EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6456 ISSN 1307-5543 – ejpam.com Published by New York Business Global A New Approach to Vague Soft Rough Topological Spaces Raed Hatamleh1, Haitham Qawaqneh2, Nasir Odat1, Abdallah Al-Husban3, Arif Mehmood4,∗, Alaa M. Abd El-latif5, Walid Abdelfattah5, M. I. Elashiry5, Abdelhalim Hasnaoui5 1 Department of Mathematics, Faculty of Science, Jadara University, P.O. Box 733, Irbid 21110, Jordan 2 Al-Zaytoonah University of Jordan, Amman 11733, Jordan. 3 Department of Mathematics, Faculty of Science and Technology, Irbid National University, P.O. Box: 2600 Irbid, Jordan 4 Department of Mathematics, Institute of Numerical Sciences, Gomal University, Dera Ismail Khan 29050, KPK, Pakistan 5 Department of Mathematics, College of Science, Northern Border University, Arar 91431, Saudi Arabia Abstract. In this particular piece of work, the new hybrid concept of vague soft rough set theory is introduced. It is a combination of rough set theory, soft set theory, and vague set the- ory. Based on this new concept, some definitions and operations are introduced. Furthermore, lower and upper vague soft approximations, vague soft rough positive, vague soft negative, and vague soft boundaries are discussed with examples. In addition, several theorems are presented in terms of lower and upper vague soft approximations, supported by examples for better un- derstanding. Finally, an entirely new mathematical structure known as the vague soft rough topological structure is introduced. The related definitions of open sets, closed sets, closure, and interior, as well as their relationships in vague soft rough topological spaces, are addressed. To enhance understanding of this study, numerous examples are provided. 2020 Mathematics Subject Classifications: 03E72, 06E25, 54A40, 54H05 Key Words and Phrases: Rough sets, vague soft rough sets, vague soft rough topology. vague soft rough open sets, vague soft rough close sets ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6456 Email addresses: raed@jadara.edu.jo (R. Hatamleh), h.alqawaqneh@zuj.edu.jo (H. Qawaqneh), nodat@jadara.edu.jo (N. Odat), dralhosban@inu.edu.jo (A. Al-Husban), mehdaniyal@gmail.com (A. Mehmood), alaa.ali@nbu.edu.sa (A. M. Abd El-latif), walid.abdelfattah@nbu.edu.sa (W. Abdelfattah), mustafa.elashiry@nbu.edu.sa (M. I. Elashiry), abdllhalim.hasanawa@nbu.edu.sa (A. Hasnaoui) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6456 2 of 18 1. Introduction Zadeh [1] introduced the idea of fuzzy set theory (FST) . Pawlak [2] originated rough set theory (RST). Molodtsov [3] established soft set theory (SST). Maji et al. [4] strengthened SST more influential by showing its applications in practical problems. Maji et al. [5] bridged the holes that exist in [2]. Work on the journey continued be- cause the idea of SST was still in its infancy. Pei and Miao [6] and Chen [7] polished the work of Maji et al. The (IVNSR), which is an advancement over relations such as soft, fuzzy soft, intuitionistic fuzzy soft set theory (IFST), etc., was developed by Broumi et al. [8]. The most valuable structure in mathematics, known as soft topology (ST), was introduced by Cagman et al. [9]. The identical structure was also worked on by Shabir and Naz [10]. Developed ST with additional points known as soft points by Bayramov and Gunduz [11]. The idea of IFST was developed by Atanassov [12]. Untouched find- ings remained in some cases. Fundamental concepts were touched upon and the concept of intuitionistic fuzzy topology (IFT) was introduced by Bayramov and Gunduz [13]. Type-2 soft sets are a complex structure that Hayat et al. [14] explored. Traditional relationship between a vertex and its neighbors was established by Hayat et al. [15]. The concept of soft set was introduced by Hayat et al. [16] along with TOPSIS and the Shannon entropy. The notion of bipolar soft sets and its foundations were developed by Shabir and Naz [17]. A new access to the bipolar soft set was established by Karaaslan and Karatas [18]. Bipo- lar soft topology was organized by Ozturk [19]. Numerous soft semi-compact spaces of the journal type were discussed by Al-Shami et al. [20]. Soft pre-open set was used by Al-Shami and