EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 5969 ISSN 1307-5543 – ejpam.com Published by New York Business Global Supra ϵ-open Sets: Features, Operators and Applications Alaa M. Abd El-latif1, Radwan Abu-Gdairi2, A. A. Azzam3,4, Husham M. Attaalfadeel1,∗, Shaaban M. Shaaban5, M. Aldawood3, Khaled A. Aldwoah6 1 Mathematics Department, College of Science, Northern Border University, Arar 91431, Saudi Arabia 2 Mathematics Department, Faculty of Science, Zarqa University, Zarqa 13132, Jordan 3 Department of Mathematics, Faculty of Science and Humanities, Prince Sattam Bin Abdulaziz University, Alkharj 11942, Saudi Arabia 4 Department of Mathematics, Faculty of Science, New Valley University, Elkharga 72511, Egypt 5 Center for Scientific Research and Entrepreneurship, Northern Border University, Arar 73213, Saudi Arabia 6 Department of Mathematics, Faculty of Science, Islamic University of Madinah, Medinah, Saudi Arabia Abstract. In supra topological spaces, we provide supra ϵ-open sets, an extremely broad class of open sets. We demonstrate that, the previously comparable concepts of supra regular (respectively, α-, semi-, pre-, b-, β-, and R-) open sets are contained in this new category of open sets. To further illustrate the key concepts discussed in the study, we have included a geometric topological diagram [see Diagram 1]. Also, we outline this class’s primary characteristics. Specifically, we show that our new category forms a supra topology rather than a topological space. Utilizing our recently introduced category of supra open sets, we define new kinds of operators called supra ϵ-interior (closure, accumulation, exterior, and boundary, respectively). Moreover, we highlight the deviations between these new operators and their corresponding operators. Furthermore, we also give some key examples and counterexamples to illustrate the importance of our new operators. In addition, we highlight the advantages and distinctions of our work in comparison to similar studies in the field. 2020 Mathematics Subject Classifications: 54A05, 54C10, 54C08. Key Words and Phrases: Supra ϵ-open set; Supra ϵ-interior operator; Supra ϵ-closure operator; Partition; Applications ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5969 Email addresses: alaa.ali@nbu.edu.sa (A. M. Abd El-Latif), rgdairi@zu.edu.jo (R. Abu-Gdairi), aa.azzam@psau.edu.sa (A. A. Azzam), Husham.Alhassan@nbu.edu.sa (H. M. Attaalfadeel),shabaan27@gmail.com (S. M. Shaaban), m.aldawood@psau.edu.sa (M. Aldawood), aldwoah@yahoo.com (K. A. Aldwoah) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) Abd El-latif et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5969 2 of 19 1. Introduction In the past few decades, a significant focus of topological, supra topological, and soft topological research has been the examination of various generalized open, supra open, and soft open set types as well as their structural characteristics. Semi-open sets, and semi-continuity of mappings were first proposed by Levine [1] in 1963. Njasta [2] then presented his approach of α-open sets in 1965. Mashhour et al. [3] developed the notion of pre-open set to analyze the pre-continuous mappings. The notion of β-open sets was proposed by Abd-El-Monsef et al. [4] in 1983 as a way to study β-continuous mappings. In [5, 6], the notion of b-open sets was thoroughly examined. Piotrowski [7] defined somewhat open sets to preseent somewhat continuity as stated in [8]. The notion of somewhere dense sets was proposed in [9, 10]. Additional characteristics of