for some t \in T & h \in Y . \\ \Leftrightarrow & h \in (h_{t}) |_{Y} for some t \in T \\ \Leftrightarrow & h \in {\tilde{\lor}}_{t \in T} {(h_{t}) |}_{Y} \end{matrix}$ ∴ ${({\tilde{\lor}}_{t \in T} h_{t}) |}_{Y} \approx {\tilde{\lor}}_{t \in T} {(h_{t}) |}_{Y}$

in the similar manner (2) and (3) follows...□

4. Continuity, Connectedness and Compactness

This section gives the notion of a continuous hesitant fuzzy mapping and studies its properties. The notions of connectedness and compactness in hesitant fuzzy topological spaces are studied and their behaviour under continuous mappings is explored.

Definition 4.1. (Extension Principle). Let f be a function from X into Y, and let h₁ ∈ HF (X) and h₂ ∈ HF (Y). Define the hesitant fuzzy subsets $\tilde{f} (h_{1}) \in HF (Y)$ and ${\tilde{f}}^{- 1} (h_{2}) \in HF (Y)$ by ∀y ∈ Y, $\tilde{f} (h_{1}) (y) = {\begin{matrix} \tilde{\lor} {h_{1} (x) | x \in X, f (x) = y}; if f^{- 1} (y) \neq φ, \\ {0}, otherwise \end{matrix}$ and $\forall x \in x, {\tilde{f}}^{- 1} (h_{2}) (x) = h_{2} (f (x)) .$ , Then $\tilde{f} (h_{1})$ is called the image of h₁ under $\tilde{f}$ and ${\tilde{f}}^{- 1} (h_{2})$ is called the pre image of h₂ under $\tilde{f}$ .

Theorem 4.2. Let f be a function from X into Y and g a function from Y into Z. Then the following assertions hold.

$h_{1} ⪯ h_{2} \Rightarrow \tilde{f} (h_{1}) ⪯ \tilde{f} (h_{2}) \forall h_{1}, h_{2} \in HF (X)$

$h_{1} ⪯ h_{2} \Rightarrow {\tilde{f}}^{- 1} (h_{1}) ⪯ {\tilde{f}}^{- 1} (h_{2}) \forall h_{1}, h_{2} \in HF (Y)$

${\tilde{f}}^{- 1} (\tilde{f} (h_{1})) ≽ h_{1} \forall h_{1} \in HF (X)$ . In particular if f is a surjection, then ${\tilde{f}}^{- 1} (\tilde{f} (h_{1})) = h_{1} \forall h_{1} \in HF (X) .$

$\tilde{f} ({\tilde{f}}^{- 1} (h_{2})) ⪯ h_{2} \forall h_{2} \in HF (X)$ . In particular if f is a surjection, then $\tilde{f} ({\tilde{f}}^{- 1} (h_{2})) = h_{2} \forall h_{2} \in HF (Y) .$

$\tilde{f} (h_{1}) ⪯ h_{2} \Leftrightarrow h_{1} ⪯ {\tilde{f}}^{- 1} (h_{2}) \forall h_{1} \in HF (X)$ and h₂ ∈ HF (Y) .

$\tilde{g} (\tilde{f} (h_{1})) = (\tilde{g} \circ \tilde{f}) (h_{1}) \forall h_{1} \in HF (X)$ and ${\tilde{f}}^{- 1} ({\tilde{g}}^{- 1} (h_{3})) = (\tilde{g} \circ \tilde{f})^{- 1} (h_{3}) \forall h_{3} \in HF (Z) .$

Proof.

