EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 4, 2024, 4147-4163 ISSN 1307-5543 – ejpam.com Published by New York Business Global Pythagorean Fuzzy Soft Somewhat Continuous Functions A. A. Azzam1,2,∗, M. Aldawood1, Radwan Abu-Gdairi3 1 Mathematics Department, Faculty of Science and Humanities, Prince Sattam Bin Abdulaziz University, Alkharj 11942, Saudi Arabia 2 Mathematics Department, Faculty of science, New Valley University, Elkharga 72511, Egypt 3 Mathematics Department, Faculty of Science, Zarqa University, Zarqa 13132, Jordan Abstract. In this work, we introduce the concept of Pythagorean fuzzy soft somewhat open sets utilizing the Pythagorean fuzzy soft interior operator, extending its application to Pythagorean fuzzy soft topological spaces. This study aims to enhance decision-making processes in future- assisted economies by addressing the limitations of existing fuzzy set theories. We investigate the distinctive properties of Pythagorean fuzzy soft somewhat open sets as a subclass of Pythagorean fuzzy soft somewhere dense sets. Additionally, we explore Pythagorean fuzzy soft somewhat meta- morphism’s within the context of Pythagorean fuzzy soft somewhat continuous functions, offering new insights into their topological invariant. Through detailed analysis and examples, we demon- strate the applicability of these concepts in various scientific and engineering problems. This work provides a comprehensive framework for understanding and utilizing Pythagorean fuzzy soft sets in complex decision-making scenarios. Finally, we compare various relationships across some generalizations of Pythagorean fuzzy soft continuous functions. 2020 Mathematics Subject Classifications: 54C08, 03E99, 54C10, 03E72 Key Words and Phrases: Fuzzy soft, Pythagorean fuzzy soft, somewhat open set, Pythagorean fuzzy soft somewhat open sets 1. Introduction Many researchers in the fields of economics, engineering, medicine, and other sci- ences face the daily challenge of lacking sufficient data to make decisions due to the emergence of new problems in our daily lives that did not previously exist and for which innovative and modern approaches are needed to find solutions. A topology is an im- portant branch of mathematics called rubber geometry that helps solve these problems. As a result, scientists are trying to expand the topological space in order to help with ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5430 Email addresses: aa.azzam@psau.edu.sa (A. A. Azzam), rgdairi@zu.edu.jo (R. Abu-Gdairi) https://www.ejpam.com 4147 Copyright: © 2024 The Author(s). (CC BY-NC 4.0) A. A. Azzam, M. Aldawood, R. Abu-Gdairi / Eur. J. Pure Appl. Math, 17 (4) (2024), 4147-4163 4148 everyday problems related to the environment, economy, health, and even human needs. To get beyond these obstacles, a number of theories have been put forward, similar to the 1999 introduction of the notion of soft sets (brevity Sss) by Molodtsov [23] and which has been used in a number of sectors. The character of parameter sets is central to the notion of Sss, it offers an extensive structure for modeling ambiguous data. In a short time, this essentially advances the topic of soft set (brevity Ss) theory. The theoretical basis of the Ss theory has been extensively examined by Maji et al. [22] and Azzam et al. [10–12]. In addition, Radwan and et al. [1] proposed soft ditopological spaces to achieve nearly soft β-open sets. To address this problem, Zadeh [32] developed the fuzzy set (brevity Fs) theory. Following Fs theory concept for various specific purposes, Higher order and nonclassical fuzzy sets (abbreviated Fss) have been presented. Atanassov [8] established the idea of the intuitionistic fuzzy set (abbreviated IFs). A growth of theory Fs that addresses both membership and non-membership values (abbreviated m-values and n-m-values consequently) [7, 17]. Yager invented the Pythagorean fuzzy set (abbre- viated PyFs) in two thousand thirteen, which is an additional extension of Fs and IFs [30]. Numerous applications in the scientific and social sciences have been made possible by this set theory. For instance, the work by Akram and et al. [2] introduced significant