this notion were examined in [11]. Recently, Alqahtani and Abd El-latif [12], generalized almost all the previous notions by introducing the approach of N -open sets, in 2024. Mashhour et al. [13], presented the notion of supra open sets which consider the basic building blocks of supra topology ((abbreviated, STS)). They expanded on some basic topological concepts, including the continuity and separation axioms, as well as interior and closure operators. The notions of of supra α- [14] (respectively, pre- [15], b- [16], β- [17], R- [18], and semi- [19]) open sets have been presented and their primary characteristics have been presented. More operators on supra topological spaces [20–22] have been introduced. In the field of generalized soft open sets [23, 24], generalized soft continuity [25], soft semi-open sets and soft semi irresolute soft mappings [26, 27], various types of soft open sets and continuity [28], soft somewhere dense sets[29], and nearly soft β-open sets [30] , have been provided. Subsequent studies on soft continuity were carried out [31, 32]. The concept of the soft ideal was initially presented in [33]. Soft compactness [34], soft connectedness [35], soft generalized open sets [36–38], generalized (fuzzy) soft rough sets [39, 40], and soft separation axioms [41] are just a few of the topological properties that are generalized using this concept. El-Sheikh et al. proposed the concept of supra soft topological spaces [42]. Addi- tionally, they presented the notions of supra soft pre- (respectively, α, semi, β, and γ-) open sets. Subsequent research has examined numerous generalized supra soft operators through supra soft-b-open sets [43], supra soft-δi-open sets [44, 45], supra soft somewhere dense sets [46], and supra soft somewhat open sets [47]. We aim in this paper to present the approach of supra ϵ-open sets to supra topological spaces. Also, we go over the connections between our novel approach and the earlier related studies. In order to properly demonstrate the main ideas covered in the paper, we have included a geometric topological diagram [see Diagram 1]. SOregular(χ) −→SO(χ) −→ SαO(χ) −→ SSO(χ) −→ SβO(χ) −→ SRO(χ) ↓ ↓ ↗ ↓ SPO(χ) −→ SBO(χ) −→ SOϵ(χ) DIAGRAM 1. The connections between the new category and other preceding studies. Abd El-latif et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5969 3 of 19 Moreover, we provide the main features of this new category. In particular, we show that supra ϵ-open sets in an STS and their subspaces generally do not relate to one another. In addition to, the intersection of finite numbers of supra ϵ-open sets is not supra ϵ-open, generally. After that, we investigate new types of operators known as supra ϵ-interior (closure, accumulation, exterior, and boundary, respectively) operator using our recently established category of supra open sets. In order to demonstrate the significance of our new operators, we also provide some important examples and counterexamples. 2. Preliminaries and background Let (χ, ν) be an STS, the categories of supra (respectively, regular-, pre-, semi, β-, α-, b-, and R-) open sets will represented by SO(χ) (respectively, SOregular(χ), SPO(χ), SSO(χ), SβO(χ) ,SαO(χ), SBO(χ), and SRO(χ)) across this paper. Definition 1. [13] The collection ν ⊆ P (χ) is called supra topology (or STS) on χ if ν contains χ and ∅ and closed under arbitrary union. Also, if G ∈ ν, then G is called supra open set and Gc is called supra closed set. Moreover, SO(χ) will denote the class of all supra open sets. Definition 