$h_{1} ⪯ h_{2} \Rightarrow s (h_{1} (x)) \leq s (h_{2} (x)) \forall x \in X$ (4.0.1)

$\begin{matrix} \tilde{f} (h_{1}) (y) & = & \tilde{\lor} {h_{1} (x) | x \in X, f (x) = y} \\ ⪯ & \tilde{\lor} {h_{2} (x) | x \in X, f (x) = y} from (2.1) \\ = & \tilde{f} (h_{2} (y)) \end{matrix}$

${\tilde{f}}^{- 1} (h_{1}) (x) = h_{1} (f (x)) ⪯ h_{2} (f (x)) = {\tilde{f}}^{- 1} (h_{2}) (x)$

$\begin{matrix} {\tilde{f}}^{- 1} (\tilde{f} (h_{1})) (x) & = & \tilde{f} (h_{1}) (f (x)) \\ = & \tilde{\lor} {h_{1} (x^{'}) | x^{'} \in X, f (x^{'}) = f (x)} \\ ≽ & h_{1} \forall x \in X \end{matrix}$

In particular if $\tilde{f}$ is an injection then ${\tilde{f}}^{- 1} (\tilde{f} (h_{1})) (x) = \tilde{\lor} {h_{1} (x^{'}) | x^{'} \in X, f (x^{'}) = f (x)} = h_{1} (x)$

Let h₂ ∈ HF (Y) then $\begin{matrix} \tilde{f} ({\tilde{f}}^{- 1} (h_{2})) (y) & = & \tilde{\lor} {{\tilde{f}}^{- 1} (h_{2} (x)) | x \in X, f (x) = y} \\ = & \tilde{\lor} {h_{2} (f (x)) | x \in X, f (x) = y} \\ = & {\begin{matrix} h_{2} (y), & for y \in f (X) \\ {0}, & otherwise \end{matrix} \\ = & h_{2} (y) \forall y \in Y \end{matrix}$ Thus $\tilde{f} ({\tilde{f}}^{- 1} (h_{2})) (y) = h_{2} (y) \forall y \in Y$ if f is a surjection.

$\tilde{f} (h_{1}) ⪯ h_{2} \Rightarrow {\tilde{f}}^{- 1} (\tilde{f} (h_{1})) ⪯ \tilde{f} (h_{2}) \forall y \in Y$ from (2)

$\Rightarrow h_{1} ⪯ {\tilde{f}}^{- 1} (\tilde{f} (h_{1})) ⪯ \tilde{f} (h_{2})$ from (3)

$\Rightarrow h_{1} ⪯ \tilde{f} (h_{2})$

Conversely, $h_{1} ⪯ {\tilde{f}}^{- 1} (h_{2})$ from (1)

$\Rightarrow \tilde{f} (h)_{1} ⪯ \tilde{f} ({\tilde{f}}^{- 1} (h_{2}))$

$\Rightarrow \tilde{f} (h_{1}) ⪯ \tilde{f} ({\tilde{f}}^{- 1} (h_{2})) ⪯ h_{2}$ from (4)

$\Rightarrow \tilde{f} (h_{1}) ⪯ h_{2}$

Consider any h ∈ HF (X) and any z ∈ Z. Then $\begin{matrix} \tilde{g} (\tilde{f} (h)) (z) & = & \tilde{\lor} {\tilde{f} (h) (y) | y \in Y, g (y) = z} \\ = & \tilde{\lor} {\tilde{\lor} {h (x) | x \in X, f (x) = y} | y \in Y, g (y) = z} \\ = & \tilde{\lor} {h (x) | x \in X, (g \circ f) (x) = z} \\ = & (\tilde{g} \circ \tilde{f}) (h) (z) \forall z \in Z \end{matrix}$ Further, ∀h′ ∈ HF (Z) and ∀x ∈ X, $\begin{matrix} ((\tilde{g} \circ \tilde{f})^{- 1} (h^{'}) (x)) & = & h^{'} (\tilde{g} (\tilde{f} (x))) \\ = & {\tilde{g}}^{- 1} (h^{'}) (\tilde{f} (x)) \\ = & {\tilde{f}}^{- 1} ({\tilde{g}}^{- 1} (h^{'})) (x) \end{matrix}$

□

Definition 4.3. Continuous Mappings: Let $(\tilde{X}, τ_{1})$ and $(\tilde{Y}, τ_{2})$ be two hesitant fuzzy topological spaces. $\tilde{f} : HF (X) \to HF (Y)$ be a hesitant fuzzy mapping. We say that $\tilde{f}$ is a hesitant fuzzy continuous mapping from HF (X) to HF (Y) or the hesitant fuzzy map $\tilde{f}$ is continuous if ${\tilde{f}}^{- 1} : HF (Y) \to HF (X)$ maps every open subset of HF (Y) to an open set in HF (X) i.e., $\forall h \in τ_{2}, {\tilde{f}}^{- 1} (h) \in τ_{1}$ .