advancements in fuzzy subsets. Similarly, Azzam [9] explored its applications in social sciences, while Cuong and et al. [15] and Garg [16] provided critical insights into its mathematical underpinnings. Further contributions by Garg [18] and Yager [30] expanded its practical applications, and Zadeh [32] developed foundational theories that underpin the current study. Olgun et al. [25] suggested and studied Pythagorean fuzzy topological spaces (abbreviated PyFTSs) in 2019. Independent definitions of soft (generic) topology were provided in two thousand eleven by Cağman et al. [13] and Shabir and Naz [27]. Nazmul and Samanta [24] provided a definition of soft continuity (abbreviated SC) of functions in 2013. Next, a number of SC and soft openness generalizations functions that were documented in the literature. PyFTS was first described in [25] and Pythagorean fuzzy soft topological space (PyFSTS) [7, 26]. In recent years, the need for advanced fuzzy set theories has grown significantly due to the increasing complexity of problems in economics, engineering, and decision-making processes. The motivation behind this study is to address these challenges by developing a robust framework using Pythagorean fuzzy soft sets. Our main contribution lies in defining and exploring the properties of Pythagorean fuzzy soft somewhat open sets and their applications in topological spaces. This approach provides a more nuanced understanding of fuzzy environments, enabling more precise modeling of uncertainty and imprecision in real-world scenarios. Following this quick introduction, we will review some preliminary principles in Part 2. Part 3 then introduces the idea that Pythagorean fuzzy soft somewhat open (brevity PyFSsw-open) sets and looks at how it relates to a few soft open set assumptions. The objectives of Part 4 is to examine PyFSsw-C functions, which are stronger than PyFSsw-dense C, but weaker than soft semicontinuous. We wrap up and offer some suggestions for next works in Part 5. A. A. Azzam, M. Aldawood, R. Abu-Gdairi / Eur. J. Pure Appl. Math, 17 (4) (2024), 4147-4163 4149 2. Preliminaries Various fundamental ideas and symbols will be used in the sequel are contained in this part. We will henceforth refer to an original universe X, a collection of parameters η, an exponential set of X (℘(X)), soft topology ST , a soft topological space STS, picture fuzzy set PFs, positive membership function pmf , negative membership function nmf , and continuous C. Definition 1. [32] A membership function ξD(x) that assigns a real number in the range [0, 1] to each point in X characterizes a fuzzy set D in X. The ”grade of m” of x in D is indicated by the value of ξD(x) at x. Definition 2. [29] Let X represent the universe, then the set D = {(x, ξD(x), ψD(x)) : x ∈ X} is referred to as IFs of X, ξD : X → [0, 1] and ψD : X → [0, 1] are referred to as x’s pmf in X, and within X, x has a nmf effectively under the circumstances 0 ≤ ξD(x) + ψD(x) ≤ 1, ∀x ∈ X. Definition 3. [15] Assume X is the universe setting, then the set D = {(x, ξD(x), υD(x), ψD(x)) : x ∈ X} is referred to as PFs of X, ξD : X → [0, 1], ψD : X → [0, 1] and ψD : Ω → [0, 1] the degrees of positive, neutral, and negative m of x in X, as well as their respective conditions 0 ≤ ξD(x) + υD(x) +ψD(x)) ≤ 1, ∀x ∈ X, are designated accordingly. Definition 4. [29] Let X represent the cosmos, then the set D = {(x, ξ(x), ψ(x)) : x ∈ X} is named PyFs of X, ξ : X → [0, 1] and ψ : X → [0, 1] are referred to the degree of pmf of x in X and nmf degree of x in X effectively under the circumstances 0 ≤ ξ2 + ψ2 ≤ 1, ∀x ∈ X. Definition 5. [29] Let D1 = {(x, ξD1(x), ψD1(x)) : x ∈ X} and D2 = {(x, ξD2(x), ψD2(x)) : x ∈ X} are two FyFs on X, then i) D1 ⊓D2 = {(x, ξD1(x) ∧ ξD2(x), ξD1(x) ∨ ξD2(x) : x ∈ X}, ii) D1 ⊔D2 = {(x, ξD1(x) ∨ ξD2(x), ξD1(x) ∧ ξD2(x) : x ∈ X}, iii) D1 ⊑ D2 if and only if ξD1(x) ≤ ξD2(x), ψD1(x) ≥ ψD2(x) : x ∈ X. Definition 6. [25] PyFTS is the PyF family τ that subsets of a non-empty set X if i) 0X , and 1X belong to τ , ii) We have x1 ⊓ x2 belong to τ for any pair x1, x2 ∈ τ , iii) We have ⊔ixi belong to τ for any xi ∈ τ . Definition 7. [23] When ξ : η → ℘(X) is a (crisp) map, then a Ss over