2. [13] For the subset K of an STS (χ, ν), the int(K) or K◦ (respectively, cl(K) or K, and b(K)) will denote the supra interior (respectively, closure, and boundary) of K, where int(K) = ∪{G : G ∈ ν and G ⊆ K}, cl(K) = ∩{N : N ∈ νc and K ⊆ N}, and b(K) = cl(K)\int(K). Theorem 1. [13] Regarding a subset T of an STS (χ, ν), we have (1) cl(T c) = [int(T )]c. (2) int(T c) = [cl(T )]c. Definition 3. [15–19] Let H be a subset of an STS (χ, ν). Then, (1) If H = int(cl(H)), then H ∈ SOregular(χ). (2) If H ⊆ int(cl(H)), then H ∈ SPO(χ). (3) If H ⊆ cl(int(H)), then H ∈ SSO(χ). (4) If H ⊆ int(cl(int(H))), then H ∈ SαO(χ). (5) If H ⊆ cl(int(cl(H))), then H ∈ SβO(χ). (6) If H ⊆ cl(int(H))∪̃int(cl(H)), then H ∈ SBO(χ). (7) If int(cl(H)) ̸= ∅, then H ∈ SRO(χ). (8) If int(cl(H)) = ∅, then H ∈ SND(χ). Abd El-latif et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5969 4 of 19 Definition 4. For the subset K of an STS (χ, ν), the class νK = {K ∩G : G ∈ ν} defines an STS on K, and it is called a subspace of (χ, ν). 3. Supra ϵ-open sets and relationships This part begins by presenting the definitions of supra ϵ-open and supra ϵ-closed sets and the properties based on them. We show that, this new category of supra open sets includes the previously comparable concepts of supra regular (α-, semi-, pre-, b-, β-, and R-) open sets. In addition, we have included a geometric topological diagram [see Diagram 1] to further demonstrate the basic concepts covered in the study. Moreover, we discuss the main features of this class. In particular, we demonstrate that instead of forming a topological space, our new category forms a supra topology. Definition 5. Let H be a subset of an STS (χ, ν). Then, H is called supra ϵ-open set if either H = ∅ or H ⊆ { b(H) ∪H ◦ , H ∈ SRO(χ), b(H), H ∈ SND(χ) and b(H) is infinite. Also, Hc is called supra ϵ-closed-set. The category of all supra ϵ-open (closed) sets will be indicated by SOϵ(χ) (SCϵ(χ)). Proposition 1. Every singleton {c} subset of an STS (χ, ν) is either supra ϵ-open or supra nowhere dense. Proof. Let {c} ⊈ SND(χ). Then, {c} ◦ ̸= ∅̃ and so {c} ∈ SRO(χ). Hence, {c} ∈ SOϵ(χ). On the other way, suppose that {c} ̸∈ SOϵ(χ), then {c} ⊈ {c} ◦ ∪b({c}) and so {c} ⊈ {c} ◦ . Therefore, {c} ∈ SND(χ). Remark 1. For the subset K of an STS (χ, ν), we have: (1) If K is a non-empty finite, closed and nowhere dense set, then K ̸∈ SOϵ(χ). (2) If K is a infinite and nowhere dense set, then K ∈ SOϵ(χ). Remark 2. As the authors demonstrated in [18], each supra regular (respectively, α-, semi-, pre -, b-, and β-) open set is a supra-R-open. Consequently from Definition 5, the reader can notice that, they are all supra ϵ-open. The following counterexample will demonstrate that our perspective on the aforemen- tioned remark is non-reversible, generally. Example 1. Consider the supra topology ν = {∅, T ⊆ R : −1 ∈ T or 0 ∈ T}, on the set of real numbers R. Regarding the set of natural numbers N, we have N ◦ = N◦ = ∅, and hence N ̸∈ SRO(χ). However, b(N) = N is infinite. Hence, N ∈ SOϵ(R). Abd El-latif et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5969 5 of 19 Corollary 1. The next implications are hold for an STS (χ, ν), which are not reversible. SOregular(χ) −→SO(χ) −→ SαO(χ) −→ SSO(χ) −→ SβO(χ) −→ SRO(χ) ↓ ↓ ↗ ↓ SPO(χ) −→ SBO(χ) −→ SOϵ(χ) DIAGRAM 1. The connections between the new category and other preceding studies. Supra ϵ-open sets in an STS (χ, ν) and their subspaces generally do not relate to one another, as shown in the upcoming example. Example 2. Let ν = {χ, ∅, {i, l}, {i, j, k}, {i, j, l}} be