Hypersoft sets on nearness approximation space and its topology

Abstract

Hypersoft set theory represents an advanced version to soft set theory, offering enhanced capabilities for addressing uncertainty. By combining hypersoft set theory with nearness approximation spaces, a novel mathematical model known as near hypersoft set emerges. This hybrid model enables improved decision-making accuracy. In this study, our focus is on selecting an object from a product containing a function parameter set described by a distinct Cartesian feature with multiple arguments. Furthermore, we define fundamental features and topology on this set.

Keywords

Soft sets near sets near soft sets hypersoft set near hypersoft set

1 Introduction

Various models have been developed to handle uncertainty, each designed for specific problem scenarios. As a result, different set concepts have been introduced. Pawlak [1] introduced the concept of rough sets, which involves examining sets based on objects and their features. Building upon this concept, Peters [2, 3] introduced near sets, which explore sets that are in proximity to each other in terms of these features. Another notable set concept is the soft set, proposed by Molodtsov [4], which has been extensively studied in both theory and practice [4-9]. The soft set concept offers flexibility in defining objects, allowing researchers to utilize various parameters according to their specific needs. Consequently, it greatly facilitates decision-making processes and finds wide-ranging applications in diverse fields [10]. Chen [11] focused on parameter reduction in soft sets and investigated problem implementations. Subsequently, Babita [12, 13] introduced the concept of soft set relation as a sub-soft set of the Cartesian product of soft sets. They also explored related concepts such as equivalent soft set relation, partition, composition, and function. By replacing the function in soft sets with a multi-row function defined by the Cartesian product of parameter sets, a complex state of the parameter set is obtained. This concept, known as hypersoft set, was introduced by Smarandache [14] in 2018. Since then, it has been extensively studied in both theory and practice [15-18]. Hypersoft set extends the concept of soft set to incorporate n different sets of parameters with multiple independent variables [14]. Hypersoft set is more adaptable than soft set and proves to be highly effective in decision-making problems. The generalization of soft sets provided by hypersoft set has attracted considerable attention, leading to investigations by researchers such as Saeed et al. [19] and Abbas et al. [20], who described and studied various operations on hypersoft sets. Moreover, hypersoft sets have recently found successful applications in numerous fields [10, 21-25].

On the other hand, Feng and Li [26] introduced a new notion by combining the concepts of soft set and near set. Similarly, Tasbozan et al. [27] integrated the concepts of near set and soft set. These concepts have further been developed and applied in the field of topology [28, 29]. Specifically, the concept of near soft set arises based on the existence of equivalence classes in soft sets within nearness approximation spaces. Accordingly, if the equivalence classes in the soft set exhibit similarity, these sets can be considered close to each other.

In this study, we provide the necessary definitions in the initial part and subsequently introduce the concept of near hypersoft sets by incorporating set characteristics that are in proximity to hypersoft sets. The set of parameters is expressed in a more complex manner as a Cartesian product. By employing this new parameter set, we can impose parameter restrictions within the nearness approximation space in the context of hypersoft set, which is a complex structure. Consequently, objects in the hypersoft set can be constrained to desired properties. By establishing the hypersoft set structure in nearness approximation spaces, we develop its topology and examine specific properties within this space. In the topology of this set, referred to as near hypersoft set, concepts such as set interior and neighborhood closure are investigated.

2 Preliminary

2.1 Soft sets, near soft sets and hyper soft set

Let $O$ be an objects set, $F$ be a set of parameters that define properties on objects and $P (O)$ is the power set of $O$ .

Definition 1. A soft set(SS) over $O$ is a represented by (T, D), where D ⊆ T and $T : D \to P (O)$ [7].

Definition 2. Consider a nearness approximation space $NAS = (O, F, \sim_{Dr}, N_{r}, υ_{N_{r}})$ , where D is a non-empty subsets of $F$ and (T, D) is a SS over $O$ . Then, lower and upper near approximation operators are defined as follows: $\begin{matrix} N_{r} * ((T, D)) \\ = & (N_{r} * (T (k) = \cup {x \in O : [x]_{Dr} \subseteq T (k)}, D)) \end{matrix}$ and $\begin{matrix} N_{r}^{*} ((T, D)) \\ = & (N_{r}^{*} (T (k) = \cup {x \in O : [x]_{Dr} \cap T (k) \neq \emptyset}, D)) . \end{matrix}$ The SS N_r ((T, D)) satisfying ${Bnd}_{N_{r} (D)} ((T, D)) \geq 0$ is called a near soft set(NSS) [27].

Definition 3. Let C₁, C₂, . . . , C_n be pairwise disjoint parameters, D_i ⊆ C_i for i = 1, 2, . . . , n and T be a mapping given by $T : D = D_{1} \times D_{2} \times . . . \times D_{n} \to P (O)$ . A pair (T, D₁ × D₂ × . . . × D_n) is refered to as a hypersoft set (HSS) over $O$ and (T, D) = {(β, T (β)) : β ∈ D} is a hypersoft set (T, D) [15].

Definition 4. Let τ be the collection of HSS over $O$ . τ is called a hypersoft topology on $O$ if the following conditions are satisfied:

(1) (∅ , D), $(O, D)$ ϵτ,

(2) If (T₁, D) , (T₂, D) ∈ τ then (T₁, D) ∩ (T₂, D) ∈ τ,

(3) If (T_i, D) ∈ τ then $\underset{i}{\cup} (T_{i}, D) \in τ$ .

Then $(O, τ, D)$ is called a hypersoft topological space over $O$ [15].

3 Hyper soft sets on near aproximation space

Definition 5. Let $O$ be a set of objects, D₁, D₂, . . . , D_n be disjoint pairwise parameters, $NAS = (O, F, \sim_{Dr}, N_{r}, υ_{N_{r}})$ be a nearness approximation space, D_i be a non-empty subsets of $F$ , D = D₁ × D₂ × . . . × D_n, q_i ∈ D_i, i = 1, 2, . . . , 2ⁿ and [x] _{(q_i)_r} be equivalence classes denoted by the subscript r for the cardinality of the restricted subset (q_i) _r. Consider a hypersoft set (T, D) over $O$ , where T is a mapping given by $T : D_{1} \times D_{2} \times . . . \times D_{n} \to P (O)$ and $P (O)$ represents the power set of $O$ . Then, the lower and upper near approximation operators are defined as follows: $\begin{matrix} N_{r} * ((T, D)) \\ = & (N_{r} * (T (q) = \cup {x \in O : [x]_{(q_{i})_{r}} \subseteq T (q)}, D)) \end{matrix}$ and $\begin{matrix} N_{r}^{*} ((T, D)) \\ = & (N_{r}^{*} (T (q) = \cup {x \in O : [x]_{(q_{i})_{r}} \cap T (q) \neq \emptyset}, D)) . \end{matrix}$ The near hyper soft set $(N_{*}^{*} HSS)$ denoted by $N_{*}^{*} H (T, D)$ is the N_r ((T, D)) satisfying ${Bnd}_{N_{r} (D)} ((T, D)) \geq 0 .$

Example 6. Let $O = {y_{1}, y_{2}, y_{3}, y_{4}, y_{5}}$ represent a set of five individuals and $D = {D_{1}, D_{2}} \subseteq F = {D_{1}, D_{2}, D_{3}, D_{4}}$ be a set of parameters, where D₁, D₂, D₃, D₄ represent tall, IQ, weight and strong, respectively. Instance values of the q_i, i = 1, 2, 3, 4 functions are as follows:

$\begin{matrix} [y_{1}]_{D_{1}} & = & {y_{1}, y_{4}}, [y_{2}]_{D_{1}} = {y_{2}, y_{3}}, [y_{5}]_{D_{1}} = {y_{5}}, \\ [y_{1}]_{D_{2}} & = & {y_{1}, y_{4}}, [y_{2}]_{D_{2}} = {y_{2}, y_{5}}, [y_{3}]_{D_{2}} = {y_{3}}, \end{matrix}$

$\begin{matrix} [y_{1}]_{D_{1}, D_{2}} & = & {y_{1}, y_{4}}, \\ [y_{2}]_{D_{1}, D_{2}} & = & {y_{2}}, \\ [y_{3}]_{D_{1}, D_{2}} & = & {y_{3}}, \\ [y_{5}]_{D_{1}, D_{2}} & = & {y_{5}} . \end{matrix}$ Let (K, D) be defined as $(K, D) = ((D_{1}, {y_{1}, y_{4}}), (D_{2}, {y_{3}, y_{5}}))$ representing a SS. Then, $N_{*} ((K, D)) = (K_{*} (D_{1}), D) = {(D_{1}, {y_{1}, y_{4}})}$ for q₂ ∈ D and $N^{*} ((K, D)) = {(D_{2}, {y_{2}, y_{3}, y_{5}})}$ for D₁, D₂ ∈ D.

