EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 6020 ISSN 1307-5543 – ejpam.com Published by New York Business Global Novel Types of Supra Functions Inspired by Supra ϵ-Open Sets Alaa M. Abd El-latif1, Radwan Abu-Gdairi2, A. A. Azzam3,4, F. A. Gharib1,∗, Husham M. Attaalfadeel1, Walid Abdelfattah1, Shaaban M. Shaaban5, M. Aldawood3 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 Abstract. Using the concept of supra ϵ-open sets, this manuscript discusses and investigates new forms of supra continuity. More specifically, we introduce the concept of supra ϵ-continuous functions, which built upon the previous types of weaker forms of such notions. The relationships between our new class and existing previous supra continuity notions were examined using the diagram in Figure 1. Furthermore, the essential features of this concept are analyzed, as well as its analogous circumstances. Additionally, the notions of supra ϵ-irresolute functions and supra ϵ∗-cts functions were introduced. Moreover, we prove that the composition of supra ϵ-irresolute function and supra ϵ-cts function (respectively, supra ϵ-cts function and cts function is supra ϵ-cts, two supra ϵ-irresolute functions) is supra ϵ-cts (respectively, supra ϵ-cts, supra ϵ-irresolute). Also, we provide three new approaches for supra functions named supra ϵ-open functions, supra ϵ-closed functions, and supra ϵ-homeomorphism functions. We conclude with a detailed discussion of their key characteristics and provide several essential examples. 2020 Mathematics Subject Classifications: 54A05, 54C10, 54C08 Key Words and Phrases: Supra ϵ-open set, Supra ϵ-continuous functions, Supra ϵ-irresolute functions, supra ϵ-homeomorphism functions ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.6020 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), fatouh.gharib@nbu.edu.sa (F. A. Gharib), Husham.Alhassan@nbu.edu.sa (H. M. Attaalfadeel), walid.abdelfattah@nbu.edu.sa (W. Abdelfattah), shabaan27@gmail.com (S. M. Shaaban), m.aldawood@psau.edu.sa (M. Aldawood) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. M. Abd El-latif et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6020 2 of 17 1. Introduction The study of different types of generalized continuous functions, supra continuous functions, and soft continuous functions and their structural properties has been a major area of topological, supra topological, and soft topological research in the last several decades. Levine [1] initially introduced semi-open sets and semi-continuity of functions in 1963. Then, in 1965, Njasta [2] introduced his α-open sets approach. The concept of pre-open set was proposed by Mashhour et al. [3] in order to investigate pre-continuous functions. Abd-El-Monsef et al. [4] introduced the concept of β-open sets in 1983 as a means of studying β-continuous functions. The concept of b-open sets was studied in detail in [5, 6]. According to [7], Piotrowski [8] defined relatively open sets to present somewhat continuity. In [9, 10], the concept of somewhere dense sets was presented. Other aspects of this concept were studied in [11]. In 2024, Alqahtani and Abd El-latif [12] introduced the N -open sets approach, which generalized nearly all of the previously proposed concepts. The concept of supra open sets, which take into account the fundamental components of supra topology (abbreviated, STS), was introduced by Mashhour et al. [13]. The continuity and separation axioms, as well as interior and closure operators, were among the fundamental