EL-Shafei [21] to handle soft compact and Lindelof spaces. Feng etal. [22] made a marriage of rough sets with soft sets. Soft set theory is used, for the first time, to generalize Pawak’s rough set model. The authors presented the fundamental properties of SRA and SRS. New types of soft sets such as full soft sets, the concept of soft rough equal relations is presented and concerning properties are examined in between the lines. Li et al,[23] supposed a new kind of SS. Based on them, the authors, proposed soft ap- proximations and explored their characteristics. Soft rough sets are defined and their topological structures are obtained. At the end, the relationship between SRS and topologies has been investigated carefully. Liu et al. [24] proposed a new hybrid mod- els that combined together fuzzy set, soft sets and rough sets. The beauty of these models lie in the fact that it reduces the uncertainties. These models are verified by providing excellent examples. Liu et al. [25] suggested extremely a new hybrid model called N-soft rough sets, which is attachment of rough sets with N-soft sets. On top of that, approximation operators and some useful properties relative to N- soft rough approximation space are exhibited. Alkhazaleh and Marei [26] pointed out some flaws in [22] and showed that these models do work with application of real-life problems. These models were modified and the verification was addressed with real-life problems. Safty et al. [27] presented a new technique for creating a soft approximation as a modification and generalization of Zhaowen et al. approach. Comparisons were pictured out between our approach and previous study. Besides, an application on corona virus has been pre- sented. On the top of that, construction of algorithm and proposed model was made and R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6456 3 of 18 its application to decision-making problem were pictured out. Zhang et al. [28] installed the concept of IFS and IFSRS. Demirtas et al. [29] presented the notion of inverse soft rough sets by applying the concept of inverse soft sets and soft rough sets. Besides, different approaches were used to discuss the relationships between them. On the top of that development of algorithm was also sorted out to apply it to decision making prob- lems. F. Afzal et al. [30] characterized of bipolar vague soft s-open Sets. Alkhazaleh and Salleh [33] introduced the concept of FSES and in continuation, launched the basic operations related to this new theory. This reference [33] became source of motivation for this study. Abd El-latif [34] introduced a generalized fuzzy soft rough model. The fuzzy soft IP-upper and fuzzy soft IP-lower approximations are two novel fuzzy soft rough approximations that are offered along with their associated qualities. Further- more, in comparison to what is currently known in the literature, the author was able to lower the fuzzy soft border region. The structure of fuzzy soft ideal rough topologies caused by fuzzy soft sets and fuzzy ideals was finally determined by the author. Atef et al. [35] introduced the concept of complementary soft neighborhoods and presents three types of covering soft rough set (CSR) models. Soft computing techniques intro- duced in modelling complex data structures have developed tremendously in the past few years. To illustrate this point, we could take several examples of using deep learning in medical image analysis [36], the investigation of IoT-based learning techniques [37], or the development of fractional-order PID controllers to manage the robotic systems [38]. Furthermore, conformable fractional Pareto distribution has found usage in modelling the uncertain systems [? ]. This effort is fueled by such developments, and our contri- bution suggests an alternative, a new direction on vague, soft, rough topological spaces that seeks to represent better the imprecise and vague information that is of dynamical space of decision-making. It looks at these models’ fundamental characteristics and how they relate to one another. The article also presents the △-topological spaces (△− TS) approach to CSR, which examines topological features and their interactions, including △-open sets, △-closed