The pair (K, D) is a NSS since it satisfies ${Bnd}_{N} (σ) \geq 0 .$

Table 1
Properties of objects

y₁ y₂ y₃ y₄ y₅

b₁₁ 1 0 0 1 0

b₁₂ 1 0 0 1 0.5

b₂₁ 0 1 0 0 1

b₂₂ 0 1 1 0 1

	y₁	y₂	y₃	y₄	y₅
b₁₁	1	0	0	1	0
b₁₂	1	0	0	1	0.5
b₂₁	0	1	0	0	1
b₂₂	0	1	1	0	1

Let b₁₁, b₁₂, b₂₁, b₂₂ stand for taller than 1.70, shorter than 1.70, greater than 100, less than 100 respectively where D₁ = {b₁₁, b₁₂}, D₂ = {b₂₁, b₂₂} and

$\begin{matrix} D & = & D_{1} \times D_{2} \\ = & {(b_{11}, b_{21}), (b_{11}, b_{22}), (b_{12}, b_{21}), (b_{12}, b_{22})} . \end{matrix}$

Let D = D₁ × D₂, q_i ∈ D where $\begin{matrix} q_{1} & = & (b_{11}, b_{21}), \\ q_{2} & = & (b_{11}, b_{22}), \\ q_{3} & = & (b_{12}, b_{21}), \\ q_{4} & = & (b_{12}, b_{22}), \\ q_{i} & = & {q_{1}, q_{2}, q_{3}, q_{4}} \end{matrix}$ and T is a mapping given by $T : D_{1} \times D_{2} \to P (O)$ $\begin{matrix} T (q_{1}) & = & {y_{1}, y_{4}}, \\ T (q_{2}) & = & {y_{2}, y_{3}} \end{matrix}$ can be written. Then

$\begin{array}{l} (T, D) = (T, D_{1} \times D_{2}) \\ = {((b_{11}, b_{21}), {y_{1}, y_{4}}), ((b_{11}, b_{22}), {y_{2}, y_{3}})} \\ = {(q_{1}, {y_{1}, y_{4}}), (q_{2}, {y_{2}, y_{3}})} \end{array}$

is a $N_{*}^{*} HSS$ where

$\begin{matrix} [y_{1}]_{q_{1}} & = & {y_{1}, y_{4}}, [y_{2}]_{q_{1}} = {y_{2}, y_{5}}, [y_{3}]_{q_{1}} = {y_{3}}, \\ [y_{1}]_{q_{2}} & = & {y_{1}, y_{4}}, [y_{2}]_{q_{2}} = {y_{2}, y_{3}, y_{5}}, \\ [y_{1}]_{q_{3}} & = & {y_{1}, y_{4}}, [y_{2}]_{q_{3}} = {y_{2}}, \\ [y_{3}]_{q_{3}} & = & {y_{3}}, [y_{5}]_{q_{3}} = {y_{5}}, \\ [y_{1}]_{q_{4}} & = & {y_{1}, y_{4}}, [y_{2}]_{q_{4}} = {y_{2}, y_{3}}, [y_{5}]_{q_{4}} = {y_{5}} \end{matrix}$

and

$\begin{matrix} N_{r} * ((T, D)) \\ = & (N_{r} * (T (q) = \cup {x \in O : [x]_{(q_{i}) r} \subseteq T (q)}, D)), \\ N_{r} * ((T, D = D_{1} \times D_{2})) \\ = & (N_{r} * (T (q_{1}) = [y_{1}]_{q_{1}} = {y_{1}, y_{4}} \subseteq T (q_{1}) = {y_{1}, y_{4}}) \\ = & {(q_{1}, {y_{1}, y_{4}})}, \\ N_{r}^{*} ((T, D = D_{1} \times D_{2})) \\ = & (N_{r}^{*} (T (q_{2}) = \cup {x \in O : [x]_{q_{2}} \cap T (q_{2}) \neq \emptyset}, D)) \\ = & {(q_{1}, {y_{1}, y_{4}}), (q_{2}, {y_{2}, y_{3,} y_{5}})} . \end{matrix}$

Then (T, D) hyper soft set is a near hyper soft set $(N_{*}^{*} HSS)$ since Bnd_{N_r(D)} ((T, D)) ≥0 is satisfied.

Definition 7. Let $N_{*}^{*} H (T, A)$ and $N_{*}^{*} H (L, D)$ be a $N_{*}^{*} HSS$ defined over a common universe $O$ . We define $N_{*}^{*} H (T, A)$ as a near hypersoft subset(NHSs) of $N_{*}^{*} H (L, D)$ , if the following conditions hold:

(1) A ⊆ D,

(2) T (β) ⊆ L (β) for all β ∈ A.

We denote this relationship as $N_{*}^{*} H (T, A) \subseteq N_{*}^{*} H (L, D)$ . $N_{*}^{*} H (T, A)$ is considered a $N_{*}^{*} HS$ superset of $N_{*}^{*} H (L, D)$ , if $N_{*}^{*} H (L, D)$ is a $N_{*}^{*} HS$ subset of $N_{*}^{*} H$ (T, A).

Definition 8. Let $T^{c} : D \to P (O)$ be a mapping such that $T^{c} (β) = O ∖ T (β)$ for all β ∈ D. Then $N_{*}^{*} H (T, D)^{c} = N_{*}^{*} H (T^{c}, D)$ is called the complement of $N_{*}^{*} HSS$ .

Definition 9. If for all β∈ D, T (β) = ∅ then $N_{*}^{*} H (T, D)$ is referred to as a relative null $N_{*}^{*} HSS$ , denoted as (∅ , D).

Definition 10. A $N_{*}^{*} H (T, D)$ defined over $O$ is considered a relative whole $N_{*}^{*} HSS$ , denoted as $(O, D)$ , if for all β ∈ D, $T (β) = O$ .

Definition 11. The difference of $N_{*}^{*} HSS$ $N_{*}^{*} H (T, A)$ and $N_{*}^{*} H (L, D)$ , defined over a common universe $O$ , is defined as $N_{*}^{*} H (T, A) ∖ N_{*}^{*} H (L, D) = N_{*}^{*} H (M, C),$ where C = A ∩ D and for all β ∈ C,M (β) = T (β) ∖ L (β). $N_{*}^{*} H (M, C)$ is also considered a $N_{*}^{*} HSS$ .

Definition 12. The union of $N_{*}^{*} HSS$ $N_{*}^{*} H (T, A)$ and $N_{*}^{*} H (L, D)$ , defined over a common universe $O$ , is defined as $N_{*}^{*} H (T, A) \cup N_{*}^{*} H (L, D) = N_{*}^{*} H (M, C)$ , where C = A ∩ D and for all β ∈ C, M (β) = T (β) ∪ L (β). $N_{*}^{*} H (M, C)$ is also a $N_{*}^{*} HSS$ .

