EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6407 ISSN 1307-5543 – ejpam.com Published by New York Business Global Various Types of Supra ϵ-Separation Axioms and Relationships M. Aldawood 1, Alaa M. Abd El-latif2, Khaled A. Aldwoah3, A. A. Azzam1,4, Abdelhalim Hasnaoui2,∗, M. I. Elashiry2, Husham M. Attaalfadeel2, Enas H. Elkordy1,5 1 Department of Mathematics, Faculty of Science and Humanities, Prince Sattam Bin Abdulaziz University, Alkharj 11942, Saudi Arabia 2 Department of Mathematics, College of Science, Northern Border University, Arar 91431, Saudi Arabia 3 Department of Mathematics, Faculty of Science, Islamic University of Madinah, Medinah, Saudi Arabia 4 Department of Mathematics, Faculty of Science, New Valley University, Elkharga 72511, Egypt 5 Mathematics and Computer Science Department, Faculty of Science, Beni-Suef University, Beni Suef, Egypt Abstract. This manuscript presents a new weaker version of supra septarian axioms based on supra ϵ-open sets, along with its essential features, which are called supra-ϵ-Tj-space, j = 0, 1, 2, in the framework of supra topological spaces (or STSs). We give comprehensive explanations of each type of them, backed up by several examples and counterexamples that highlight the significance of our original approaches. We also provide a diagram that outlines these relationships. Additionally, we present the supra ϵ-symmetric property and supra difference property and study their effects on these version of supra-ϵ-septarian axioms. In especial, we show that the two concepts of supra- ϵ-T0-space and supra-ϵ-T1-space are the same for any STS that fulfills supra ϵ-symmetric property. Finally, we study the supra topological and supra hereditary properties for each of the previously discussed approaches. In particular, we show that the property of being a supra-ϵ-Tj-space, where j = 0, 1, 2, is a supra-hereditary (topological) property. 2020 Mathematics Subject Classifications: 54A05, 54C10, 54D10 Key Words and Phrases: Supra ϵ-separation axioms, Supra-ϵ-Hausdorff-space, Supra ϵ-symmetric property ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6407 Email addresses: m.aldawood@psau.edu.sa (M. Aldawood), alaa.ali@nbu.edu.sa, alaa−8560@yahoo.com (A. M. Abd El − latif), aldwoah@yahoo.com (K. A. Aldwoah), aa.azzam@psau.edu.sa (A. A. Azzam), Abdllhalim.hasanawa@nbu.edu.sa (A. Hasnaoui), Mustafa.elashiry@nbu.edu.sa (M. I. Elashiry), Husham.Alhassan@nbu.edu.sa (H. M. Attaalfadeel), e.elkordy@psau.edu.sa (E. H. Elkordy) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. Aldawood et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6407 2 of 15 1. Introduction The investigation of different kinds of generalized open, supra open, and soft open sets and their fundamental features has played a significant role in topological, supra topological, and soft topological research over the past few decades. In 1963, Levine [1] first proposed semi-open sets (continuity). After two years, Njasta [2] presented the concept of α-open sets. In 1982, Mashhour et al. [3] provided the concept of pre open sets (pre continuity). The notion of β-open sets ( β-continuity) was presented by Abd-El-Monsef et al. [4] in 1983. The definition of b-open sets was studied in detail in [5, 6] in 1966. Based on [7], Piotrowski [8] presented the notion of somewhat open sets (continuity). The