EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 4, 2024, 3043-3060 ISSN 1307-5543 – ejpam.com Published by New York Business Global Plithogenic Crisp Hypersoft Topology Nehmat K. Ahmed1, Osama T. Pirbal1,∗ 1 Mathematics Department, College of Education, Salahaddin University-Erbil, Kurdistan, Iraq Abstract. In this paper, we deal with the plithogenic crisp hypersoft set. This notion is more adaptable than the hypersoft set and more suited to challenges involving decision-making. Con- sequently, the topology defined by the collection of this type of set will be of great importance. Through this paper, first we redefine the set operations on this type of set (set theoretic). Then, we introduce plithogenic crisp hypersoft topological spaces, which are defined over an initial uni- versal set with a fixed set of parameters. The plithogenic crisp hypersoft set considers the degree of appurtenance of the elements with respect to the attribute system. Further, the notions of plithogenic crisp hypersoft open sets, plithogenic crisp hypersoft closed sets, plithogenic crisp hypersoft neighborhood, plithogenic crisp hypersoft limit point, and plithogenic crisp hypersoft subspace are introduced, and their basic properties are investigated. Finally, we introduce the concepts of plithogenic crisp hypersoft closure and plithogenic crisp hypersoft interior. 2020 Mathematics Subject Classifications: 54C50 Key Words and Phrases: Hypersoft sets, Plithogenic hypersoft sets, Plithogenic crisp hypersoft sets, Plithogenic crisp hypersoft topology. 1. Introduction In 1999, Molodtsov [18] introduced the concept of a soft set to deal with the diffi- cult problems in economics, engineering, and the environment, where no mathematical methods could effectively deal with the many types of uncertainty. Biswas et al. in [15] introduced various operators for soft sets (see [3, 5, 8, 25]). Also, Finite soft-open sets introduced by Abd El-latif et al. [4] and Supra finite soft-open sets and applications to operators and continuity introduced by Arar et al. [26] . It is known that topology is a branch of mathematics that has numerous applications in the physical and computer sciences. Topology is the study of the qualitative properties of particular objects, known as topological spaces, which are invariant under specific transformations, known as con- tinuous mappings. Open sets are commonly used to describe these characteristics. By replacing open sets with more general ones, the concept of topological space is frequently ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5401 Email addresses: nehmat.ahmed@su.edu.krd (N. K. Ahmed), osama.pirbal@su.edu.krd (O. T. Pirbal) https://www.ejpam.com 3043 Copyright: © 2024 The Author(s). (CC BY-NC 4.0) N. K. Ahmed, O. T. Pirbal / Eur. J. Pure Appl. Math, 17 (4) (2024), 3043-3060 3044 generalized. A classic example of this form of generalization is fuzzy topology, proposed by Chang [10] and later fuzzy topology introduced by Lowen [14]. Topological structures on soft sets, in a similar manner, are more generalized methods that can be used to measure the similarities and differences between the objects in a universe that are soft sets (for more, see [11, 12, 24]). In 2018, Smarandache [27] expanded the concept of soft set to a hypersoft set by sub- stituting the function with a multi-argument function described in the Cartesian product with a different set of parameters. This concept is more adaptable than the soft set and more useful when it comes to making decisions. Researchers have been drawn to the hy- persoft set structure because it is better suited to decision-making difficulties than the soft set structure. The fundamentals of hypersoft sets are studied by Siddique et al. [23] (also see [22]). The idea of hypersoft sets is combined with topology by Musa and Asaad [20], in which they introduced hypersoft topological spaces, which are defined over an initial universe with a fixed set of parameters. Furthermore, Continuity and compactness via hypersoft open sets introduced by Asaad and Musa [6], Connectedness on hypersoft topo- logical spaces by Musa and Asaad [19] and Hypersoft separation axioms by Asaad and Musa [7] are developed the field of topology. Also, an innovative extension of hypersoft sets and their applications [21] is introduced by Mohammed et al. In the same paper [27], the author developed the concept of hypersoft into a concept called plithogenic hypersoft set. Furthermore, in the plithogenic crisp hypersoft set, there is a degree (0 or 1) of appurtenance of an element x to the set with respect to each attribute