topological notions they elaborated on. The concepts of supra α [14] (pre- [15], b- [16], β- [17], R- [18], and semi- [19]) open sets have been introduced, along with their main features. Several types of soft open sets and soft continuity have been provided in the field of generalized soft open sets [20, 21], generalized soft continuity [22], soft semi-open sets [23, 24], several types of soft continuity [25], soft somewhere dense sets [26], and nearly soft β-open sets [27]. More research on soft continuity was later conducted [28, 29]. In [30], the notion of the soft ideal was first introduced. Later, Fatouh and Abd El-latif [31] generalized this notion using soft semi-open sets. After that, this concept is used to generalize several types of topological properties, involving soft compactness [32], soft connectedness [33], soft generalized open sets [34–36], soft separation axioms [37], and generalized soft rough sets [38, 39]. Recently, some lower soft separation axioms [40] and Some applications of soft δ-closed sets [41] were presented. The notion of supra soft topological spaces (SSTs) was put forth by El-Sheikh et al. [42]. Additionally, the concepts of supra soft pre- (respectively, α-, semi, β, and γ-) open sets were presented. The approach of supra ϵ-open sets in supra topological spaces (STSs) was introduced by Abd El-latif et al.[43]. They also discussed the relationships between their novel approach and previous relevant research. Additionally, they supplied this new category’s primary characteristics. Furthermore, in general, the intersection of finite numbers of supra ϵ-open sets is not such. They then used their previously defined category of supra open sets to study new kinds of operators called supra ϵ-interior (closure, accumulation, exterior, and boundary, respectively). Several generalized supra soft operators have been studied in later studies using supra soft-b-open sets [44], supra generalized closed soft sets inspired by soft ideals [45], supra soft sw-open sets [46], supra soft δi-open sets [47, 48], supra soft somewhere dense sets [49], and separation axioms via supra soft topological spaces [50]. A. M. Abd El-latif et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6020 3 of 17 We continue studying the features of supra topological spaces in this paper. In par- ticular, we present and discuss novel types of supra continuity. Building on the earlier types of weaker forms of such conceptions, we established the notion of supra ϵ-continuous functions. Figure 1’s diagram was used to analyze the connections between our new class and earlier supra continuity concepts. Figure 1: The connections between Supra-ϵ-cts functions and other preceding studies We also introduced the concepts of supra ϵ-irresolute functions and supra ϵ∗-cts func- tions and provide their essential features in detail. Furthermore, we present novel ap- proaches for supra functions, which we call supra ϵ-open functions, supra ϵ-closed func- tions, and supra ϵ-homeomorphism functions. Finally, several essential examples were provided with a detailed discussion of their key characteristics. 