Definition 4.4. Let (X, τ₁) and (Y, τ₂) be two hesitant fuzzy topological spaces. A hesitant fuzzy mapping $\tilde{f} : HF (X) \to HF (Y)$ is called continuous at a hesitant fuzzy point ${\tilde{y}}_{A} \in HF (X)$ if ${\tilde{f}}^{- 1} (h) \in τ_{1}$ for some h ∈ τ₂ such that ${\tilde{y}}_{A} \in h$

Proposition 4.5. Let $(\tilde{X}, τ_{1})$ and $(\tilde{Y}, τ_{2})$ be two hesitant fuzzy topological spaces.

$\tilde{f} : HF (X) \to HF (Y)$ be a hesitant fuzzy continuous mapping. If h ∈ HF (Y) is closed then ${\tilde{f}}^{- 1} (h)$ is closed in HF(X).

Proof. $h \in τ_{2}^{c} \Rightarrow h^{c} \in τ_{2} \Rightarrow {\tilde{f}}^{- 1} (h^{c}) \in τ_{1}$ , since $\tilde{f}$ is continuous.

Now, ${\tilde{f}}^{- 1} (h^{c}) \in τ_{1}$ i.e., ${\tilde{f}}^{- 1} (h^{c}) = ({\tilde{f}}^{- 1} (h))^{c} \in τ_{1}$

$\Rightarrow {\tilde{f}}^{- 1} (h) \in τ_{1}^{c}$ . Hence we have that if $h \in τ_{2}^{c}$ then ${\tilde{f}}^{- 1} (h) \in τ_{1}^{c}$ □

Theorem 4.6. Let (X, τ₁) and (Y, τ₂) be two hesitant fuzzy topological spaces. $\tilde{f} : HF (X) \to HF (Y)$ be a hesitant fuzzy mapping. Then the following are equivalent

$\tilde{f}$ is continuous

$\forall h \in τ_{2}^{c}, {\tilde{f}}^{- 1} (h) \in τ_{1}^{c}$

$\forall h_{1} \in HF (X), \tilde{f} (\bar{h_{1}}) ⪯ (\bar{\tilde{f} (h_{1})})$

$\forall h_{2} \in HF (Y), \bar{{\tilde{f}}^{- 1} (h_{2})} ⪯ {\tilde{f}}^{- 1} (\bar{h_{2}})$

$\forall h_{2} \in HF (Y), {\tilde{f}}^{- 1} (h_{2}^{\circ}) ⪯ (\tilde{f} (h_{2}))^{\circ}$

Proof. (i)⇒(ii). $h \in τ_{2}^{c} \Rightarrow h^{c} \in τ \Rightarrow {\tilde{f}}^{- 1} (h^{c}) \in τ$ , since $\tilde{f}$ is continuous. $\Rightarrow ({\tilde{f}}^{- 1} (h^{c}))^{c} \in τ^{c}$

Now, $({\tilde{f}}^{- 1} (h^{c}))^{c} (x) = (h^{c} (\tilde{f} (x)))^{c} = (h^{c})^{c} (\tilde{f} (x)) = h (\tilde{f} (x)) = {\tilde{f}}^{- 1} (h) (x)$

$\Rightarrow {\tilde{f}}^{- 1} (h) \in τ_{1}^{c}$

(ii)⇒(i) can be proved similarly.

(iii)⇒(iv) Given $\tilde{f} (\bar{h_{1}}) ⪯ (\bar{\tilde{f} (h_{1})})$ and we have $\tilde{f} ({\tilde{f}}^{- 1} (h_{2})) ⪯ h_{2}$ .