value. The idea of a plithogenic hypersoft set has many applications (see [2, 9, 13, 16, 17]). In [1], Murtaza et al. studied basic operations on hypersoft sets and hypersoft point together some basic properties of plithogenic α hypersoft sets where α can take any value in the (Crisp, Fuzzy, Intuitionistic Fuzzy and Neutrosophic) sets. When these two advanced concepts—plithogenic logic by Smarandache [28] and hypersoft topology by Musa and Asaad [20] are combined, they form the basis for the plithogenic crisp hypersoft topology. This emerging field allows for the modeling of topological spaces where each point can be characterized by a set of attributes, with each attribute having a degree of belongingness that is influenced by multiple, possibly conflicting criteria. In essence, plithogenic crisp Hypersoft topology provides a robust framework for analyzing and interpreting complex systems where uncertainty, indeterminacy, and multi-criteria decision-making play crucial roles. This framework has potential applications in areas such as decision theory, artificial intelligence, and data science, where traditional topological concepts may fall short in capturing the intricacies of real-world phenomena. Our work is organized as follows: Sections 2 and 3 contain some basic definitions related to the hypersoft set and the plithogenic hypersoft set that are required in our work. In Section 4, we introduce plithogenic crisp hypersoft topological spaces, which are defined over an initial universe set with a fixed set of parameters, and investigate the concepts of plithogenic crisp hypersoft neighborhood and plithogenic crisp hypersoft limit points. In Section 5, the notions of plithogenic crisp hypersoft closure and plithogenic crisp hypersoft interior are introduced. One thing that must be mentioned is that in this paper we have redefined the definitions that relate to plithogenic crisp hypersoft set in N. K. Ahmed, O. T. Pirbal / Eur. J. Pure Appl. Math, 17 (4) (2024), 3043-3060 3045 both aspects: symbolic and expression, and this is for the sake of the study. 2. Preliminaries Since the plithogenic crisp hypersoft is the extension of hypersoft set, we put the basic definitions here. Definition 1. [18] Let U be a universe of discourse, P (U) the power set of U , and A a set of attributes. Then, the pair (F,U) where F : A→ P (U) is called a soft set over U . Definition 2. [20] Let U be a universal set and P (U) be the power set of U . Let ψ= {r1, r2, . . . ,rn} be a set of n-distinct attributes with attribute value sets respectively as E1, E2, . . . ,En, where Ei∩Ej=ϕ for i̸=j and i, j∈{1, 2, . . . ,n}. Also, let Di be the nonempty subset of Ei for each i∈{1, 2, . . . ,n} and Vψ=D1×D2×· · ·×Dn. The pair (Γ,Vψ) where Γ:Vψ→P (U) is called a hypersoft (in short, HS) set. That is, (Γ,Vψ)={(α,Γ(α)) :α∈Vψ}. Definition 3. [23] Let (Γ1, Fψ) and (Γ2, Hψ) be two hypersoft sets over U . Then (Γ1, Fψ) is a hypersoft subsets of (Γ2, Hψ) if: (i) Fψ⊆Hψ, and (ii) Γ1(α)⊆Γ2(α), ∀α∈Fψ. We write (Γ1, Fψ) ∼ ⊆ (Γ2, Hψ) . And (Γ1, Fψ) is said to be a hypersoft superset of (Γ2, Hψ), if (Γ2, Hψ) is a hypersoft subset of (Γ1, Fψ). We write it as (Γ1, Fψ) ∼ ⊇ (Γ2, Hψ). Definition 4. [23] Two hypersoft sets (Γ1, Fψ) and (Γ2, Hψ) over a common universe U are said to be hypersoft equal if (Γ1, Fψ) is a hypersoft subset of (Γ2, Hψ) and (Γ2, Hψ) is a hypersoft subset of (Γ1, Fψ). Definition 5. [23] Let ψ= {r1, r2, . . . ,rn} be a set of parameters (attributes). The NOT set of ψ denoted by ¬ψ is defined by ¬ψ= {¬r1,¬r2, . . . ,¬rn} where ¬ri= notri for i∈{1, 2, . . . ,n}. Definition 6. [23] Let U be a universal set. The complement of a hypersoft set (Γ, Vψ) is denoted by (Γ, Vψ) c and is defined by (Γ, Vψ) c = (Γc, Vψ) where Γc : Vψ → P (U) is a mapping given by Γc(α) = U \ Γ(α), for all α ∈ Vψ. Definition 7. [20] Let U be a universal set. A hypersoft set (Γ,Vψ) over U is said to be a null hypersoft set and denoted by (ϕ, Vψ), if for all α∈Vψ,Γ(α) =ϕ. Definition 8. [20] Let U be a universal set. A hypersoft set (Γ,Vψ) over U is said to be a whole hypersoft set and denoted by (U, Vψ), if for all α∈Vψ,Γ(α) =U . Definition 9. [20] Difference of two hypersoft sets (Γ1, Fψ) and (Γ2, Hψ) over a universe U is a hypersoft set (Γ, Vψ) where Vψ = Fψ∩Hψ and for all α ∈ Vψ, Γ(α) = Γ1(α)\Γ2(α). We write (Γ1, Fψ) \ (Γ2, Hψ) = (Γ, Vψ). N. K. Ahmed, O. T. Pirbal / Eur. J. Pure Appl. Math, 17 (4) (2024), 3043-3060 3046 Definition 10. [20] Union of two hypersoft sets (Γ1, Fψ) and (Γ2, Hψ) over a universe U is a hypersoft set (Γ,Vψ) where Vψ=Fψ∪Hψ and for all α∈Vψ,Γ(α) =Γ1(α)∪Γ2(α) and will be written as (Γ1, Fψ) ∼ ⊔ (Γ2, Hψ)= (Γ,Vψ). Definition 11. [20] Intersection of two hypersoft sets (Γ1, Fψ) and (Γ2, Hψ) over a uni- verse U , is a hypersoft set (Γ, Vψ) where Vψ = Fψ ∩ Hψ and for all α ∈ Vψ, Γ(α) = Γ1(α) ∩ Γ2(α) and will be written as (Γ1, Fψ) ∼ ⊓ (Γ2, Hψ) = (Γ, Vψ). 