an STS on χ = {i, j, k, l}. Regarding the set W = {k, l}, we have νW = {W, ∅, {k}, {l}}. The set {k} is supra ϵ-open in (W, νW ) whereas {k} is not supra ϵ-open in (χ, ν). Definition 6. For the subset K of an STS (χ, ν), the class νK = {K ∩G : G ∈ SOϵ(χ)} defines an STS on K, and it is called an ϵ-subspace of (χ, ν). Proposition 2. Let (W, νW ) be an ϵ-subspace of an STS (χ, ν) and F be a subset of χ. Then, F ∈ SCϵ(W ) if and only if there is B ∈ SCϵ(χ) such that F =W ∩B. Proof. Obvious. Theorem 2. If Y ∈ SBO(χ) such that int(Y ) = ∅, for a proper subset Y of an STS (χ, ν), then both of Y and Y c are supra ϵ-open. Proof. Let Y ∈ SBO(χ), then Y ⊆ cl(int(Y ))∪̃int(cl(Y )). Since int(Y ) = ∅, Y ⊆ int(cl(Y )). Hence, Y ∈ SRO(χ). Given Remark 2, Y ∈ SOϵ(χ). Furthermore, we have [int(Y )]c = cl(Gc) = ∅c = χ. This implies, int(cl(Gc)) = int(χ) = χ ̸= ∅ and so Gc ∈ SRO(χ). Thus, Gc ∈ SOϵ(χ). Proposition 3. Each supra neighbourhood of any point in an STS (χ, ν) is supra ϵ-open. Proof. Suppose that S is a supra neighbourhood for x ∈ χ. Then, ∃ G ∈ ν such that x ∈ G ⊆ S. Hence, G ⊆ cl(G) ⊆ cl(S) and so S ∈ SRO(χ). Therefore, S ⊆ S ◦ ∪ b(S) = cl(S). Thus, S ∈ SOϵ(χ). Remark 3. Generally, the converse of Proposition 3 is untrue. Let ν = {χ, ∅, {1, 2}, {1, 3, 4}} be an STS on χ = {1, 2, 3, 4}. Then, {1} ∈ SOϵ(χ), however {1} is not supra neighbour- hood for any point in χ. Theorem 3. (1) If ψ = {Hȷ, ȷ ∈ π} ⊆ SOϵ(χ), then ⋃ ȷ∈πHȷ ∈ SOϵ(χ). (2) If ψ = {Hȷ, ȷ ∈ π} ⊆ SCϵ(χ)), then ⋂̃ ȷ∈πHȷ ∈ SCϵ(χ). Abd El-latif et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5969 6 of 19 Proof. (1) Let ψ = {Hȷ, ȷ ∈ π} ⊆ SOϵ(χ). If for all ȷ ∈ π, Hȷ = ∅, then we get our result. Now, if some members of ψ are non-empty, then we have to cases. Case (1): If ⋃ ȷ∈πHȷ ∈ SND(χ), then ⋃ ȷ∈πHȷ ◦ ⊆ ⋃ ȷ∈πHȷ ◦ = ∅. This means, Hȷ ∈ SND(χ) for each ȷ ∈ π, which follows b(Hȷ) is infinite for all ȷ ∈ π and hence b( ⋃ ȷ∈πHȷ) = ⋃ ȷ∈π b(Hȷ) is infinite. Therefore, we get our result. Case (2): If ( ⋃ ȷ∈πHȷ ∈ SRO(χ), then ( ⋃ ȷ∈πHȷ ◦ ) ̸= ∅. Hence, ⋃ ȷ∈πHȷ ⊆ ( ⋃ ȷ∈πHȷ ◦ ) ∪ b( ⋃ ȷ∈πHȷ) = ( ⋃ ȷ∈πHȷ). Therefore, ⋃ ȷ∈πHȷ ∈ SOϵ(χ). (2) By a similar way to (1). Remark 4. The next example shall prove that: (1) The intersection of finite numbers of supra ϵ-open sets is not supra ϵ-open, generally. (2) The union of finite numbers of supra ϵ-closed sets is not supra ϵ-closed, generally. Example 3. Let ν = {χ, ∅, {200, 300}, {100, 300}} be an STS on χ = {100, 200, 300}. Then, A = {100, 300} and B = {100, 200} are supra ϵ-open sets, however A ∩B = {100} is not supra ϵ-open. Also, C = {200} and D = {300} are supra ϵ-closed sets, however C ∪D = {200, 300} is not supra ϵ-closed. Remark 5. It is evident from Theorem 3 and Remark 4 that our new category does not form a topological space and instead forms a supra topology. 4. Applications of supra ϵ-open for new supra operators The objective of this section, is to outline novel kinds of operators, called supra ϵ- interior (respectively, closure, accumulation, exterior, and boundary) operator, using our new category of supra open sets. The primary characteristics of every operator are listed. We also give the distinctions between these new operators and the operators that corre- spond to them. Furthermore, in order to demonstrate the significance of our new operators, we provide several essential examples and counterexamples. Abd El-latif et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5969 7 of 19 Definition 7. For the subset K of an STS (χ, ν), the intϵ(K) will denote the supra ϵ-interior