Hence $\tilde{f} (\bar{{\tilde{f}}^{- 1} (h_{2})}) ⪯ \bar{\tilde{f} ({\tilde{f}}^{- 1} (h_{2}))} ⪯ \bar{h_{2}}$

Now we have ${\tilde{f}}^{- 1} \tilde{f} (h) ⪯ h$

$\Rightarrow \bar{{\tilde{f}}^{- 1} (h_{2})} ⪯ {\tilde{f}}^{- 1} \tilde{f} (\bar{{\tilde{f}}^{- 1} (h_{2})}) ⪯ {\tilde{f}}^{- 1} (\bar{h_{2}})$

(iv)⇒(v) Given $\forall h_{2} \in HF (Y), \bar{{\tilde{f}}^{- 1} (h_{2})} ⪯ {\tilde{f}}^{- 1} (\bar{h_{2}})$

We have that $(h^{o})^{c} \approx \bar{(h^{c})}$ and $\tilde{f} (h^{c}) \approx (\tilde{f} (h))^{c}$

Hence $(({\tilde{f}}^{- 1} (h_{2}))^{o})^{c} \approx \bar{({\tilde{f}}^{- 1} (h_{2}))^{c}} \approx \bar{({\tilde{f}}^{- 1} (h_{2}^{c}))} ⪯ {\tilde{f}}^{- 1} (\bar{h_{2}^{c}}) \approx ({\tilde{f}}^{- 1} (h_{2}^{o}))^{c}$

$\Rightarrow (({\tilde{f}}^{- 1} (h_{2}))^{o})^{c} ⪯ ({\tilde{f}}^{- 1} (h_{2}^{o}))^{c}$

$\Rightarrow ({\tilde{f}}^{- 1} (h_{2}))^{o} ≽ {\tilde{f}}^{- 1} (h_{2}^{o})$

(v)⇒(i) We have ${\tilde{f}}^{- 1} (h_{2}^{\circ}) ⪯ (\tilde{f} (h_{2}))^{\circ}$

Let h₂ ∈ τ₂ then

${\tilde{f}}^{- 1} (h_{2}) \approx {\tilde{f}}^{- 1} (h_{2}^{o}) ⪯ ({\tilde{f}}^{- 1} (h_{2}))^{o} ⪯ {\tilde{f}}^{- 1} (h_{2})$ i.e., ${\tilde{f}}^{- 1} (h_{2}) ⪯ ({\tilde{f}}^{- 1} (h_{2}))^{o}$ and ${\tilde{f}}^{- 1} (h_{2}) ≽ ({\tilde{f}}^{- 1} (h_{2}))^{o}$

So, ${\tilde{f}}^{- 1} (h_{2}) \approx ({\tilde{f}}^{- 1} (h_{2}))^{o}$

$\Rightarrow {\tilde{f}}^{- 1} (h_{2}) \in τ_{1}$ since $({\tilde{f}}^{- 1} (h_{2}))^{o}$ is open

(i)⇒(iii) We have $h_{1} ⪯ {\tilde{f}}^{- 1} \bar{(\tilde{f} (h_{1}))}$ By Theorem 4 (iii)

Now $\bar{(\tilde{f} (h_{1}))}$ is closed and ${\tilde{f}}^{- 1} \bar{(\tilde{f} (h_{1}))}$ is closed since $\tilde{f}$ is continuous.

$\Rightarrow \bar{h_{1}} ⪯ {\tilde{f}}^{- 1} \bar{(\tilde{f} (h_{1}))}$

$\Rightarrow \tilde{f} (\bar{h_{1}}) ⪯ \bar{(\tilde{f} (h_{1}))}$ □

Definition 4.7. Let (X, τ) be a hesitant fuzzy topological space and h ∈ HF (X). A separation of h is a pair h₁, h₂ ∈ HF (X) ; h₁, h₂ ⪯ h such that $\bar{h_{1}} \tilde{\land} h_{2} \approx h_{0}$ and $h_{1} \tilde{\land} \bar{h_{2}} \approx h_{0}$ . h is called connected if there does not exist such a separation {h₁, h₂}, h₁, h₂notapproxh₀, for h such that $h \approx h_{1} \tilde{\lor} h_{2}$ .