approach of somewhere dense sets (or sd-sets) was proposed in [9, 10]. Additional facets of this idea were examined in [11]. Alqahtani introduced the approach of F-open [12]. The N -open sets approach was presented by Alqahtani and Abd El-latif [13] in 2024, and it is generalized almost all of the earlier concepts. Alghamdi et al. provided new types of operators in context of primal topological spaces [14]. The concept of supra open sets, which take into account the fundamental components of supra topology (or STS), was introduced by Mashhour et al. [15]. Among the basic topological concepts they developed were the separation axioms, continuity, and closure (interior) operators. Along with their key characteristics, the notions of supra semi- [16] (R- [17], β- [18], b- [19], pre- [20], and α [21]) open sets have been presented. Abd El-latif et al. [22] proposed the concept of supra ϵ-open sets in STSs. He and his coauthors [23] used this concept to investigate new forms of supra continuity. The field of broadly applicable soft open sets [24, 25], soft semi-open sets [26, 27], various kinds of soft continuity [28, 29], soft sd-sets [30, 31], and nearly soft β-open sets [32] has produced a variety of soft open sets and soft continuity. More studies on soft continuity were conducted later [33, 34]. In [35], the idea of the soft ideal was first introduced. After thet, Fatouh et al. [36] used soft semi-open sets to generalize this idea. Soft compactness [37], soft connectedness [38], soft generalized open sets [39], soft open sets via soft ideals [40, 41], soft separation axioms [42], generalized soft rough sets [43, 44], and congruence representations via soft ideals [45] are some of the topological characteristics that this concept is then used to generalize. Certain applications of soft δ-closed sets [46] and certain lower soft separation axioms [47] were recently introduced. The definition of supra soft topological space was introduced by El-Sheikh et al. [48]. Later research has examined several kinds of generalized supra soft operators using supra soft-b-open sets [49], supra generalized closed soft sets in terms of soft ideals [50, 51], supra soft sw-open sets [52], supra soft δi-open sets [53, 54], and soft separation axioms [55]. Recently, Abd El-latif et al. used the notion of supra soft sd-sets [56, 57] to present novel kinds of soft connectedness [58] and several types of compactness and connectedness [59, 60]. Alqahtani et al. presented the notion of soft nodec spaces [61, 62]. Our aim of this work is to provide new types of generalized separation axioms. In spe- cial, we present three new types of separation axioms inspired by supra ϵ-open sets named supra-ϵ-T0-space, supra-ϵ-T1-space, and supra-ϵ-T2-space. We give in-depth explanations of each of them, backed up by several examples and counterexamples that highlight the M. Aldawood et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6407 3 of 15 significance of our original ideas. Additionally, as shown in figure 1, we provide a diagram that summarizes their links and connections to earlier research. supra-T2-space =⇒ supra-T1-space =⇒ supra-T0-space ⇓ ⇓ ⇓ supra-ϵ-T2-space =⇒ supra-ϵ-T1-space =⇒ supra-ϵ-T0-space Diagram 1. The connections among various types of separation axioms in the context of STSs inspired by supra ϵ-open sets Finally, we assess the supra topological and supra hereditary properties for each of the concepts. In particular, we show that the property of being a supra-ϵ-Tj-space, where j = 0, 1, 2, is a supra-hereditary (topological) property. 