X is a pair (ξ, η) = {(a, η(a)) : a ∈ η}. Instead of writing the soft set (ξ, η), we write ξη. Ssη(X) or for all Sss on X, the class is represented by just Ss(X). When A ⊑ η, then SsA(X) will serve as its symbol. Definition 8. [6] The term for a Ss ξη on X is: (1) a soft element if ξ(a) = {x} for every a ∈ η, for x ∈ X, {x}η is used to represent A. A. Azzam, M. Aldawood, R. Abu-Gdairi / Eur. J. Pure Appl. Math, 17 (4) (2024), 4147-4163 4150 it(maybe soon x). (2) a soft point if for every a ̸= á, ξ(a) = {x} and ξ(á) = ϕ for every a ∈ η and x ∈ X. It’s indicated by pxa. If x ∈ ξ(a), then the expression pxa ∈ ξ(a). Definition 9. [5] A soft set Xη − ξη (or simply ξcη is the complement of ξη, ξ c : η → ℘(X) is defined as ξc(a) = X − ξ(a) for every a ∈ η. Definition 10. [23] The term for a soft subset ξη over X is null for any a ∈ η if ξ(a) = ϕ, and absolute if ξ(a) = X. Both empty and absolute SSs are denoted by ϕη and Xη, respectively. It is evident that ϕcη = Xη and Xc η = ϕη. Definition 11. [22] Assume C,D ⊑ η. If C ⊑ D and ξ(a) ⊑ G(a) for each a ∈ C, then GC is a soft subset of HD (written as GC ⊑ HD). If GC ⊑ HD and HD ⊑ GC , we refer to GC soft equates to HD. Maji et al. [22] defined the soft union and soft overlap of two Sss with respect to arbitrary subsets of η. However, as noted by Ali et al. [5], these definitions are imprecise and ambiguous. Consequently, we adhere to the definitions provided by Ali et al. [5] and M. Terepeta [28]. Definition 12. [27] A subfamily τ of Ssη is said to be a ST on X if (i) Xη and ϕη elements in τ , (ii) τ owns the finite intersection of sets from τ , and (iii) τ owns any union of sets from τ . We refer to (X, τ, η) as a STS on X. τ ’s elements are known as soft open sets, while their complements are known as soft closed sets. Definition 13. [27] Suppose Zη is a non-null soft subset of (X, τ, η). In that case, (Z, τZ , η) represents a soft subspace of (X, τ, η), A soft relative topology on Z is denoted by τZ = {Gη ⊓ Zη : Gη ∈ τ}. Definition 14. [27] ξη is a soft subset of (X, τ, η). Denoted by intξη, the largest soft open set contained in ξη is the soft interior of ξη. The soft closure of ξη is clξη, which is the smallest soft closed set containing ξη. Definition 15. The terms ”soft dense,” ”soft co-dense,” ”soft semiopen [14],” ”soft β-C [31],” ”soft somewhat open [4],” and ”soft somewhere dense [3],” if ”cl(Gη) = Xη,” ”int(Gη) = ϕη,” ”Gη ⊑ cl(int(Gη)),” ”Gη ⊑ cl(int(clGη)), ” ”int(Gη) ̸= ϕη,” ”int(cl(Gη)) ̸= ϕη,” respectively ”referring to the different states of a soft subset GE of (X, τ, η). (We compel ϕη to be soft somewhere dense in order to improve the connectivity between these soft sets). A. A. Azzam, M. Aldawood, R. Abu-Gdairi / Eur. J. Pure Appl. Math, 17 (4) (2024), 4147-4163 4151 Definition 16. Suppose (X, τ, η) and (Z, ρ, ή) be STS. A soft function ξ : (X, τ, η) → (Z, ρ, ή) is called i) SC [24] (resp., soft semi-C [20], soft SD-C [4], soft β-C [31]) if every soft open subset of (Z, ρ, ή) has a soft open as its inverse image (resp., soft semiopen, soft somewhere dense, β-open) subset of (X, τ, η). ii) soft open [23] (resp., soft semiopen [20], soft SD-open [4], soft β-open [31]) if the image of each soft open subset of (X, τ, η) is a soft open (resp., soft semiopen, soft somewhere dense, β-open) subset of (Z, ρ, ή). iii) If it is one to one soft open and SC from (X, τ, η) onto (Z, ρ, ή), then it is a soft homeomorphism [24]. The reader is referred to [19] for a definition of soft functions spanning collections of all Sss. From here on, we refer to ”soft function” when we use the term ”function.” Definition 17. [29] The PyFSS may be expressed as a collection of ordered pairs (ξ̃, η̃) = {(a, {(x, ξξ̃(a)(x), ψξ̃(a)(x)) : a ∈ η̃}} because it is not a set but rather a specified unit of certain components of the set PyF (X̃), where ξξ̃(a)(x) and ψξ̃(a)(x) are the pmfs and nmfs, successively. If x ∈ X̃, 0 ≤ ξ2 ξ̃(a) (x) + ψ2 ξ̃(a) (x) ≤ 1. We introduced the idea of PyFSTS and looked into its properties in more detail. Let PyF (X̃, η̃) and X̃ represent, respectively, the family of PyFSs on X̃ and the origin of the universal set. Definition 18. [7] A void PyFSSs (or 0̃) is defined as a