Definition 13. The intersection of $N_{*}^{*} HSS$ $N_{*}^{*} H (T, A)$ and $N_{*}^{*} H (L, D)$ , defined over a common objects $O$ , is defined as $N_{*}^{*} H (T, A) \cap N_{*}^{*} H (L, D) = N_{*}^{*} H (M, C)$ where C = A ∩ D and for all β ∈ C, M (β) = T (β) ∩ L (β). $N_{*}^{*} H (M, C)$ is also a $N_{*}^{*} HSS$ .

Definition 14. Let $N_{*}^{*} H (T, D)$ be a $N_{*}^{*} HSS$ over $O$ and Y≠ ∅ be a subset of $O$ . The $N_{*}^{*} HSS$ of $N_{*}^{*} H (T, D)$ over T, denoted as $N_{*}^{*} H$ (T_Y, D) is defined as T_Y (β)= T ∩ Y (β) for each β ∈ D. In other words $N_{*}^{*} H (T_{Y}, D) = N_{*}^{*} H (T, D) \cap N_{*}^{*} H (Y, D)$ .

Definition 15. Let $τ_{N_{*}^{*} H}$ be the collection of $N_{*}^{*} HSS$ over $O$ . If the following conditions satisfies:

(1) $N_{*}^{*} H (\emptyset, D)$ , $N_{*}^{*} H (O, D)$ $ϵ τ_{N_{*}^{*} H}$ ,

(2) If $N_{*}^{*} H (T_{1}, D), N_{*}^{*} H (T_{2}, D) \in τ_{N_{*}^{*} H}$ , then $N_{*}^{*} H (T_{1}, D), N_{*}^{*} H (T_{2}, D) \in τ_{N_{*}^{*} H}$ ,

(3) If $N_{*}^{*} H (T_{i}, D) \in τ_{N_{*}^{*} H}$ then $\underset{i}{\cup} (N_{*}^{*} H (T_{i}, D) \in τ_{N_{*}^{*} H}$ ,

then $(O, τ_{N_{*}^{*} H}, D)$ is referred as a near hypersoft topological space $(N_{*}^{*} HSTS)$ over $O$ and the members of $τ_{N_{*}^{*} H}$ are called near hyper soft open $(N_{*}^{*} HSOS)$ sets in $O$ .

Example 16. Let $O = {y_{1}, y_{2}, y_{3}, y_{4}, y_{5}}$ represent a set of five individuals and D = {D₁, D₂} as in Example 6. $N_{*}^{*} H (T_{1}, D)$ and $N_{*}^{*} H (T_{2}, D)$ are $N_{*}^{*} HSS$ over $O$ , defined as follows: $N_{*}^{*} H (T_{1}, D) = {(q_{1}, {y_{1}, y_{4}}), (q_{2}, {y_{2}, y_{3}})},$ and $N_{*}^{*} H (T_{2}, D) = {(q_{1}, {y_{1}, y_{4}}), (q_{2}, {y_{2}, y_{3}, y_{5}})} .$ Thus, we write $\begin{matrix} τ_{N_{*}^{*} H} = \\ {N_{*}^{*} H (\emptyset, D), N_{*}^{*} H (O, D), N_{*}^{*} H (T_{1}, D), \\ N_{*}^{*} H (T_{2}, D)} . \end{matrix}$ Thus, the collection $τ_{N_{*}^{*} H}$ forms a $N_{*}^{*} HSTS$ on $O$ .

Definition 17. Let $(O, τ_{N_{*}^{*} H}, D)$ be a $N_{*}^{*} HSTS$ over $O$ . If $N_{*}^{*} H (T, D)^{c} ϵ τ_{N_{*}^{*} H}$ , then $N_{*}^{*} H (T, D)$ is referred to as a near hypersoft closed set $(N_{*}^{*} HSCS)$ in $O$ .

Definition 18. Let $O$ be an initial universe, D be the set of parameters, ${τ_{1}}_{N_{*}^{*} H}$ $= {(\emptyset, D), (O, D)}$ , and ${τ_{2}}_{N_{*}^{*} H}$ be the collection of all $N_{*}^{*} HSS$ s that can be defined over $O$ . ${τ_{1}}_{N_{*}^{*} H}$ and ${τ_{2}}_{N_{*}^{*} H}$ are referred to as the $N_{*}^{*} HS$ indiscrete topology and $N_{*}^{*} HS$ discrete topology on $O$ , respectively.

Definition 19. Let $(O,$ ${τ_{1}}_{N_{*}^{*} H}, D)$ and $(O,$ ${τ_{2}}_{N_{*}^{*} H}, D)$ be $N_{*}^{*} HSTS$ over $O$ . If ${τ_{1}}_{N_{*}^{*} H} \subseteq {τ_{2}}_{N_{*}^{*} H}$ , then ${τ_{2}}_{N_{*}^{*} H}$ is said to be finer than ${τ_{1}}_{N_{*}^{*} H}$ .

Definition 20. Let $(O,$ $τ_{N_{*}^{*} H}, D)$ be a $N_{*}^{*} HSTS$ over $O$ , $N_{*}^{*} H (T, D)$ be a $N_{*}^{*} HSS$ over $O$ and n∈ $O$ . Then $N_{*}^{*} H (T, D)$ is said to be a $N_{*}^{*} HS$ neighborhood $(N_{*}^{*} HSn)$ of n if there exists a $N_{*}^{*} H (L, D)$ that is a $N_{*}^{*} HSOS$ such that n∈ $N_{*}^{*} H (L, D) \subseteq N_{*}^{*} H (T, D)$ .

Proposition 21. Let $(O,$ $τ_{N_{*}^{*} H}, D)$ be a $N_{*}^{*} HSTS$ over $O$ , then

(1) If $N_{*}^{*} H (T, D)$ is a $N_{*}^{*} HSn$ of n∈ $O$ , then n∈ $N_{*}^{*} H (T, D)$ ,

(2) Each n∈ $O$ has a $N_{*}^{*} HSn$ ,

(3) If $N_{*}^{*} H (T, D)$ and $N_{*}^{*} H (L, D)$ are $N_{*}^{*} HSn$ s of some n∈ $O$ , then $N_{*}^{*} H (T, D)$ ∩ $N_{*}^{*} H (L, D)$ is also a $N_{*}^{*} HSn$ of n,

(4) If $N_{*}^{*} H (T, D)$ is a $N_{*}^{*} HSn$ of n∈ $O$ and $N_{*}^{*} H (T, D) \subseteq N_{*}^{*} H (L, D)$ , then $N_{*}^{*} H (L, D)$ is also a $N_{*}^{*} HSn$ of n∈ $O$ .

Proof. (1) This follows directly from the definition 20.

(2) For any n∈ $O$ , n∈ $N_{*}^{*} H (O, D)$ and since $N_{*}^{*} H (O, D) \in τ_{N_{*}^{*} H}$ , so n∈ $N_{*}^{*} H (O, D) \subseteq$ $N_{*}^{*} H (O, D)$ which means that $N_{*}^{*} H (O, D)$ is a $N_{*}^{*} HSn$ of n.

(3) Let $N_{*}^{*} H (T, D)$ and $N_{*}^{*} H (G, D)$ be the $N_{*}^{*} HSn$ of n∈ $O$ . Then there exist $N_{*}^{*} H (T_{1}, D)$ , $N_{*}^{*} H (T_{2}, D)$ ∈ $τ_{N_{*}^{*} H}$ such that $n \in N_{*}^{*} H (T_{1}, D) \subseteq$ $N_{*}^{*} H (T, D)$ and n∈ $N_{*}^{*} H (T_{2}, D) \subseteq$ $N_{*}^{*} H (G, D)$ . Since n∈ $N_{*}^{*} H (T_{1}, D)$ and $n \in N_{*}^{*} H (T_{2}, D)$ , it implies that n∈ $N_{*}^{*} H (T_{1}, D) \cap$ $N_{*}^{*} H (T_{2}, D)$ and $N_{*}^{*} H (T_{1}, D) \cap N_{*}^{*} H (T_{2}, D)$ ∈ $τ_{N_{*}^{*} H}$ .