2. Preliminaries and background Definition 1. [15] Supra topology (or STS) on λ is the family ϑ ⊆ 2λ which contains λ and ∅ and closed under arbitrary union. Additionally, H and Hc are referred to as supra open and supra closed sets, respectively, if H ∈ ϑ. Also, SO(λ) will also be used to indicate the class of all supra open sets. Furthermore, ϑ is referred to as an associated STS with σ if σ ⊂ ϑ for a given topology σ. Definition 2. [15] The ints(H) (cls(H), frs(H)) will indicate the supra interior (closure, boundary) for a subset H of an STS (λ, ϑ), where ints(H) = ∪{C : C ∈ ϑ and C ⊆ H}, cls(H) = ∩{D : D ∈ ϑc and H ⊆ D} and frs(H) = cls(H)\ints(H). Definition 3. [17] Let E be a subset of an STS (λ, ϑ). If ints(cls(E)) ̸= ∅, then E ∈ SRO(λ). Also, if ints(cls(E)) = ∅, then E ∈ SND(λ). Definition 4. [22] Regarding the subset S of an STS (λ, ϑ), the family ϑS = {S ∩ J : J ∈ ϑ} defines an STS on S, which is referred to as a supra subspace of (λ, ϑ). Definition 5. [22] A subset E of an STS (λ, ϑ) is referred to as supra ϵ-open set if either E = ∅ or E ⊆ { frs(E) ∪ ints(cls(E)), E ∈ SRO(λ), frs(E), E ∈ SND(λ) and frs(E) is infinite. Additionally, Ec is referred to as supra ϵ-closed-set. Furthermore, all supra ϵ-open (respectively, supra ϵ-closed) sets will be classified by SOϵ(λ) (respectively, SCϵ(λ)). Definition 6. [22] The intsϵ(S) (clsϵ(S)) will indicate the supra ϵ-interior (closure) of S for a subset S of an STS (λ, ϑ), where M. Aldawood et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6407 4 of 15 intsϵ(S) = ∪{J : J ∈ SOϵ(λ) and J ⊆ S} and clsϵ(S) = ∩{N : N ∈ SCϵ(λ) and S ⊆ N} Theorem 1. [22] If we consider a subset T of an STS (λ, ϑ) with σ ⊂ ϑ, we have that (1) clsϵ(T c) = [intsϵ(T )] c. (2) intsϵ(T c) = [clsϵ(T )] c. (3) int(T ) ⊆ ints(T ) ⊆ intsϵ(T ), where int(T ) refers the interior of T w.r.t σ. (4) clsϵ(T ) ⊆ cls(T ) ⊆ cl(T ), where cl(T ) refers the closure of T w.r.t σ. Definition 7. [22] Let T be a subset of an STS (λ, ϑ) with an arbitrary point s ∈ λ. If each supra ϵ-open set Js containing s, we have that [T\{s}] ∩ Js ̸= ∅, then s is referred to as a supra ϵ-accumulation point of T . The notation accϵ(T ) will represent the set of all supra ϵ-accumulation points of T . 3. Separation axioms inspired by supra ϵ-open sets and relationships In this section, we present three new types of separation axioms inspired by supra ϵ-open sets named supra-ϵ-T0-space, supra-ϵ-T1-space, and supra-ϵ-Hausdorff-space. We provide thorough descriptions of each of them. Specifically, we explore sufficient conditions for several analogous linkages between them and generally illustrate their key characteris- tics. Moreover, we propose a diagram [see diagram 1] that summarizes their relationships. Furthermore, we introduce the supra ϵ-symmetric property and demonstrate that, for any STS that satisfies it, the two approaches of supra-ϵ-T0-space and supra-ϵ-T1-space are identical. Definition 8. An STS (λ, ϑ) is said to be (1) Supra-ϵ-T0-space if for each two distinct points there is a supra-ϵ-open set including one but excluding the other. (2) Supra-ϵ-T1-space if for each two distinct points