PyFSSs(ξ̃, η̃) over X̃ if and only if ∀a ∈ η̃, (ξ̃, η̃)(a) = (0̃, 1̃), where 0̃, 1̃ are the pmf and the value of the nmfs, the null and absolute, respectively PyFSs Pythagorean over X̃. Definition 19. [7] An absolute PyFSSs, or(1̃), is a PyFSSs(ξ̃, η̃) over X̃ if and only if ∀a ∈ η̃, (ξ̃, η̃)(a) = (0̃, 1̃), where 0̃, 1̃ are the pmf and the value of the nmfs, the null and absolute, respectively, of the absolute and null function. Definition 20. [7] Let Ω̃ ⊑ PyF (X̃, η̃), at hence, Ω̃ is claimed to be a PyFSTS if i) Ω̃ includes 0̃ and 1̃ as members, ii) Any two PyFSS that intersect in Ω̃ are related to Ω̃, ii) Any number of PyFSS in Ω̃ that is united belongs to Ω̃, It is argued that the triple (X̃, Ω̃, η̃) is a PyFSTS over X̃. ∗. All Ω̃ members are considered to be Ω̃-open PyFSS. ∗∗. A Ω̃-closed PyFSS is considered to be the complement of a Ω̃-open. A. A. Azzam, M. Aldawood, R. Abu-Gdairi / Eur. J. Pure Appl. Math, 17 (4) (2024), 4147-4163 4152 3. Pythagorean fuzzy soft somewhat open sets We create key properties and introduce the concept of PyFSsw-open sets in this section. We provide examples to show the relationships between PyFS semiopen and PyFS somewhere dense sets, as well as various generalizations of PyFSsw-open sets. Definition 21. A subset Gη̃ of a PyFSTS (X̃, Ω̃, η̃) is claimed to be PyFSsw-open if int(Gη̃) ̸= ϕη̃ or Gη̃ is null. PyFSsw-closed is the complement of PyFSsw-open set. That is, a set ξη̃ is PyFSsw-closed if cl(ξη̃) ̸= Hη̃ or ξη̃ = X̃η̃. Remark 1. Let (X̃, Ω̃, η̃) be a PyFSTS. i) If and only if there is a PyFS-open set Uη̃ that ϕη̃ ̸= Uη̃ ⊑ Gη̃, The non-null set Gη̃ over X̃ is PyFSsw-open. ii) If ξη̃ is a PyFSsw-closed set that Hη̃ ⊑ ξη̃ ̸= X̃η̃, then a valid set Hη̃ over X̃ is PyFSsw-closed. Proposition 1. i) Each superset of a PyFSsw-open set is PyFSsw-open. ii) Each subset of a PyFSsw-closed set is PyFSsw-closed. Proof. Obvious. Proposition 2. A non-null PyFSs is PyFSsw-open if and only if it is a PyFS neigh- borhood of a PyFS point. Proof. Let Gη̃ be a PyFSsw-open set that isn’t null. Next, there exists a PyFS open set Uη̃, where ϕη̃ ̸= Uη̃ ⊑ Gη̃. As a result, Gη̃ is every soft point in Uη̃’s soft neighborhood. Let Gη̃, on the other hand, be the PyFS neighborhood of a PyF soft point pxa. After that, Uη̃ is PyF softly opened so that pxa ∈ Uη̃ ⊑ Gη̃. As a result, we get intGη̃ ̸= ϕη̃, as needed. Proposition 3. A union of PyFSsw-open sets is PyFSsw-open. Proof. Suppose that {Gβ η̃ : β ∈ Λ} is the collection of PyFSsw-open subsets of a PyFSTS (X̃, Ω̃, η̃). At hence, int(∪β∈ΛG β η̃ ) ⊒ ∪β∈Λint(G β η̃ ) ̸= ϕη̃. Thus ∪β∈ΛG β η̃ is PyFSsw-open. Corollary 1. The intersection of PyFSsw-closed sets is PyFSsw-closed. As demonstrated by the example that follows, the intersection of two PyFSsw-open sets need not be PyFSsw-open. Example 1. Let η̃ = {a1, a2, a3, a4} be the parameters or characteristics set and As the reference set, let X̃ = {x1, x2, x3} represent the applicants who have been recommended for promotion, in which a1 denotes intelligence, a2 experience, a3 attitude, and a4 competence. Let D1 = {a1, a2} ⊑ η̃, D2 = {a2} ⊑ η̃. Next, two PyFSsw(ξ̃1, D1) and (ξ̃2, D2) are examined. These are represented as follows: A. A. Azzam, M. Aldawood, R. Abu-Gdairi / Eur. J. Pure Appl. Math, 17 (4) (2024), 4147-4163 4153 (ξ̃1, D1) = {(a1, ξ̃1(a1)), (a2, ξ̃1(a2))}, and (ξ̃2, D2) = {(a2, ξ̃2(a2))}, where ξ̃1(a1)) = {x1 = (0.5, 0.6), x2 = (0.4, 0.7), x3 = (0.1, 0.7)}, ξ̃1(a2)) = {x1 = (0.3, 0.2), x2 = (0.6, 0.5), x3 = (0.2, 0.7)}, ξ̃2(a2)) = {x1 = (0.8, 0.4), x2 = (0.8, 0.3), x3 = (0.4, 0.3)} Ω̃1 = {1̃, 0̃, (ξ̃1, D1)} and Ω̃2 = {1̃, 0̃, (ξ̃1, D1), (ξ̃2, D2)}} are two PyFSTSs and Ω̃ = {1̃, 0̃, (ξ̃1, D1), (ξ̃2, D2)} is a PyFST over X̃, (ξ̃1, D1) ⊓ (ξ̃2, D2) ̸= ϕη̃ but int((ξ̃1, D1) ⊓ (̃ξ̃2, D2)) = ϕη̃. There are several examples when the intersection of a PyFSsw-open set with another PyFS open, PyFS closed, or PyFS dense set is not a PyFSsw-open set. The following result shows when the intersection of PyFSsw-open and PyFS open sets is a PyFSsw-open set. Definition 22. A PyFSTS (X̃, Ω̃, η̃) is named i) PyFS separable if it has a countable PyFS dense subset. ii) PyFS hyperconnected if any pair of non-null PyFS open subsets intersect. Proposition 