sets, △-interior, △-closure, △-boundary, △-neighborhoods, and △-limit points. Lastly, utilizing the constructed topologies, a method is suggested to handle uncertainty and resolve Multi-Group Decision Making (MGDM) problems. Ali. Et al. [40] developed new kinds of soft rough sets models by using the concept of near open sets, where the accuracy of approximations is enhanced significantly This article introduced the concept of near soft rough approximations, called ”JSR-approximations,” for each J ∈ {P, S, γ, α, β}, generalizing several previously introduced concepts. It dis- cusses the properties and relationships of these approximations and compares them with earlier methods. An algorithm is provided for decision-making problems, and its perfor- mance is tested on hypothetical data to compare it with existing methods. The layout of this paper is systematized as follows, Section 2, implies some basic defini- tions including soft set, rough set, vague soft sets and some operations on vague soft sets which are necessary for the up-coming section. In section 3, lower and UVSA, vague soft rough positive, vague soft negative and vague soft boundary zone are addressed with examples. In addition to this, few theorems are presented in terms of lower and upper vague soft approximations sense and these theorems are supported with examples. In R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6456 4 of 18 section 4, we defined and studied, vague soft rough topology. Few results are studied on the basis of interior and closure and the linked between interior and closures is also exhibited in few theorems. For better understanding these results are supported with examples. On the top of that, future work and the conclusion of this research work is summarized in section 5. 2. Preliminaries In this section, we present some basic definitions. Definition 1. [21]. Let X be an initial universe, E be a set parameter and P (X) denotes the power set of X. A pair (F,E) is named as soft set , with mapping given by F : E −→ P (X). Definition 2. [23]. Let X be initial universe, E be a parameter’s set. Lower, upper and boundary approximation of E are define as Ř(E) = ⋃ x∈X (Ř(x) : Ř(x) ⊆ E) Ř−(E) = ⋃ x∈X (Ř(x) : Ř(x) ∩ E ̸= ϕ) and BŘ(E) = Ř−(E)\Ř−(E) with Ř ⊆ X ×X which indicates over information about element of X. Definition 3. [32]. A vague set F on universe of discourse X is defined as, F = {(x, µF (x), wF (x)) : x ∈ X}, Where µ,w : X −→]− 0,+1[ and ≤ µF (x), wF (x) ≤ 2+. Definition 4. [31]. Let X be initial universe set and E be a set of parameters. Let FS(X) denotes set of all vague set then, a vague soft set (F̃ , E) over X is a set defined by a set valued function F representing a mapping F̃ : E −→ FS(X) where F̃ is called approximate function of vague soft set (F̃ , E). That is (F̃ , E) = {(e, (x, µF̃ (e)(x), wF̃ (e)(x)) : x ∈ X), e ∈ E} With µF̃ (e)(x), wF̃ (e)(x) ∈ [0, 1], respectively called truth membership and falsity-membership function ofF̃ (e). Since supremum of each µ,wis1 so inequality 0 ≤ µF̃ (e)(x)+wF̃ (e)(x)) ≤ 2 We denoted set of all vague soft sets of X by FSS(X,E). Definition 5. [31] Let (F̃ , E) be a vague soft set over universe set X. The complement of (F̃ , E) is denoted by (F̃ , E)c and is defined by: (F̃ , E)c = {(e, (x,wF̃ (e)(x), µF̃ (e)(x)) : x ∈ X), e ∈ E} Obvious that, ((F̃ , E)c)c = (F̃ , E) R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6456 5 of 18 Definition 6. [31] Let(F̃1, E) and (F̃2, E) be two vague soft sets. (F̃1, E) is said to be a vague soft subset of (F̃2, E) if µF̃1(e) (x) ≤ µF̃2(e) (x), wF̃1(e) (x) ≥ wF̃2(e) (x) ∀e ∈ E,∀x ∈ X It is denoted by, (F̃1, E) ⊆ (F̃2, E). (F̃1, E) is said to be vague soft equal to (F̃2, E) if (F̃1, E) is a vague soft subset of (F̃2, E) and (F̃2, E) is a vague soft subset of (F̃1, E). It is denoted by, (F̃1, E) = (F̃2, E). Definition 7. [32] Let(F̃1, E) and (F̃2, E) be two vague soft sets. Then their union is denoted by (F̃1, E) ∪ (F̃2, E) = (F̃3, E) and is defined by: (F̃3, E) = {(e, (x, µF̃ (e)(x), wF̃ (e)(x)) : x ∈ X), e ∈ E} Where. µF̃ (e)(x) = max{µF̃1(e) (x), µF̃2(e) (x)} wF̃ (e)(x) = min{wF̃1(e) (x), wF̃2(e) (x)} Definition 8. [32] Let(F̃1, E) and (F̃2, E) be two vague soft sets. Then their intersection