of K, where intϵ(K) = ∪{G : G ∈ SOϵ(χ) and G ⊆ K}. The proof of the next lemma is obvious from Definition 7, so it is omitted. Lemma 1. For the subsets K and I of an STS (χ, ν), we have the following: (1) u ∈ intϵ(K) ⇔ if there is I ∈ SOϵ(χ) such that u ∈ I ⊆ K. (2) K ∈ SOϵ(χ) ⇔ intϵ(K) = K . Theorem 4. For the supra ϵ-interior operator intϵ : P (χ) −→ P (χ) and E ∈ P (χ), we have intϵ(E) =  ∅, E ∈ SND(χ) and b(E) is finite. E ∩ b(E), E ∈ SND(χ) and b(E) is infinite. E, E ∈ SRO(χ). Proof. Assume contrary that, s ∈ E, where as E ∈ SND(χ) and b(E) is finite. Given Lemma 1 (1), there is I ∈ SOϵ(χ) such that s ∈ I ⊆ E. Since E ∈ SND(χ), I ∈ SND(χ) and so s ∈ b(I) ⊆ b(E). Given I ∈ SOϵ(χ), b(I) is infinite, then b(E) is also infinite, which contradicts our assumption. Hence, intϵ(E) = ∅. Now, Assume contrary that s ∈ E, where as E ∈ SND(χ) and b(E) is infinite. By the same technique, we can get intϵ(E) ⊆ E ∩ b(E) (1) On the other way, assume that s ∈ E ∩ b(E). Since E ∈ SND(χ) and b(E) is infinite, E ∈ SOϵ(χ), given Definition 7. By Lemma 1 (2), s ∈ E = intϵ(E). Hence, E ∩ b(E) ⊆ intϵ(E) (2) From Eqs (1) and (2), intϵ(E) = E ∩ b(E). Finally, If E ∈ SRO(χ), given Definition 7 and Lemma 1 (2), intϵ(E) = E. In the example that follows, we illustrate the previously mentioned theorem. Example 4. Consider the sets B = {0, 1, 2}, C = {2, 3, 4, 5} and the natural number set N, in Example 1, we have (1) N ∈ SND(χ) and b(N) is infinite, and hence intϵ(N) = N ∩ b(N) = N. (2) B ∈ SRO(χ), and so intϵ(B) = B. (3) C ∈ SND(χ) and b(C) is finite, and then intϵ(C) = ∅. Theorem 5. For the subsets K and I of an STS (χ, ν), we have the following: (1) If K ⊆ I, then intϵ(K) ⊆ intϵ(I). Abd El-latif et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5969 8 of 19 (2) int(K) ⊆ intϵ(K). Proof. (1) Suppose that u ∈ intϵ(K). Given Theorem 4, eitherK ∈ SND(χ) and b(K) is infinite or K ∈ SRO(χ), and in both situations, results in intϵ(K) = K. Hence, u ∈ intϵ(I), and therefore intϵ(K) ⊆ intϵ(I). (2) suppose that u ∈ int(K), then there exists G ∈ ν such that u ∈ G ⊆ K. Given (1), G ∈ SOϵ(χ) and u ∈ intϵ(G) = G ⊆ intϵ(K). Thus, u ∈ intϵ(K). Theorem 6. Let (χ, ν) be an STS and V,U ∈ P (χ). Then, (1) intϵ(χ) = χ and intϵ(∅) = ∅. (2) intϵ(V ) ⊆ (V ). (3) intϵ(intϵ(V )) = intϵ(V ). (4) intϵ[V ∩ U ] ⊆ intϵ(V ) ∩ intϵ(U). (5) intϵ(V ) ∪ intϵ(U) ⊆ intϵ[V ∪ U ]. Proof. Follows from Definition 7. Remark 6. The equality of Theorem 5 and Theorem 6 parts (2), (4) and (5) are not satisfied as shall shown in the provided counterexamples. Examples 1. Consider the sets B = {0, 1, 2, 3}, C = {2, 3, 4, 6}, D = {−1, 2, 4, 5} and the natural number set N, in Example 1, we have: (1) intϵ(C) = ∅ ⊆ intϵ(B) = B, however C ⊈ B. (2) intϵ(N) = N ⊈ int(N) = ∅. (3) C ⊈ intϵ(C) = ∅. (4) intϵ(B) ∩ intϵ(D) = {2} ⊈ intϵ[B ∩D] = intϵ({2}) = ∅. (5) intϵ(B∪C) = intϵ({0, 1, 2, 3, 4, 6}) = {0, 1, 2, 3, 4, 6} ⊈ intϵ(B)∪intϵ(C) = {0, 1, 2, 3}. Definition 8. Let C ∈ P (χ) be a subset of an STS (χ, ν), then clϵ(C) will denote the supra ϵ-closure of C, where clϵ(C) = ∩{N : N ∈ SCϵ(χ) and C ⊆ N}. Theorem 7. Given a subset J of an STS (χ, ν), then J has the following characteristics: Abd El-latif et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5969 9 of 19 (1) clϵ(J) = J ⇔ J ∈ SCϵ(χ)). (2) u ∈ clϵ(J) ⇔ J ∩G ̸= ∅ for each Gu ∈ SOϵ(χ). (3) clϵ(J) ⊆ cl(J). Proof. (1) Follows from Definition 8. (2) ”Necessity” Assume that there is Gu ∈ SOϵ(χ) such that J ∩ G = ∅, whereas u ∈ clϵ(J). Then, J ⊆ Gc. Given (1), clϵ(J) ⊆ Gc and u ̸∈ Gc. Therefore, u ̸∈ clϵ(J), which is a