Therefore, we have n∈ $N_{*}^{*} H (T_{1}$ , D)∩ $N_{*}^{*} H (T_{2}, D)$ ⊆ $N_{*}^{*} H (T_{1}, D) \cap$ $N_{*}^{*} H (T_{2}, D)$ . Thus, $N_{*}^{*} H (T_{1}, D) \cap$ $N_{*}^{*} H (T_{2}, D)$ is a $N_{*}^{*} HSn$ of n.

(4) Let $N_{*}^{*} H (T, D)$ be a $N_{*}^{*} HSn$ of n∈ $O$ and $N_{*}^{*} H (T, D) \subseteq$ $N_{*}^{*} H (G, D)$ . By definition 20, $N_{*}^{*} H (T_{1}, D)$ is a $N_{*}^{*} HSOS$ such that n∈ $N_{*}^{*} H (T_{1}, D)$ $\subseteq N_{*}^{*} H (T, D)$ ⊆ $N_{*}^{*} H (G, D)$ . Thus, n∈ $N_{*}^{*} H (T_{1}, D)$ ⊆ $N_{*}^{*} H (G, D)$ . Therefore, $N_{*}^{*} H (G, D)$ is a $N_{*}^{*} HSn$ of n. ■ Definition 22. Let $(O,$ $τ_{N_{*}^{*} H}, D)$ be a $N_{*}^{*} HSTS$ over $O$ and $Y \neq \emptyset \subseteq O$ . Then $τ_{N_{*}^{*} H_{T}} =$ { $N_{*}^{*} H (T_{Y}, D)$ : $N_{*}^{*} H (T, D)$ $\in τ_{N_{*}^{*} H}}$ is a $N_{*}^{*} HS$ topology on Y.

Definition 23. Let $(O,$ $τ_{N_{*}^{*} H}, D)$ be a $N_{*}^{*} HSTS$ and $N_{*}^{*} H (T, D)$ be a $N_{*}^{*} HSS$ over $O$ . Then $\begin{matrix} \bar{N_{*}^{*} H (T, D)} & = & \cap {N_{*}^{*} H (L, D) : N_{*}^{*} H (G, D)^{c} \in \\ τ_{N_{*}^{*} H}, N_{*}^{*} H (T, D) \subseteq N_{*}^{*} H (L, D)} \end{matrix}$ is called the near hypersoft closure $(N_{*}^{*} HSCl)$ of $N_{*}^{*} H (T, D)$ .

Proposition 24. Let $(O,$ $τ_{N_{*}^{*} H}, D)$ be a $N_{*}^{*} HS$ space and $N_{*}^{*} H (T, D)$ be a $N_{*}^{*} HSS$ over $O$ . Then

(1) $\bar{N_{*}^{*} H (T, D)}$ is the smallest $N_{*}^{*} HSCS$ containing $N_{*}^{*} H (T, D)$ ,

(2) $N_{*}^{*} H (T, D)$ is a $N_{*}^{*} HSCS$ if and only if $N_{*}^{*} H (T, D)$ $= \bar{N_{*}^{*} H (T, D)}$ .

Proof. (1)It is obvious from the definition 23.

(2) Let $N_{*}^{*} H (T, D)$ be a $N_{*}^{*} HSCS$ . So, $N_{*}^{*} H (T, D)$ is the smallest $N_{*}^{*} HSCS$ over $O$ , contains $N_{*}^{*} H (T, D)$ and hence $N_{*}^{*} H (T, D)$ $= \bar{N_{*}^{*} H (T, D)}$ . Contrarily, suppose that $N_{*}^{*} H (T, D)$ $= \bar{N_{*}^{*} H (T, D)}$ . By (1), $\bar{N_{*}^{*} H (T, D)}$ is a $N_{*}^{*} HSCS$ , so $N_{*}^{*} H (T, D)$ is a $N_{*}^{*} HSCS$ over $O$ . ■ Proposition 25. Let $(O,$ $τ_{N_{*}^{*} H}, D)$ be a $N_{*}^{*} HSTS$ over $O$ and $N_{*}^{*} H (T_{1}, D)$ , $N_{*}^{*} H (T_{2}, D)$ be $N_{*}^{*} HSS$ over $O$ . Then

(1) $\bar{N_{*}^{*} H (\emptyset, D)} = N_{*}^{*} H (\emptyset, D)$

and

$\bar{N_{*}^{*} H (O, D)} = N_{*}^{*} H (O, D)$ ,

(2) $N_{*}^{*} H (T_{1}, D) \subseteq \bar{N_{*}^{*} H (T_{1}, D)}$ ,

(3) $N_{*}^{*} H (T_{1}, D) \subseteq N_{*}^{*} H (T_{2}, D)$

implies

$\bar{N_{*}^{*} H (T_{1}, D)} \subseteq \bar{N_{*}^{*} H (T_{2}, D)}$ ,

(4) $\bar{(N_{*}^{*} H (T_{1}, D) \cup N_{*}^{*} H (T_{2}, D))}$ = $\bar{N_{*}^{*} H (T_{1}, D)}$ ∪ $\bar{N_{*}^{*} H (T_{2}, D)}$ ,

(5) $\bar{(N_{*}^{*} H (T_{1}, D) \cap N_{*}^{*} H (T_{2}, D))}$ ⊆ $\bar{N_{*}^{*} H (T_{1}, D)}$ ∩ $\bar{N_{*}^{*} H (T_{2}, D)}$ ,

(6) $\bar{\bar{N_{*}^{*} H (T_{1}, D)}}$ = $\bar{N_{*}^{*} H (T_{1}, D)}$ .

Proof. (1) and (2) are obvious from the definition 23.

(3) From (2), $N_{*}^{*} H (T_{2}, D)$ ⊆ $\bar{N_{*}^{*} H (T_{2}, D)}$ . Since $N_{*}^{*} H (T_{1}, D) \subseteq$ $N_{*}^{*} H (T_{2}, D)$ , we have $N_{*}^{*} H (T_{1}, D)$ $\subseteq \bar{N_{*}^{*} H (T_{2}, D)}$ . But $\bar{N_{*}^{*} H (T_{2}, D)}$ is a $N_{*}^{*} HSCS$ . $\bar{N_{*}^{*} H (T_{2}, D)}$ is a $N_{*}^{*} HSCS$ containing $N_{*}^{*} H (T_{1}, D)$ . Since $\bar{N_{*}^{*} H (T_{1}, D)}$ is the smallest $N_{*}^{*} HSCS$ over $O$ containing $N_{*}^{*} H (T_{1}, D)$ , so we have $\bar{N_{*}^{*} H (T_{1}, D)}$ ⊆ $\bar{N_{*}^{*} H (T_{2}, D)}$ .

(4) Since $N_{*}^{*} H (T_{1}, D) \subseteq N_{*}^{*} H (T_{1}, D) \cup N_{*}^{*} H (T_{2}, D)$ and $N_{*}^{*} H (T_{2}, D) \subseteq N_{*}^{*} H (T_{1}, D) \cup N_{*}^{*} H (T_{2}, D),$ from (3), we have $\bar{N_{*}^{*} H (T_{1}, D)} \subseteq \bar{N_{*}^{*} H (T_{1}, D) \cup N_{*}^{*} H (T_{2}, D)}$ and $\bar{N_{*}^{*} H (T_{2}, D)} \subseteq \bar{N_{*}^{*} H (T_{1}, D) \cup N_{*}^{*} H (T_{2}, D)} .$ Hence, $\begin{matrix} \bar{N_{*}^{*} H (T_{1}, D)} \cup \bar{N_{*}^{*} H (T_{2}, D)} \\ \subseteq \bar{N_{*}^{*} H (T_{1}, D) \cup N_{*}^{*} H (T_{2}, D)} . \end{matrix}$

Now, since $\bar{N_{*}^{*} H (T_{1}, D)}$ and $\bar{N_{*}^{*} H (T_{2}, D)}$ are $N_{*}^{*} HSCS$ s, $\bar{N_{*}^{*} H (T_{1}, D)}$ ∪ $\bar{N_{*}^{*} H (T_{2}, D)}$ is also $N_{*}^{*} HSC$ . Also, $N_{*}^{*} H (T_{1}, D) \subseteq \bar{N_{*}^{*} H (T_{1}, D)}$ and $N_{*}^{*} H (T_{2}, D) \subseteq \bar{N_{*}^{*} H (T_{2}, D)}$ implies that $\begin{matrix} N_{*}^{*} H (T_{1}, D) \cup N_{*}^{*} H (T_{2}, D) \\ \subseteq \bar{N_{*}^{*} H (T_{1}, D)} \cup \bar{N_{*}^{*} H (T_{2}, D)} . \end{matrix}$ Thus, $\bar{N_{*}^{*} H (T_{1}, D)}$ ∪ $\bar{N_{*}^{*} H (T_{2}, D)}$ is a $N_{*}^{*} HSC$ containing $N_{*}^{*} H (T_{1}, D)$ ∪ $N_{*}^{*} H (T_{2}, D)$ .

Since $\bar{N_{*}^{*} H (T_{1}, D) \cup N_{*}^{*} H (T_{2}, D)}$ is the smallest $N_{*}^{*} HSCS$ containing $N_{*}^{*} H (T_{1}, D)$ ∪ $N_{*}^{*} H (T_{2}, D)$ , we have $\begin{matrix} \bar{N_{*}^{*} H (T_{1}, D) \cup N_{*}^{*} H (T_{2}, D)} \\ \subseteq \bar{N_{*}^{*} H (T_{1}, D)} \cup \bar{N_{*}^{*} H (T_{2}, D)} . \end{matrix}$ Hence,

$\begin{matrix} \bar{N_{*}^{*} H (T_{1}, D) \cup N_{*}^{*} H (T_{2}, D)} \\ = & \bar{N_{*}^{*} H (T_{1}, D)} \cup \bar{N_{*}^{*} H (T_{2}, D)} . \end{matrix}$

(5) Since

$N_{*}^{*} H (T_{1}, D)$ ∩ $N_{*}^{*} H (T_{2}, D)$ ⊆ $N_{*}^{*} H (T_{1}, D)$

and

$N_{*}^{*} H (T_{1}, D)$ ∩ $N_{*}^{*} H (T_{2}, D)$ ⊆ $N_{*}^{*} H (T_{2}, D)$ , then $\bar{N_{*}^{*} H (T_{1}, D) \cap N_{*}^{*} H (T_{2}, D)} \subseteq \bar{N_{*}^{*} H (T_{1}, D)}$ and $\bar{N_{*}^{*} H (T_{1}, D) \cap N_{*}^{*} H (T_{2}, D)} \subseteq \bar{N_{*}^{*} H (T_{2}, D)} .$ Therefore, $\begin{matrix} \bar{N_{*}^{*} H (T_{1}, D) \cap N_{*}^{*} H (T_{2}, D)} \\ \subseteq \bar{N_{*}^{*} H (T_{1}, D)} \cap \bar{N_{*}^{*} H (T_{2}, D)} . \end{matrix}$ (6) Since $\bar{N_{*}^{*} H (T_{1}, D)}$ is a $N_{*}^{*} HSCS$ , therefore from (2), we have $\bar{\bar{N_{*}^{*} H (T_{1}, D)}} = \bar{N_{*}^{*} H (T_{1}, D)} .$ ■ Example 26. Let us consider the near hypersoft topological space $τ_{N_{*}^{*} H}$ in example 16 and let $N_{*}^{*} H (T_{1}, D)$ be near hypersoft set defined as follows: $N_{*}^{*} H (T_{1}, D) = {(q_{1}, {y_{1}, y_{4}}), (q_{2}, {y_{2}, y_{3}})} .$ Then

$\begin{matrix} τ_{N_{*}^{*} H} & = & {N_{*}^{*} H (\emptyset, D), N_{*}^{*} H (O, D), N_{*}^{*} H (T_{1}, D), \\ N_{*}^{*} H (T_{2}, D)}, \end{matrix}$ $\begin{matrix} C_{N_{*}^{*} H} & = & {N_{*}^{*} H (\emptyset, D), N_{*}^{*} H (O, D), N_{*}^{*} HSCS (T_{1}, D), \\ N_{*}^{*} HSCS (T_{2}, D)} . \end{matrix}$

are near hypersoft open set and near hypersoft closed sets, respectively. Thus, we write

$\begin{matrix} N_{*}^{*} HSCS (T_{1}, D) & = & {(q_{1}, {y_{2}, y_{3}, y_{5}}), (q_{2}, {y_{1}, y_{4}, y_{5}})}, \\ N_{*}^{*} HSCS (T_{2}, D) & = & {(q_{1}, {y_{2}, y_{3}, y_{5}}), (q_{2}, {y_{1}, y_{4}})} . \end{matrix}$

Hence $\bar{N_{*}^{*} H (T_{1}, D)} = N_{*}^{*} H (O, D) .$

Definition 27. Let $(O,$ $τ_{N_{*}^{*} H}, D)$ be a $N_{*}^{*} HSTS$ over $O$ , $N_{*}^{*} H (T, D)$ be a $N_{*}^{*} HSS$ over $O$ and n∈ $O$ . If there exists a $N_{*}^{*} H (L, D)$ which is a $N_{*}^{*} HSOS$ such that $n \in N_{*}^{*} H (L, D)$ ⊆ $N_{*}^{*} H (T, D)$ , then n is said to be a $N_{*}^{*} HS$ interior point of $N_{*}^{*} H (T, D)$ .

Definition 28. Let $(O,$ $τ_{N_{*}^{*} H}, D)$ be a $N_{*}^{*} HSTS$ over $O$ . Then $\begin{matrix} N_{*}^{*} H (T, D)^{\circ} & = & \cup {N_{*}^{*} H (L, D) | N_{*}^{*} H (L, D) \in \\ τ_{N_{*}^{*} H}, N_{*}^{*} H (L, D) \subseteq N_{*}^{*} H (T, D)} \end{matrix}$ is called $N_{*}^{*} HS$ interior of $N_{*}^{*} HSS$ .

Proposition 29. Let $(O,$ $τ_{N_{*}^{*} H}, D)$ be a $N_{*}^{*} HSTS$ and $N_{*}^{*} H (T, D)$ be a $N_{*}^{*} HSS$ over $O$ . Then

(1) $N_{*}^{*} H (T, D)^{\circ}$ is the largest $N_{*}^{*} HSOS$ contained in $N_{*}^{*} H (T, D)$ ,

(2) $N_{*}^{*} H (T, D)$ is a $N_{*}^{*} HSOS$ if and only if $N_{*}^{*} H (T, D)$ $= N_{*}^{*} H (T, D)^{\circ}$ .

Proof. (1) It is obvious from the definition 28.