Theorem 4.8. Let (X, τ) be a hesitant fuzzy topological space and h, h₁, h₂, h₃ ∈ HF (X). Let h₁, h₂ be a separation of h, then $h_{3} \tilde{\land} h_{1}$ and $h_{3} \tilde{\land} h_{2}$ are separated.

Proof. $h_{3} \tilde{\land} h_{1} ⪯ h_{1} \Rightarrow \bar{h_{3} \tilde{\land} h_{1}} ⪯ \bar{h_{1}}$ , and we have $h_{3} \tilde{\land} h_{2} ⪯ h_{2}$ .

Hence $(\bar{h_{3} \tilde{\land} h_{1}}) \tilde{\land} (h_{3} \tilde{\land} h_{2}) ⪯ \bar{h_{1}} \tilde{\land} h_{2} \approx h_{0}$ as h is separated by h₁ and h₂.

similarly, $(h_{3} \tilde{\land} h_{1}) \tilde{\land} \bar{(h_{3} \tilde{\land} h_{2})} \approx h_{0}$

therefore $h_{3} \tilde{\land} h_{1}$ and $h_{3} \tilde{\land} h_{2}$ are separated.□

Definition 4.9. $\tilde{f} : (X, τ_{1}) \to (Y, τ_{2})$ is called a hesitant fuzzy homeomorphism if it is bijective, continuous and open. We say that (X, τ₁) and (Y, τ₂) are homeomorphic if there exists a hesitant fuzzy homeomorphism $\tilde{f}$ from (X, τ₁) to (Y, τ₂).

Theorem 4.10. Let (X, τ₁) and (Y, τ₂) be two hesitant fuzzy topological spaces, $\tilde{f} : (X, τ_{1}) \to (Y, τ_{2})$ be a hesitant fuzzy mapping, X₀ ⊂ X. Then $\tilde{f} : (X, τ_{1}) \to (Y, τ_{2})$ is continuous ⇒ ${\tilde{f} |}_{X_{0}} : (X_{0}, {τ_{1} |}_{X_{0}}) \to (f [X_{0}], {τ_{2} |}_{f [X_{0}]})$ is continuous. Here ${\tilde{f} |}_{X_{0}} = \tilde{f} (h) \forall h \in X_{0}$

Proof. Let h ∈ τ₂|_f[X₀] then ∃h′ ∈ τ₂ such that h = h′|_f[X₀]. We have that $\tilde{f}$ is continuous.

Now, $({\tilde{f} |}_{X_{0}})^{- 1} (h) = h [({\tilde{f} |}_{X_{0}})] = ({h^{'} |}_{f [X_{0}]}) [\tilde{f} |_{X_{0}}]$ $= {h^{'} (\tilde{f}) |}_{X_{0}} = {{\tilde{f}}^{- 1} (h^{'}) |}_{X_{0}} \in {τ_{1} |}_{X_{0}}$ $\tilde{f}$ is continuous $\Rightarrow \tilde{f} (h^{'})$ is open for h′ ∈ τ₂ $\Rightarrow {\tilde{f} (h^{'}) |}_{X_{0}}$ is open in τ₁|_{X
₀}□

Theorem 4.11. Let (X, τ) be a hesitant fuzzy topological space, h, h₁, h₂ ∈ HF (X). If h₁ and h₂ are separated $(h ⪯ h_{1} \tilde{\lor} h_{2})$ and h is connected then either h ⪯ h₁ or h ⪯ h₂.