is denoted by (F̃1, E) ∩ (F̃2, E) = (F̃3, E) and is defined by: (F̃3, E) = {(e, (x, µF̃ (e)(x), wF̃ (e)(x)) : x ∈ X), e ∈ E} Where. µF̃ (e)(x) = min{µF̃1(e) (x), µF̃2(e) (x)} wF̃ (e)(x) = max{wF̃1(e) (x), wF̃2(e) (x)} Definition 9. [32] Let(F̃1, E) and (F̃2, E) be two vague soft sets over the universe set X. Then, (F̃1, E) \ (F̃2, E) = (F̃3, E) and is introduced as (F̃3, E) = (F̃1, E) ∩ (F̃2, E)c: (F̃ , E) = {(e, ⟨x, µF̃ (e)(x), wF̃ (e)(x)⟩ : x ∈ X), e ∈ E} Where. µF̃ (e)(x) = min{µF̃1(e) (x), µF̃2(e) (x)} wF̃ (e)(x) = max{wF̃1(e) (x), wF̃2(e) (x)} Definition 10. [32] 1. A vague soft set (F̃ , E) over the universe set X is said to be a null vague soft set if: µF̃3(e) (x) = 0, wF̃3(e) (x) = 1, ∀e ∈ E,∀x ∈ X. It is denoted by 0(X,E). 2. A vague soft set (F̃ , E) is said to be an absolute vague soft set if: µF̃3(e) (x) = 1, wF̃3(e) (x) = 0, , ∀e ∈ E,∀x ∈ X. It is symbolized as 1(X,E). R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6456 6 of 18 3. Characterization of Few Structures in Terms of Vague Soft Rough Sets In this section, lower and upper vague soft approximations, vague soft rough positive, vague soft negative and vague soft boundary zone are addressed with examples. In addition to this, few theorems are presented in terms of lower and upper vague soft approximations sense and these theorems are supported with examples. Definition 11. Let X be a non-empty universal set and E be a set of parameters. . Let FS(X) be the set of all vague set over X. Then (F̃ , E) is a vague soft set over the universe set X with F̃ is a mapping given by F̃ : E −→ FS(X). Then (X, F̃ , E) is vague soft approximation (VSA) space. The lower and upper vague soft approximation of ή ⊆ FSS(X,E) concerning (X, F̃ , E) are denoted by apr FS (ή) and aprFS(ή) respectively, define by; apr FS (ή) = {( e, ( x µ ei (x), wei(x) ) ) , ∀ei ∈ E,∀x ∈ X } aprFS(ή) = {( e, ( x µei(x), wei(x) ) ) , ∀ei ∈ E,∀x ∈ X } µ ei (x) = {∧µei(x) : µei(x) ∈ ή ∩ (F̃i, E);∀(F̃i, E) ⊆ ή,∀ei ∈ E,∀x ∈ X} wei(x) = {∨wei(x) : wei(x) ∈ ή ∩ (F̃i, E);∀(F̃i, E) ⊆ ή,∀ei ∈ E,∀x ∈ X} µei(x) = {∨µei(x) : µei(x) ∈ ή ∪ (F̃i, E);∀(F̃i, E) ⊆ ή,∀ei ∈ E,∀x ∈ X} wei(x) = {∧wei(x) : wei(x) ∈ ή ∪ (F̃i, E);∀(F̃i, E) ⊆ ή,∀ei ∈ E,∀x ∈ X} Where ∧ and ∨ mean min and max operators, respectively. Since apr FS (ή) and aprFS(ή) are two V SS of FSS(X,E). If apr FS (ή) = aprFS(ή) then ή is said to be vague soft definable set; otherwise it is V SRS. Example 1. Let’s assume X = {x1, x2, x3, x4} represents the students in a classroom or program. Let E = {e1, e2, e3} represent education-related attributes or criteria. We consider following vague soft sets; (F̃1, E) = [( (e1, (x1, 3× 10−1, 7× 10−1)), (x3, 6× 10−1, 4× 10−1), (x4, 7× 10−1, 2× 10−1) ) ,( (e2, (x2, 6× 10−1, 5× 10−1)), (x3, 4× 10−1, 7× 10−1), (x4, 2× 10−1, 2× 10−1) ) ] (F̃2, E) = ((e1, (x1, 8× 10−1, 5× 10−1)), (x3, 3× 10−1, 3× 10−1) ) ,( (e2, (x1, 8× 10−1, 2× 10−1)), (x3, 7× 10−1, 2× 10−1) ) ,( (e3, (x2, 2× 10−1, 5× 10−1)), (x4, 7× 10−1, 2× 10−1) )  (F̃3, E) =  ( (e1, (x3, 8× 10−1, 3× 10−1)), (x4, 6× 10−1, 3× 10−1) ) ,( (e2, (x2, 7× 10−1, 3× 10−1), (x3, 8× 10−1, 1× 10−1), (x4, 7× 10−1, 1× 10−1)) ) ,( (e3, (x2, 4× 10−1, 2× 10−1), (x3, 8× 10−1, 3× 10−1), (x4, 6× 10−1, 2× 10−1)) )  R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6456 7 of 18 ή =  ((e1, (x1, 7× 10−1, 4× 10−1), (x2, 8× 10−1, 2× 10−1), (x3, 8× 10−1, 2× 10−1), (x4, 9× 10−1, 1× 10−1))),( (e2, (x2, 7× 10−1, 3× 10−1), (x3, 8× 10−1, 1× 10−1), (x4, 7× 10−1, 1× 10−1)) ) ,( (e3, (x2, 4× 10−1, 2× 10−1), (x3, 8× 10−1, 3× 10−1), (x4, 6× 10−1, 2× 10−1)) )  The lower and upper vague soft approximation of ή are calculated as; apr FS (ή) = [ ( (e1, (x1, 3× 10−1, 7× 10−1)) ) ,( (e2, (x2, 4× 10−1, 7× 10−1)), (x3, 4× 10−1, 7× 10−1) )] aprFS(ή) =  ((e1, (x1, 8× 10−1, 3× 10−1), (x2, 8× 10−1, 2× 10−1), (x3, 8× 10−1, 2× 10−1), (x4, 9× 10−1, 1× 10−1))),( (e2, (x2, 7× 10−1, 3× 10−1), (x3, 8× 10−1, 1× 10−1), (x4, 6× 10−1, 1× 10−1)) ) ,( (e3, (x2, 4× 10−1, 2× 10−1), (x3, 8× 10−1, 3× 10−1), (x4, 6× 10−1, 1× 10−1)) )  Remark 1. For any ή ⊆ F̃SS(X,E), the sets of POSFS(ή) = apr FS (ή), FEGFS(ή) = aprFS(ή) ve BndFS(ή) = apr FS (ή)\aprFS(ή) are called vague soft rough positive, vague soft rough boundary regions of considered set (ή), respectively. Example 