4. In a PyFS hyperconnected space (X̃, Ω̃, η̃), a PyFSsw-open set is the intersection of two PyFSsw-open sets. Proof. The evidence is easy to understand if one of the two PyFSsw-open sets is null. Assume that there are two PyFSsw-open sets, Gη̃ and Hη̃. Next, int(Gη̃) = Uη̃ ̸= ϕη̃ and int(Hη̃) = Vη̃ ̸= ϕη̃ are obtained. Now, int(Gη̃ ⊓ Hη̃) = int(Gη̃) ⊓ int(Hη̃) = Uη̃ ⊓ Vη̃. Then, Uη̃ ⊓Vη̃ ̸= ϕη̃ since (X̃, Ω̃, η̃) is a PyFS hyperconnected. Hence, int(Gη̃ ⊓Hη̃) ̸= ϕη̃, and we achieve the intended outcome. Corollary 2. In a PyFS hyperconnected space (X̃, Ω̃, η̃), the intersection of PyFSsw- open and PyFS open sets is a PyFSsw-open. Corollary 3. A PyFS topology is formed by the family of PyFSsw-open subsets of a PyFS hyperconnected space (X̃, Ω̃, η̃). Lemma 1. Suppose Gη̃ and Hη̃ are subsets of (X̃, Ω̃, η̃). If Gη̃ is sw-open and Hη̃ is a PyFS dense over X̃, then Gη̃ ⊓Hη̃ is PyFSsw-open over X̃. Proof. Since intH(Gη̃ ⊓ Hη̃) = intH(Gη̃) ⊓ Hη̃ ⊒ int(Gη̃) ⊓ Hη̃ ̸= ϕη̃, hence Gη̃ ⊓ Hη̃ is PyFSsw-open over X̃. Lemma 2. Assume that Gη̃ ⊑ Yη̃ and that (Ỹ , Ω̃Y , η̃) is a PyFS open subspace of (X̃, Ω̃, η̃). If and only if Gη̃ is PyFSsw-open over X̃, then it is also PyFSsw-open over Ỹ . A. A. Azzam, M. Aldawood, R. Abu-Gdairi / Eur. J. Pure Appl. Math, 17 (4) (2024), 4147-4163 4154 Proof. Let’s say that Gη̃ is PyFSsw-open over Ỹ . It is possible to have a PyFS open set Uη̃ over Ỹ such that ϕη̃ ̸= Uη̃ ⊑ Gη̃. Because Yη̃ is PyFS open over X̃, Uη̃ is also PyFS open over X̃. As a result, Gη̃ is PyFSsw-open over X̃. In contrast, let’s say that Gη̃ is PyFSsw-open over X̃. This is equivalent to intX̃(Gη̃) ̸= ϕη̃. According to Theorem 2 in [19], and Remark 3.2, intX̃(Gη̃) ⊑ intỸ (Gη̃) , hence Gη̃ is PyFSsw-open over Ỹ . If Yη̃ is PyFS dense in X̃, as the following example demonstrates, then the previous result is not valid. Example 2. Suppose X̃ = {x1, x2, x3, x4}, η = {a1, a2}, and Ω̃ = {0̃, Fη̃, Gη̃, Hη̃, 1̃}, where Fη̃ = {(a1, {x2, x4}), (a2, {x1, x2})} Gη̃ = {(a1, X̃), (a2, {x3, x4})} Hη̃ = {(a1, {x2, x4}), (a2, ϕη̃)} Let Ỹ = {x2, x3} at hence, Ω̃Y = {0̃, Iη̃, Jη̃,Kη̃, 1̃}, where Iη̃ = {(a1, {x2}), (a2, {x2})} Jη̃ = {(a1, Ỹ ), (a2, {x3})} Kη̃ = {(a1, {x2}), (a2, ϕη̃)} Ỹη̃ = {(a1, {x2, x4}), (a2, {x2, x4})}. Over the PyFS dense set Ỹ , the set Iη̃ is PyFSsw-open, but not over X̃. Lemma 3. Suppose Gη̃ that a subset of (X̃, Ω̃, η̃). Hence, Gη̃ is PyFS semiopen if and only if cl(Gη̃) = cl(int(Gη̃)). Proof. Suppose Gη̃ is PyFS semiopen, that Gη̃ ⊑ cl(int(Gη̃)), and then cl(Gη̃) ⊑ cl(int(Gη̃)). For the opposite side of inclusion, there is always int(Gη̃) ⊑ Gη̃. So, cl(Gη̃) = cl(int(Gη̃)). In contrast, let’s say that cl(Gη̃) = cl(int(Gη̃)), but Gη̃ ⊑ cl(Gη̃) always, at hence Gη̃ ⊑ cl(int(Gη̃)). So, Gη̃ is PyFS semiopen. Lemma 4. Consider Gη̃ as a non-null subset of (X̃, Ω̃, η̃). Hence, Gη̃ is PyFS semiopen if int(Gη̃) ̸= ϕη̃. Proof. Suppose otherwise that, if Gη̃ is a non-null soft semiopen set with int(Gη̃) = ϕη̃, then Gη̃ = ϕη̃ is implied by Lemma 3.14 since cl(Gη̃) = ϕη̃. Inconsistency. Remark 2. Since int(Gη̃) = int(cl(Gη̃)) for each PyFS Gη̃ in a PyFSTs (X̃, Ω̃, η̃), so each PyFSsw-open set is PyFS somewhere dense. The following figure depicts the relation between different extensions of PyFS open sets. As demonstrated below, none of these implications can, in general, be replaced by equivalency. A. A. Azzam, M. Aldawood, R. Abu-Gdairi / Eur. J. Pure Appl. Math, 17 (4) (2024), 4147-4163 4155 Figure 1: The relationships between some generalizations of PyFS open sets. Example 3. Think about PyFST over X̃ that Example 3.7. The PyFSs over X̃ is not PyFSsw-open, meaning it is not PyFS semiopen, but is PyFSβ-open, meaning it is PyFS somewhere dense. However, it is evident that the set {(a1, ξ̃1(a1)), (a2, ξ̃1(a2))} is not PyFS semiopen, but rather PyFSsw-open. Lemma 5. Suppose Gη̃ that a non-null subset of (X̃, Ω̃, η̃). Then cl(Gη̃)⊓Hη̃ ⊑ cl(Gη̃⊓Hη̃) for all PyFS open set Hη̃ on X̃. Lemma 6. Assume that Gη̃, Hη̃ is a subset of (X̃, Ω̃, η̃). Gη̃ ⊓Hη̃ is PyFS semiopen over X̃ if Gη̃ is PyFS open and Hη̃ is PyFS semiopen. Proof. Suppose that Gη̃ is PyFS open and Hη̃ is PyFS semiopen. Then there’s a PyFS open set. Uη̃ over X̃ with Uη̃ ⊑ Hη̃ ⊑ cl(Uη̃). Now