Proof. By Theorem 4.8 we have $h \tilde{\land} h_{1}$ and $h \tilde{\land} h_{2}$ are separated. Since $h ⪯ h_{1} \tilde{\lor} h_{2}$ , we have $h \approx h \tilde{\land} (h_{1} \tilde{\lor} h_{2}) \approx (h \tilde{\land} h_{1}) \tilde{\lor} (h \tilde{\land} h_{2})$ (4.2) Now h is connected and $h \tilde{\land} h_{1}$ and $h \tilde{\land} h_{2}$ are separated. Hence either $h \tilde{\land} h_{1} \approx h_{0}$ or $h \tilde{\land} h_{2} \approx h_{0}$ , otherwise we will arrive at a contradiction. Let $h \tilde{\land} h_{1} \approx h_{0}$ then from (4.0.2) we have $h \approx h_{0} \tilde{\lor} (h \tilde{\land} h_{2}) \approx h \tilde{\land} h_{2}$ . So h ⪯ h₂. Similarly if $h \tilde{\land} h_{1} \approx h_{0} \Rightarrow h ⪯ h_{1}$ □

Corollary 4.12. Let (X, τ) be a hesitant fuzzy topological space, h, h₁, h₂ ∈ HF (X). If h₁ and h₂ are separated $(h ⪯ h_{1} \tilde{\lor} h_{2})$ and h is connected then either $h \tilde{\land} h_{1} \approx h_{0}$ or $h \tilde{\land} h_{2} \approx h_{0}$ .

Theorem 4.13. Let (X, τ) be a hesitant fuzzy topological space. h ∈ HF (X) be connected, h₁ ∈ HF (X) such that $h ⪯ h_{1} ⪯ \bar{h}$ . Then h is connected.

Proof. Let h be connected, and $h ⪯ h_{1} ⪯ \bar{h}$ . Suppose $h_{1} \approx h_{2} \tilde{\lor} h_{3}$ is a separation of h₁, $\Rightarrow h ⪯ h_{2} \tilde{\lor} h_{3}$ as h ⪯ h₁. Then by Theorem 4.11 we have h ⪯ h₂ or h ⪯ h₃. Suppose that h ⪯ h₂, then $\bar{h} ⪯ \bar{h_{2}}$ . Since $\bar{h_{2}} \tilde{\land} h_{3}$ we have $h_{1} \tilde{\land} h_{3} \approx h_{0}$ as $h_{1} \approx h_{2} \tilde{\lor} h_{3}$ .

Now we have $h_{1} ⪯ \bar{h}$ and $\bar{h} ⪯ \bar{h_{2}} \Rightarrow h_{1} ⪯ \bar{h_{2}}$ . In addition we have $h_{1} \approx h_{2} \tilde{\lor} h_{3}$ and $\bar{h_{2}} \tilde{\land} h_{3} \approx h_{0}$ from which we can conclude that h₃ ≈ h₀. Hence there does not exist any separation for h⇒h is connected.□

Theorem 4.14. Let (X, τ₁) and (Y, τ₂) be hesitant fuzzy topological spaces. $\tilde{f} : (X, τ_{1}) \to (Y, τ_{2})$ be a hesitant fuzzy continuous mapping. Then $\tilde{f} (h)$ is connected.

Proof. Suppose $\tilde{f} (h) \approx h_{1} \tilde{\lor} h_{2}$ where h₁ and h₂ are separated. By Theorem 4.2 we have ${\tilde{f}}^{- 1} (\tilde{f} (h)) ≽ h$ . i.e., $h ⪯ {\tilde{f}}^{- 1} (\tilde{f} (h)) \approx {\tilde{f}}^{- 1} (h_{1} \tilde{\lor} h_{2}) \approx {\tilde{f}}^{- 1} (h_{1}) \tilde{\lor} {\tilde{f}}^{- 1} (h_{2})$ .