2. We consider Example 1. Then we can write vague soft rough positive, negative and boundary region as follow; POSFS(ή) = apr FS (ή) = [ ( (e1, (x1, 5× 10−1, 5× 10−1)) ) ,( (e2, (x3, 4× 10−1, 7× 10−1)), (e1, (x4, 2× 10−1, 2× 10−1)) )] FEGFS(ή) = aprFS(ή) =  ((e1, (x1, 8× 10−1, 3× 10−1), (x2, 8× 10−1, 2× 10−1), (x3, 8× 10−1, 2× 10−1), (x4, 9× 10−1, 1× 10−1))), ((e2, (x2, 7× 10−1, 3× 10−1), (x3, 8× 10−1, 1× 10−1), (x4, 6× 10−1, 1× 10−1))), ((e3, (x2, 4× 10−1, 2× 10−1), (x3, 8× 10−1, 1× 10−1), (x4, 6× 10−1, 1× 10−1)))  BndFS(ή) = aprFS(ή)\aprFS (ή) =  ((e1, (x1, 8× 10−1, 3× 10−1), (x2, 8× 10−1, 2× 10−1), (x3, 8× 10−1, 2× 10−1), (x4, 9× 10−1, 1× 10−1))), ((e2, (x2, 7× 10−1, 3× 10−1), (x3, 7× 10−1, 4× 10−1), (x4, 2× 10−1, 2× 10−1))), ((e3, (x2, 4× 10−1, 2× 10−1), (x3, 8× 10−1, 3× 10−1), (x4, 6× 10−1, 1× 10−1)))  Obviously, since it is clear aprFS(ή) ̸= apr FS (ή). So ή is vague set in approximation space (X, F̃ , E). R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6456 8 of 18 Theorem 1. Let (F̃ , E) be a vague soft set over X. (X, F̃ , E) be a VSRA space and ή, κ ∈ FSS(X,E), then we have (i) apr FS (ή) ⊆ ή ⊆ aprFS(ή). (ii) apr FS (0(X,E)) = 0(X,E), aprFS(1(X,E)) = 1(X,E). (iii) If ή ⊆ κ, then apr FS (ή) ⊆ apr FS (κ). (iv) If ή ⊆ κ, then aprFS(ή) ⊆ aprFS(κ). (v) If ή ∩ κ, then apr FS (ή) ∩ apr FS (κ). (vi) If ή ∩ κ, then aprFS(ή) ∩ aprFS(κ). (vii) If ή ∩ κ, then apr FS (ή) ∩ apr FS (κ). (viii) If ή ∪ κ, then aprFS(ή) ∪ aprFS(κ). Proof. (i). From definition 11, it is seen apr FS (ή) ⊆ ή. Also from definition of vague soft upper approximation, ∀(F̃i, E) ∩ ή ̸= (0(X,E)), µ, v, w ∈ ή ∪ (F̃i, E) Hence ή ⊆ aprFS(ή). Thus aprFS (ή) ⊆ ή ⊆ aprFS(ή). (ii). From definition11, proof of (ii) is straightforward. (iii). Let ή ⊆ κ and (F̃i, E) ⊆ (ή), i = (1, 2). Then apr FS (ή) = ή ∩ (∩2 i=1)(F̃i, E). In addition (F̃i, E) ⊆ (ή) then (F̃i, E) ⊆ (κ). Hence, (κ) = κ ∩ (∩2 i=1)(F̃i, E) ⇒ apr FS (ή) ⊆ apr FS (κ) (iv). Let ή ⊆ κ and (F̃i, E) ⊆ (ή) ̸= 0, i = (1, 2). Then aprFS(ή) = ή ∪ (∪2 i=1)(F̃i, E). In addition (F̃i, E) ⊆ (ή) then (F̃i, E) ⊆ (κ). Hence, (κ) = κ ∪ (∪2 i=1)(F̃i, E) ⇒ aprFS(ή) ⊆ aprFS(κ) (v). Let xei(α,γ) ∈ apr FS (ή∩κ), there exist (F̃ , E), such that xei(α,γ) ∈ (F̃ , E) ⊆ apr FS (ή∩ κ), xei(α,γ) ∈ (F̃ , E) ⊆ ή and xei(α,γ) ∈ (F̃ , E) ⊆ κ, so xei(α,γ) ∈ apr FS (ή) and (F̃ , E) ⊆ apr FS (κ), ⇒ xei(α,γ) ∈ apr FS (ή) ∩ apr FS (κ) Thus, apr FS (ή ∩ κ) ⊆ apr FS (ή) ∩ apr FS (κ). (vi). Let xei(α,γ) ∈ aprFS(ή ∩ κ), ∀ (F̃ , E), such that, xei(α,γ) ∈ (F̃ , E) ∩ (ή ∩ κ) ̸= 0(X,E), (F̃ , E) ∩ (ή) ̸= 0(X,E) and (F̃ , E) ∩ (κ) ̸= 0(X,E). So, xei(α,γ) ∈ aprFS(ή) and xei(α,γ) ∈ aprFS(κ) ⇒ aprFS(ή) ∩ aprFS(κ) R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6456 9 of 18 (vii). Let xei(α,γ) /∈ apr FS (ή∩κ), there exist (F̃ , E), such that xei(α,γ) ∈ (F̃ , E) ⊊ apr FS (ή∩ κ), ⇒ (F̃ , E) ⊊ ή and xei(α,γ) ∈ (F̃ , E) ⊊ κ. Therefore, xei(α,γ) /∈ apr FS (ή) and xei(α,γ) /∈ apr FS (κ), ⇒ xei(α,γ) /∈ apr FS (ή) ∩ apr FS (κ) Thus, apr FS (ή ∩ κ) ⊆ apr FS (ή) ∪ apr FS (κ). (viii). Let xei(α,γ) ∈ aprFS(ή ∩ κ), ∀ (F̃ , E), such that, xei(α,γ) ∈ (F̃ , E) ∪ (ή ∩ κ) ̸= 0(X,E), it folllows that (F̃ , E) ∩ (ή) ̸= 0(X,E)and(F̃ , E) ∩ (κ) ̸= 0(X,E). So, xei(α,γ) ∈ aprFS(ή) or xei(α,γ) ∈ aprFS(κ). Hence, ⇒ xei(α,γ) ∈ aprFS(ή) ∩ aprFS(κ) . Thus, aprFS(ή ∩ κ) ⊆ aprFS(ή) ∪ aprFS(κ). Theorem 2. Let (F̃ , E) be a vague soft set over X. (X, F̃ , E) and ή, κ ⊆ FSS(X,E). Then following characteristics followed, (i) apr FS [apr FS (ή)] = apr FS (ή). (ii) aprFS [aprFS(ή)] ⊇ aprFS(ή). (iii) aprFS(ή) ⊇ aprFS [aprFS(ή)]. (iv) aprFS [aprFS(ή)] ⊇ apr FS (ή). (v) apr FS (ήc) ⊇ [aprFS(ή)] c. Proof. (i). Let xei(α,γ) ∈ apr FS (ή). Then we have, xei(α,γ) ∈ (F̃ , E) ⊆ apr FS (ή). So, xei(α,γ) ∈ apr FS [apr FS (ή)]. So, apr FS [apr FS (ή)] = apr FS (ή). From theorem1 apr FS (ή) ⊆ ή. Using (iii) of theorem3.5, we obtain apr FS [apr FS (ή)] ⊆ apr FS (ή). Hence, apr FS [apr FS (ή)] = apr FS (ή). (ii). Let H = aprFS(ή). Using (i) of theorem 1, we got H ⊆ aprFS(ή). Hence, aprFS [aprFS(ή)] ⊇ aprFS(ή). (iii). Let H = aprFS(ή). Using (i) of theorem1, we got apr FS (ή) ⊆ H. Hence, apr FS [aprFS(ή)] ⊇ aprFS(ή). (iv). Let H = apr FS (ή). Using (i) of theorem1, we got aprFS(ή) ⊇ H. Hence, aprFS(ή) ⊆ apr FS [apr FS (ή)]aprFS(ή). (v). Let