ν1, ν2 ∈ λ, then there are two supra-ϵ- open subsets µ1 and µ2 of λ, such that ν1 ∈ µ1, ν2 /∈ µ1, and ν1 /∈ µ2, ν2 ∈ µ2. (3) Supra-ϵ-T2-space ”supra-ϵ-Hausdorff space” if for each two distinct points ν1, ν2 ∈ λ, then there are two disjoint supra-ϵ-open subsets µ1 and µ2 of λ, such that ν1 ∈ µ1 and ν2 ∈ µ2. Theorem 2. (1) Any supra-ϵ-Tj-space is supra-ϵ-Tj−1, j = 1, 2. (2) Any supra-Tj-space is supra-ϵ-Tj, j = 1, 2. Proof. It is clear from Definition 8 and form the fact that every supra open set is supra ϵ-open. M. Aldawood et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6407 5 of 15 Remark 1. The converse of Theorem 2 is not hold as the upcoming examples will demon- strate. Example 1. (1) Let ϑ = {λ, ∅, {b}} be an STS on λ = {v, b, n}. Then we have that SOϵ(λ) = {λ, ∅, {v, b}, {b, n}, {b}}. It easy to check that, λ is supra-ϵ-T0-space, how- ever λ is not supra-ϵ-T1, since v ̸= n ∈ λ, however there are not two supra-ϵ-open subsets of λ separate them. (2) Let ϑ = {λ, ∅, {a, s}, {a, s, d}, {a, d}, {s, d}} be an STS on λ = {a, s, d, f}. Then we have that SOϵ(λ) = {λ, ∅, {a, s}, {a, d}, {s, d}, {a, s, d}, {a, s, f}, {a, d, f}, {s, d, f}}. It follows that, λ is supra-ϵ-T1-space, however λ is not supra-ϵ-T2, since d ̸= f ∈ λ, however there are not two disjoint supra-ϵ-open subsets of λ separates them. . (3) In (1), we have that λ is supra-ϵ-T0-space, however λ is not supra-T0, since v ̸= n ∈ λ, however there are not two supra open subsets of λ separate them. (4) Let ϑ = {λ, ∅, {a, s}, {d, f}, {a, d}, {s, f}, {s, d}, {a, s, d}, {a, s, f}, {a, d, f}} be an STS on λ = {a, s, d, f}. Then we have that SOϵ(λ) = ϑ. It follows that, λ is supra-ϵ-T1- space, however λ is not supra-T1, since {a} ̸∈ ϑc. Proposition 1. For an STS (λ, ϑ), the following implications are held, which are not reversible, depending on the previously mentioned results. supra-T2-space =⇒ supra-T1-space =⇒ supra-T0-space ⇓ ⇓ ⇓ supra-ϵ-T2-space =⇒ supra-ϵ-T1-space =⇒ supra-ϵ-T0-space Diagram 1. The connections among various types of separation axioms in the context of STSs inspired by supra ϵ-open sets Theorem 3. Any STS (λ, ϑ) is supra-ϵ-T0-space. Proof. Let λ be an STS and let ν1 ̸= ν2 in λ. This follows that, either λ \ {ν1} ∈ SRO(λ) or λ \ {ν1} ∈ SND(λ). If λ \ {ν1} ∈ SND(λ), then λ \ {ν1} ≠ λ, and so λ \ {ν1} ∈ ϑc; and hence {ν1} ∈ ϑ. Therefore, {ν1} is a supra ϵ-open set containing ν1, but not ν2. In addition, if λ \ {ν1} ∈ SRO(λ), then λ \ {ν1} ⊆ λ \ {ν1} ◦ ∪ frs(λ \ {ν1}) = λ \ {ν1} is a supra ϵ-open set containing ν2, but not ν1. Consequently, λ is supra-ϵ-T0-space. Lemma 1. Any infinite subset of an STS (λ, ϑ) is supra-ϵ-open. Proof. Let H be any infinite subset of an STS (λ, ϑ). Then, H is either H ∈ SRO(λ) or H ∈ SND(λ). If H ∈ SND(λ), then H ⊆ H ◦ ∪ frs(H) = frs(H), frs(H) is infinite. Hence, H ∈ SOϵ(λ). Also, if H ∈ SRO(λ), then H ⊆ H ◦ ∪ frs(H) = H ∈ SOϵ(λ). Theorem 4. Any supra-T1-space is supra-ϵ-T2. M. Aldawood et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6407 6 of 15 Proof. Let (λ, ϑ) be a supra-T1-space and let ν1 ̸= ν2 in v. If λ is finite, then λ is the discrete space, hence λ is supra-ϵ-T2-space. Now, assume that λ is infinite. Then, there exist two infinite disjoint K and H subsets of λ, containing ν1 and ν2, respectively. By Lemma 1, K and H are supra-ϵ-open sets. Therefore, λ is supra-ϵ-T2-space. Corollary 