Since $\tilde{f}$ is continuous By Theorem 4.6 we have $\bar{{\tilde{f}}^{- 1} (h_{1})} ⪯ {\tilde{f}}^{- 1} (\bar{h_{1}})$ .

ie., $\bar{{\tilde{f}}^{- 1} (h_{1})} \tilde{\land} {\tilde{f}}^{- 1} (h_{2}) ⪯ {\tilde{f}}^{- 1} (\bar{h_{1}}) \tilde{\land} {\tilde{f}}^{- 1} (h_{2}) \approx {\tilde{f}}^{- 1} (h_{1} \tilde{\land} h_{2}) \approx {\tilde{f}}^{- 1} (h_{0}) \approx h_{0}$ .

Similarly ${\tilde{f}}^{- 1} (h_{1}) \tilde{\land} \bar{{\tilde{f}}^{- 1} (h_{2})} \approx h_{0}$ . So ${\tilde{f}}^{- 1} (h_{1})$ and ${\tilde{f}}^{- 1} (h_{2})$ are separated. Hence by Theorem 4 we have $h ⪯ {\tilde{f}}^{- 1} (h_{1})$ or $h ⪯ {\tilde{f}}^{- 1} (h_{2})$ .

Suppose that $h ⪯ {\tilde{f}}^{- 1} (h_{1})$ , then $\tilde{f} (h) ⪯ \tilde{f} ({\tilde{f}}^{- 1} (h_{1}))$ . ⇒h₂ ≈ h₀ since $\tilde{f} (h) \approx h_{1} \tilde{\lor} h_{2}$ . This is a contradiction since we have assumed that $\tilde{f} (h) \approx h_{1} \tilde{\lor} h_{2}$ and h₁ and h₂ are separations of $\tilde{f} (h)$ . i.e., h₁, h₂≄h₀ and $\bar{h_{1}} \tilde{\land} h_{2} \approx h_{0}$ and $h_{1} \tilde{\land} \bar{h_{2}} \approx h_{0}$ . Hence $\tilde{f} (h)$ is connected.□

Definition 4.15. Let (X, τ) be a hesitant fuzzy topological space and α ∈ [0, 1). A family of hesitant fuzzy subsets $C \subset HF (X)$ is called an α-cover if for every x ∈ X, $\exists h \in C$ such that $h (x) ≽ {\tilde{x}}_{α}$ . $C$ is called an open α-cover if $C \subset τ$ and $C$ is an α-cover. $C_{0} \subset HF (X)$ is called a sub α-cover of $C$ if $C_{0} \subset C$ and $C_{0}$ is an α-cover.

Definition 4.16. For a given α ∈ [0, 1), (X, τ) is called α-compact if every open α-cover has a finite sub α-cover.

Example 4.17. Let X = [0, 1]; h_n : X ⟶ P [0, 1] for $n \in ℕ \ {1}$ . $h_{n} (x) = (\frac{1}{2} - \frac{1}{n}, \frac{1}{2} + \frac{1}{n}) \cap [0, 1]; n \in ℕ \ {1}$ Let (X, τ_n) be the hesitant fuzzy topological space generated by ${h_{0}, h_{X}} \cup {h_{n} | n \in ℕ \ {1}}$ . Then (X, τ_n) is α-compact for all α ∈ [0, 1], since for $α \in [0, \frac{1}{2}]$ any h_n will cover (X, τ) and for $α \in (\frac{1}{2}, 1]$ the only open cover is h_X.

Let X = [c, d] and a ∈ (0, 1). $h_{t}^{a} (x) = {\begin{matrix} (a - \frac{1}{2}, a + \frac{1}{2}) & if x = t \\ {0} & if x \neq t \end{matrix}$ Let (X, τ_t) be the hesitant fuzzy topological space generated by ${h_{0}, h_{X}} \cup {h_{t}^{a} | t \in X}$ . Then (X, τ_t) is α-compact iff α = 0 or a < α ≤ 1.

Theorem 4.18. Let (X, τ₁) and (Y, τ₂) be two hesitant fuzzy topological spaces. Let $\tilde{f} : HF (X) \to HF (Y)$ be a hesitant fuzzy continuous map. If X is α-compact then $\tilde{f} (X)$ is α-compact in (Y, τ₂).