contradiction. ”Sufficient” Assume contrary that, u ̸∈ clϵ(J), then there is V ∈ SCϵ(χ) such that u ̸∈ V and J ⊆ V , and so u ∈ V c and V c ∩ J = ∅, where V c ∈ SOϵ(χ), which is a contradiction. (3) Assume that u ̸∈ cl(J). Then, J ∩ G = ∅, for some Gu ∈ ν. Hence, J ∩ G = ∅, for some Gu ∈ SOϵ(χ). Given (2), u ̸∈ clϵ(J). Theorem 8. For the supra ϵ-closure operator clϵ : P (χ) −→ P (χ) and E ∈ P (χ), we have clϵ(E) =  χ, Ec ∈ SND(χ) and b(Ec) is finite. E, Ec ∈ SND(χ) and b(Ec) is infinite. E, Ec ∈ SRO(χ). Proof. Much like the proof of Theorem 4. Abd El-latif et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5969 10 of 19 The relationship between the supra ϵ-closure operator and the supra ϵ-closure operator is examined in the forthcoming theorem. Theorem 9. Regarding a subset T of an STS (χ, ν), we have (1) clϵ(T c) = [intϵ(T )] c. (2) intϵ(T c) = [clϵ(T )] c. Proof. (1) Suppose that u ̸∈ [intϵ(T )] c. Then, u ∈ intϵ(T ). This implies that, ∃ G ∈ SOϵ(χ) such that u ∈ G ⊆ T , given Lemma 1, which follows T c ∩G = ∅. Hence, u ̸∈ clϵ(T c) from Theorem 7 (2). Thus, clϵ(T c) ⊆ [intϵ(T )] c (3) Now, assume that u ̸∈ clϵ(T c). Given Theorem 7 (2), there is Gu ∈ SOϵ(χ) such that T c ∩G = ∅ That means, u ∈ G ⊆ T Given Lemma 1, u ∈ intϵ(T ), and so u ̸∈ [intϵ(T )] c Therefore, [intϵ(T )] c ⊆ clϵ(T c) (4) From Eqs (3) and (4), clϵ(T c) = [intϵ(T )] c. (2) Through a method akin to (1). Proposition 4. Regarding subsets J and I of an STS (χ, ν). Then, Abd El-latif et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5969 11 of 19 (1) clϵ(∅) = ∅ and clϵ(χ) = χ. (2) J ⊆ clϵ(J). (3) clϵ(clϵ(J)) = clϵ(J). (4) If J ⊆ (I), then clϵ(J) ⊆ clϵ(I). (5) clϵ(J ∩ I) ⊆ clϵ(J) ∩ clϵ(I). (6) clϵ(J) ∪ clϵ(I) ⊆ clϵ(J ∪ I). Proof. Straightforward. Remark 7. The inclusions of part (3) in Theorem 7 and parts (2), (4), (5) and (6) in Proposition 4 are proper as the upcoming examples will demonstrate. Examples 2. Let ν = {χ, ∅, {2, 3}, {1, 3}} be an STS on χ = {1, 2, 3}. Consider the sets A = {1, 3}, C = {2}, D = {3} and E = {2, 3}. We have (1) cl(D) = χ ⊈ clϵ(D) = D. (2) clϵ(E) = χ ⊈ E. (3) E ⊈ A whereas clϵ(E) = χ ⊆ clϵ(A) = χ. (4) clϵ(A) ∩ clϵ(E) = χ ⊈ clϵ[A ∩ E] = clϵ(D) = D. (5) clϵ(C ∪D) = clϵ(E) = χ ⊈ clϵ(C) ∪ clϵ(D) = E. Definition 9. Given a subset T of an STS (χ, ν) with arbitrary point s ∈ χ. Then, s called a supra ϵ-accumulation point of T if each supra ϵ-open set Gs, we have [T\{s}] ∩G ̸= ∅. The set of all supra ϵ-accumulation points of T will denoted by accϵ(T ). Theorem 10. Let (χ, ν) be an STS and T ∈ P (χ). Then, (1) accϵ(T ) ⊆ acc(T ). (2) accϵ(T ) ⊆ T ⇔ T is a proper supra ϵ-closed set. Proof. (1) Let’s pretend that s ̸∈ acc(T ), then ∃ Gs ∈ ν such that [T\{s}] ∩ G = ∅. Then, Gs ∈ SOϵ(χ) such that [T\{s}] ∩G = ∅. Hence , s ̸∈ accϵ(T ). Abd El-latif et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5969 12 of 19 (2) (⇒) Pretend that s ̸∈ T for a proper subset T . Considering the condition, s ̸∈ accϵ(T ) and so there is Gs ∈ SOϵ(χ) such that [Gs\{s}] ∩ T = ∅. Since s ̸∈ T , Gs ∩ T = ∅ and so s ̸∈ clϵ(T ). Hence, clϵ(T ) ⊆ T . Nevertheless, we have T ⊆ clϵ(T ). Therefore, T = clϵ(T ). Thus, T is a proper supra ϵ-closed set. (⇐) Let s ̸∈ T for a proper supra ϵ-closed set T and then s ∈ T c for T c ∈ SOϵ(χ). Since T ∩ [T c\{s}] = ∅ for T c ∈ SOϵ(χ), s ̸∈ accϵ(T ). Therefore, accϵ(T ) ⊆ T . Proposition 5. Let (χ, ν) be an STS and T,H ∈ P (χ), then (1) If T ⊆ H, then accϵ(T ) ⊆ accϵ(H). (2) accϵ[T ∩H] ⊆ accϵ(T ) ∩ accϵ(H). (3) accϵ(T ) ∪ accϵ(H) ⊆ accϵ[T ∪H]. Proof. Follows from Definition 9 and Theorem 10. Remark 