1. Any infinite STS (λ, ϑ) is supra-ϵ-T2. Proof. It follows from Lemma 1. Definition 9. [63] If H ∈ ψ implies that H\{ν} ∈ ψ for ν ∈ λ, then a subfamily ψ ⊆ 2λ in a nonempty set λ is considered to have the difference property. Proposition 2. Any STS (λ, ϑ) has the difference property for the category SOϵ(λ) is supra-ϵ-T1-space. Proof. Let ν1 ̸= ν2 ∈ λ. Given the difference property for λ ∈ SOϵ(λ), we have that λ\{ν1} ∈ SOϵ(λ) and λ\{ν2} ∈ SOϵ(λ) which separate ν1 and ν2. Consequently, λ is supra-ϵ-T1-space. Theorem 5. For any STS (λ, ϑ), the following are equivalent: (1) λ is supra-ϵ-T0-space; (2) For each ν1 ̸= ν2 ∈ λ, clsϵ({ν1}) ̸= clsϵ({ν2}); (3) For each ν ∈ λ, accϵ({ν}) = ∪{G : G ∈ SCϵ(λ)}. Proof. (1) ⇒ (2) Let ν1 ̸= ν2 ∈ λ. Given (1), there is a supra-ϵ-open D set including one point (say ν1) but excluding the other. This follows that, ν1 ∈ D and D∩{ν2} = ∅. Hence, ν1 ̸∈ clsϵ({ν2}), however ν1 ∈ clsϵ({ν1}). Thus, clsϵ({ν1}) ̸= clsϵ({ν2}). (2) ⇒ (3) Let ω ∈ accϵ({ν}), then ω ̸= ν and ω ∈ accϵ({ν}) ∪ {ν} = clsϵ({ν}). Hence, ω ∈ clsϵ{ω} ⊆ clsϵ({ν}) = accϵ({ν}) ∪ {ν}. Since ω ̸∈ {ν}, ω ∈ accϵ({ν}). Therefore, ω ∈ clsϵ{ω} ⊆ accϵ({ν}), and consequently accϵ({ν}) = ∪{clsϵ{ω} : ω ∈ accϵ({ν})}. (3) ⇒ (1) Let ν1 ̸= ν2 ∈ λ. Then, either ν2 ∈ accϵ({ν1}) or ν2 ̸∈ accϵ({ν1}). If ν2 ∈ accϵ({ν1}), then there is H ∈ SCϵ(λ) such that ν2 ∈ H ⊆ accϵ({ν1}). Since ν1 ̸∈ accϵ({ν1}, ν1 ̸∈ H, and so ν1 ∈ Hc and ν2 ̸∈ Hc, Hc ∈ SOϵ(λ). Thus, λ is supra-ϵ- T0-space. Additionally, if ν2 ̸∈ accϵ({ν1}), then there is K ∈ SCϵ(λ) such that ν2 ∈ K and ν1 ̸∈ K. Thus, λ is supra-ϵ-T0-space. Definition 10. A subset S of an STS (λ, ϑ) is called supra ϵ-dense if clsϵ(S) = λ. Corollary 2. If (λ, ϑ) is a supra-ϵ-T0-space, then there is at most a supra ϵ-dense singleton set in λ. M. Aldawood et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6407 7 of 15 Proof. It is follows from Theorem 5. Theorem 6. For any STS (λ, ϑ), the following are equivalent: (1) λ is supra-ϵ-T1-space; (2) For each ν ∈ λ, {ν} ∈ SCϵ(λ); (3) ∩{H : H ∈ SOϵ(λ) and C ⊆ H} = C; (4) For each ν ∈ λ, accϵ({ν}) = ∅. Proof. (1) ⇒ (2) Let ν ∈ λ. We prove that {ν}c ∈ SOϵ(λ), so let ω ∈ {ν}c. Then, ω ̸= ν. Given (1), there is Hω ∈ SOϵ(λ) such that ω ∈ Hω and ν ̸∈ Hω. Hence, ω ∈ Hω ⊆ {ν}c, and consequently {ν}c ∈ SOϵ(λ). Therefore, we get the desired outcome. (2) ⇒ (3) Let K ⊆ λ. Given (2), for all ν ∈ Kc we have that {ν} ∈ SCϵ(λ). Then, {ν}c ∈ SOϵ(λ). Hence, K ⊆ {H : H ∈ SOϵ(λ) and K ⊆ H} ⊆ {{ν}c : ν ∈ Kc} ⊆ K. Therefore, K = {H : H ∈ SOϵ(λ) and K ⊆ H}. (3) ⇒ (4) Assume the contrary that accϵ({ν}) ̸= ∅ for some ν ∈ λ, then there is ω ̸= ν such that ω ∈ accϵ({ν}). Hence, for every supra ϵ-open set Gω containing ω we have that [Gω\{ω}] ∩ {ν} ≠ ∅ and so ν ∈ Gω\{ω}. This means that, every supra ϵ-open set Gω containing ω also contains ν. Therefore, ∩{Gω : Gω ∈ SOϵ(λ) and {ω} ⊆ Gω} ̸= {ω}, which contradicts (3). Thus, for each ν ∈ λ, accϵ({ν}) = ∅. (4) ⇒ (1) Let ν1 ̸= ν2 ∈ λ. Given (4), accϵ({ν1}) = ∅ and accϵ({ν2}) = ∅ which follows clϵ({ν1}) = {ν1} ∪ accϵ({ν1}) = {ν1} and clϵ({ν2}) = {ν2} ∪ accϵ({ν2}) = {ν2}. Hence, {ν1}c and {ν2}c ∈ SOϵ(λ) which separate ν1 and ν2. Consequently, λ is supra-ϵ-T1-space. Definition 11. The space (λ, ϑ) is referred to as supra ϵ-symmetric if ν1 ∈ clϵ({ν2}) demonstrates that ν2 ∈ clϵ({ν1}) for ν1 ̸= ν2 ∈ λ. Theorem 7. Every supra ϵ-symmetric and