(2) Let $N_{*}^{*} H (T, D)^{\circ}$ be a $N_{*}^{*} HSOS$ . Then $N_{*}^{*} H (T, D)^{\circ}$ is the largest $N_{*}^{*} HSO$ subset of $N_{*}^{*} H (T, D)$ . But from (1), $N_{*}^{*} H (T, D)^{\circ}$ is the largest $N_{*}^{*} HSO$ subset of $N_{*}^{*} H (T, D)$ . Hence, $N_{*}^{*} H (T, D)$ = $N_{*}^{*} H (T, D)^{\circ}$ . Conversely, let $N_{*}^{*} H (T, D)$ = $N_{*}^{*} H (T, D)^{\circ}$ . By (1), $N_{*}^{*} H (T, D)^{\circ}$ is a $N_{*}^{*} HSOS$ and therefore $N_{*}^{*} H (T, D)$ is also $N_{*}^{*} HSOS$ . ■ Proposition 30. Let $(O,$ $τ_{N_{*}^{*} H}, D)$ be a $N_{*}^{*} HS$ space over $O$ and let $N_{*}^{*} H (T_{1}, D), N_{*}^{*} H (T_{2}, D)$ be $N_{*}^{*} HSS$ over $O$ . Then

(1) $N_{*}^{*} H (\emptyset, D)^{\circ}$ $= N_{*}^{*} H (\emptyset, D)$ and $N_{*}^{*} H (O, D)^{\circ}$ $= N_{*}^{*} H (O, D)$ ,

(2) $N_{*}^{*} H (T_{1}, D)^{\circ} \subseteq N_{*}^{*} H (T_{1}, D)$ ,

(3) $N_{*}^{*} H (T_{1}, D) \subseteq N_{*}^{*} H (T_{2}, D)$ implies $N_{*}^{*} H (T_{1}, D)^{\circ} \subseteq N_{*}^{*} H (T_{2}, D)^{\circ}$ ,

(4) $N_{*}^{*} H (T_{1}, D)^{\circ} \cap N_{*}^{*} H (T_{2}, D)^{\circ}$ $= (N_{*}^{*} H (T_{1}, D) \cap N_{*}^{*} H (T_{2}, D))^{\circ}$ ,

(5) $N_{*}^{*} H (T_{1}, D)^{\circ} \cup N_{*}^{*} H (T_{2}, D)^{\circ} \subseteq$ $(N_{*}^{*} H (T_{1}, D) \cup N_{*}^{*} H (T_{2}, D))^{\circ}$ ,

(6) $(N_{*}^{*} H (T_{1}, D)^{\circ})^{\circ} =$ $N_{*}^{*} H (T_{1}, D)^{\circ}$ .

Proof. (1) It is obvious from the definition 28.

(2) Let n∈ $N_{*}^{*} H (T_{1}, D)^{\circ}$ , then n is a $N_{*}^{*} HS$ interior point of $N_{*}^{*} H (T_{1}, D)$ . This implies that $N_{*}^{*} H (T_{1}, D)$ is a $N_{*}^{*} HSn$ of n. Then n∈ $N_{*}^{*} H (T_{1}, D)$ . Hence, $N_{*}^{*} H (T_{1}, D)^{\circ}$ $\subseteq N_{*}^{*} H (T_{1}, D)$ .

(3) Let $n \in N_{*}^{*} H (T_{1}, D)^{\circ}$ . Then n is a $N_{*}^{*} HS$ interior point of $N_{*}^{*} H (T_{1}, D)$ and so $N_{*}^{*} H (T_{1}, D)$ is a $N_{*}^{*} HSn$ of n. Since $N_{*}^{*} H (T_{1}, D)$ ⊆ $N_{*}^{*} H (T_{2}, D)$ , $N_{*}^{*} H (T_{2}, D)$ is also is a $N_{*}^{*} HSn$ of n. This implies that n∈ $N_{*}^{*} H (T_{2}, D)^{\circ}$ . Thus, $N_{*}^{*} H (T_{1}, D)^{\circ} \subseteq N_{*}^{*} H (T_{2}, D)^{\circ} .$ (4) Since $\begin{matrix} N_{*}^{*} H (T_{1}, D) \cap N_{*}^{*} H (T_{2}, D) \\ \subseteq N_{*}^{*} H (T_{1}, D) \end{matrix}$ and $N_{*}^{*} H (T_{1}, D) \cap N_{*}^{*} H (T_{2}, D) \subseteq N_{*}^{*} H (T_{2}, D),$ from (3),

$(N_{*}^{*} H (T_{1}, D) \cap N_{*}^{*} H (T_{2}, D))^{\circ} \subseteq N_{*}^{*} H (T_{1}, D)^{\circ}$ and $(N_{*}^{*} H (T_{1}, D) \cap N_{*}^{*} H (T_{2}, D))^{\circ} \subseteq N_{*}^{*} H (T_{2}, D)^{\circ} .$ This implies that $(N_{*}^{*} H (T_{1}, D) \cap N_{*}^{*} H (T_{2}, D))^{\circ}$ ⊆ $N_{*}^{*} H (T_{1}, D)^{\circ} \cap N_{*}^{*} H (T_{2}, D)^{\circ}$ . Let $n \in N_{*}^{*} H (T_{1}, D)^{\circ} \cap N_{*}^{*} H (T_{2}, D)^{\circ}$ . Then $n \in N_{*}^{*} H (T_{1}, D)^{\circ}$ and $n \in N_{*}^{*} H (T_{2}, D)^{\circ}$ . Hence n is a $N_{*}^{*} HS$ interior point of each of the near hypersoft sets $N_{*}^{*} H (T_{1}, D)$ and $N_{*}^{*} H (T_{2}, D)$ . It follows that $N_{*}^{*} H (T_{1}, D)$ and $N_{*}^{*} H (T_{2}, D)$ are $N_{*}^{*} HSn$ of n so that their intersection $N_{*}^{*} H (T_{1}, D) \cap N_{*}^{*} H (T_{2}, D)$ is also is a $N_{*}^{*} HSn$ of n. Hence, n∈ $(N_{*}^{*} H (T_{1}, D) \cap N_{*}^{*} H (T_{2}, D))^{\circ}$ . Thus, $\begin{matrix} N_{*}^{*} H (T_{1}, D)^{\circ} \cap N_{*}^{*} H (T_{2}, D)^{\circ} \\ \subseteq (N_{*}^{*} H (T_{1}, D) \cap N_{*}^{*} H (T_{2}, D))^{\circ} . \end{matrix}$ Therefore, $\begin{matrix} N_{*}^{*} H (T_{1}, D)^{\circ} \cap N_{*}^{*} H (T_{2}, D)^{\circ} \\ = (N_{*}^{*} H (T_{1}, D) \cap N_{*}^{*} H (T_{2}, D))^{\circ} . \end{matrix}$ (5) From (3), $N_{*}^{*} H (T_{1}, D)$ ⊆ $N_{*}^{*} H (T_{1}, D) \cup$ $N_{*}^{*} H (T_{2}, D)$ implies $N_{*}^{*} H (T_{1}, D)^{\circ} \subseteq (N_{*}^{*} H (T_{1}, D) \cup N_{*}^{*} H (T_{2}, D))^{\circ}$ and $N_{*}^{*} H (T_{2}, D)$ ⊆ $N_{*}^{*} H (T_{1}, D) \cup N_{*}^{*} H (T_{2}, D)$ implies $N_{*}^{*} H (T_{2}, D)^{\circ} \subseteq (N_{*}^{*} H (T_{1}, D) \cup N_{*}^{*} H (T_{2}, D))^{\circ} .$ Hence, $N_{*}^{*} H (T_{1}, D)^{\circ} \cup N_{*}^{*} H (T_{2}, D)^{\circ}$ ⊆ $(N_{*}^{*} H (T_{1}, D) \cup N_{*}^{*} H (T_{2}, D))^{\circ} .$

(6) From $N_{*}^{*} H (T_{1}, D)^{\circ}$ is the $N_{*}^{*} HSOS$ . Hence, from (2) of the same proposition, $(N_{*}^{*} H (T_{1}, D)^{\circ})^{\circ}$ $= N_{*}^{*} H (T_{1}, D)^{\circ}$ . ■ Definition 31. Let $(O,$ $τ_{N_{*}^{*} H}, D)$ be a $N_{*}^{*} HSTS$ over $O$ and $N_{*}^{*} H (T, D)$ be a $N_{*}^{*} HSOS$ over $O$ . Then $N_{*}^{*} H (T, D)^{\circ}$ ⊆ $N_{*}^{*} H (T, D) \subseteq$ $\bar{N_{*}^{*} H (T, D)}$ .