8. In Proposition 5, the reverse inclusions aren’t hold in general, as demonstrated by the upcoming examples. Examples 3. In Examples 2, consider the sets A = {1, 3} and B = {1, 2}. We have (1) accϵ(B) = {3} ⊆ accϵ(A) = {2, 3}, whereas B ⊈ A. (2) accϵ(A) ∩ accϵ(B) = {3} ⊈ accϵ[A ∩B] = ∅. (3) accϵ[A ∪B] = χ ⊈ accϵ(A) ∪ accϵ(B) = {2, 3}. Lemma 2. Given a subset T of an STS (χ, ν) with arbitrary point x ∈ χ. Then, x ∈ accϵ(T ) if and only if x ∈ accϵ(T\{x}). Proof. Follows from Lemma 5. Theorem 11. For any subset Z of an STS (χ, ν). (1) Z ∈ SCϵ(χ)) if and only if accϵ(Z) ⊆ Z. (2) Z ∪ accϵ(Z) ∈ SCϵ(χ). Proof. (1) Assume that Z is a supra ϵ-closed set and z ̸∈ Z. Then, Zc is a supra ϵ-open set with z ∈ Zc, and hence [Z\{z}] ∩ Zc = ∅. Therefore, z ̸∈ accϵ(Z), and thus accϵ(Z) ⊆ Z. Presently, we prove that Z ∈ SCϵ(χ)) which sufficient to prove that Zc ∈ SOϵ(χ)). So, let z ∈ Zc. Then, z ̸∈ Z. Given the condition, z ̸∈ accϵ(Z), and hence [Z\{z}]∩ Gz ̸= ∅, for some supra ϵ-open set Gz. Since z ̸∈ Z, Z ∩Gz ̸= ∅ and thus Gz ⊆ Zc. Therefore, Zc ∈ SOϵ(χ)), and consequently Z ∈ SCϵ(χ)). (2) Suppose that s ̸∈ Z ∪ accϵ(Z) and so s ̸∈ Z and s ̸∈ accϵ(Z). Hence, there is Gs ∈ SOϵ(χ) such that Abd El-latif et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5969 13 of 19 [Z\{s}] ∩Gs = ∅ and then s ̸∈ clϵ(Z). Hence, clϵ(Z) ⊆ Z ∪ accϵ(Z) (5) Now, suppose that s ̸∈ clϵ(Z). Given Theorem 7 (2), Nu ∩ Z ̸= ∅ for some Nu ∈ SOϵ(χ). Since, Z ⊆ clϵ(Z), s ̸∈ Z and hence [Z\{s}]∩Nu = ∅. Hence, s ̸∈ accϵ(Z). Therefore, Z ∪ accϵ(Z) ⊆ clϵ(Z) (6) According to Eqs 5 and 6, Z∪accϵ(Z) = clϵ(Z). Given Theorem 7 (1), Z∪accϵ(Z) ∈ SCϵ(χ). Corollary 2. Given a subset Z of an STS (χ, ν). Then, clϵ(Z) = Z ∪ accϵ(Z). Proof. It is derived from Theorem 11. Definition 10. If s ∈ [clϵ(Z)\intϵ(Z)] for an arbitrary point s and subset Z of an STS (χ, ν), then s is called a supra-ϵ-boundary point of Z. The supra-ϵ-boundary set of Z is the set of all upper-so-boundary points of Z, and it is represented by bϵ(Z). Also, the supra-ϵ-exterior of Z is also represented by extϵ(Z), where extϵ(Z) = intϵ(Z c). Theorem 12. Regarding a subset J of an STS (χ, ν), we have (1) bϵ(J) = clϵ(J) ∩ [intϵ(J)] c = clϵ(J) ∩ clϵ(Jc) = [intϵ(J)∪̃extϵ(J)]c. (2) bϵ(J) = bϵ(J c). Proof. (1) [intϵ(J)∪̃extϵ(J)]c = [intϵ(J)] c ∩ [intϵ(J c)]c = clϵ(J) ∩ [intϵ(J)] c from Theorem 9 (1) = clϵ(J) ∩ clϵ(Jc) = clϵ(J)\intϵ(J) = bϵ(J). (2) bϵ(J c) = clϵ(J c) ∩ [intϵ(J c)]c = [intϵ(J)] c ∩ clϵ(J) = bϵ(J). Theorem 13. Regarding a subset J of an STS (χ, ν), we have (1) clϵ(J) = intϵ(J)∪̃bϵ(J). (2) clϵ(J) = J∪̃bϵ(J). (3) intϵ(J) = J\bϵ(J). Proof. Abd El-latif et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5969 14 of 19 (1) intϵ(J)∪̃bϵ(J) = intϵ(J)∪̃[clϵ(J) ∩ [intϵ(J)] c] from Theorem 12 (1) = [intϵ(J)∪̃clϵ(J)] ∩ [intϵ(J)∪̃[intϵ(J)]c] = clϵ(J) ∩ χ = clϵ(J). (2) By a similar way to (1). (3) J\bϵ(J) = J ∩ [clϵ(J) ∩ [intϵ(J)] c]c = J ∩ [[clϵ(J)] c∪̃[intϵ(J)]] = [J ∩ [clϵ(J)] c]∪̃[J ∩ intϵ(J)] = ∅∪̃intϵ(J) = intϵ(J). Proposition 6. Regarding a subset J of an STS (χ, ν), the class {bϵ(J), intϵ(J), extϵ(J)} forms a partition for χ. Proof. bϵ(J) ∪ intϵ(J) ∪ extϵ(J) = [clϵ(J) ∩ [intϵ(J)] c] ∪ intϵ(J) ∪ [clϵ(J)] c = χ. Moreover, bϵ(J) ∩ intϵ(J) ∩ extϵ(J) = [clϵ(J) ∩ [intϵ(J)] c] ∩ intϵ(J) ∩ [clϵ(J)] c = ∅. Proposition 7. Regarding subsets T and J of an STS (χ, ν), we have (1) bϵ[intϵ(T )] ⊆ bϵ(T ). (2) bϵ[clϵ(T )] ⊆ bϵ(T ). (3) bϵ[T ∪ J ] ⊆ bϵ(T ) ∪ bϵ(J). (4) bϵ[T ∩ J ] ⊆ bϵ(T ) ∪ bϵ(J). Proof. (1) bϵ[intϵ(T )] = clϵ(intϵ(T )) ∩ [intϵ(intϵ(T ))] c ⊆ clϵ(T ) ∩ [intϵ(T )] c = bϵ(T ). (2) bϵ[clϵ(T )] = clϵ(clϵ(T )) ∩ [intϵ(clϵ(T ))] c = clϵ(T ) ∩ clϵ[clϵ(T )]c ⊆ clϵ(T ) ∩ [intϵ(T )] c = bϵ(T ). (3)-(4) Follows from Theorem 12. Remark 9. The inclusions of Proposition 7 are proper as shown in the next example. Example 5. Regarding the sets A = {1, 3}, C = {2}, D = {3} and E = {2, 3}, in Example 2, we have: (1) bϵ(C) = C ⊈ bϵ[intϵ(C)] = bϵ(∅) = ∅. (2) bϵ(E) = χ\{2, 3} = {1} ⊈ bϵ[clϵ(E)] = bϵ(χ) = ∅. (3) bϵ(A) ∪ bϵ(E) = {1, 2} ∪ {2} = {1, 2} ⊈ bϵ[A ∪ E] = bϵ(χ) = ∅. Abd El-latif et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5969 15 of 19 (4) bϵ(A) ∪ bϵ(E) = {1, 2} ⊈ bϵ[A ∩ E] = bϵ({3}) = ∅. Proposition 8. The following holds for a subset H of an STS (χ, ν): (1) bϵ(H) ∩H = ∅ if and only if H ∈ SOϵ(χ). (2) bϵ(H) ⊆ H if and only if H is a supra ϵ-closed set. (3) bϵ(H) = ∅ if and only if H is both supra ϵ-closed and supra ϵ-open set. Proof. (1) “ ⇒ ” Let bϵ(H) ∩ (H) = ∅, then [clϵ(H) ∩ [intϵ(H)]c] ∩ (H) = [intϵ(H)]c ∩H = ∅. Hence, H ⊆ intϵ(H). But, we have intϵ(H) ⊆ H. Thus, intϵ(H) = H and so H ∈ SOϵ(χ), given Proposition 1. “ ⇐ ” Obvious. (2) Clear. (3) “ ⇒ ” Assume that bϵ(H) = ∅, then clϵ(H)∩[intϵ(H)]c = ∅. Hence, clϵ(H) ⊆ intϵ(H). However, we have that intϵ(H) ⊆ clϵ(H). Thus, intϵ(H) = clϵ(H). Therefore, H is both supra ϵ-closed and supra ϵ-open set, given Theorem 6 (2) and Proposition 4 (2). “ ⇐ ” Obvious Theorem 14. bϵ(H) ∈ SCϵ(χ) for a subset H of an STS (χ, ν). Proof. If either clϵ(H) = χ or clϵ(H c) = χ, given Theorem 12 (1), we get the our proof. If clϵ(H) ̸= χ and clϵ(H c) ̸= χ, given Theorem 3 (2), bϵ(H) = clϵ(H) ∩ clϵ(Hc) ∈ SCϵ(χ). 5. Conclusion In this project, we introduce a novel weaker form of supra open sets, named supra ϵ- open sets and provide its essential features. The notions of supra regular ( α-, semi-, pre-, b-, β-, and R-) open sets, which were previously similar, are shown to be included in this new supra open set category. Moreover, we provide new types of operators named supra ϵ-interior (closure, accumulation, exterior, and boundary, respectively) using our recently established category of supra open sets. We also describe the differences between these new operators and their corresponding operators. Moreover, we prove that supra ϵ-interior Abd El-latif et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5969 16 of 19 operator, supra ϵ-boundary operator and supra ϵ-exterior operator form a partition for χ. Finally, we complement our investigations with many examples and counterexamples that highlight the significance of our innovative operators. Further research on the theoretical aspects of these generalized concepts might be conducted from the specific approaches presented in this work by examining the following topics: • Study some topological properties inspired by the specific approaches presented in this work, like supra continuity (separation axioms, connectedness, and compact- ness). • Examine whether these notions, in particular the separation axiom, may be applied to information systems. • Apply these approaches to soft ideal topological spaces [33, 48, 49], and supra soft topological spaces [42]. Acknowledgements The authors extend their appreciation to the Deanship of Scientific Research at North- ern Border University, Arar, KSA for funding this research work through the project number ”NBU-FFR-2025-1153-01”. Also, this study is supported via funding from Prince Sattam bin Abdulaziz University project number (PSAU/2025/R/1446) and this research is funded by Zarqa University Jordan. 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