supra-ϵ-T0-space (λ, ϑ) is supra-ϵ-T1. Proof. Let ν1 ̸= ν2 ∈ λ. Since λ is supra-ϵ-T0-space, there is a supra-ϵ-open set H including one point (say ν1) but excluding the other and so ν1 ̸∈ clϵ({ν2}). Given λ is supra ϵ-symmetric, ν2 ̸∈ clϵ({ν1}). Therefore, [clϵ({ν1})]c and [clϵ({ν2})]c are two supra-ϵ-open sets which separate ν1 and ν2. Thus, λ is supra-ϵ-T1-space. Corollary 3. Every supra ϵ-symmetric space is supra-ϵ-T0-space if and only if it is supra- ϵ-T1. M. Aldawood et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6407 8 of 15 Proof. It is direct from Theorem 2 and Theorem 7. Theorem 8. For any STS (λ, ϑ), the following are equivalent: (1) λ is supra-ϵ-T2-space; (2) ψ = {(ν, ν) : ν ∈ λ} is supra-ϵ-closed subset of λ× λ; (3) ∩{Hν : Hν ∈ SCϵ(λ)} = {ν}, for each ν ∈ λ. Proof. (1) ⇒ (2) Let (ν1, ν2) ∈ λ× λ\ψ, then ν1 ̸= ν2. Given (1), there are two disjoint supra- ϵ-open subsets µ1 and µ2 of λ, such that ν1 ∈ µ1 and ν2 ∈ µ2. Hence, (ν1, ν2) ∈ µ1 × µ1 ⊆ λ × λ\ψ, and thus λ × λ\ψ is a supra supra-ϵ-neighborhood for each of its points. Therefore, ψ is supra-ϵ-closed subset of λ× λ. (2) ⇒ (1) Assume that ψ = {(ν, ν) : ν ∈ λ} is supra-ϵ-closed subset of λ × λ and ν1 ̸= ν2 ∈ λ, then λ × λ\ψ is supra-ϵ-open set including (ν1, ν2). Hence, there are A,B ∈ SOϵ(λ) such that (ν1, ν2) ∈ A × B ⊆ λ × λ\ψ. Therefore, A,B ∈ SOϵ(λ) separate ν1 and ν2 with A ∩B = ∅. Thus, λ is supra-ϵ-T2-space. (1) ⇒ (3) Let λ be a supra-ϵ-T2-space. Then, for any ν1 ̸= ν2 ∈ λ, there are two disjoint supra-ϵ-open subsets µ1 and µ2 of λ, such that ν1 ∈ µ1 and ν2 ∈ µ2. This implies that ν1 ∈ clϵ(µ1) ⊆ µc2, µ c 2 ∈ SCϵ(λ) which containing ν1 but not ν2. Hence, ∩{µc2 : ν1 ∈ µc2 ∈ SCϵ(λ)} = {ν1}. (3) ⇒ (1) Let ν1 ̸= ν2 ∈ λ. Given (3), ∩{Hν1 : Hν1 ∈ SCϵ(λ)} = {ν1}. This means that, there is Hν1 ∈ SCϵ(λ) including ν1 but not ν2. Hence, there is Oν1 ∈ SOϵ(λ) such that ν1 ∈ clϵ(Oν1) ⊆ Hν1 , and consequently Oν1 , [clϵ(Oν1)] c ∈ SOϵ(λ) separate ν1 and ν2 with [clϵ(Oν1)] c ∩Oν1 = ∅. Therefore, λ is supra-ϵ-T2-space. 4. More features of supra ϵ-separation axioms Herein, we study the supra hereditary property and supra topological property for each aforementioned notion. In special, we show that he property of being a supra-ϵ-Tj-space, j = 0, 1, 2, is a supra hereditary property. Moreover, we show that he property of being a supra-ϵ-Tj-space, j = 0, 1, 2, is a supra topological property under special types of supra-ϵ- functions. Definition 12. For the subset L of an STS (λ, ν), the class νL = {L ∩O : O ∈ SOϵ(λ)} defines an STS on L, and it is called a supra-ϵ-subspace of (λ, ν). M. Aldawood et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6407 9 of 15 Since Definition 12 provides a clear evidence for the next two propositions, their proofs are excluded. Proposition 3. Let (U, νU ) be an supra ϵ-subspace of an STS (λ, ν) and V be a subset of λ. Then, (clϵ(V ))νU = U ∩ clϵ(V ). Proposition 4. Let (U, νU ) be an supra ϵ-subspace of an STS (λ, ν) and V be a subset of λ. Then, V ∈ SCϵ(U) if and only if there is N ∈ SCϵ(λ) such that V = U ∩N . Theorem 9. Every supra-ϵ-subspace of supra-ϵ-Tj-space is supra-ϵ-Tj, j = 0, 1, 2. Proof. The other cases are evidently contained in the case of j = 2, which we prove. Assume that (χ, ϑχ) is a supra subspace of supra-ϵ-T2-space (λ, ϑ) and ν1 ̸= ν2 ∈ χ ⊆ λ. Given λ is supra-ϵ-T2, then there are two disjoint supra-ϵ-open