Proposition 32. Let $(O,$ $τ_{N_{*}^{*} H}, D)$ be a $N_{*}^{*} HSTS$ over $O$ and $N_{*}^{*} H (T_{1}, D), N_{*}^{*} H (T_{2}, D)$ be $N_{*}^{*} HSS$ over $O$ . Then

(1) $(\bar{N_{*}^{*} H (T_{1}, D)})^{c} =$ $(N_{*}^{*} H (T_{1}, D)^{c})^{\circ}$ ,

(2) $(N_{*}^{*} H (T_{1}, D)^{\circ})^{c}$ = $(\bar{N_{*}^{*} H (T_{1}, D)^{c}})$ ,

(3) $\bar{N_{*}^{*} H (T_{1}, D)}$ = $((N_{*}^{*} H (T_{1}, D)^{c})^{\circ})^{c}$ ,

(4) $N_{*}^{*} H (T_{1}, D)^{\circ}$ = $(\bar{N_{*}^{*} H (T_{1}, D)^{c}})^{c}$ ,

(5) $(N_{*}^{*} H (T_{1}, D) ∖ N_{*}^{*} H (T_{2}, D)^{\circ})^{\circ}$ ⊆ $N_{*}^{*} H (T_{1}, D)^{\circ}$ ∖ $N_{*}^{*} H (T_{2}, D)^{\circ}$ .

Proof. (1) From the definition 23, we get

$\begin{matrix} \bar{(N_{*}^{*} H (T_{1}, D))} & = \cap {(N_{*}^{*} H (G, D) | (N_{*}^{*} H (G, D)^{c} \in \\ τ_{N_{*}^{*} H}, (N_{*}^{*} H (G, D) \supseteq (N_{*}^{*} H (T_{1}, D)}, \end{matrix}$ $\begin{matrix} (\bar{N_{*}^{*} H (T_{1}, D)})^{c} & = \cap [{(N_{*}^{*} H (G, D) | (N_{*}^{*} H (G, D)^{c} \in \\ τ_{N_{*}^{*} H}, (N_{*}^{*} H (G, D) \supseteq (N_{*}^{*} H (T_{1}, D)}]^{c}, \end{matrix}$ $\begin{matrix} (\bar{N_{*}^{*} H (T_{1}, D)})^{c} & = \cup {((N_{*}^{*} H (G, D)^{c} : (N_{*}^{*} H (G, D)^{c} \in \\ τ_{N_{*}^{*} H}, (N_{*}^{*} H (G, D)^{c} \subseteq N_{*}^{*} H (T_{1}, D)^{c}} \\ = (N_{*}^{*} H (T_{1}, D)^{c})^{\circ} . \end{matrix}$ (2)From the definition 8, we get $\begin{matrix} (N_{*}^{*} H (T_{1}, D))^{c} & = {(N_{*}^{*} H (G, D) : (N_{*}^{*} H (G, D) \in \\ τ_{N_{*}^{*} H}, (N_{*}^{*} H (G, D) \subseteq N_{*}^{*} H (T_{1}, D)}, \end{matrix}$ $\begin{matrix} ((N_{*}^{*} H (T_{1}, D))^{\circ})^{c} & = [\cup {(N_{*}^{*} H (G, D) : (N_{*}^{*} H (G, D) \in \\ τ_{N_{*}^{*} H}, (N_{*}^{*} H (G, D) \subseteq N_{*}^{*} H (T_{1}, D)}]^{c}, \end{matrix}$ $\begin{matrix} (N_{*}^{*} H (T_{1}, D)^{\circ})^{c} & = \cap {(N_{*}^{*} H (G, D)^{c} : (N_{*}^{*} H (G, D) \in \\ τ_{N_{*}^{*} H}, N_{*}^{*} H (T_{1}, D)^{c} \subseteq (N_{*}^{*} H (G, D)^{c}} \\ = (\bar{N_{*}^{*} H (T_{1}, D)^{c}}) . \end{matrix}$

(3) It can be proved similarly from the definitions 8, 23 and 28.

(4) It can be proved similarly from the definitions 8, 23 and 28.

(5) From the definitions 8 and 28, we get $\begin{matrix} (N_{*}^{*} H (T_{1}, D) ∖ N_{*}^{*} H (T_{2}, D)^{\circ})^{\circ} \\ = & (N_{*}^{*} H (T_{1}, D) \cap N_{*}^{*} H (T_{2}, D)^{c})^{\circ} \\ = & N_{*}^{*} H (T_{1}, D)^{\circ} \cap (N_{*}^{*} H (T_{2}, D)^{c})^{\circ} \\ = & N_{*}^{*} H (T_{1}, D)^{\circ} \cap (N_{*}^{*} H (T_{2}, D))^{c} \\ = & N_{*}^{*} H (T_{1}, D)^{\circ} ∖ N_{*}^{*} H (T_{2}, D) \\ \subseteq & N_{*}^{*} H (T_{1}, D)^{\circ} ∖ N_{*}^{*} H (T_{2}, D)^{\circ} . \end{matrix}$

■ Example 33. Let us consider the near hypersoft topological space $τ_{N_{*}^{*} H}$ and $N_{*}^{*} H (T_{1}, D)$ near hypersoft set in example 26. $N_{*}^{*} H (T_{1}, D)^{\circ} = N_{*}^{*} H (T_{1}, D)$

Definition 34. Let $(O,$ $τ_{N_{*}^{*} H}, D)$ be a $N_{*}^{*} HSTS$ over $O$ , then $N_{*}^{*} H (T, D)$ which is a $N_{*}^{*} HS$ boundary of $N_{*}^{*} HSS$ is denoted by $N_{*}^{*} H (T, D)^{b}$ and is defined as $N_{*}^{*} H (T, D)^{b} = \bar{N_{*}^{*} H (T, D)} \cap \bar{N_{*}^{*} H (T, D)^{c}} .$

Proposition 35. Let $(O,$ $τ_{N_{*}^{*} H}, D)$ be a $N_{*}^{*} HSTS$ over $O$ and $N_{*}^{*} H (T, D)$ be a $N_{*}^{*} HSS$ over $O$ . Then

(1) $N_{*}^{*} H (T, D)^{b}$ ⊆ $\bar{N_{*}^{*} H (T, D)}$ ,

(2) $N_{*}^{*} H (T, D)^{b}$ ⊆ $\bar{N_{*}^{*} H (T, D)} ∖$ $N_{*}^{*} H (T, D)^{\circ}$ ,

(3) $N_{*}^{*} H (T, D)^{\circ} = N_{*}^{*} H (T, D) ∖ N_{*}^{*} H (T, D)^{b},$

(4) $(N_{*}^{*} H (T, D)^{\circ})^{b}$ $\subseteq N_{*}^{*} H (T, D)^{b}$ ,

(5) $(\bar{N_{*}^{*} H (T, D)})^{b}$ ⊆ $N_{*}^{*} H (T, D)^{b}$ .