2. Preliminaries and background Let (λ, ϑ) be an STS, the classes of supra (respectively, regular-, pre-, semi, β-, α-, b-, and R-) open sets will represented by SO(λ) (respectively, SOregular(λ), SPO(λ), SSO(λ), SβO(λ) , SαO(λ), SBO(λ), and SRO(λ)). Also, the classes of supra (respectively, regular- , semi, pre-, β-, b-, α-, and R-) continuous functions will represented by supra (respectively, regular-, semi, pre-, β-, b-, α-, and R-) cts, through this paper. Definition 1. [13] The collection ϑ ⊆ P (λ) is called supra topology (or STS) on λ if ϑ contains λ and ∅ and closed under arbitrary union. Also, if G ∈ ϑ, then G is called supra open set and Gc is called supra closed set. Moreover, if ν ⊂ ϑ for a given topology ν, then ϑ is called an associated STS with ν. Definition 2. [13] Regarding a subset K of an STS (λ, ϑ), the ints(K) or K◦ (respec- tively, cls(K) or K, and b(K)) will refer to the supra interior (respectively, closure, and boundary) of K, where ints(K) = ∪{G : G ∈ ϑ and G ⊆ K}, cls(K) = ∩{N : N ∈ ϑc and K ⊆ N}, and b(K) = cls(K)\ints(K). Theorem 1. [13] Regarding a subset T of an STS (λ, ϑ), we have (1) cls(T c) = [ints(T )]c. A. M. Abd El-latif et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6020 4 of 17 (2) ints(T c) = [cls(T )]c. Definition 3. [15–19] Let H be a subset of an STS (λ, ϑ). Then, (1) If H = ints(cls(H)), then H ∈ SOregular(λ). (2) If H ⊆ ints(cls(H)), then H ∈ SPO(λ). (3) If H ⊆ cls(ints(H)), then H ∈ SSO(λ). (4) If H ⊆ ints(cls(ints(H))), then H ∈ SαO(λ). (5) If H ⊆ cls(ints(cls(H))), then H ∈ SβO(λ). (6) If H ⊆ cls(ints(H))∪̃ints(cls(H)), then H ∈ SBO(λ). (7) If ints(cls(H)) ̸= ∅, then H ∈ SRO(λ). (8) If ints(cls(H)) = ∅, then H ∈ SND(λ). Definition 4. [13] Regarding the subset K of an STS (λ, ϑ), the class ϑK = {K ∩G : G ∈ ϑ} defines an STS on K, and it is called a subspace of (λ, ϑ). Definition 5. [43] Let H be a subset of an STS (λ, ϑ). Then, H is called supra ϵ-open set if either H = ∅ or H ⊆ { b(H) ∪H ◦ , H ∈ SRO(λ), b(H), H ∈ SND(λ) and b(H) is infinite. Also, Hc is called supra ϵ-closed-set. The category of all supra ϵ-open (respectively, supra ϵ-closed) sets will be indicated by SOϵ(λ) (respectively, SCϵ(λ)). Theorem 2. [43] Every supra (respectively, α-, semi-, b-, regular, pre-, β-, R-) open set is supra ϵ-open. Definition 6. [43] For the subset K of an STS (λ, ϑ), the intsϵ(K) will denote the supra ϵ-interior of K, where intsϵ(K) = ∪{G : G ∈ SOϵ(λ) and G ⊆ K}. Theorem 3. [43] For the supra ϵ-interior operator intsϵ : P (λ) −→ P (λ) and E ∈ P (λ), we have intsϵ(E) =  ∅, E ∈ SND(λ) and b(E) is finite. E ∩ b(E), E ∈ SND(λ) and b(E) is infinite. E, E ∈ SRO(λ). Definition 7. [43] Let C ∈ P (λ) be a subset of an STS (λ, ϑ), then clsϵ(C) will denote the supra ϵ-closure of C, where A. M. Abd El-latif et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6020 5 of 17 clsϵ(C) = ∩{N : N ∈ SCϵ(λ) and C ⊆ N}. Theorem 4. [43] For the supra ϵ-closure operator clsϵ : P (λ) −→ P (λ) and E ∈ P (λ), we have clsϵ(E) =  λ, Ec ∈ SND(λ) and b(Ec) is finite. E, Ec ∈ SND(λ) and b(Ec) is infinite. E, Ec ∈ SRO(λ). Theorem 5. [43] Regarding a subset T of an STS (λ, ϑ), we have (1) clsϵ(T c) = [intsϵ(T )] c. (2) intsϵ(T c) = [clsϵ(T )] c. (3) int(T ) ⊆ ints(T ) ⊆ intsϵ(T ). (4) clsϵ(T ) ⊆ cls(T ) ⊆ cl(T ). Definition 8. [43] Given a subset T of an STS (λ, ϑ) with arbitrary point s ∈ λ. Then, s called a supra ϵ-accumulation point of T if all 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 ). Definition 9. [43] If s ∈ [clsϵ(Z)\intsϵ(Z)] for an arbitrary point s and oft subset Z of an STS (λ, ϑ), 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 upper-so-exterior of Z is also represented by extϵ(Z), where extϵ(Z) = intsϵ(Z c). 