Uη̃⊓Gη̃ ⊑ Hη̃⊓Gη̃ ⊑ cl(Uη̃)⊓Gη̃. By Lemma 3.18, Uη̃ ⊓Gη̃ ⊑ Hη̃ ⊓Gη̃ ⊑ cl(Uη̃ ⊓Gη̃) and since Uη̃ ⊓Gη̃ is PyFS open, then Hη̃ ⊓Gη̃ is PyFS semiopen over X̃. Lemma 7. Assume that Gη̃, Hη̃ is a subset of (X̃, Ω̃, η̃). Gη̃ ⊓Hη̃ is PyFS semiopen over Gη̃ if Gη̃ is PyFS open and Hη̃ is PyFS semiopen. Proof. Utilizing the same procedures as in the lemma proof above, apply the assertion that cl(Uη̃) ⊓Gη̃ = clGη̃(Uη̃). Lemma 8. A subset Gη̃ of (X̃, Ω̃, η̃) is PyFS semiopen if and only if Gη̃⊓Uη̃ is PyFSsw open for each PyFS open set Uη̃ over X̃. A. A. Azzam, M. Aldawood, R. Abu-Gdairi / Eur. J. Pure Appl. Math, 17 (4) (2024), 4147-4163 4156 Proof. The first part follows because each PyFS semiopen set is PyFSsw open and because the intersection of a PyFS semiopen set with a PyFS open set is semiopen according to Lemma 3.19. On the other hand, suppose that pxa ∈ Gη̃ and that for any PyFS open set Uη̃ over X̃, Gη̃ ⊓ Uη̃ is PyFSsw open. That is int(Gη̃ ⊓ Uη̃) ̸= ϕη̃. But ϕη̃ ̸= int(Gη̃ ⊓ Uη̃) = int(Gη̃) ⊓ int(Uη̃) = int(Gη̃) ⊓ Uη̃, that is p x a ∈ cl(int(Gη̃)) and then Gη̃ ⊑ cl(int(Gη̃)). This demonstrates Gη̃’s PyFS semiopenness. Lemma 9. Consider Fη̃. is a subset of (X̃, Ω̃, η̃). If Fη̃ is PyFS semiclosed and PyFS somewhere dense, it is PyFSsw open. Proof. It may be inferred directly from Lemma 3.15 that Fη̃ is semiclosed if and only if int(cl(Fη̃))=int(Fη̃). 4. PyFSsw-continuous functions This part focuses on outlining the ideas behind PyFSsw C functions, also known as PyFSsw C, and providing several characterizations of them. Furthermore, we demon- strate its connections to various forms of PyFS continuity. In conclusion, we obtain certain findings about hyperconnected and PyFS separable spaces. Definition 23. Consider (X̃, Ω̃1, η̃1) and (Ỹ , Ω̃2, η̃2) are a PyFSTSs. If every PyFS open set over Ỹ has an inverse image that is also PyFSsw open over X̃, then the function f : (X̃, Ω̃1, η̃1) → (Ỹ , Ω̃2, η̃2) is considered PyFSsw-C. Remark 3. A function f : (X̃, Ω̃1, η̃1) → (Ỹ , Ω̃2, η̃2) is PyFSsw-C if each pxa ∈ X̃ and each PyFS open set Vη̃2 over Ỹ ⊒ f(pxa), there is a PyFSsw open set Uη̃ on X̃ ⊒ pxa that f(Uη̃) ⊑ Vη̃2. Based on Figure 1, we deduce that The ramifications shown in the above graphic are all irreversible. Example 4. Let X̃ = {x1, x2, x3}, η = {a1, a2}, and Ω̃ = {0̃, Fη̃, Gη̃, 1̃}, where Fη̃ = {(a1, {x2}), (a2, {x2})} Gη̃ = {(a1, {x1, x3}), (a2, {x1, x3})} and Ω̃1 = {0̃, Hη̃, 1̃} where Hη̃ = {(a1, X̃}), (a2, {x1, x2})}. Let f : (X̃, Ω̃1, η̃) → (X̃, Ω̃2, η̃) be the PyFS identity function. At hence, f is PyFSsw-C but not PyFSsw-semicontinuous. Example 5. Let X̃ = ℜ be the set of real numbers and η = {a} be a collection of parameters. Let Ω̃ be the PyFST on ℜ generated by {(a, ξ(a)) : (x1, x2) ∈ ℜ;x1 < x2}. Define a PyFS function f : (X̃, Ω̃, η̃) → (X̃, Ω̃, η̃) by A. A. Azzam, M. Aldawood, R. Abu-Gdairi / Eur. J. Pure Appl. Math, 17 (4) (2024), 4147-4163 4157 Figure 2: The relationships between some generalizations of PyFS continuous. f(x) =  x if x /∈ {0̃, 1̃}η̃, 0 if x = 1, 1 if x = 0. Given that every PyFS basic open set has an inverse image that also contains another PyFS basic open, one can simply demonstrate that f is PyFSsw-C (and hence, PyFS SD-C), since its PyFS interior cannot be null. However, f is not PyFSβ-C. Let Gη̃ = {(a, (−ε, ε))} be the PyFS open set, with ε < 1. Therefore f−1(Gη̃) = {(a, (−ε, 0))} ⊔ {(a, (0, ε))} ⊔ {(a, {1})}. But cl(int((cl(f−1(Gη̃))) = {(a, [−ε, ε])} and so f−1(Gη̃) ⊈ cl(int((cl(f−1(Gη̃))). As a result, f is not PyFS semicontinuous and cannot be PyFSβ-C. Example 6. Consider the PyFSTS (X̃, Ω̃, η̃) as described in Example 4.4. Define f : (X̃, Ω̃, η̃) → (X̃, Ω̃, η̃) as follows: f(x) = { 0 if x /∈ Qη̃, 1 if x ∈ Qη̃. In such case, f is not PyFSsw-continuous but soft SD-continuous. Any PyFS open set with only one element is its inverse image, and Qη̃ is not a PyFSsw-open set over X̃. A. A. Azzam, M. Aldawood, R. Abu-Gdairi / Eur. J. Pure Appl. Math, 17 (4) (2024), 4147-4163 4158 Definition 24. We introduce the following for a subset Gη̃ of a PyFSTS (X̃, Ω̃, η̃): 1-clsw(Gη̃) = ⊓{Fη̃ : Fη̃ is PyFSsw-closed over X̃ and Gη̃ ⊑ Fη̃}. 