subsets µ1 and µ2 of λ, such that ν1 ∈ µ1 and ν2 ∈ µ2. Hence, ν1 ∈ µ1 ∩ χ and ν2 ∈ µ2 ∩ χ such that [µ1 ∩ χ] ∩ [µ2 ∩ χ] = χ ∩ [µ1 ∩ µ1] = χ ∩ ∅ = ∅ and µ1 ∩ χ, µ2 ∩ χ ∈ ϑχ. Therefore, χ is supra-ϵ-T2-space. Definition 13. A function Λϵ : (λ1, σ1) → (λ2, σ2) with ϑ1 as an associated STS with σ1 is said to be a supra ϵ-continuous (abbreviate: supra ϵ-cts) if Λ−1 ϵ (G) ∈ SOϵ(λ1) for each G ∈ σ2. Theorem 10. If Λϵ : (λ1, σ1) → (λ2, σ2) is an injective supra ϵ-cts function with ϑ1 as an associated STS with σ1 such that (λ2, σ2) is Tj-space, then (λ1, ϑ1) is a supra-ϵ- Tj , j = 0, 1, 2. Proof. The other cases are evidently contained in the case of j = 2, which we prove. Let ν1 ̸= ν2 ∈ λ1. Since Λϵ is injective, there are ζ1 ̸= ζ2 ∈ λ2 such that Λϵ(ν1) = ζ1 and Λϵ(ν2) = ζ2. Since (λ2, σ2) is T2-space, there are two disjoint open subsets µ1 and µ2 of λ2, such that ζ1 ∈ µ1 and ζ2 ∈ µ2. Given Λϵ is supra ϵ-cts, Λ −1 ϵ (µ1) and Λ−1 ϵ (µ2) are two disjoint supra-ϵ-open subsets of λ1 containing ν1, ν2, respectively. Therefore, (λ1, ϑ1) is supra-ϵ-T2. Definition 14. A function Λϵ : (σ1, ν1) → (σ2, ν2) with ϑ1, ϑ2 associated STSs with σ1, σ2, respectively, is said to be supra ϵ-irresolute if Λ−1 ϵ (D) ∈ SOϵ(λ1) for each D ∈ SOϵ(λ2). Proposition 5. If Λϵ : (λ1, σ1) → (λ2, σ2) is an injective supra ϵ-irresolute function with ϑ1, ϑ2 associated STSs with σ1, σ2 respectively, such that (λ2, σ2) is supra-ϵ-Tj-space, then (λ1, ϑ1) is a supra-ϵ-Tj , j = 0, 1, 2. Proof. The other cases are evidently contained in the case of j = 2, which we prove. Let ν1 ̸= ν2 ∈ λ1. Since Λϵ is injective, there are ζ1 ̸= ζ2 ∈ λ2 such that Λϵ(ν1) = ζ1 and Λϵ(ν2) = ζ2. Since (λ2, σ2) is supra-ϵ-T2, there are two disjoint supra-ϵ-open subsets µ1 and µ2 of λ2, such that ζ1 ∈ µ1 and ζ2 ∈ µ2. Given Λϵ is supra ϵ-irresolute, Λ−1 ϵ (µ1) and Λ−1 ϵ (µ2) are two disjoint supra-ϵ-open subsets of λ1 containing ν1, ν2, respectively. Therefore, (λ1, ϑ1) is supra-ϵ-T2. M. Aldawood et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6407 10 of 15 Definition 15. A function Λϵ : (λ1, σ1) → (λ2, σ2) with ϑ1, ϑ2 associated STSs with σ1, σ2 respectively, is said to be supra ϵ (ϵ∗)-open if Λϵ(U) ∈ SOϵ(λ2) for each U ∈ σ1 (U ∈ SOϵ(λ1)). Theorem 11. Let Λϵ : (λ1, σ1) → (λ2, σ2) be a function with ϑ2 as an associated STS with σ2 and α ⊆ λ1, then Λϵ is supra ϵ-open if and only if Λϵ(int(α)) ⊆ intsϵ [Λϵ(α)] ∀ α ⊆ λ1. Proof. “ ⇒ ” Let Λϵ be a supra ϵ-open function and α ⊆ λ1. Since int(α) ⊆ α, Λϵ(int(α)) ⊆ Λϵ((α)), which leads to Λϵ(int(α)) = intsϵ [Λϵ(int(α))] ⊆ intsϵ [Λϵ((α))], given Λϵ is supra ϵ-open. “ ⇐ ” Suppose that α ∈ σ1. Considering the condition, Λϵ(α) = Λϵ(int(α)) ⊆ intsϵ [Λϵ(α)] . However, we have that intsϵ [Λϵ(α)] ⊆ Λϵ(α). Hence, intsϵ [Λϵ(α)] = Λϵ(α). Therefore, Λϵ(α) ∈ SOϵ(λ2), and consequently Λϵ is a supra ϵ-open function. Proposition 6. Let Λϵ : (λ1, σ1) → (λ2, σ2) be a function with ϑ1, ϑ2 associated STSs with σ1, σ2 respectively, and α ⊆ λ1, then Λϵ is supra ϵ∗-open if and only if Λϵ(int s ϵ(α)) ⊆ intsϵ [Λϵ(α)] ∀ α ⊆ λ1. Proof. It is similar to the proof of Theorem 11. Theorem 12. The image of each Tj-space is a supra-ϵ-Tj under a bijective supra ϵ-open function, j = 0, 1, 2. Proof. The other cases are evidently contained in the case of j = 2, which we prove. Let Λϵ : (λ1, σ1) → (λ2, σ2) with ϑ1, ϑ2 associated STSs with σ1, σ2 respectively, be a