Proof. Let $C$ be an α-cover of f (X). Then for every $y = f (x) \in f (X) \exists h \in C$ such that $h (f (x)) ≽ {\tilde{x}}_{α}$ $\Rightarrow {\tilde{f}}^{- 1} (h (x)) ≽ {\tilde{x}}_{α}$ . Hence $\tilde{f} (C) = {{\tilde{f}}^{- 1} (h) : h \in C}$ is an α-cover of X since $\forall x \in X \exists {\tilde{f}}^{- 1} (h) \in \tilde{f} (C)$ such that ${\tilde{f}}^{- 1} (h (x)) ≽ {\tilde{x}}_{α}$ .

X is α-compact $\Rightarrow \tilde{f} (h)$ has a finite sub α-cover ${{\tilde{f}}^{- 1} (h_{1}), {\tilde{f}}^{- 1} (h_{2}), . . ., {\tilde{f}}^{- 1} (h_{n})}$ . Hence {h₁, h₂, . . . , h_n} is a finite sub α -cover of h. Now for y = f (x) ∃ k such that ${\tilde{f}}^{- 1} (h_{k}) (x) ≽ {\tilde{x}}_{α} \Rightarrow h_{k} (f (x)) ≽ {\tilde{x}}_{α}$ . Hence the result.□

Theorem 4.19. Let (X, τ) be a hesitant fuzzy topological space. Let Y be a subspace of X and τ_Y the hesitant fuzzy subspace topology on Y. The hesitant fuzzy topological space (Y, τ_Y) is α-compact iff for every α-covering $C \subset τ$ of Y there exists a finite sub α-cover $C_{0} \subset C$ such that $C_{0}$ covers Y.

Proof. Suppose that (Y, τ_Y) is compact and $C = {h_{β}}_{β \in J}$ is an α-covering of Y where h_β ∈ τ. Then the collection {h_β|_Y} _β∈J is a covering of Y by open sets in τ_Y. Hence {h_{β
₁}|_Y, h_{β
₂}|_Y, . . . , h_{β
_n}|_Y} covers Y. Then {h_{β
₁}, h_{β
₂}, . . . , h_{β
_n}} is a subcollection of $C$ that covers Y.

Conversely, we have for every α-covering $C \subset τ$ of Y there exists a finite sub α-cover $C_{0} \subset C$ such that $C_{0}$ covers Y.

Let $C_{1} = {h_{β}^{'}} \subset τ_{Y}$ be a covering of Y. For each β choose h_β ∈ HF (X) such that $h_{β}^{'} = h_{β} |_{Y}$ . The $C = {h_{β}} \subset τ_{Y}$ is an α-covering of Y ⇒ there exists a finite sub collection {h_{β
₁}, h_{β
₂}, . . . , h_{β
_n}} that covers Y. Then ${h_{β_{1}}^{'}, h_{β_{2}}^{'}, . . ., h_{β_{n}}^{'}}$ is a subcollection of $C_{1}$ that covers Y.□

5. Conclusions

In this paper, an attempt is made to develop a topological structure in the context of hesitant fuzzy sets using score based intersection and union. Other than the introduction of the structure, main contributions in the paper are the following: 1) Closure and interior are defined and their properties are investigated. 2) Continuous mappings between hesitant fuzzy topological spaces are introduced and a characterization is obtained. 3) The notions of connectedness and compactness in hesitant fuzzy topological spaces introduced and their behaviour under continuous mappings is studied. On the basis of the introduced concepts, the paper opens up the scope for studying separation axioms, covering properties and metrizability in the case of hesitant fuzzy topological spaces in future. Similar topological studies in the context of other hybrid structures involving fuzzy sets like Pythagorean fuzzy sets [39], Picture fuzzy sets [40], orthopair fuzzy sets [41] is also possible. Development of entropy and similarity measures for hybrid structures involving fuzzy sets, Pythagorean fuzzy sets, picture fuzzy sets, orthopair fuzzy sets, and their applications in multi criteria decision making problems is also a fruitful area for future work.

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