3. Set-theoretic Operations on Plithogenic Crisp Hypersoft Sets and Their Properties The results of this section is appear in [1], but we have redefined. Definition 12. Let UP be a universal set and ψ= {r1, r2, . . . ,rn} be a set of n-distinct attributes with attribute value sets respectively as E1, E2, . . . ,En, where Ei∩Ej=ϕ for i̸=j and i, j∈{1, 2, . . . ,n}. Also, let Di be the nonempty subset of Ei for each i∈{1, 2, . . . ,n} and Vψ=D1×D2×· · ·×Dn. The triple (Γ,C, V ψ)PC is called a Plithogenic crisp hypersoft (in short, PCHS) set where Γ:Vψ→P (UP ) and C : P (UP ) ×Di → {0, 1}, for all x ∈ P (UP ), for each i∈{1, 2, . . . ,n}. That is, (Γ,C, Vψ)PC = {< (β) , {x (C (x, di))} ; β ∈ Vψ and x ∈ Γ(β)} >}. Note that for x /∈ Γ(β) , C(x, di) = 0 for each i ∈ {1, 2, ..., n} . The set of all the PCHS sets over UP will be denoted as PPC(UP ). Definition 13. A PCHS set (Γ, C, Vψ)PC over UP is called a null plithogenic crisp hyper- soft (in short, null PCHS) set if ∀β ∈ Vψ, C(x, di) = 0PC for each i ∈ {1, 2, . . . , n} and for all x ∈ UP . The null PCHS set will be denoted by (Φ, C, Vψ)PC . Definition 14. A PCHS set (Γ, C, Vψ)PC is called a whole plithogenic crisp hypersoft (in short, whole PCHS) set if ∀β ∈ Vψ, C(x, di) = 1PC for each i ∈ {1, 2, . . . , n} and for all x ∈ UP . The whole PCHS set will be denoted by (Ψ, C, Vψ)PC . Definition 15. Let (Γ1, C1, Fψ)PC and (Γ2, C2, Hψ)PC be two PCHS sets over UP . Then (Γ1, C1, Fψ)PC is a PCHS subset of (Γ2, C2, Hψ)PC if Fψ ⊆ Hψ and Γ1 (β) ⊆ Γ2(β) for all β ∈ Fψ and C1 (x, di) ≤ C2 (x, di), for each i∈{1, 2, . . . ,n} and for all x ∈ Γ1 (β). And it will be denoted by (Γ1, C1, Fψ)PC ≍ ⊑ (Γ2, C2, Hψ)PC . Thus (Γ1, C1, Fψ)PC and (Γ2, C2, Hψ)PC are equal, if (Γ1, C1, Fψ)PC ≍ ⊑ (Γ2, C2, Hψ)PC and (Γ2, C2, Hψ)PC ≍ ⊑ (Γ1, C1, Fψ)PC and denoted by (Γ1, C1, Fψ)PC ≍ = (Γ2, C2, Hψ)PC . Definition 16. Let (Γ1, C1, Fψ)PC and (Γ2, C2, Hψ)PC be two PCHS sets over UP . The intersection of (Γ1, C1, Fψ)PC and (Γ2, C2, Hψ)PC is a PCHS set as defined the follows: Let β ∈ Fψ ∩Hψ, then : (Γ1, C1, V ψ)PC ≍ ⊓ (Γ2, C2, Hψ)PC ≍ = {< β, {x(Min{C1(x, di), C2(x, di)})} >}. Definition 17. Let (Γ1, C1, Fψ)PC and (Γ2, C2, Hψ)PC be two PCHS sets over UP . The union of (Γ1, C1, Fψ)PC and (Γ2, C2, Hψ)PC is a PCHS set as defined the follows: Let N. K. Ahmed, O. T. Pirbal / Eur. J. Pure Appl. Math, 17 (4) (2024), 3043-3060 3047 β ∈ Fψ ∪Hψ, then : (Γ1, C1, Fψ)PC ≍ ⊔(Γ2, C2, Hψ)PC ≍ =  {< (β) , {x (C1 (x, di))} >} if β ∈ Fψ \Hψ {< (β) , {x (C2 (x, di))} >} if β ∈ Hψ \ Fψ {< (β) , {x (Max{C1 (x, di) , C2 (x, di)} >} if β ∈ Hψ ∩ Fψ Definition 18. Let (Γ1, C1, Fψ)PC and (Γ2, C2, Hψ)PC be two PCHS sets over UP . The PCHS difference of (Γ1, C1, Fψ)PC and (Γ2, C2, Hψ)PC is denoted by (Γ, C, V ψ)PC where V ψ = Fψ ∩Hψ and for all β ∈ Vψ , Vψ(β) = Fψ(β) \Hψ(β) . We write (Γ, C, V ψ)PC ≍ = (Γ1, C1, Fψ)PC ≍ \ (Γ2, C2, Hψ)PC . Definition 19. The complement of a PCHS set (Γ,C, Vψ)PC = {< (β) , {x (C (x, di))} ; β ∈ Vψ and x ∈ Γ(β)} >} is denoted by ( Γ,C, V ψ )c PC and defined by (Γ, C, Vψ) c PC ≍ = {< (β) , {x (C (x, di)) c} ;β ∈ Vψ and for all x ∈ UP }. That is, if C(x, di) = 0, then (C(x, di)) c = 1 or the reverse. Proposition 1. Let ( Γ,C, V ψ ) PC be a PCHS set over UP . Then the following are true: (i) (Γ,C, Vψ)PC ≍ ⊔ (Φ,C, V ψ) PC ≍ = ( Γ,C, V ψ ) PC (ii) ( Γ,C, V ψ ) PC ≍ ⊓ (Φ,C, V ψ) PC ≍ = (Φ,C, V ψ)PC (iii) ( Γ,C, V ψ ) PC ≍ ⊔ (Ψ,C, V ψ) PC ≍ = (Ψ,C, V ψ)PC (iv) ( Γ,C, V ψ ) PC ≍ ⊓ (Ψ,C, V ψ) PC ≍ = ( Γ,C, V ψ ) PC (v) (Ψ,C, V ψ) PC ≍ \ (Γ,C, Vψ)PC ≍ = ( Γ,C, V ψ )c PC (vi) (Γ,C, Vψ)PC ≍ ⊔ ( Γ,C, V ψ )c PC ≍ = (Ψ,C, V ψ) PC (vii) (Γ,C, Vψ)PC ≍ ⊓ ( Γ,C, V ψ )c PC ≍ = (Φ,C, V ψ) PC . Proof. Straightforward. Definition 20. A PCHS set (Γ,C, Vψ)PC is said to be a PCHS point, if range (Γ) ={x} and ∃i ∈ {1, 2, . . . ,n} such that C (x, di) = 1 and will be denote by PCP (β,x) where β ∈ Vψ. Proposition 2. Let (Γ,C, Vψ)PC , (Γ1, C1, Fψ)PC and (Γ2, C2, Hψ)PC be PCHS sets over UP . Then the following hold: (i) If (Γ,C, Vψ)PC is not a null PCHS point, then (Γ,C, Vψ)PC contains at least one non-null PCHS point. N. K. Ahmed, O. T. Pirbal / Eur. J. Pure Appl. Math, 17 (4) (2024), 3043-3060 3048 (ii) (Γ1, C1, Fψ)PC ≍ ⊑ (Γ2, C2, Hψ)PC ⇐⇒ PcP (β,x) ∈ (Γ1, C1, Fψ)PC implies that PcP (β,x) ∈ (Γ2, C2, Hψ)PC . (iii) PcP (β,x) ∈ (Γ1, C1, Fψ)PC ≍ ⊔ (Γ2, C2, Hψ)PC ⇐⇒ PcP (β,x) ∈ (Γ1, C1, Fψ)PC or PcP (β,x) ∈ (Γ2, C2, Hψ)PC . (iv) PcP (β,x) ∈ (Γ1, C1, Fψ)PC ≍ ⊓ (Γ2, C2, Hψ)PC ⇐⇒ PcP (β,x) ∈ (Γ1, C1, Fψ)PC and PcP (β,x) ∈ (Γ2, C2, Hψ)PC . (v) PcP (β,x) ∈ (Γ1, C1, Fψ)PC ≍ \ (Γ2, C2, Hψ)PC ⇐⇒ PcP (β,x) ∈ (Γ1, C1, Fψ)PC and PcP (β,x) /∈ (Γ2, C2, Hψ)PC . Proof. Straightforward. Remark 1. (i) If C (x, di) = 0PC for all i∈{1, 2, . . . ,n}, then PCP (β,x) is called null PCHS point and denoted by PCP (β,x,0PC). Also, if ( Γ,C, V ψ ) PC is null PCHS set, then it can be considered as a null PCHS point. (ii) ( Γ,C, V ψ ) PC ≍ = ≍ ⊔ { PCP (β,x); PCP (β,x) ∈ (Γ,C, Vψ)PC } 4. Plithogenic Crisp Hypersoft Topology Definition 21. Let ( Γ,C, V ψ ) PC be a PCHS set over UP and x ∈ UP . Then x ∈( Γ,C, V ψ ) PC if x ∈ Γ(β) for all β ∈ Vψ and C(x, di) ̸= 0 for some i ∈ {1, 2, ..., n}. Otherwise, x /∈ ( Γ,C, V ψ ) PC . Definition 22. Let Y be a non-empty subset of UP . Then (Y, C, Vψ)PC denoted a PCHS set over UP and defined by Y (β) = Y , for all β ∈ Vψ. Definition 23. Let (Γ, C, Vψ)PC be a PCHS set over UP , and let Y be a non-empty subset of UP . The sub-PCHS set of (Γ, C, Vψ)PC over Y is denoted by (ΓY , C, Vψ)PC and is defined as ΓY (β) = Y ∩ Γ(β) for each β ∈ Vψ. That is, (ΓY , C, Vψ)PC ≍ = (Y,C, Vψ)PC ≍ ⊓ (Γ, C, Vψ)PC . Definition 24. Let τPC be the collection of PCHS sets over UP . Then τPC is said to be plithogenic crisp hypersoft (in short, PCHS) topology over UP , if the following holds: (i) (Φ,C, V ψ) PC , (Ψ,C, V ψ) PC belong to τPC , (ii) The intersection of any two PCHS sets in τPC belongs to τPC , (iii) The union of any number of PCHS sets in τPC belongs to τPC . N. K. Ahmed, O. T. Pirbal / Eur. J. Pure Appl. Math, 17 (4) (2024), 3043-3060 3049 Then (UP , τPC , Vψ) is called a plithogenic crisp hypersoft topological (in short, PCHST) space over UP . Also, the members of (Up, τPC , Vψ) are said to be plithogenic crisp hypersoft (in short, PCHS) open sets over UP . Definition 25. In a PCHST space (Up, τPC , Vψ) a PCHS set ( Γ,C, V ψ ) PC over UP is said to be plithogenic crisp hypersoft (in short, PCHS) closed set if its complement belongs to τPC . Example 1. Let UP = {x1, x2, x3, x4}, E1 = {e1, e2}, E2 = {e3}, E3 = {e4}. Let Vψ = E1×E2×E3 and (α) = (e1, e3, e4) and (β) = (e2, e3, e4). Define the following PCHS sets: (Γ1, C1, Vψ)PC = {< (α), {x1 (1, 0, 1)} >,< (β) , {x2 (1, 0, 1)} >} (Γ2, C2, Vψ)PC = {< (α), {x1 (1, 0, 1)} >,< (β) , 1PC >} (Γ3, C3, Vψ)PC = {< (α), 1PC >,< (β) , {x2 (1, 0, 1)} >}. Then the collection: τPC = {(Φ, C, Vψ)PC , (Γ1, C1, Vψ)PC , (Γ2, C2, Vψ)PC , (Γ3, C3, Vψ)PC , (Ψ, C, Vψ)PC} forms a PCHS topology over UP . Remark 2. Let (Up, τPC , Vψ) be a PCHST space over UP . Then the following holds: (i) (Φ,C, V ψ)PC , (Ψ,C, V ψ)PC are PCHS closed sets over UP . (ii) The intersection of any number of PCHS closed sets is PCHS closed set over UP . (iii) The union of any two number of PCHS closed sets is PCHS closed set over UP . Definition 26. Let UP be the plithogenic crisp universal set. Then (i) τ IPC = {(Φ, C, Vψ)PC , (Ψ, C, Vψ)PC} is called Plithogenic crisp hypersoft indiscrete (in short, PCHSI) topology over UP and ( Γ, τ IPC , Vψ ) is called plithogenic crisp hy- persoft indiscrete topological (in short, PCHSIT) space over UP . (ii) τDPC = PPC(UP ) is called Plithogenic crisp hypersoft discrete (in short, PCHSD) topology over UP and ( Γ, τDPC , Vψ ) is called plithogenic crisp hypersoft discrete (in short, PCHSDT) topological space over UP . Definition 27. Let (Γ, τPC1 , Vψ) and (Γ, τPC2 , Vψ) be two PCHST spaces over UP . If τPC1 ≍ ⊑ τPC2, then τPC2 is said to be finer than τPC1. If τPC2 ≍ ⊑ τPC1, then τPC1 is said to be finer than τPC2. If τPC1 ≍ ⊑ τPC2 or τPC2 ≍ ⊑ τPC1, then τPC1 and τPC2 are said to be comparable PCHS topologies over UP . Proposition 3. Let (Γ, τPC , Vψ) and (Γ, τ∗PC , Vψ) be two PCHST spaces over UP , then( Γ, τPC ≍ ⊓ τ∗PC , Vψ ) is a PCHS topological space over UP . Proof. N. K. Ahmed, O. T. Pirbal / Eur. J. Pure Appl. Math, 17 (4) (2024), 3043-3060 3050 (i) Clearly (Φ,C, V ψ) PC , (Ψ,C, V ψ) PC∈ τPC ≍ ⊓ τ∗PC . (ii) Let (Γ1, C1, Vψ)PC , (Γ2, C2, Vψ)PC∈τPC ≍ ⊓ τ∗PC , then (Γ1, C1, Vψ)PC , (Γ2, C2, Vψ)PC∈τPC and (Γ1, C1, Vψ)PC , (Γ2, C2, Vψ)PC∈τ ∗ PC . Since (Γ1, C1, Vψ)PC ≍ ⊓ (Γ2, C2, Vψ)PC∈τPC and (Γ1, C1, Vψ)PC ≍ ⊓ (Γ2, C2, Vψ)PC∈τ ∗ Pc, so (Γ1, C1, Vψ)PC ≍ ⊓ (Γ2, C2, Vψ)PC∈τPC ≍ ⊓ τ∗PC . (iii) Let {(Γi, Ci, Vψ)PC ; i ∈ I} be a family of PCHS sets in τPC ≍ ⊓ τ∗PC . Then (Γi, Ci, Vψ)PC∈τPC and (Γi, Ci, Vψ)PC∈τ ∗ PC for each i ∈ I, so ≍⊔ i∈I (Γi, Ci, Vψ)PC ∈ τPC and ≍⊔ i∈I (Γi, Ci, Vψ)PC∈τ ∗ PC . Therefore, ≍⊔ i∈I (Γi, Ci, Vψ)PC∈τPC ≍ ⊓ τ∗Pc. Thus, τPC ≍ ⊓ τ∗PC forms a PCHS topology over UP and ( Γ, τPC ≍ ⊓ τ∗PC , Vψ ) is a PCHST