xei(α,γ) /∈ apr FS (ήc) and xei(α,γ) /∈ (F̃ , E). We have that, (F̃ , E) ⊆ ήc and (F̃ , E) ∩ ήc = 0(X,E). Thus (F̃ , E) ∩ ήc ̸= 0(X,E), where xei(α,γ) ∈ aprFS(ή c) but xei(α,γ) /∈ [apr FS (ή)]c. Therefore apr FS (ήc) ⊇ [aprFS(ή)] c. R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6456 10 of 18 4. Vague Soft Rough Topological Spaces (VSRTS) in Term of Open Sets In this section, VSRTS with some important properties is studied. Few results are studied on the basis of interior and closure and the linked between interior and closures is also exhibited in few theorems. For better understanding these results are supported with examples. Definition 12. Let X be a universe and (X, F̃ , E) be a VSAS. Then, vague soft rough topology is defined as; τ̃FR(ή) = ( 0(X,E), 1(X,E), aprFS (ή), aprFS(ή), BndFS(ή) ) Where ή ∈ FSS(X,E) and tildeτFR(ή) satisfies the following conditions (i) 0(X,E) and 1(X,E) belongs to tildeτFR(ή). (ii) Union of members of any number of tildeτFR(ή) belongs to tildeτFR(ή). (iii) Intersection of members of finite number of tildeτFR(ή) belongs to tildeτFR(ή). The topology defined by tildeτFR(ή) on X is called vague soft rough topology on X and [X, τ̃FR(ή), E] is said to be vague soft rough topological space. Example 3. Let X = {x1, x2, x3, x4} : representing students and E = {e1, e2, e3} : representing educational attributes. Then we can construct the following vague soft sets as; (F̃1, E) = [ ( (e1, (x1, 4× 10−1, 8× 10−1), (x3, 6× 10−1, 5× 10−1)) ) ,( (e2, (x1, 5× 10−1, 4× 10−1), (x3, 3× 10−1, 5× 10−1), (x4, 3× 10−1, 5× 10−1)) )] (F̃2, E) = [( (e1, (x1, 7× 10−1, 3× 10−1), (x2, 7× 10−1, 4× 10−1), (x4, 4× 10−1, 8× 10−1)) ) ,( (e2, (x1, 7× 10−1, 1× 10−1), (x3, 4× 10−1, 1× 10−1)) ) ] (F̃3, E) =  ( (e1, (x1, 7× 10−1, 1× 10−1), (x3, 4× 10−1, 1× 10−1)) ) ,( (e2, (x1, 7× 10−1, 5× 10−1), (x3, 4× 10−1, 3× 10−1)) ) ,( (e3, (x1, 5× 10−1, 4× 10−1), (x3, 4× 10−1, 1× 10−1), (x4, 3× 10−1, 3× 10−1)) )  For another vague soft set ή ⊂ FSS(X,E) with respect to (X, F̃ , E); ή =  ((e1, (x1, 7× 10−1, 2× 10−1), (x2, 9× 10−1, 1× 10−1), (x3, 6× 10−1, 3× 10−1), (x4, 5× 10−1, 2× 10−1))), ((e2, (x2, 6× 10−1, 1× 10−1), (x2, 8× 10−1, 3× 10−1), (x3, 5× 10−1, 1× 10−1), (x4, 4× 10−1, 1× 10−1))),( (e3, (x1, 7× 10−1, 1× 10−1), (x2, 5× 10−1, 1× 10−1), (x4, 3× 10−1, 2× 10−1)) )  R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6456 11 of 18 The lower and upper vague soft approximation of ή are calculated as apr FS (ή) = [( (e1, (x3, 5× 10−1, 5× 10−1)), (e1(x2, 3× 10−1, 5× 10−1)) )] aprFS(ή) =  ((e1, (x1, 7× 10−1, 2× 10−1), (x2, 9× 10−1, 1× 10−1), (x3, 6× 10−1, 3× 10−1), (x4, 5× 10−1, 2× 10−1))), ((e2, (x2, 6× 10−1, 1× 10−1), (x2, 8× 10−1, 3× 10−1), (x3, 5× 10−1, 1× 10−1), (x4, 4× 10−1, 1× 10−1))),( (e3, (x1, 7× 10−1, 1× 10−1), (x2, 5× 10−1, 1× 10−1), (x4, 3× 10−1, 2× 10−1)) )  BndFS(ή) =  ((e1, (x1, 7× 10−1, 2× 10−1), (x2, 9× 10−1, 1× 10−1), (x3, 5× 10−1, 5× 10−1), (x4, 5× 10−1, 2× 10−1))), ((e2, (x2, 6× 10−1, 1× 10−1), (x2, 8× 10−1, 3× 10−1), (x3, 5× 10−1, 1× 10−1), (x4, 4× 10−1, 3× 10−1))),( (e3, (x1, 7× 10−1, 1× 10−1), (x2, 5× 10−1, 1× 10−1), (x4, 3× 10−1, 2× 10−1)) )  Then τ̃FR(ή) = ( 0(X,E), 1(X,E), aprFS (ή), aprFS(ή), BndFS(ή) ) is VSRT on X. Definition 13. Let (X, τ̃FR(ή), E) be a VSRTS. Then, each number of τ̃FR(ή) are vague soft rough open sets (VSROSs). A vague soft rough set is said to be a vague soft closed set if its complement belong to X, τ̃FR(ή). Theorem 3. Consider (X, τ̃FR(ή), E) as VSRTS. Then, (i) 0(X,E) and 1(X,E) are vague soft rough closed sets (VSROSs). (ii) The intersection of any number of vague soft rough closed sets( VSRCSs) is a vague soft rough closed set over X. Finite union of VSRCS is VSRCS over X. Proof. Trivial. Definition 14. Let (X, τ̃FR(ή), E) be VSRTS and ω ∈ FSS(X,E). Then τ̃FR(ή) = ( ω ∩ (F̃i, E) : (F̃i, E) ∈ τ̃FR(ή)fori ∈ I ) is vague soft rough subspace topology on ω and (X, τ̃FR(ή), E) is vague soft rough subspace of [X, τ̃FR(ή), E]. Definition 15. Let (X, τ̃FR(ή), E) be a vague soft rough topological space over X and ω ∈ FSS(X,E). Then the vague soft rough interior of ω is vague soft union of all vague soft open subsets of ω and we denoted it as IntFSR(ω).Clearly, IntFSR(ω) is the largest vague soft rough open set contained by (ω). R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6456 12 of 18 Example 4. Let consider example 4.2. Suppose that any ω ∈ FSS(X,E) be defined as follow; ω =  ((e1, (x1, 8× 10−1, 1× 10−1), (x2, 9× 10−1, 1× 10−1), (x3, 6× 10−1, 4× 10−1), (x4, 6× 10−1, 1× 10−1))), ((e2, (x2, 6× 10−1, 1× 10−1), (x2, 9× 10−1, 2× 10−1), (x3, 6× 10−1, 1× 10−1), (x4, 5× 10−1, 2× 10−1))), ((e3, (x1, 8× 10−1, 1× 10−1), (x2, 5× 10−1, 3× 10−1), (x4, 6× 10−1, 1× 10−1), (x4, 4× 10−1, 2× 10−1)))  Then 0(X,E), 1(X,E), aprFS (ή), aprFS(ή), BndFS(ή ⊆ ω Therefore, IntFSR(ω) = 0(X,E) ∪ apr FS (ή) ∪BndFS(ή = BndFS(ή Theorem 4. Let (X, τ̃FR(ή), E) be a vague soft rough topological space over X and ω,£ ∈ FSS(X,E). Then (i) IntFSR[0(X,E)] = 0(X,E) and IntFSR[1(X,E)] = 1(X,E), (ii) IntFSR(ω) ⊆ ω, (iii) ω is a vague soft rough open set ⇔ IntFSR(ω) = ω, (iv) ω ⊆ £ ⇒ IntFSR(ω) ⊆ IntFSR(£), (v) IntFSR(ω) ∩ IntFSR(£) = IntFSR(ω ∩£), (vi) IntFSR(ω) ∪ IntFSR(£) = IntFSR(ω ∪£). Proof. (i) and (ii) are obvious. (iii). Suppose that IntFSR(ω) = ω. Since IntFSR(ω) is a vague soft rough open set ω is a vague soft rough open set. Conversely, if ω is a vague soft rough open set, then the largest vague set rough open set contained in ω is ω itself. Thus IntFSR(ω) = ω. (iv). Suppose that ω ⊆ £. By the condition (ii), IntFSR(ω) ⊆ ω ⊆ £. Since IntFSR(ω) is the largest vague a in ω and so IntFSR(ω) ⊆ IntFSR(£). (v). Since ω ∩£ ⊆ ω and ω ∩£ ⊆ £ then IntFSR(ω ∩£) ⊆ IntFSR(ω) and IntFSR(ω ∩ £) ⊆ IntFSR(£). Hence IntFSR(ω ∩ £) ⊆ IntFSR(ω) ∩ IntFSR(£). On the other hand, since IntFSR(ω) ⊆ IntFSR(ω) and IntFSR(£) ⊆ IntFSR(£) then IntFSR(ω) ∩ IntFSR(£) ⊆ ω ∩ £. Besides IntFSR(ω ∩ £) ⊆ ω ∩ £ and it is the largest vague soft rough open set. Therefore, IntFSR(ω)∩IntFSR(£) ⊆ IntFSR(ω∩£). Thus IntFSR(ω)∩ IntFSR(£) = IntFSR(ω ∩£). (vi). Since ω ⊆ ω∪£ and £ ⊆ ω∪£ then IntFSR(ω) ⊆ IntFSR(ω∪£) and IntFSR(£) ⊆ IntFSR(ω ∪£). Therefore, IntFSR(ω) ∪ IntFSR(£) = IntFSR(ω ∪£). Definition 16. Let (X, τ̃FR(ή), E) be a vague soft rough topological space over X and ω ∈ FSS(X,E). Then the vague soft rough closure of ω is vague soft intersection of all vague soft closed subsets of ω and we denoted it as clFSR(ω). R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6456 13 of 18 Example 5. Let consider the vague soft rough topology τ̃FR(ή). Suppose that any ω ∈ FSS(X,E) be defined as follow; ω =  ((e1, (x1, 1× 10−1, 8× 10−1, ), (x2, 1× 10−1, 9× 10−1), (x3, 2× 10−1, 7× 10−1), (x4, 2× 10−1, 6× 10−1))), ((e2, (x2, 1× 10−1, 7× 10−1), (x2, 2× 10−1, 9× 10−1), (x3, 1× 10−1, 7× 10−1), (x4, 1× 10−1, 7× 10−1))), ((e3, (x1, 1× 10−1, 8× 10−1), (x2, 3× 10−1, 6× 10−1), (x4, 1× 10−1, 6× 10−1), (x4, 2× 10−1, 4× 10−1)))  Obviously (0(X,E)) c, (1(X,E)) c, (apr FS (ή))c, (aprFS(ή)) c, (BndFS(ή) c are all vague soft rough closed sets over (X, τ̃FR(ή), E). We can calculate these sets as follows, (0(X,E)) c = 0(X,E), (1(X,E)) c = 1(X,E) (apr FS (ή))c =  ((e1, (x1, 1× 10−1, 0× 10−1, ), (x2, 1× 10−1, 1× 10−1), (x3, 5× 10−1, 5× 10−1), (x4, 1× 10−1, 0× 10−1))), ((e2, (x2, 1× 10−1, 0× 10−1), (x2, 1× 10−1, 0× 10−1), (x3, 1× 10−1, 0× 10−1), (x4, 5× 10−1, 3× 10−1))), ((e3, (x1, 1× 10−1, 0× 10−1), (x2, 1× 10−1, 0× 10−1), (x4, 1× 10−1, 0× 10−1), (x4, 1× 10−1, 0× 10−1)))  (aprFS(ή)) c =  ((e1, (x1, 7× 10−1, 2× 10−1, ), (x2, 1× 10−1, 9× 10−1), (x3, 3× 10−1, 6× 10−1), (x4, 2× 10−1, 5× 10−1))), ((e2, (x2, 1× 10−1, 6× 10−1), (x2, 3× 10−1, 8× 10−1), (x3, 1× 10−1, 5× 10−1), (x4, 1× 10−1, 6× 10−1))), ((e3, (x1, 1× 10−1, 7× 10−1), (x2, 1× 10−1, 0× 10−1), (x4, 1× 10−1, 8× 10−1), (x4, 2× 10−1, 3× 10−1)))  (BndFS(ή)) c =  ((e1, (x1, 7× 10−1, 2× 10−1, ), (x2, 1× 10−1, 9× 10−1), (x3, 5× 10−1, 5× 10−1), (x4, 2× 10−1, 5× 10−1))), ((e2, (x2, 1× 10−1, 6× 10−1), (x2, 3× 10−1, 8× 10−1), (x3, 1× 10−1, 5× 10−1), (x4, 3× 10−1, 4× 10−1))), ((e3, (x1, 1× 10−1, 7× 10−1), (x2, 1× 10−1, 0× 10−1), (x4, 1× 10−1, 8× 10−1), (x4, 2× 10−1, 3× 10−1)))  Then (ω) ⊆ (0(X,E)) c, (1(X,E)) c, (apr FS (ή))c, (aprFS(ή)) c, (BndFS(ή) c. Therefore, clFSR(ω) ⊆ (0(X,E)) c ∩ (apr FS (ή))c ∩ (aprFS(ή)) c ∩ (BndFS(ή) c = (aprFS(ή) c Theorem 5. Let (X, τ̃FR(ή), E) be a vague soft rough topological space over X and ω,£ ∈ FSS(X,E). Then R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6456 14 of 18 (i) clFSR[0(X,E)] = 0(X,E) and clFSR[1(X,E)] = 