Proof. (1) From $N_{*}^{*} H (T, D)^{b} = \bar{N_{*}^{*} H (T, D)} \cap \bar{N_{*}^{*} H (T, D)^{c}},$ hence, $N_{*}^{*} H (T, D)^{b} \subseteq \bar{N_{*}^{*} H (T, D)} .$

(2) From the definition 34, we get $\begin{matrix} N_{*}^{*} H (T, D)^{b} & = & \bar{N_{*}^{*} H (T, D)} \cap \bar{(N_{*}^{*} H (T, D)^{c})} \\ = & \bar{N_{*}^{*} H (T, D)} \cap (N_{*}^{*} H (T, D)^{\circ})^{c} \\ = & \bar{N_{*}^{*} H (T, D)} ∖ N_{*}^{*} H (T, D)^{\circ} . \end{matrix}$

(3) From the definition 34, we get $\begin{matrix} N_{*}^{*} H (T, D) ∖ N_{*}^{*} H (T, D)^{b} \\ = & N_{*}^{*} H (T, D) \cap (N_{*}^{*} H (T, D)^{b})^{c} \\ = & N_{*}^{*} H (T, D) \cap (N_{*}^{*} H (T, D)^{\circ} \cup N_{*}^{*} H (T, D)) \\ = & (N_{*}^{*} H (T, D) \cap N_{*}^{*} H (T, D)^{\circ}) \\ \cup & (N_{*}^{*} H (T, D) \cap N_{*}^{*} H (T, D)) \\ = & N_{*}^{*} H (T, D)^{\circ} \cup N_{*}^{*} H (\emptyset, D) \\ = & N_{*}^{*} H (T, D)^{\circ} . \end{matrix}$

(4)From the definition 34, we get $\begin{matrix} (N_{*}^{*} H (T, D)^{\circ})^{b} & = & \bar{N_{*}^{*} H (T, D)^{\circ}} \cap \bar{(N_{*}^{*} H (T, D)^{\circ})^{c}} \\ = & \bar{N_{*}^{*} H (T, D)^{\circ}} \cap \bar{\bar{N_{*}^{*} H (T, D)^{c}}} \\ \subseteq & \bar{N_{*}^{*} H (T, D)^{\circ}} \cap \bar{N_{*}^{*} H (T, D)^{c}} \\ = & N_{*}^{*} H (T, D)^{b} . \end{matrix}$ (5)From the definition 34, we get $\begin{matrix} (\bar{N_{*}^{*} H (T, D)})^{b} & = & \bar{\bar{N_{*}^{*} H (T, D)}} \cap \bar{(\bar{N_{*}^{*} H (T, D)})^{c}} \\ \subseteq & \bar{N_{*}^{*} H (T, D)} \cap \bar{N_{*}^{*} H (T, D)^{c}} \\ = & N_{*}^{*} H (T, D)^{b} . \end{matrix}$ ■ Example 36. Let us consider the near hypersoft topological space $τ_{N_{*}^{*} H}$ in example 16 and let $N_{*}^{*} H (T_{1}, D)$ , $N_{*}^{*} H (T_{2}, D)$ , $N_{*}^{*} H (T_{3}, D)$ be near hypersoft sets defined as follows:

$\begin{matrix} N_{*}^{*} H (T_{1}, D) & = & {(q_{1}, {y_{1}, y_{4}}), (q_{2}, {y_{2}, y_{3}})} \\ N_{*}^{*} H (T_{2}, D) & = & {(q_{1}, {y_{1}, y_{4}}), (q_{2}, {y_{2}, y_{3,} y_{5}})} \\ N_{*}^{*} H (T_{3}, D) & = & {(q_{1}, {y_{1}, y_{4}}), (q_{2}, {y_{1}, y_{2}, y_{3}})} . \end{matrix}$ Then

are near hypersoft open set and near hypersoft closed sets, respectively. Thus, we write

$\begin{matrix} N_{*}^{*} H (T_{1}, D)^{c} & = & N_{*}^{*} HSCS (T_{1}, D) \\ = & {(q_{1}, {y_{2}, y_{3}, y_{5}}), (q_{2}, {y_{1}, y_{4}, y_{5}})}, \\ N_{*}^{*} H (T_{2}, D)^{c} & = & N_{*}^{*} HSCS (T_{2}, D) \\ = & {(q_{1}, {y_{2}, y_{3}, y_{5}}), (q_{2}, {y_{1}, y_{4}})} . \end{matrix}$ Hence

$\begin{matrix} \bar{N_{*}^{*} H (T_{1}, D)} & = & N_{*}^{*} H (O, D), \\ \bar{N_{*}^{*} H (T_{2}, D)} & = & N_{*}^{*} H (O, D), \\ \bar{N_{*}^{*} H (T_{3}, D)} & = & N_{*}^{*} H (O, D), \end{matrix}$ $\begin{matrix} N_{*}^{*} H (T_{1}, D)^{\circ} & = & N_{*}^{*} H (T_{1}, D), \\ N_{*}^{*} H (T_{2}, D)^{\circ} & = & N_{*}^{*} H (T_{2}, D), \\ N_{*}^{*} H (T_{3}, D)^{\circ} & = & N_{*}^{*} H (T_{1}, D) \end{matrix}$ and $\begin{matrix} N_{*}^{*} H (T_{1}, D)^{b} & = & N_{*}^{*} H (O, D) \cap N_{*}^{*} H (T_{1}, D)^{c} \\ = & N_{*}^{*} HSCS (T_{1}, D), \\ N_{*}^{*} H (T_{2}, D)^{b} & = & N_{*}^{*} H (O, D) \cap N_{*}^{*} H (T_{2}, D)^{c} \\ = & N_{*}^{*} HSCS (T_{2}, D), \\ N_{*}^{*} H (T_{3}, D)^{b} & = & \bar{N_{*}^{*} H (T_{3}, D)} \cap \bar{N_{*}^{*} H (T_{3}, D)^{c}} \\ = & N_{*}^{*} HSCS (T_{1}, D) \end{matrix}$ where $N_{*}^{*} H (T_{3}, D)^{c} = {(q_{1}, {y_{2}, y_{3}, y_{5}}), (q_{2}, {y_{4}, y_{5}})} .$ For $N_{*}^{*} H (T_{3}, D)$ , we can write $\begin{matrix} N_{*}^{*} H (T_{3}, D)^{b} & \subseteq & \bar{N_{*}^{*} H (T_{3}, D)} ∖ N_{*}^{*} H (T_{3}, D)^{\circ}, \\ N_{*}^{*} HSCS (T_{1}, D) & \subseteq & N_{*}^{*} H (O, D) ∖ N_{*}^{*} H (T_{1}, D), \\ N_{*}^{*} HSCS (T_{1}, D) & \subseteq & N_{*}^{*} HSCS (T_{1}, D) . \end{matrix}$ Similarly, the accuracy of all propositions in the article can be demonstrated.

4 Conclusions

In this study, we have explored hyper-soft sets in nearness approximation spaces and introduced the concept of near hypersoft sets. By considering the Cartesian product of features, we can identify objects that possess specific features from a larger set of objects. This clustering approach enables the identification of objects with similar properties, allowing us to select the best products that meet our desired complex features. Consequently, we have gained the ability to make informed choices based on our specific needs. Moreover, we have established a topology on this space and derived several definitions and propositions.

By conducting a thorough investigation into hypersoft sets within approximation spaces, we have successfully introduced the concept of near hypersoft sets. Utilizing the Cartesian product of features, we have devised a method to identify objects possessing specific features among a diverse range of objects. This clustering approach has facilitated the grouping of objects with similar properties, enabling us to select the most suitable products that align with our desired complex features. As a result, we have gained the capability to make targeted choices according to our specific requirements. Additionally, we have established a topology within this space and formulated various definitions and propositions.

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Hypersoft sets on nearness approximation space and its topology

Abstract

Keywords

1 Introduction

2 Preliminary

2.1 Soft sets, near soft sets and hyper soft set

3 Hyper soft sets on near aproximation space

Table 1 Properties of objects y1 y2 y3 y4 y5 b11 1 0 0 1 0 b12 1 0 0 1 0.5 b21 0 1 0 0 1 b22 0 1 1 0 1

References

Table 1
Properties of objects

y₁ y₂ y₃ y₄ y₅

b₁₁ 1 0 0 1 0

b₁₂ 1 0 0 1 0.5

b₂₁ 0 1 0 0 1

b₂₂ 0 1 1 0 1