2-intsw(Gη̃) = ⊔{Oη̃ : Oη̃ is PyFSsw-open over X̃ and Oη̃ ⊑ Gη̃}. Proposition 5. Consider (X̃, Ω̃1, η̃1) and (Ỹ , Ω̃2, η̃2) that a PyFSTSs. The function f : (X̃, Ω̃1, η̃1) → (Ỹ , Ω̃2, η̃2) can be represented by the following functions: 1- f is PyFSsw-C, 2- f−1(Fη̃2) is PyFSsw-closed set over X̃, for every PyFS closed set Fη̃2 over Ỹ , 3- f(clsw(Gη̃)) ⊑ cl(f(Gη̃)) for every set Gη̃ on X̃, 4- clsw(f −1(Hη̃2)) ⊑ f−1(cl(Hη̃2)), for every set Hη̃2 on Ỹ , 5- f−1(int(Hη̃2)) ⊑ intsw(f −1(Hη̃2)), for every set Hη̃2 on Ỹ . Proof. Straightforward. Definition 25. Assume that PyF (X̃, η̃1) and PyF (Ỹ , η̃2) be PyFSSs and let Dη̃1 ∈ (X̃, η̃1). The restriction of f : PyF (X̃, η̃1) → PyF (Ỹ , η̃2) is the FyFS function fDη̃1 : PyF (X̃, η̃1) → PyF (Ỹ , η̃2) defined by fDη̃1 (P x a ) = f(P x a ) for all P x a ∈ Dη̃1. a PyFS function’s expansion f is a PyFS function of g, meaning that f restricts g. Theorem 1. Consider (X̃, Ω̃1, η̃1) and (Ỹ , Ω̃2, η̃2) that a PyFSTSs, and let dη̃1 be a PyFS dense subspace over X̃. If f : (X̃, Ω̃1, η̃1) → (Ỹ , Ω̃2, η̃2) is PyFSsw-C over X̃, then f | dη̃1 is PyFSsw-C over d. Proof. Straightforward (with the aid of Lemma 3.12). Theorem 2. Let (X̃, Ω̃1, η̃1) and (Ỹ , Ω̃2, η̃2) be a PyFSTSs, and let f : (X̃, Ω̃1, η̃1) → (Ỹ , Ω̃2, η̃2) be a function and {Gβ η̃1 : β ∈ Λ} be a PyFS open cover of X̃. At hence, f is PyFSsw-C, if f | Gβ η̃1 is PyFSsw-C for each β ∈ Λ. Proof. Suppose Vη̃2 is a PyFS open set across Ỹ . By presumption, (f | Gβ η̃1 )−1(Vη̃2) is PyFSsw open over Gβ η̃1 . By Lemma 3.13, (f | Gβ η̃1 )−1(Vη̃2) is PyFSsw open over X̃ foe all β ∈ Λ. But f−1(Vη̃2) = ⊔β∈Λ[(f | Gβ η̃1 )−1(Vη̃2)], this is the union of PyFSsw open sets, and f−1(Vη̃2) is PyFSsw open over X̃. f is hence PyFSsw-C. Theorem 3. Let (X̃, Ω̃1, η̃1) and (Ỹ , Ω̃2, η̃2) be a PyFSTSs, and let Uη̃1 be a PyFS open set over X̃. If f : (Ũ , Ω̃1, η̃1) → (Ỹ , Ω̃2, η̃2) is a PyFSsw-C function that f(Uη̃1) is FyS dense over Ỹ , then PyFSsw-C is the extension function of each f over X̃. Proof. Let Vη̃2 be a (non-null) PyFS open set on Ỹ and let g be an extension of f . If g−1(Vη̃2) = ϕη̃1 , then g is simply PyFSsw-C. Let g−1(Vη̃2) ̸= ϕη̃1 . By density of f(Uη̃1), f(Uη̃1) ⊓ Vη̃2 ̸= ϕη̃2 it suggests that Uη̃1 ⊓ f−1(Vη̃2) ̸= ϕη̃1 . Therefore f−1(Vη̃2) ̸= ϕη̃1 . Presumably, a non-null PyFS open set Wη̃1 on U exists such that Wη̃1 = Wη̃1 ⊓ Uη̃1 ⊑ f−1(Vη̃2) ⊓ Uη̃1 = g−1(Vη̃2) ⊓ Uη̃1 ⊑ g−1(Vη̃2). Since Wη̃1 is a PyFS open set over X̃ according to Lemma 3.13, ϕη̃1 ̸= Wη̃1 ⊑ g−1(Vη̃2). Consequently, across X̃, g is PyFSsw-C. A. A. Azzam, M. Aldawood, R. Abu-Gdairi / Eur. J. Pure Appl. Math, 17 (4) (2024), 4147-4163 4159 Theorem 4. Consider (X̃, Ω̃1, η̃1) and (Ỹ , Ω̃2, η̃2) be a PyFSTSs. A function f : (Ũ , Ω̃1, η̃1) → (Ỹ , Ω̃2, η̃2) is a PyFS-semicontinuous if and only if f | Wη̃1 is sw-C for all PyFS open set Wη̃1 over X̃. Proof. Let f be a PyFS-semicontinuous, Wη̃1 is any PyFS open set on X̃. Let Gη̃2 be a PyFS open set on Ỹ . then f−1(Gη̃2) is PyFS semiopen and from Lemma 3.19, (f | Wη̃1) −1(Gη̃2) = f−1(Gη̃2) ⊓Wη̃1 is PyFS semiopen over W . Then f | Wη̃1 is PyFS- semicontinuous and hence PyFSsw-C. Conversely, Let f | Wη̃1 is sw-C for all PyFS open set Wη̃1 over X̃, and Hη̃2 be PyFS open set over Ỹ . Then (f | Wη̃1) −1(Hη̃2) = f−1(Hη̃2) ⊓Wη̃1 is PyFSsw-open over W . Since Wη̃1 is a PyFSsw-open over X̃ by Lemma 3.12, f−1(Hη̃2) ⊓ Wη̃1 is a PyFSsw- open over X̃ and so, by Lemma 3.22, f−1(Hη̃2) is PyFS semiopen over X̃. Thus f is PyFS-semicontinuous. Theorem 5. Consider (X̃, Ω̃1, η̃1) and (Ỹ , Ω̃2, η̃2) be a PyFSTS. The function f : (Ũ , Ω̃1, η̃1) → (Ỹ , Ω̃2, η̃2) can be represented by the following function: 1- f is PyFSsw-continuous, 2-There is a non-null PyFS open set Wη̃1 on X̃ that Wη̃1 ⊑ f−1(Vη̃2), for any PyFS open set f−1(Vη̃2) on Y with f−1(Vη̃2) ̸= ϕη̃1, 3-There is a proper PyFS closed Kη̃1 on X̃ that f−1(Fη̃2) ⊑ Kη̃1, for any PyFS closed set Fη̃2 on Y with f−1(Fη̃2) ̸= X̃η̃1, 4- f(dη̃1) is PyFS dense over f(X̃) for any PyFS dense set dη̃1 over X̃. Proof. 1 ⇒ 2 The definition of sw-continuity and Remark 3.2. 2 ⇒ 3 Given a PyFS closed set Fη̃2 over Ỹ , f−1(Fη̃2) ̸= X̃η̃1 . f −1(Ỹη̃2 \ Fη̃2) ̸= ϕη̃1 indi- cates that Ỹη̃2 \Fη̃2 is PyFS open over Ỹ . A PyFS open set Wη̃1 over X̃ exists according to (2) in such a way that ϕη̃1 ̸=Wη̃1 ⊑ f−1(Ỹη̃2 \Fη̃2) = X̃η̃1 \f−1(Fη̃2). This suggests that f−1(Fη̃2) ⊑ X̃η̃1 \Wη̃1 ̸= X̃η̃1 . Kη̃1 is a proper PyFS closed set that meets the necessary property if Kη̃1 = X̃η̃1 |Wη̃1 . 