space over UP . Remark 3. The union of two PCHS topologies on UP may not be a PCHS topology on UP . See the next example. Example 2. Let UP = {x1, x2, x3, x4}, E1 = {e1, e2}, E2 = {e3}, E3 = {e4}. Let Vψ = E1×E2×E3 and (α) = (e1, e3, e4) and (β) = (e2, e3, e4). Define the following PCHS sets:( Γ1, C1, V ψ ) PC = {< (α), {x3 (1, 1, 0) , x4 (1, 1, 1)} >, < (β) , {x2 (1, 1, 1) , x3 (1, 1, 1)} >}( Γ2, C2, V ψ ) PC = {< (α), {x1 (1, 1, 1) , x2 (1, 1, 1) , x3 (1, 1, 1)} >, < (β), {x1 (1, 1, 1) , x4 (1, 1, 1)} >}( Γ3, C3, V ψ ) PC = {< (α), {x3 (1, 1, 0)} >, < (β), 0PC >} (Γ∗ 1, C ∗ 1 , Vψ)PC = {< (α), {x1 (1, 0, 1)} >,< (β), {x2 (1, 1, 0)} >} (Γ∗ 2, C ∗ 2 , Vψ)PC = {< (α), {x1 (1, 0, 0)} >,< (β), {x2 (1, 0, 0)} >} (Γ∗ 3, C ∗ 3 , Vψ)PC = {< (α), {x1 (1, 0, 0)} >,< (β), {x2 (1, 1, 0)} >}. Then the collections: τPC = { (Φ,C, V ψ) PC , (Γ1, C1, Vψ)PC , (Γ2, C2, Vψ)PC , (Γ3, C3, Vψ)PC , (Ψ,C, V ψ) PC } and τ∗PC = { (Φ,C, V ψ) PC , (Γ∗ 1, C ∗ 1 , Vψ)PC , (Γ∗ 2, C ∗ 2 , Vψ)PC , (Γ∗ 3, C ∗ 2 , Vψ)PC , (Ψ,C, V ψ) PC } forms a PCHS topological spaces on UP . Now, we see that: (Γ∗ 3, C ∗ 3 , Vψ)PC ≍ ⊔(Γ3, C3, Vψ)PC = {< (α), {x1 (1, 1, 0) , x3 (1, 1, 0)} >,< (β), {x2 (1, 1, 0)} >} not in τPC ≍ ⊔ τ∗PC . Hence, τPC ≍ ⊔ τ∗PC does not form a PCHS topology over UP . Definition 28. Let (Up, τPC , Vψ) be a PCHST space over UP , ( Γ,C, V ψ ) PC be a PCHS set over UP and x∈UP . Then ( Γ,C, V ψ ) PC is said to be a PCHS neighborhood of x if there exists a PCHS open set ( Γ∗, C∗, V ψ ) PC such that x∈ ( Γ∗, C∗, V ψ ) PC ≍ ⊑ ( Γ,C, V ψ ) PC . Remark 4. Let (Up, τPC , Vψ) be a PCHST space over UP , then: N. K. Ahmed, O. T. Pirbal / Eur. J. Pure Appl. Math, 17 (4) (2024), 3043-3060 3051 (i) If ( Γ,C, V ψ ) PC is a PCHS neighborhood of x∈UP , then x ∈ ( Γ,C, V ψ ) PC . (ii) Each x∈UP has a PCHS neighborhood. (iii) If ( Γ,C, V ψ ) PC and ( Γ∗, C∗, V ψ ) PC are PCHS neighborhoods of some x∈UP , then( Γ,C, V ψ ) PC ≍ ⊓ ( Γ,C, V ψ ) PC is also a PCHS neighborhood of x. (iv) If ( Γ,C, V ψ ) PC is a PCHS neighborhood of x∈UP and ( Γ,C, V ψ ) PC ≍ ⊑ ( Γ∗, C∗, V ψ ) PC , then ( Γ∗, C∗, V ψ ) PC is also a PCHS neighborhood of x∈UP . Remark 5. Let Let (Up, τPC , Vψ) be a PCHST space over UP and ( Γ,C, V ψ ) PC ∈ τPC . Then for any x in image of Γ (β) for β ∈ Vψ, we have x ∈ ( Γ,C, V ψ ) PC ≍ ⊑ ( Γ,C, V ψ ) PC and so ( Γ,C, V ψ ) PC is a PCHS neighborhood of each its points. But the converse of it may not be true. See the next example. Example 3. Consider the PCHST space (UP , τPC , Vψ) in Example 2. Now, cosider the following PCHS set: (Γ∗, C∗, Vψ)PC = {< (α), {x1 (1, 1, 0) , x3 (1, 1, 0) , x4 (1, 1, 0)} >,< (β) , {x2 (1, 1, 0) , x3 (1, 1, 0) >} is a PCHS neighborhood of each of its points, but it is not a PCHS open set. Definition 29. Let (Up, τPC , Vψ) be a PCHST space over UP and Let ( Γ,C, V ψ ) PC be a PCHS set over UP . A point x ∈ UP is called a PCHS limit point of ( Γ,C, V ψ ) PC if( Γ,C, V ψ ) PC ≍ ⊓ [( Γ∗, C∗, V ψ ) PC ≍ \ {x} ] ≍ ̸= (Φ,C, V ψ) PC for every PCHS open set ( Γ∗, C∗, V ψ ) PC containing x. The set of all PCHS limit points of ( Γ,C, V ψ ) PC is denoted by ( Γ,C, V ψ )d PC . Proposition 4. Let (Up, τPC , Vψ) be a PCHS space over UP and let ( Γ1, C1, V ψ ) PC ,( Γ2, C2, V ψ ) PC be two PCHS sets over UP . Then: (i) ( Γ1, C1, V ψ ) PC ≍ ⊑ ( Γ2, C2, V ψ ) PC implies ( Γ1, C1, V ψ )d PC ≍ ⊑ ( Γ2, C2, V ψ )d PC . (ii) [( Γ1, C1, V ψ ) PC ≍ ⊓ ( Γ2, C2, V ψ ) PC ]d ≍ ⊑ ( Γ1, C1, V ψ )d PC ≍ ⊓ ( Γ2, C2, V ψ )d PC . (iii) [( Γ1, C1, V ψ ) PC ≍ ⊔ ( Γ2, C2, V ψ ) PC ]d ≍ = ( Γ1, C1, V ψ )d PC ≍ ⊔ ( Γ2, C2, V ψ )d PC . Proof. (i) Let x ∈ ( Γ1, C1, V ψ )d PC , so that x is a PCHS limit point of ( Γ1, C1, V ψ ) PC , then, it follows that ( Γ1, C1, V ψ ) PC ≍ ⊓ [( Γ∗, C∗, V ψ ) PC ≍ \ {x} ] ≍ =(Φ,C, V ψ) PC for every PCHS open set ( Γ∗, C∗, V ψ ) PC containing x. But since ( Γ1, C1, V ψ ) ≍ ⊑ ( Γ2, C2, V ψ ) PC , it follows that ( Γ2, C2, V ψ ) PC ≍ ⊓ [( Γ∗, C∗, V ψ ) PC ≍ \ {x} ] ≍ =(Φ,C, V ψ) PC . Thus, x ∈( Γ2, C, V ψ )d PC . Therefore, ( Γ1, C1, V ψ )d PC ≍ ⊑ ( Γ2, C2, V ψ )d PC . N. K. Ahmed, O. T. Pirbal / Eur. J. Pure Appl. Math, 17 (4) (2024), 3043-3060 3052 (ii) Since ( Γ1, C1, V ψ ) PC ≍ ⊓ ( Γ2, C2, V ψ ) PC ≍ ⊑ ( Γ1, C1, V ψ ) PC and( Γ1, C1, V ψ ) PC ≍ ⊓ ( Γ2, C2, V ψ ) PC ≍ ⊑ ( Γ2, C2, V ψ ) PC , then by part (i) follows that [( Γ1, C1, V ψ ) PC ≍ ⊓ ( Γ2, C2, V ψ ) PC ]d ≍ ⊑ ( Γ1, C1, V ψ )d PC and [( Γ1, C1, V ψ ) PC ≍ ⊓ ( Γ2, C2, V ψ ) PC ]d ≍ ⊑ ( Γ2, C1, V ψ )d PC . Hence [( Γ1, C1, V ψ ) PC ≍ ⊓ ( Γ2, C2, V ψ ) PC ]d ≍ ⊑ ( Γ1, C1, V ψ )d PC ≍ ⊓ ( Γ2, C2, V ψ )d PC . (iii) Since ( Γ1, C1, V ψ ) PC ≍ ⊑ ( Γ1, C1, V ψ ) PC ≍ ⊔ ( Γ2, C2, V ψ ) PC and ( Γ2, C2, V ψ ) PC ≍ ⊑( Γ1, C1, V ψ ) PC ≍ ⊔ ( Γ2, C2, V ψ ) PC , by part (i) we have ( Γ1, C1, V ψ )d PC ≍ ⊑ [( Γ1, C1, V ψ ) PC ≍ ⊔ ( Γ2, C2, V ψ ) PC ]d and ( Γ2, C2, V ψ )d PC ≍ ⊑[( Γ1, C1, V ψ ) PC ≍ ⊔ ( Γ2, C2, V ψ ) PC ]d . So, ( Γ1, C1, V ψ )d PC ≍ ⊔ ( Γ2, C2, V ψ )d PC ≍ ⊑ [( Γ1, C1, V ψ ) PC ≍ ⊔ ( Γ2, C2, V ψ ) PC ]d . Now, Let x∈ [( Γ1, C1, V ψ ) PC ≍ ⊔ ( Γ2, C2, V ψ ) PC ]d , Then [( Γ1, C1, V ψ ) PC ≍ ⊔ ( Γ2, C2, V ψ ) PC ] ≍ ⊓ [( Γ∗, C∗, V ψ ) PC \≍ {x} ] ≍ ̸=(Φ,C, V ψ) PC for every PCHS set ( Γ∗, C∗, V ψ ) PC containing x. Therefore, ( Γ1, C1, V ψ ) PC ≍ ⊓ [( Γ∗, C∗, V ψ ) PC ≍ \ {x} ] ≍ = (Φ,C, V ψ) PC or ( Γ2, C2, V ψ ) PC ≍ ⊓[( Γ∗, C∗, V ψ ) PC ≍ \ {x} ] ≍ = (Φ,C, V ψ) PC . Thus, x∈ ( Γ1, C1, V ψ )d PC or x∈ ( Γ2, C2, V ψ )d PC and then x∈ ( Γ1, C1, V ψ )d PC ≍ ⊔ ( Γ2, C2, V ψ )d PC . Therefore, ( Γ1, C1, V ψ )d PC ≍ ⊔ ( Γ2, C2, V ψ )d PC ≍ ⊒[( Γ1, C1, V ψ ) PC ≍ ⊔ ( Γ2, C2, V ψ ) PC ]d . Hence, [( Γ1, C1, V ψ ) PC ≍ ⊔ ( Γ2, C2, V ψ ) PC ]d ≍ =( Γ1, C1, V ψ )d PC ≍ ⊔ ( Γ2, C2, V ψ )d PC . Remark 6. The converse of above the proposition part (ii), may not be true. See the next example. Example 4. Consider the PCHST space (UP , τPC , Vψ) in Example 2. Let ( Γ4, C4, V ψ ) PC and ( Γ5, C5, V ψ ) PC be two PCHS sets defined as follows:( Γ4, C4, V ψ ) PC = {< (α), 0PC >, < (β), {x4 (1, 1, 0)} >}( Γ5, C5, V ψ ) PC = {< (α), {x2 (1, 1, 0)} >, < (β), {x3 (1, 1, 0)} >} Now, ( Γ4, C4, V ψ )d PC ≍ ⊓ ( Γ5, C5, V ψ )d PC ≍ = {x1, x2, x4} but [( Γ4, C4, V ψ ) PC ≍ ⊓ ( Γ5, C5, V ψ ) PC ]d ≍ = (Φ,C, V ψ) PC . Hence, [( Γ1, C, V ψ ) PC ≍ ⊓ ( Γ2, C, V ψ ) PC ]d ≍ ̸= ( Γ1, C, V ψ )d PC ≍ ⊓ ( Γ2, C, V ψ )d PC . N. K. Ahmed, O. T. Pirbal / Eur. J. Pure Appl. Math, 17 (4) (2024), 3043-3060 3053 Definition 30. Let (Up, τPC , Vψ) be a PCHST space over UP and Y be a non-empty subset of UP . Then τPCY = {( ΓY, C, V ψ ) PC ∣∣∣ (Γ,C, V ψ ) PC ∈τPC } is said to be the relative PCHS topology on Y and (Y, τPCY , Vψ) is called a PCHS subspace of (Up, τPC , Vψ). One can verify that τPCY is a PCHS topology on Y . Example 5. Consider the PCHST space (UP , τPC , Vψ) in Example 2. and Definition 23. Let Y = {x2, x3}, then:( ΓY1 , C1, V ψ ) PC = {< (α), {x3 (1, 1, 0)} >, < (β) , {x2 (1, 1, 1) , x3 (1, 1, 1)} >}( ΓY2 , C2, V ψ ) PC = {< (α), {x2 (1, 1, 1) , x3 (1, 1, 1)} >, < (β), 0PC >}.( ΓY3 , C3, V ψ ) PC = {< (α), {x3 (1, 1, 0)} >, < (β), 0PC >}. Then, τPCY = { (Φ,C, V ψ) PC , (ΓY1 , C1, Vψ)PC , (ΓY2 , C2, Vψ)PC , (ΓY3 , C3, Vψ)PC , (ΨY,C, V ψ)PC } Proposition 5. Let (Y, τPCY , Vψ) be a PCHS subspace of PCHST space (Up, τPC , Vψ) and( ΓY, C, V ψ ) PC be a PCHS open set in Y . If ( Y,C, V ψ ) PC ∈ τPC , then ( ΓY, C, V ψ ) PC ∈ τPC . Proof. Let ( ΓY, C, V ψ ) PC be a PCHS open set in Y , then there exist a PCHS open set ( Γ,C, V ψ ) PC in UP such that ( ΓY, C, V ψ ) PC ≍ = ( Y,C, V ψ ) PC ≍ ⊓ ( Γ,C, V ψ ) PC . Now, if( Y,C, V ψ ) PC ∈ τPC , then ( Y,C, V ψ ) PC ≍ ⊓ ( Γ,C, V ψ ) PC ∈ τPC . Hence, ( ΓY, C, V ψ ) PC ∈ τPC . Proposition 6. Let (Y, τPCY , Vψ) and (Z, τPCZ , Vψ) be two PCHS subspace of (UP , τPC , Vψ) and let Y ⊆ Z. Then (Y, τPCY , Vψ) is a PCHS subspace of (Z, τPCZ , Vψ). Proof. Let ( ΓY, C, V ψ ) PC be a PCHS open set in Y , then there exists a PCHS open set ( Γ,C, V ψ ) PC in UP such that ( ΓY, C, V ψ ) PC ≍ = (Y, τPCY , Vψ) ≍ ⊓ ( Γ,C, V ψ ) PC , or equiv- alently, for each β ∈ Vψ, ΓY (β) = Y ⊓ Γ(β). Since Y ⊑ Z, then Y = Y ⊓ Z. Now, ΓY (β) = Y ⊓ΓZ (β) = (Y ⊓Z)⊓Γ (β) = Y ⊓ΓZ (β). Hence, (Y, τPCY , Vψ) is a PCHS sub- space of (Z, τPCZ , Vψ). 5. PCHS Closure and PCHS Interior Definition 31. Let (UP , τPC , Vψ) be a PCHST space and ( Γ,C, V ψ ) PC be a PCHS set over UP . The intersection of all PCHS closed supersets of ( Γ,C, V ψ ) PC is called the PCHS closure of ( Γ,C, V ψ ) PC and is denoted by ( Γ,C, V ψ ) PC . In other words:( Γ,C, V ψ ) PC ≍ = ≍ ⊓ {( Γ∗, C∗, V ψ ) PC ∣∣∣∣ (Γ∗,C∗, V ψ )c PC ∈ τPC , ( Γ, C, V ψ ) PC ≍ ⊑ ( Γ∗,C∗, V ψ ) PC } . Proposition 7. Let (UP , τPC , Vψ) be a PCHST space and ( Γ,C, V ψ ) PC be a PCHS set over UP , then: (i) ( Γ,C, V ψ ) PC is the smallest PCHS closed set containing ( Γ,C, V ψ ) PC . N. K. Ahmed, O. T. Pirbal / Eur. J. Pure Appl. Math, 17 (4) (2024), 3043-3060 3054 (ii) ( Γ,C, V ψ ) PC is a PCHS closed sets if and only if ( Γ,C, V ψ ) PC ≍ = ( Γ,C, V ψ ) PC . Proof. (i) Obvious. (ii) Let ( Γ,C, V ψ ) PC be a PCHS closed set. So, ( Γ,C, V ψ ) PC itself is the the smallest PCHS closed set over UP containing ( Γ,C, V ψ ) PC , hence, ( Γ,C, V ψ ) PC ≍ = ( Γ,C, V ψ ) PC . Conversely, let ( Γ,C, V ψ ) PC ≍ = ( Γ,C, V ψ ) PC , by part (i), ( Γ,C, V ψ ) PC is a PCHS closed set, so ( Γ,C, V ψ ) PC is a PCHS closed set over UP . Proposition 8. Let (UP , τPC , Vψ) be a PCHS topological space and let ( Γ1, C1, V ψ ) PC ,( Γ2, C2, V ψ ) PC be two PCHS set over UP , then: (i) (Φ,C, V ψ) PC ≍ = (Φ,C, V ψ) PC and(Ψ,C, V ψ) PC ≍ = (Ψ,C, V ψ) PC . (ii) ( Γ1, C1, V ψ ) PC ≍ ⊑ ( Γ1, C1, V ψ ) PC . (iii) ( Γ1, C1, V ψ ) PC ≍ ⊑ ( Γ2, C2, V ψ ) PC implies ( Γ1, C1, V ψ ) PC ≍ ⊑ ( Γ2, C2, V ψ ) PC . (iv) ( Γ1, C1, V ψ ) PC ≍ ⊔ ( Γ2, C2, V ψ ) PC ≍ = ( Γ1, C1, V ψ ) PC ≍ ⊔ ( Γ2, C2, V ψ ) PC . (v) ( Γ1, C1, V ψ ) PC ≍ ⊓ ( Γ2, C2, V ψ ) PC ≍ ⊑ ( Γ1, C1, V ψ ) PC ≍ ⊓ ( Γ2, C2, V ψ ) PC . (vi) ( Γ1, C1, V ψ ) PC ≍ = ( Γ1, C1, V ψ ) PC . Proof. (i) Obvious. (ii) By proposition 7(i), ( Γ1, C1, V ψ ) PC is the smallest PCHS closed set containing( Γ1, C1, V ψ ) PC , so it follows ( Γ1, C1, V ψ ) PC ≍ ⊑ ( Γ1, C1, V ψ ) PC . (iii) By part (ii), ( Γ2, C2, V ψ ) PC ≍ ⊑ ( Γ2, C2, V ψ ) PC . Since ( Γ1, C2, V ψ ) PC ≍ ⊑ ( Γ2, C2, V ψ ) PC , we have ( Γ1, C1, V ψ ) PC ≍ ⊑ ( Γ2, C2, V ψ ) PC , but ( Γ2, C2, V ψ ) PC is a PCHS closed set containing ( Γ1, C1, V ψ ) PC and since ( Γ1, C1, V ψ ) PC is the smallest PCHS closed set over UP containing ( Γ1, C, V ψ ) PC , so it follows that ( Γ1, C1, V ψ ) PC ≍ ⊔ ( Γ2, C2, V ψ ) PC . N. K. Ahmed, O. T. Pirbal / Eur. J. Pure Appl. Math, 17 (4) (2024), 3043-3060 3055 (iv) Since ( Γ1, C1, V ψ ) PC ≍ ⊑ ( Γ1, C1, V ψ ) PC ≍ ⊔ ( Γ2, C2, V ψ ) PC and ( Γ2, C2, V ψ ) PC ≍ ⊑( Γ1, C1, V ψ ) PC ≍ ⊔ ( Γ2, C2, V ψ ) PC , by part (iii), we have( Γ1, C1, V ψ ) PC ≍ ⊑ ( Γ1, C1, V ψ ) PC ≍ ⊔ ( Γ2, C1, V ψ ) PC and( Γ2, C2, V ψ ) PC ≍ ⊑ ( Γ1, C1, V ψ ) PC ≍ ⊔ ( Γ2, C2, V ψ ) PC . Hence, ( Γ1, C1, V ψ ) PC ≍ ⊔ ( Γ2, C2, V ψ ) PC . ≍ ⊑ ( Γ1, C1, V ψ ) PC ≍ ⊔ ( Γ2, C2, V ψ ) PC . Now, since ( Γ1, C1, V ψ ) PC and ( Γ2, C2, V ψ ) PC are PCHS closed sets over UP , then ( Γ1, C1, V ψ ) PC ≍ ⊔( Γ2, C2, V ψ ) PC is also PCHS closed set. Also, ( Γ1, C1, V ψ ) PC ≍ ⊑ ( Γ1, C1, V ψ ) PC and( Γ2, C2, V ψ ) PC ≍ ⊑ ( Γ2, C2, V ψ ) PC it implies that ( Γ1, C1, V ψ ) PC ≍ ⊔ ( Γ2, C2, V ψ ) PC ≍ ⊑( Γ1, C2, V ψ ) PC ≍ ⊔ ( Γ2, C2, V ψ ) PC . Thus, ( Γ1, C, V ψ ) PC ≍ ⊔ ( Γ2, C, V ψ ) PC is a PCHS closed set containing ( Γ1, C1, V ψ ) PC ≍ ⊔ ( Γ1, C1, V ψ ) PC . Since ( Γ1, C2, V ψ ) PC ≍ ⊔ ( Γ2, C2, V ψ ) PC is the smallest PCHS closed set contain- ing ( Γ1, C1, V ψ ) PC ≍ ⊔ ( Γ2, C2, V ψ ) PC , we have ( Γ1, C1, V ψ ) PC ≍ ⊔ ( Γ2, C2, V ψ ) PC ≍ ⊑( Γ1, C1, V ψ ) PC ≍ ⊔ ( Γ2, C2, V ψ ) PC . Hence, ( Γ1, C1, V ψ ) PC ≍ ⊔ ( Γ2, C2, V ψ ) PC ≍ = ( Γ1, C1, V ψ ) PC ≍ ⊔( Γ2, C2, V ψ ) PC . (v) Since ( Γ1, C1, V ψ ) PC ≍ ⊓ ( Γ2, C2, V ψ ) PC ≍ ⊑ ( Γ1, C1, V ψ ) PC and ( Γ2, C2, V ψ ) PC ≍ ⊓( Γ1, C1, V ψ ) PC ≍ ⊑ ( Γ2, C2, V ψ ) PC . Therefore, ( Γ1, C1, V ψ ) PC ≍ ⊓ ( Γ2, C2, V ψ ) PC ≍ ⊑( Γ1, C1, V ψ ) PC ≍ ⊓ ( Γ2, C2, V ψ ) PC . (vi) Since ( Γ1, C1, V ψ ) PC is a PCHS closed set, by proposition 7(ii), it follows that( Γ1, C1, V ψ ) PC ≍ = ( Γ1, C1, V ψ ) PC . Remark 7. The equality of above proposition part (v) does not hold in general. See the next example. Example 6. Consider the PCHST space (UP , τPC , Vψ) in Example 2. Define (Γ4, C4, Vψ)PC and (Γ5, C5, Vψ)PC as the follow: (Γ4, C4, Vψ)PC= {< (α), {x1 (0, 1, 0)} >,< (β), {x4 (1, 1, 1)} >} and (Γ5, C5, Vψ)PC = {< (α), {x2 (1, 1, 1)} >,< (β), {x1 (1, 0, 1)} >}. Then: (Γ4, C4, Vψ)PC ≍ = (Γ1, C1, Vψ) c PC and (Γ5, C5, Vψ)PC ≍ = (Γ1, C1, Vψ) c PC . Now, (Γ4, C4, Vψ)PC ≍ ⊓(Γ5, C5, Vψ)PC ≍ = (Γ1, C1, Vψ) c PC but (Γ4, C4, Vψ)PC ≍ ⊓ (Γ5, C5, Vψ)PC ≍ = (Φ,C, V ψ) PC and (Γ1, C1, Vψ) c PC ≍ ̸= (Φ,C, V ψ) PC . N. K. Ahmed, O. T. Pirbal / Eur. J. Pure Appl. Math, 17 (4) (2024), 3043-3060 3056 Definition 32. Let (UP , τPC , Vψ) be a PCHST space over UP , ( Γ,C, V ψ ) PC be a PCHS set and x ∈ UP . Then x is said to be a PCHS interior point of ( Γ,C, V ψ ) PC if there exist a PCHS open set ( Γ∗, C∗, V ψ ) PC such that x ∈ ( Γ∗, C∗, V ψ ) PC ≍ ⊑ ( Γ,C, V ψ ) PC . Definition 33. Let (UP , τPC , Vψ) be a PCHST space over UP . Then the PCHS interior of PCHS set ( Γ,C, V ψ ) PC over UP is denoted by ( Γ,C, V ψ )o PC and is defined as the union of all PCHS open sets contained in ( Γ,C, V ψ ) PC . In other words:( Γ,C, V ψ )o PC ≍ = ≍ ⊔ {( Γ∗, C∗, V ψ ) PC ∣∣∣∣ (Γ∗, C∗, V ψ ) PC ∈ τPC , ( Γ∗, C∗, V ψ ) PC ≍ ⊑ ( Γ,C, V ψ ) PC } Proposition 9. Let (UP , τPC , Vψ) be a PCHST space and let ( Γ,C, V ψ ) PC be a PCHS set over UP . Then: (i) ( Γ,C, V ψ )o PC is the largest PCHS open set contained in ( Γ,C, V ψ ) PC . (ii) ( Γ,C, V ψ ) PC is a PCHS open set if and only if ( Γ,C, V ψ ) PC ≍ = ( Γ,C, V ψ )o PC . Proof. (i) Follows from the definition. (ii) Let ( Γ,C, V ψ ) PC be a PCHS open set. Then ( Γ,C, V ψ ) PC is surely identical with the largest PCHS open subset of ( Γ,C, V ψ ) PC , but by part (i), ( Γ,C, V ψ )o PC is the largest PCHS open subset of ( Γ,C, V ψ ) PC . Hence, ( Γ,C, V ψ ) PC ≍ = ( Γ,C, V ψ )o PC . Conversely, let ( Γ,C, V ψ ) PC ≍ = ( Γ,C, V ψ )o PC , by part (i), ( Γ,C, V ψ )o PC is a PCHS open set. Therefore, ( Γ,C, V ψ ) PC is also a PCHS open set. Proposition 10. Let (UP , τPC , Vψ) be a PCHST space over UP and let ( Γ1, C, V ψ ) PC ,( Γ2, C, V ψ ) PC be two PCHS sets over UP , then: (i) ( Φ,C, V ψ )o PC ≍ = (Φ,C, V ψ) PC and ( Ψ,C, V ψ )o PC ≍ = (Ψ,C, V ψ) PC . (ii) ( Γ1, C1, V ψ )o PC ≍ ⊑ ( Γ1, C1, V ψ ) PC . (iii) ( Γ1, C1, V ψ ) PC ≍ ⊑ ( Γ2, C2, V ψ ) PC implies ( Γ1, C1, V ψ )o PC ≍ ⊑ ( Γ2, C2, V ψ )o PC . (iv) ( Γ1, C1, V ψ )o PC ≍ ⊓ ( Γ2, C2, V ψ )o PC ≍ = [( Γ1, C1, V ψ ) PC ≍ ⊓ ( Γ2, C2, V ψ ) PC ]o . (v) ( Γ1, C1, V ψ )o PC ≍ ⊔ ( Γ2, C2, V ψ )o PC ≍ ⊑ [( Γ1, C1, V ψ ) PC ≍ ⊔ ( Γ2, C2, V ψ ) PC ]o . (vi) [( Γ1, C1, V ψ )o PC ]o ≍ = ( Γ1, C1, V ψ )o PC . N. K. Ahmed, O. T. Pirbal / Eur. J. Pure Appl. Math, 17 (4) (2024), 3043-3060 3057 Proof. (i) Obvious. (ii) Let x ∈ ( Γ1, C1, V ψ )o PC , then x is a PCHS interior point of ( Γ1, C1, V ψ ) PC and this implies that ( Γ1, C1, V ψ ) PC is PCHS neighborhood of x. Then, x ∈ ( Γ1, C1, V ψ ) PC . Hence, ( Γ1, C1, V ψ )o PC ≍ ⊑ ( Γ1, C1, V ψ ) PC . (iii) Let x ∈ ( Γ1, C1, V ψ )o PC . Then x is a PCHS interior point of ( Γ1, C1, V ψ ) PC and so( Γ1, C1, V ψ ) PC is PCHS neighborhood of x. Since ( Γ1, C1, V ψ ) PC ≍ ⊑ ( Γ2, C2, V ψ ) PC , so ( Γ2, C2, V ψ ) PC is also a PCHS neighborhood of x. This implies that x ∈ ( Γ2, C2, V ψ )o PC . Thus, ( Γ1, C1, V ψ )o PC ≍ ⊑ ( Γ2, C2, V ψ )o PC . (iv) Since [( Γ1, C1, V ψ ) PC ≍ ⊓ ( Γ2, C2, V ψ ) PC ] ≍ ⊑ ( Γ1, C2, V ψ ) PC and[( Γ1, C1, V ψ ) PC ≍ ⊓ ( Γ2, C2, V ψ ) PC ] ≍ ⊑ ( Γ2, C2, V ψ ) PC , by part (ii), we have[( Γ1, C1, V ψ ) PC ≍ ⊓ ( Γ2, C2, V ψ ) PC ]o ≍ ⊑ ( Γ1, C2, V ψ )o PC and[( Γ1, C1, V ψ ) PC ≍ ⊓ ( Γ2, C2, V ψ ) PC ]o ≍ ⊑ ( Γ2, C2, V ψ ) PC . This implies that [( Γ1, C1, V ψ ) PC ≍ ⊓ ( Γ2, C2, V ψ ) PC ]o ≍ ⊑ ( Γ1, C1, V ψ )o PC ≍ ⊓ ( Γ2, C2, V ψ )o PC . Again, let x ∈ ( Γ1, C1, V ψ )o PC ≍ ⊓ ( Γ2, C2, V ψ )o PC , then x ∈ ( Γ1, C1, V ψ )o PC and x ∈ ( Γ2, C2, V ψ )o PC . Hence, x is a PCHS interior point of each of the PCHS sets( Γ1, C1, V ψ ) PC and ( Γ2, C2, V ψ ) PC . It follows that ( Γ1, C1, V ψ ) PC and ( Γ2, C2, V ψ ) PC are PCHS neighborhood of x, so that, their intersection ( Γ1, C1, V ψ ) PC ≍ ⊓ ( Γ2, C2, V ψ ) PC is also a PCHS neighborhood of x. Hence, x ∈ [( Γ1, C1, V ψ ) PC ≍ ⊓ ( Γ2, C2, V ψ ) PC ]o . This, [( Γ1, C1, V ψ ) PC ≍ ⊓ ( Γ2, C2, V ψ ) PC ]o ≍ = ( Γ1, C1, V ψ )o PC ≍ ⊓ ( Γ2, C2, V ψ )o PC . (v) By part (iii), ( Γ1, C1, V ψ ) PC ≍ ⊑ ( Γ1, C1, V ψ ) PC ≍ ⊔ ( Γ2, C2, V ψ ) PC implies that ( Γ1, C1, V ψ )o PC ≍ ⊑[( Γ1, C1, V ψ ) PC ≍ ⊔ ( Γ2, C2, V ψ ) PC ]o and ( Γ1, C1, V ψ ) PC ≍ ⊑ ( Γ1, C1, V ψ ) PC ≍ ⊔ ( Γ2, C2, V ψ ) PC implies that ( Γ2, C2, V ψ ) PC ≍ ⊑ [( Γ1, C1, V ψ ) PC ≍ ⊔ ( Γ2, C2, V ψ ) PC ]o . Hence, ( Γ1, C1, V ψ )o PC ≍ ⊔( Γ2, C2, V ψ )o PC ≍ ⊑ [( Γ1, C1, V ψ ) PC ≍ ⊔ ( Γ2, C2, V ψ ) PC ]o . (vi) By Proposition 9(i) ( Γ1, C1, V ψ )o PC is the PCHS open set. Hence by proposition 9(ii) , [( Γ1, C1, V ψ )o PC ]o ≍ = ( Γ1, C1, V ψ )o PC . Remark 8. The equality of above proposition part (v) does not hold in general. See the next example. REFERENCES 3058 Example 7. Consider the PCHST space (UP , τPC , Vψ) in Example 2. Define (Γ4, C4, Vψ)PC and (Γ5, C5, Vψ)PC as the follow: (Γ4, C4, Vψ)PC= {< (α), {x1 (1, 0, 1) , x3 (1, 1, 1) , x4 (1, 1, 1)} >,< (β), {x2 (1, 1, 1) , x3 (1, 1, 1)} >} (Γ5, C5, Vψ)PC= {< (α), 1PC} >,< (β), {x1 (1, 1, 1) , x4 (1, 1, 1)} >}. Now, (Γ4, C4, Vψ) o PC ≍ = (Γ1, C1, Vψ)PC and (Γ5, C5, Vψ) o PC ≍ = (Γ2, C2, Vψ)PC and (Γ4, C4, Vψ) o PC ≍ ⊔(Γ5, C5, Vψ) o PC ≍ = (Ψ, C, Vψ)PC but [ (Γ4, C4, Vψ)PC ≍ ⊔ (Γ5, C5, Vψ)PC ]o ≍ ̸= (Ψ, C, Vψ)PC . Proposition 11. Let (UP , τPC , Vψ) be a PCHST space over UP and let ( Γ,C, V ψ ) PC be a PCHS set over UP . Then ( Γ,C, V ψ )o PC ≍ ⊑ ( Γ,C, V ψ ) PC ≍ ⊑ ( Γ,C, V ψ ) PC . Proof. Obvious. 6. Conclusion In this paper, we have introduced the concept of plithogenic crisp hypersoft sets and plithogenic crisp hypersoft topological spaces as an extension of the idea of hypersoft sets which are defined over an initial universal set with a fixed set of parameters. Some concepts such as plithogenic crisp hypersoft closure and plithogenic crisp hypersoft interior which are based on our definition were introduced. For future study, we can study plithogenic crisp hypersoft continuity and the most important fundamental topological properties such as plithogenic crisp hypersoft connectedness. 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