bijective supra ϵ-open function such that (λ1, ϑ1) is T2-space. Let θ1 ̸= θ2 ∈ λ2. Since Λϵ is bijective, there are ξ1 ̸= ξ2 ∈ λ1 such that Λϵ(ξ1) = θ1 and Λϵ(ξ2) = θ2. Since (λ1, σ1) is T2-space, there are two disjoint open subsets ρ1 and ρ2 of λ2, such that ξ1 ∈ ρ1 and ξ2 ∈ ρ2. Given Λϵ is supra ϵ-open, Λϵ(ρ1) and Λϵ(ρ2) are two disjoint supra-ϵ-open subsets of λ2 containing θ1, θ2, respectively. Therefore, (λ2, ϑ2) is supra-ϵ-T2. Corollary 4. The image of each supra-ϵ-Tj-space is a supra-ϵ-Tj under a bijective supra ϵ∗-open function, j = 0, 1, 2. M. Aldawood et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6407 11 of 15 Proof. It is deduced using a similar procedure to that of Theorem 12. Definition 16. A function Λϵ : (λ1, σ1) → (λ2, σ2) with ϑ1, ϑ2 associated STSs with σ1, σ2 respectively, is said to be supra-ϵ∗-homeomorphism if it is bijective supra ϵ∗-open and supra ϵ∗-cts Lemma 2. If Λϵ : (λ1, σ1) → (λ2, σ2) is supra-ϵ∗-homeomorphism function with ϑ1, ϑ2 associated STSs with σ1, σ2 respectively, then (λ2, σ2) is supra-ϵ-Tj-space if and only if (λ1, ϑ1) is supra-ϵ-Tj , j = 0, 1, 2. Proof. It is follows from Proposition 5 and Corollary 4. 5. Conclusion A new weaker version of supra septarian axioms based on supra ϵ-open sets is presented in this manuscript, along with its key characteristics named supra-ϵ-Tj-space, j = 0, 1, 2. In detail, we present three new types of separation axioms inspired by supra ϵ-open sets named supra-ϵ-T0-space, supra-ϵ-T1-space, and supra-ϵ-Hausdorff-space. We provide thor- ough descriptions of each of them supported with several examples and counterexamples that demonstrate the importance of our novel concepts. Specifically, we explore sufficient conditions for several analogous linkages between them and generally illustrate their key characteristics. Furthermore, we propose a diagram that encapsulates their connections [see figure 1]. Additionally, we present the supra ϵ-symmetric property and show that the two notions of supra-ϵ-T0-space and supra-ϵ-T1-space are the same for any STS that fulfills it. Finally, for each of the previously described concepts, we examine the supra topological and supra hereditary properties. Specifically, we demonstrate that the property of being a supra-ϵ-Tj-space, where j = 0, 1, 2, is a supra-hereditary (topological) property. From the specific approaches described in this paper, additional research on the theoretical aspects of these generalized concepts could be carried out by looking at the following subjects: • Introducing more types of septarian axioms based on supra ϵ-open sets, like supra- ϵ-completely space, supra-ϵ-regular-space, supra-ϵ-completely regular space, supra- ϵ-normal-space, supra-ϵ-T3-space, and supra-ϵ-T4-space. • Considering whether information systems can benefit from the use of these kinds of separation axioms. • Introducing theses concepts to fuzzy supra soft topological spaces [64, 65]. • Applying our new notions to rough approximations based on relations with decision making applications [66]. Conflicts of interest In relation to the publication of this work, the authors declare that they have no competing interests. M. Aldawood et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6407 12 of 15 Author contributions Each author’s contribution was equal. 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