3. New types of supra continuous functions based on supra ϵ-open sets This section refers to supra continuity using the concept of supra ϵ-open sets. To be more precise, we expanded on the earlier kinds of weaker forms of such conceptions by introducing the concept of supra ϵ-continuous functions. Figure 1, shows a diagram that was used to examine the connections between our new class and other earlier supra continuity concepts. Additionally, The fundamental characteristics of this notion are ex- amined, along with its comparable conditions. Furthermore, we presented the concepts of supra ϵ-irresolute functions and supra ϵ∗-cts functions. In addition, we prove that the composition of supra ϵ-irresolute function and supra ϵ-cts function (respectively, supra ϵ-cts function and cts function is supra ϵ-cts, two supra ϵ-irresolute functions) is supra ϵ-cts (respectively, supra ϵ-cts, supra ϵ-irresolute). Finally, several essential examples are provided. Definition 10. A function πϵ : (λ1, ν1) → (λ2, ν2) with ϑ1 as an associated STS with ν1 is said to be a supra ϵ-continuous (abbreviate: supra ϵ-cts) if π−1 ϵ (G) ∈ SOϵ(λ1) for each G ∈ ν2. A. M. Abd El-latif et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6020 6 of 17 Theorem 6. Every supra (respectively, semi-, α-, b-, pre, regular, β-, and R-) cts function is supra ϵ-cts. Proof. It is inferred from Theorem 2. Remark 1. In general, the following example demonstrates that the contrary of Theorem 6 is not valid. Example 1. Consider the two topologies ν1 = {∅, A ⊆ R : −1 ∈ A}, ν2 = {∅,R,N} and λ = {∅, T ⊆ R : −1 ∈ T or 0 ∈ T} be an associated STS with ν1 on the set of real numbers R. Consider the identity function πϵ : (R, ν1) → (R, ν2). Regarding the set of natural numbers N, we have π−1 ϵ (N) = N is a supra ϵ-open subset of R, but it is not supra R-open. Hence, πϵ is supra ϵ-cts, but it is not supra R-cts. Theorem 7. Let πϵ : (λ1, ν1) → (λ2, ν2) be a function with ϑ1 as an associated STS with ν1, then the next assertions are equivalent: (1) πϵ is supra ϵ-cts. (2) For each Z ∈ νc2, π −1 ϵ (Z) ∈ SCϵ(λ1). (3) clsϵ(π −1 ϵ (Z)) ⊆ π−1 ϵ (cl(Z)) ∀ Z ⊆ λ2. (4) πϵ(cl s ϵ(Y )) ⊆ cl(πϵ(Y )) ∀ Y ⊆ λ1. (5) π−1 ϵ (int(Z)) ⊆ intsϵ(π −1 ϵ (Z)) ∀ Z ⊆ λ2. Proof. (1) ⇒ (2) Let Z ∈ νc2, then Z c ∈ ν2. Given (1), π−1 ϵ (Zc) = [π−1 ϵ (Z)]c ∈ SOϵ(λ1). Hence, π−1 ϵ (Z) ∈ SCϵ(λ1). (2) ⇒ (3) Let Z ⊆ λ2. Since cl(Z) ∈ νc2 and given (2), π−1 ϵ (Z) ∈ SCϵ(λ1), which implies clsϵ(π −1 ϵ (Z)) ⊆ clsϵ(π −1 ϵ (cl(Z))) = π−1 ϵ (cl(Z)). Consequently, the proof is acquired. (3) ⇒ (4) Regarding πϵ(Y ) ⊆ λ2 for a subset Y ⊆ λ1, we have Y ⊆ π−1 ϵ (πϵ(Y )). Given (3), we obtain clsϵ(π −1 ϵ (πϵ(Y ))) ⊆ π−1 ϵ (cl(πϵ(Y ))). Hence, πϵ[cl s ϵ(π −1 ϵ (πϵ(Y )))] ⊆ πϵ[π −1 ϵ (cl(πϵ(Y )))] ⊆ cl(πϵ(Y )). Therefore, πϵ(cl s ϵ(Y )) ⊆ cl(πϵ(Y )). A. M. Abd El-latif et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6020 7 of 17 (4) ⇒ (5) Regarding π−1 ϵ (Zc) ⊆ λ1 for a subset Zc ⊆ λ2, and by utilizing (4), we obtain that πϵ[cl s ϵ [π −1 ϵ (Zc)]] ⊆ cl(πϵ[π −1 ϵ (Zc)]) ⊆ cl(Zc) = [int(Z)]c, from Theorem 5. Hence, π−1 ϵ [πϵ(cl s ϵ [π −1 ϵ (Zc)])] ⊆ π−1 ϵ [[int(Z)]c] = [π−1 ϵ (int(Z))]c. Therefore, clsϵ [(π −1 ϵ (Z))]c ⊆ [π−1 ϵ (int(Z))]c. Thus, π−1 ϵ (int(Z)) ⊆ [clsϵ [(π −1 ϵ (Z))]c]c = intsϵ(π −1 ϵ (Z)). (5) ⇒ (1) Regarding Z = int(Z) for a supra open set Z. Given (5), π−1 ϵ (Z) ⊆ intsϵ(π −1 ϵ (Z)). However, intsϵ(π −1 ϵ (Z)) ⊆ π−1 ϵ (Z). Therefore, intsϵ(π −1 ϵ (Z)) = π−1 ϵ (Z) ∈ SOϵ(λ1). Thus, πϵ is a supra ϵ-cts. Definition 11. A function πϵ : (λ1, ν1) → (λ2, ν2) with ϑ1, ϑ2 associated STSs with ν1, ν2, respectively, is said to be supra ϵ-irresolute (supra ϵ∗-cts ) if π−1 ϵ (D) ∈ SOϵ(λ1) for each D ∈ SOϵ(λ2) (D ∈ ϑ2). Theorem 8. (1) Every supra ϵ-irresolute function is supra ϵ∗-cts. (2) Every supra ϵ∗-cts