3 ⇒ 4 Over X̃, let dη̃1 be PyFS dense. The claim that f(dη̃1) is PyFS dense over f(X̃) must be proven. Assume that over f(X̃), c is not PyFS dense. A proper PyFS closed set Fη̃2 , exists such that f(dη̃1) ⊑ Fη̃2 ⊏ f(X̃η̃1). So, dη̃1 ⊑ f(Fη̃2). According to (3), there is a PyFS closed set Kη̃1 over X̃ such that dη̃1 ⊑ f−1(Fη̃2) ⊑ Kη̃1 ̸= X̃η̃1 . That dη̃1 is PyFS dense over X̃ is contradicted by this. Therefore, (4) is true. 4 ⇒ 1 Let Hη̃2 be a PyFS open set over Ỹ with f−1(Hη̃2) ̸= ϕη̃1 without losing gen- erality, since it is trivially PyFSsw-open if f−1(Hη̃2) = ϕη̃1 . Assume that f−1(Hη̃2) is not PyFSsw-open, i.e. int(f−1(Hη̃2)) = ϕη̃1 . At hence, cl(X̃η̃1 \ f−1(Hη̃2) = X̃η̃1 . This suggests that on X̃, X̃η̃1 \ f−1(Hη̃2) is PyFS dense. From 4, f(X̃η̃1 \ f−1(Hη̃2)) is PyFS dense over f(X̃), this means that cl(f(X̃η̃1) \ f−1(Hη̃2)) = f(X̃η̃1). This results in cl(f(X̃η̃1) \ f−1(Hη̃2) = f(X̃η̃1) \Hη̃2 = f(X̃η̃1) and so Hη̃2 = ϕη̃2 . In contrast to the selec- tion of Hη̃2 . As a result, int(f−1(H)) cannot be null. As a result, f−1(Hη̃2) is PyFSsw on X̃. Corollary 4. Consider (X̃, Ω̃1, η̃1) and (Ỹ , Ω̃2, η̃2) be a PyFSTS. The corresponding values for a one-to-one function are as follows: f : (Ũ , Ω̃1, η̃1) → (Ỹ , Ω̃2, η̃2): A. A. Azzam, M. Aldawood, R. Abu-Gdairi / Eur. J. Pure Appl. Math, 17 (4) (2024), 4147-4163 4160 a-f is PyFSsw-C, b-f(Mη̃1) is PyFS co-dense over Ỹ for any soft co-dense set Mη̃1 over X̃. This section concludes with two results about soft separable and hyperconnected space. Theorem 6. Consider (X̃, Ω̃1, η̃1) and (Ỹ , Ω̃2, η̃2) be a PyFSTSs, and f : (Ũ , Ω̃1, η̃1) → (Ỹ , Ω̃2, η̃2). If f is PyFSsw-C and (X̃, Ω̃1, η̃1) is PyFS separable, then (Ỹ , Ω̃2, η̃2) is PyFS separable. Proof. Allow dη̃1 to be a countable PyFS dense set on X̃. f(dη̃1) is clearly countable. According to f(dη̃1) is PyFS dense over f(X̃) = Ỹ . As a result, (Ỹ , Ω̃2, η̃2) is PyFS separable. Theorem 7. Let (X̃, Ω̃1, η̃1) and (Ỹ , Ω̃2, η̃2) be a PyFSTSs, and f : (Ũ , Ω̃1, η̃1) → (Ỹ , Ω̃2, η̃2). If f is PyFSsw-C and (X̃, Ω̃1, η̃1) is PyFS hyperconnected, then (Ỹ , Ω̃2, η̃2) is PyFS hyperconnected. Proof. Allow Gη̃2 , Hη̃2 be any two PyFS open sets over Ỹ with Gη̃2 ̸= Hη̃2 ̸= ϕη̃2 . Since f is PyFSsw-C, then int(f−1(Gη̃2) ̸= ϕη̃1 ̸= int(f−1(Hη̃2). But (X̃, Ω̃1, η̃1) is PyFS hyperconnected, then int(f−1(Gη̃2)) ⊓ int(f−1(Hη̃2) ̸= ϕη̃1 . If x ∈ int(f−1(Gη̃2)) ⊓ int(f−1(Hη̃2)) ⊑ f−1(Gη̃2) ⊓ f−1(Hη̃2), at hence f(x) ∈ Gη̃2 ⊓Hη̃2 . Thus (Ỹ , Ω̃2, η̃2) is PyFS hyperconnected. 5. Conclusion Numerous aspects of everyday existence are uncertain. The PyFSs theory is one the- ory developed to deal with uncertainty. This study is based on a novel mathematical structure called PyFST , which was initiated by typologists using PyFSss. In this work, we presented the idea of PyFSsw open sets as a new extension of PyFS open sets. On the one hand, the family of PyFS open to some extent sets is located between the families of PyFS semiopen sets and PyFS somewhere dense sets. The families of PyFSsw open sets and PyFSβ-open sets, on the other hand, are independent of one another. With the help of examples, these linkages have been explained and main attributes established. Then, to define PyFSsw-continuous, we used PyFSsw open sets. We defined these two functions and explored their key characteristics. Investigates some intriguing relationships in a certain PyFST in [8]. The purpose of developing these categories was to analyze the distinctions betweenPyFS homeomorphism and PyFS partly homeomorphism in terms of preserving certain PyFST features. In the following work, we intend to investigate some topological concepts such as PyFS compactness, PyFS Lindelofness, and PyFS connect- edness using PyFSsw open sets. It is also planned to investigate certain applications of PyFSsw homeomorphisms. In addition, we investigate PyFSsw open sets in the context of supra PyFSTS. This study has laid the groundwork for further exploration of PyFSs and their applications. 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