1(X,E), (ii) clFSR(ω) ⊆ ω, (iii) ω is a vague soft rough open set ⇔ clFSR(ω) = ω, (iv) clFSR[clFSR(ω)] = clFSR(ω) (v) ω ⊆ £ ⇒ clFSR(ω) ⊆ clFSR(£), (vi) clFSR(ω) ∪ IntFSR(£) = clFSR(ω ∪£), (vii) clFSR(ω) ∩ clFSR(£) = clFSR(ω ∩£). Proof. (i) and (ii) are obvious. (iii). Consider clFSR(ω) = ω. Since clFSR(ω) is a VSRCS so, ω is a VSRCS. Conversely suppose that, if ω be a VSCS over ω. Then ω is VSRC superset ω. So that, ω = clFSR(ω). (iv). Let clFSR(ω) = £. Then ω is VSRCS. Hence, by the condition (iii), clFSR(£) = £. Therefore, clFSR[clFSR(ω)] = clFSR(ω). (v). We know that ω ⊆ clFSR(ω) and £ ⊆ clFSR(£), and so ω ⊆ £ ⊆ clFSR(ω). Since clFSR(ω) is the smallest vague containing ω. So clFSR(ω) ⊆ clFSR(£). (vi). Since ω ⊆ ω ∪ £ and £ ⊆ ω ∪ £ then clFSR(ω) ⊆ clFSR(ω ∪ £) and clFSR(£) ⊆ clFSR(ω ∪ £). Hence clFSR(ω) ∪ IntFSR(£) ⊆ clFSR(ω ∪ £). On the other hand, since ω ⊆ clFSR(ω) and £ ⊆ clFSR(£) then ω ∪ £ ⊆ clFSR(ω) ∪ clFSR(£). Besides ω ∪£ ⊆ clFSR(ω ∪£) and it is the smallest vague soft rough closed set that containing ω∪£. Therefore, clFSR(ω∪£) ⊆ clFSR(ω)∪clFSR(£). Hence clFSR(ω∪£) = clFSR(ω)∪ clFSR(£). (vii). Since ω∩£ ⊆ clFSR(ω)∩ clFSR(£) and ω∩£ ⊆ clFSR(£) and clFSR(ω∩£) is the smallest vague soft rough closed set that containing ω ∩ £. Therefore, clFSR(ω ∩ £) = clFSR(ω) ∩ clFSR(£). 5. Conclusion In both pure and applied mathematics, rough set theory has many intriguing appli- cations. Our approach involves the creation of a novel theory known as vague soft rough set theory. The idea of lower and upper vague soft approximations, vague soft rough positive, vague soft negative, and vague soft border is required by this theory. All of these advancements are made and are positively handled with pertinent, thorough, and many examples. The definitions of open sets, closed sets, closure, and interior in vague soft rough topological spaces are also addressed. Finally, a completely new mathematical structure called as vague soft rough topological structure is demonstrated. Additionally, some results are discussed, and examples are provided to help the reader understand this study. We will attempt to illustrate vague soft rough topological spaces in relation to soft points of the spaces in our upcoming work. In vague soft rough topological spaces, we shall define new open sets known as pre-open sets, semi-open sets, alpha open sets, and beta open sets. R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6456 15 of 18 Building upon the foundations of vague soft rough topological spaces, future research will explore the generalization of open sets—including pre-open, semi-open, -open, and -open sets—by drawing inspiration from recent advancements in hybrid soft computing and neutrosophic topologies. Specifically, the framework of quadri-partitioned neutrosophic soft topological spaces [41] and bipolar vague soft expert sets [40] can be adapted to define these open sets, leveraging their robust capabilities for handling uncertainty and multi-attribute decision-making. Furthermore, fixed-point theorems for 3-self mappings [42] could provide powerful tools to analyze the stability and convergence of sequences within these generalized topological spaces. Additionally, weighted aggregation opera- tors developed for interval-valued Pythagorean neutrosophic sets [43] and trigonometric neutrosophic sets [44] may refine the membership and non-membership boundaries of these open sets, thereby enhancing their practical applicability in domains such as medi- cal diagnosis [40]. Finally, AI-assisted fuzzy soft relations [39] offer a promising pathway to model dynamic interactions among these sets, particularly in emerging applications like wearable health technologies. Acknowledgements We are sincerely grateful to the reviewers and the editor for their valuable and con- structive suggestions, which have significantly improved the quality of this work. Author Contributions: All authors equally contributed. Conflicts of Interest: The authors declare that there are no conflicts of interest re- garding the publication of this paper. Availability of Data and Materials: All the data and materials are provided in the manuscript. References [1] L. A. 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