function is supra ϵ-cts. Proof. It is immediately obvious from Theorem 2. Remark 2. The following examples demonstrate that the contrary of Theorem 8 is gen- erally untrue. Examples 1. (1) Consider the two topologies ν1 = {∅, A ⊆ R : −2 ∈ A}, ν2 = {∅,R,N}} on the set of real numbers R. Let ϑ1 = {∅, T ⊆ R : −2 ∈ T or 0 ∈ T} and ϑ2 = {∅,R,N, {0, 1}, {0, 2}, {0, 1, 2}} be associated STSs with ν1 and ν1, respectively, and let πϵ : (R, ν1) → (R, ν2) be the identity function. Then, we have π−1 ϵ (D) ∈ SOϵ(R) for each D ∈ ϑ2, and hence πϵ is supra ϵ ∗-cts. However, πϵ is not supra ϵ-irresolute, since {−3,−4} is supra ϵ-open set over ϑ2, but π −1 ϵ ({−3,−4}) = {−3,−4} is not supra ϵ-open over ϑ1. Therefore, πϵ is supra ϵ∗-cts, but it is not supra ϵ-irresolute. A. M. Abd El-latif et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6020 8 of 17 (2) Consider the two topologies ν1 = {λ1, ∅, {1}, {1, 2}}, ν2 = {λ2, ∅, {y, z}} on λ1 = {1, 2, 3} and λ2 = {x, y, z}, respectively. Let ϑ1 = {λ1, ∅, {1}, {1, 2}, {2, 3}} and ϑ2 = {λ2, ∅, {z}, {y, z}, {x, y}} be associated STSs with ν1 and ν1, respectively, and let πϵ : (λ1, ν1) → (λ2, ν2) be a function defined as follows: πϵ({1}) = {y}, πϵ({2}) = {x}, and πϵ({3}) = {z}. Then, we have π−1 ϵ (D) ∈ SOϵ(R) for each D ∈ ν2, and hence πϵ is supra ϵ-cts. However, πϵ is not supra ϵ∗-cts, since {z} ∈ ϑ2, but π −1 ϵ ({z}) = {3} ̸∈ SOϵ(λ1). Therefore, πϵ is supra ϵ-cts, but it is not supra ϵ∗-cts. Corollary 1. It follows from Theorem 6, Theorem 8 and [18, Reamrk 2] that we have the following implications for an STS (λ, ν), which are not reversible. Figure 1: The connections between Supra-ϵ-cts functions and other preceding studies The proofs for the next two theorems are eliminated since they could be demonstrated similarly to Theorem 7. Theorem 9. Let πϵ : (λ1, ν1) → (λ2, ν2) be a function with ϑ1 and ϑ2 as associated STSs with ν1 and ν1, respectively, then the next assertions are equivalent: (1) πϵ is supra ϵ∗-cts. (2) For each Z ∈ ϑc2, π −1 ϵ (Z) ∈ SCϵ(λ1). (3) clsϵ(π −1 ϵ (Z)) ⊆ π−1 ϵ (cls(Z)) ∀ Z ⊆ λ2. (4) πϵ(cl s ϵ(Y )) ⊆ clsϵ(πϵ(Y )) ∀ Y ⊆ λ1. (5) π−1 ϵ (intsϵ(Z)) ⊆ intsϵ(π −1 ϵ (Z)) ∀ Z ⊆ λ2. Theorem 10. Let πϵ : (λ1, ν1) → (λ2, ν2) be a function with ϑ1 and ϑ2 as associated STSs with ν1 and ν1, respectively, then the next assertions are equivalent: (1) πϵ is supra ϵ-irresolute. (2) For each Z ∈ SCϵ(λ2), π −1 ϵ (Z) ∈ SCϵ(λ1). (3) clsϵ(π −1 ϵ (Z)) ⊆ π−1 ϵ (clsϵ(Z)) ∀ Z ⊆ λ2. (4) πϵ(cl s ϵ(Y )) ⊆ clsϵ(πϵ(Y )) ∀ Y ⊆ λ1. A. M. Abd El-latif et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6020 9 of 17 (5) π−1 ϵ (intsϵ(Z)) ⊆ intsϵ(π −1 ϵ (Z)) ∀ Z ⊆ λ2. Theorem 11. Let πϵ : (λ1, ν1) → (λ2, ν2) be a supra ϵ-irresolute with ϑ1, ϑ2 associated STSs with ν1, ν2, respectively, and ψϵ : (λ2, ν2) → (λ3, ν3) be a supra ϵ-cts with ϑ3 as an associated STS with ν3, then the composition ψϵ ◦ πϵ : (λ1, ν1) → (λ3, ν3) is supra ϵ-cts. Proof. Let E ∈ ϑ3. Since ψϵ is supra ϵ-cts, ψ−1 ϵ (E) ∈ SOϵ(λ2). Given πϵ is supra ϵ-irresolute, then [ψϵ ◦ πϵ]−1(E) = π−1 ϵ [ψ−1 ϵ (E)] ∈ SOϵ(λ1). Hence, ψϵ ◦ πϵ is supra ϵ-cts. The proof of the upcoming two Corollaries is straightforward from Theorem 11. Corollary 2. The composition of supra ϵ-cts function and cts function is supra ϵ-cts. Corollary 3. The composition of two supra ϵ-irresolute functions is also supra ϵ-irresolute. Theorem 12. A function πϵ : (λ1, ν1) → (λ2, ν2) with with ϑ1, ϑ2 associated STSs with ν1, ν2, respectively, is supra ϵ-irresolute if cls(π−1 ϵ (W )) ⊆ π−1 ϵ (clsϵ(W )) ∀W ⊆ λ2. Proof. Assume that W ⊆ λ2. For π−1 ϵ (W ), taking into account the specified condition and Theorem 5 (4), we get clsϵ(π −1 ϵ (W )) ⊆ cls(π−1 ϵ (W )) ⊆ π−1 ϵ (clsϵ(W )). Given Theorem 10 (3), πϵ is supra ϵ-irresolute. Theorem 13. A function πϵ : (λ1, ν1) → (λ2, ν2) with ϑ1, ϑ2 associated STSs with ν1, ν2 , respectively, is supra ϵ-cts in the event that one of the subsequent conditions is fulfilled: (1) πϵ(cl(Y )) ⊆ clsϵ(πϵ(Y )) ∀ Y ⊆ λ1. (2) cl(π−1 ϵ (Z)) ⊆ π−1 ϵ (clsϵ(Z)) ∀ Z ⊆ λ2. (3) π−1 ϵ (intsϵ(Z)) ⊆ int(π−1 ϵ (Z)) ∀ Z ⊆ λ2. Proof. If the first condition is fulfilled, then πϵ(cl(Y )) ⊆ clsϵ(πϵ(Y )) ∀ Y ⊆ λ1. Since clsϵ(Y,Θ) ⊆ cl(Y,Θ) from Theorem 5 (4), πϵ(cl s ϵ(Y )) ⊆ πϵ(cl(Y )) ⊆ clsϵ(πϵ(Y )) ⊆ cl(πϵ(Y )). Therefore, πϵ is supra ϵ-cts according to Theorem 7 (4). If the second condition is fulfilled, then ∀ Z ⊆ λ2, then clsϵ(π −1 ϵ (Z)) ⊆ cl(π−1 ϵ (Z)) ⊆ π−1 ϵ (clsϵ(Z)) ⊆ π−1 ϵ (cl(Z)), given Theorem 5 (4). This im- plies, clsϵ(π −1 ϵ (Z)) ⊆ π−1 ϵ (cl(Z)) ∀ Z ⊆ λ2, and therefore Therefore, πϵ is supra ϵ-cts according to Theorem 7 (3). If the third condition is fulfilled, and given Theorem 5 (3) π−1 ϵ (int(Z)) ⊆ π−1 ϵ (intsϵ(Z)) ⊆ int(π−1 ϵ (Z)) ⊆ intsϵ(π −1 ϵ (Z)) ∀ Z ⊆ λ2. Hence, π−1 ϵ (int(Z)) ⊆ intsϵ(π −1 ϵ (Z)) ∀ Z ⊆ λ2. Hence, πϵ is supra ϵ-cts according to Theorem 7 (5). A. M. Abd El-latif et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6020 10 of 17 The proofs for the next two theorems are eliminated since they could be demonstrated similarly to Theorem 13. Theorem 14. A function πϵ : (λ1, ν1) → (λ2, ν2) with ϑ1, ϑ2 associated STSs with ν1, ν2 , respectively, is supra ϵ∗-cts in the event that one of the subsequent conditions is fulfilled: (1) πϵ(cl s(Y )) ⊆ clsϵ(πϵ(Y )) ∀ Y ⊆ λ1. (2) cls(π−1 ϵ (Z)) ⊆ π−1 ϵ (clsϵ(Z)) ∀ Z ⊆ λ2. (3) π−1 ϵ (intsϵ(Z)) ⊆ ints(π−1 ϵ (Z)) ∀ Z ⊆ λ2. Theorem 15. A function πϵ : (λ1, ν1) → (λ2, ν2) with ϑ1, ϑ2 associated STSs with ν1, ν2, respectively, is supra ϵ-irresolute in the event that one of the subsequent conditions is fulfilled: (1) πϵ(cl s ϵ(Y )) ⊆ clsϵ(πϵ(Y )) ∀ Y ⊆ λ1. (2) clsϵ(π −1 ϵ (Z)) ⊆ π−1 ϵ (clsϵ(Z)) ∀ Z ⊆ λ2. (3) π−1 ϵ (intsϵ(Z)) ⊆ intsϵ(π −1 ϵ (Z)) ∀ Z ⊆ λ2. 4. Supra ϵ-homeomorphism functions We present new approaches for supra functions in this section, which we call supra ϵ-open functions, supra ϵ-closed functions, and supra ϵ-homeomorphism functions. Furthermore, we show their corresponding properties in a transparent way. Moreover, for every notion, we give the analogous conditions that are required. Definition 12. A function πϵ : (λ1, ν1) → (λ2, ν2) with ϑ2 as an associated STS with ν2 is said to be: (1) Supra ϵ-open if πϵ(U) ∈ SOϵ(λ2) for each U ∈ ν1. (2) Supra ϵ-closed if πϵ(C) ∈ SCϵ(λ2) for each C ∈ νc1. Theorem 16. Let πϵ : (λ1, ν1) → (λ2, ν2) be a function with ϑ2 as an associated STS with ν2 and Y ⊆ λ1, then πϵ is supra ϵ-open if and only if πϵ(int(U)) ⊆ intsϵ [πϵ(U)] ∀ U ⊆ λ1. Proof. “ ⇒ ” Let πϵ be a supra ϵ-open function and U ⊆ λ1. Since int(U) ⊆ U , πϵ(int(U)) ⊆ πϵ((U)), which implies πϵ(int(U)) = intsϵ [πϵ(int(U))] ⊆ intsϵ [πϵ((U))]. “ ⇐ ” Assume that U ∈ ν1. Based on the presumption, πϵ(U) = πϵ(int(U)) ⊆ intsϵ [πϵ(U)] . A. M. Abd El-latif et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6020 11 of 17 But, we have intsϵ [πϵ(U)] ⊆ πϵ(U). Hence, intsϵ [πϵ(U)] = πϵ(U). Therefore, πϵ(U) ∈ SOϵ(λ2), and thus πϵ is a supra ϵ-open function. Theorem 17. Let πϵ : (λ1, ν1) → (λ2, ν2) be a function with ϑ2 as an associated STS with ν2 and Y ⊆ λ1, then πϵ is supra ϵ-closed if and only if clsϵ [πϵ(H)] ⊆ πϵ(cl(H)). Proof. ” ⇒ ” Assume that πϵ is supra ϵ-closed function and H ⊆ λ1. Since πϵ(H) ⊆ πϵ(cl(H)), clsϵ [πϵ(H)] ⊆ clsϵ [πϵ(cl(H))] = πϵ(cl(H)), given πϵ is supra ϵ-closed function. “ ⇐ ” Let H ∈ νc1. Based on the presumption, πϵ(H) ⊆ clsϵ [πϵ(H)] ⊆ πϵ(cl(H)) = πϵ(H). Hence, clsϵ [πϵ(H)] = πϵ(H). Therefore, πϵ(H) ∈ SC(λ2)Θ2, and hence πϵ is a supra ϵ-closed function. Theorem 18. Let πϵ : (λ1, ν1) → (λ2, ν2) be a bijective function with ϑ2 as an associated STS with ν2, then πϵ is supra ϵ-open function if and only if it is supra ϵ-closed. Proof. “ ⇒ ” Let R ∈ νc1, then R c ∈ ν1. Since πϵ is supra bijective ϵ-open function, [πϵ(R)] c = πϵ(R c) ∈ SOϵ(λ1). It follows that, πϵ(R) ∈ SOϵ(λ2). Therefore, πϵ is a supra ϵ-closed function. “ ⇐ ” It is followed by a comparable argument. Proposition 1. Let πϵ : (λ1, ν1) → (λ2, ν2) be a bijective function with ϑ2 as an associated STS with ν2, then the next assertions are equivalent: (1) πϵ is a supra ϵ-open function. (2) πϵ is a supra ϵ-closed function. A. M. Abd El-latif et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6020 12 of 17 (3) π−1 ϵ is a supra ϵ-cts function. Proof. (1) ⇒ (2) Direct from Theorem 18. (2) ⇒ (3) Let Z ∈ νc1. Since πϵ is supra bijective function and given (2), (π−1 ϵ )−1(Z) = πϵ(Z) ∈ SCϵ(λ2). Hence, π−1 ϵ is a supra ϵ-cts function. (3) ⇒ (1) Let S ∈ ν1. Then, πϵ(S) = (π−1 ϵ )−1(S) ∈ SOϵ(λ2), given (3). Therefore, πϵ is a supra ϵ-open function. Theorem 19. Let πϵ : (λ1, ν1) → (λ2, ν2) and ψϵ : (λ2, ν2) → (λ3, ν3) be two functions with ϑ1, ϑ2, ϑ3 associated STSs with ν1, ν2, ν3, respectively. Then (1) If ψϵ ◦ πϵ is a supra ϵ-open function and πϵ is a surjective cts function, then ψϵ is a supra ϵ-open function. (2) If ψϵ ◦ πϵ is an open function and ψϵ is an injective supra ϵ-cts function, then πϵ is a supra ϵ-open function. (3) If ψϵ ◦ πϵ is a supra open function and ψϵ is an injective supra ϵ∗-cts function, then πϵ is a supra ϵ-open function. (4) If ψϵ ◦ πϵ is a supra ϵ-open and ψϵ is an injective supra ϵ-irresolute function, then πϵ is a supra ϵ-open function. Proof. (1) Let F ∈ ν2. Since πϵ is a cts function, π−1 ϵ (F ) ∈ ν1. Given ψϵ ◦ πϵ is a supra ϵ-open function and πϵ is a surjective function, then (ψϵ ◦ πϵ)[π−1 ϵ (F )] = ψϵ[πϵ(π −1 ϵ (F ))] = ψϵ(F ) ∈ SOϵ(λ3). Therefore, ψϵ is a supra ϵ-open function. (2) Let F ∈ ν1. Since ψϵ◦πϵ is an open function, (ψϵ◦πϵ)(F ) ∈ ν3. Given ψϵ is an injective supra ϵ-cts function, then ψ−1 ϵ [ψϵ ◦ πϵ(F )] = (ψ−1 ϵ ◦ ψϵ)(πϵ(F )) = πϵ(F ) ∈ SOϵ(λ2). Therefore, πϵ is a supra ϵ-open function.function. (3) Let F ∈ ν1. Since ψϵ ◦ πϵ is a supra open function, (ψϵ ◦ πϵ)(F ) ∈ ϑ3. Given ψϵ is an injective supra ϵ∗-cts function, then ψ−1 ϵ [ψϵ ◦ πϵ(F )] = (ψ−1 ϵ ◦ ψϵ)(πϵ(F )) = πϵ(F ) ∈ SOϵ(λ2). Therefore, πϵ is a supra ϵ-open function. (4) Let F ∈ ν1. Since ψϵ◦πϵ is a supra ϵ-open function, (ψϵ◦πϵ)(F ) ∈ SOϵ(λ3). Given ψϵ is an injective supra ϵ-irresolute function, then ψ−1 ϵ [ψϵ◦πϵ(F )] = (ψ−1 ϵ ◦ψϵ)(πϵ(F )) = πϵ(F ) ∈ SOϵ(λ2). Therefore, πϵ is a supra ϵ-open function. Definition 13. A bijective function πϵ : (λ1, ν1) → (λ2, ν2) with ϑ1, ϑ2 associated STSs with ν1, ν2 , respectively, is said to be supra ϵ-homeomorphism if it is supra ϵ-cts and supra ϵ-open. A. M. Abd El-latif et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6020 13 of 17 Theorem 20. For a bijective supra ϵ-cts function πϵ : (λ1, ν1) → (λ2, ν2) with ϑ1, ϑ2 associated STSs with ν1, ν2 , respectively. The statements that follow are interchangeable: (1) πϵ is supra ϵ-homeomorphism. (2) ψ−1 sd is supra ϵ-cts. (3) πϵ is supra ϵ-closed. Proof. It is instantly evident from Definition 13 and Theorem 18. Theorem 21. A bijective function πϵ : (λ1, ν1) → (λ2, ν2) with ν1, ν2 associated STSs with ϑ1, ϑ2, respectively, is an supra ϵ-homeomorphism in the event that one of the subse- quent conditions is fulfilled: (1) πϵ(cl s ϵ(H)) ⊆ cl(πϵ(H)) and clsϵ(πϵ(H)) ⊆ πϵ(cl(H)), ∀ (H) ⊆ λ1. (2) πϵ(int(H)) ⊆ intsϵ(πϵ(H)), ∀ (H) ⊆ λ1 and ψ−1 sd (int(H)) ⊆ intsϵ(ψ −1 sd (Z,Θ2)), ∀ H ⊆ λ2. Proof. If the first condition is fulfilled, then πϵ(cl s ϵ(H)) ⊆ cl(πϵ(H)), implies πϵ is supra ϵ- cts, given Theorem 7 (4). Moreover, clsϵ(πϵ(H)) ⊆ πϵ(cl(H)), implies πϵ is supra ϵ-closed, given Proposition 17. Consequently, πϵ is supra ϵ-homeomorphism, in line with Theorem 20. If the first condition is fulfilled, then by a similar way πϵ is supra ϵ-homeomorphism, in line with Definition 13, Theorem 7 (5) and Theorem 16. 5. Conclusion Abd El-latif et al. introduced the utilization of supra ϵ-open sets to supra topological spaces[43]. Then, using their previously established category of supra open sets, they inves- tigated new types of operators known as supra ϵ-interior (closure, accumulation, exterior, and boundary, respectively). We introduce and explore new forms of supra continuity in this work. The concept of supra ϵ-continuous functions was developed by us, building on the previous sorts of weaker forms of such conceptions. Moreover, we provide a diagram to analyze the connections between our new class and earlier supra continuity concepts. Additionally, we presented the notions of supra ϵ-irresolute functions and supra ϵ-cts func- tions and thoroughly described their key characteristics. Furthermore, we propose novel approaches for supra functions, which we refer to as supra ϵ-open functions, supra ϵ-closed functions, and supra ϵ-homeomorphism functions. We also discuss the main features of each notions. From the particular methods described in this article, additional research on the theoreti- cal elements of these generalized concepts could be carried out by looking at the following subjects: Utilizing these methods in supra soft ideal topological spaces [30]. In addition, we investigate certain topological characteristics, such as supra (connectedness, separa- tion axioms, and compactness), that are motivated by specific approaches discussed in this work. A. M. Abd El-latif et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6020 14 of 17 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 num- ber ”NBU-FFR-2025-1687-01”. 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. Author Contributions Alaa M. Abd El-latif: Conceptualization, Methodology, Formal Analysis, Investiga- tion, Writing, Original Draft Preparation, Review and Editing, Funding Acquisition Radwan Abu-Gdairi: Methodology, Formal Analysis, Investigation, Writing, Original, Review and Editing, Funding Acquisition A. A. Azzam: Conceptualization, Formal Analysis, Investigation, Original Draft Prepa- ration, Review and Editing, Funding Acquisition F. A. Gharib: Conceptualization, Methodology, Investigation, Writing, Original Draft Preparation, Review and Editing Husham M. Attaalfadeel: Conceptualization, Methodology, Investigation, Writing, Original Draft Preparation, Review and Editing Walid Abdelfattah: Conceptualization, Methodology, Investigation, Writing, Original Draft Preparation, Review and Editing Shaaban M. Shaaban: Conceptualization, Methodology, Investigation, Writing, Origi- nal Draft Preparation, Review and Editing M. 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