EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 15, No. 1, 2022, 261-280 ISSN 1307-5543 – ejpam.com Published by New York Business Global Infra pre-open sets and their applications to generate new types of operators and maps Tareq M. Al-shami1∗, Hakeem A. Othman2,3 1 Department of Mathematics, Sana’a University, Sana’a, Yemen 2 Department of Mathematics, AL-Qunfudhah University college, Umm Al-Qura University, Saudi Arabia 3 Department of Mathematics, Rada’a College of Education and Science, Albaydha University, Albaydha, Yemen Abstract. Herein, we introduce the concepts of infra soft pre-open infra soft pre-closed sets which are respectively generalizations of infra soft open and infra soft closed sets. We characterize them and investigate their behaviours under infra soft homeomorphism maps and finite product of soft spaces. Then, we apply infra soft pre-open infra soft pre-closed sets to define the operators of infra pre-interior, infra pre-closure, infra pre-limit and infra pre-boundary. We discuss their main properties and show the interrelations between them. In the end, we introduce new types of soft maps using infra soft pre-open and infra soft pre-closed sets and explore their essential properties. 2020 Mathematics Subject Classifications: 54A40, 54C08, 54C99 Key Words and Phrases: Infra soft pre-open set, infra soft pre-interior points, infra soft pre- closure points, infra soft pre-continuity 1. Introduction Soft set is a new mathematical tool to address uncertainity/vagueness; it was intro- duced in 1999 by Molodtsov [34]. He proved its efficiency by applying successfully in many areas. Aktaş and Çağman [1] showed that rough set and fuzzy set, which are two approaches to handle uncertainty, may be considered soft sets. In the literature, one can note many authors have been applied soft sets to model some phenomena and problems in different disciplines such as decision-making problems [28, 38] and computer science [22]. Maji et al. [33], in 2003, formulated the basic operations and operators between soft sets like the difference between two soft sets, a complement of a soft set, and intersection and union operators. To remove anomaly appeared in their definitions and keep some ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v15i1.4275 Email addresses: tareqalshami83@gmail.com (T.M. Al-shami), haoali@uqu.edu.sa;hakim−albdoie@yahoo.com (H.A. Othman) http://www.ejpam.com 261 © 2022 EJPAM All rights reserved. T.M. Al-shami, H.A. Othman / Eur. J. Pure Appl. Math, 15 (1) (2022), 261-280 262 crisp properties in the soft set theory, Ali et al. [19] initiated new operations and operators between soft sets. Attempts were still in this path to produce new operators and relations like those introduced in [15, 36]. In 2011, Çaǧman et al. [23] and Shabir and Naz [37] applied soft sets to define a soft topology. Whereas, Çaǧman et al. defined a soft topology over an absolute soft set and different sets of parameters, Shabir and Naz defined a soft topology over a fixed set of universe and a fixed set of parameters. This article follows Shabir and Naz’ definition. The main concepts and notions of general topology were studied in soft topology such as basis [18], separation axioms [27], compactness [6, 21], connectedness [32], bioperators [20], covering properties [13, 14, 25], generalized open sets [2–4] and Bipolarity [5]. Soft topologies were generalized to various structures such as infra soft topologies [10] which is the frame of this study. The inducements of continuous investigation of infra soft topologies are that several topological features are still valid via the frame of infra soft topologies, also, easily building the examples that elucidate the interrelationships among the topological notions and concepts. In [9, 11], the authors discussed these advantages for compact and connected spaces. Some studies have en recently conducted in frame of infra soft topologies such as [7, 12, 16, 17] Extension of soft open sets was a goal of some papers. Some types of these extensions were investigated such as soft semi-open and soft pre-open sets which were presented in [24] and [30], respectively. The target of this work is to scrutinize the behaviours of soft pre-open sets via infra soft topological spaces. As we shall show many properties of soft pre-open sets are still valid for infra soft pre-open sets which offers a flexible frame (in lieu of soft topologies) to study the topological notions and the interrelationships between them. We layout the remainder of this article as following. In Sect. 2, we survey the related literature and locate the current study in its context. Sect. 3 is first of the three main sections of this study. It introduces the concept of infra soft pre-open sets and establishes its characterization. Sect. 4 is the second main section which defines and discusses the concepts of infra pre-interior, infra pre-closure, infra pre-limit and infra pre-boundary soft points of a soft set. Sect. 5 is the last main section which initiates and explores new types of soft maps namely infra soft pre-continuous, infra soft pre-open, infra soft pre-closed and infra soft pre-homeomorphism maps. Finally, Sect. 6 gives some conclusions and proposes some future works. 2. Preliminaries In this part, we recall the concepts and findings that help us to understand this article. 2.1. Soft set theory Definition 1. [34] Consider Σ as a parameters set and 2X the power set of X which is the universe. We call (Ω,Σ) a soft set over X if Ω : Σ → 2X is a crisp map. A soft set is T.M. Al-shami, H.A. Othman / Eur. J. Pure Appl. Math, 15 (1) (2022), 261-280 263 expressed as (Ω,Σ) = {(η,Ω(η)) : η ∈ Σ and Ω(η) ∈ 2X}. A class of all soft sets over X under a set of parameters Σ is symbolized by C(XΣ). Definition 2. [19] A complement of a soft set (Ω,Σ), denoted by (Ωc,Σ), provided that a map Ωc : Σ → 2X is given by Ωc(η) = X \ Ω(η) for each η ∈ Σ. Definition 3. [33] Let (Ω,Σ) be a soft set on X such that Ω(η) = ∅ (resp., Ω(η) = X) for each η ∈ Σ. Then we say that (Ω,Σ) is a null (resp., an absolute) soft set over X. The null and absolute soft sets are respectively symbolized by Φ and X̃. Definition 4. [26, 27] We call a soft set (Ω,Σ) stable (resp., finite, countable) if all components are equal (resp., finite, countable). Otherwise, we call (Ω,Σ) unstable (resp., infinite, uncountable). Definition 5. [35] We call a soft set (Ω,Σ) a soft point on X if there is η ∈ Σ such that Ω(η) = x ∈ X and Ω(η′) = ∅ for each η′ ̸= η. Henceforth, δxη denotes a soft point. Definition 6. [19] The intersection of soft sets (Ω,Σ) and (Ψ,∆) on X, symbolized by (Ω,Σ) ⋂̃ (Ψ,∆), is a soft set (Υ, T ), where T = Σ∩∆ ̸= ∅, and a map Υ : T → 2X is given by Υ(η) = Ω(η) ∩Ψ(η) for each η ∈ T . Definition 7. [33] The union of soft sets (Ω,Σ) and (Ψ,∆) on X, symbolized by (Ω,Σ) ⋃̃ (Ψ,∆), is a soft set (Υ, T ), where T = Σ ∪∆ and a map T : Σ → 2X is given as follows: Υ(η) =  Ω(η) : η ∈ Σ \∆ Ψ(η) : η ∈ ∆ \ Σ Ω(η) ∪Ψ(η) : η ∈ Σ ∩∆ Definition 8. [29] A soft set (Ω,Σ) is a subset of a soft set (Ψ,∆), symbolized by (Ω,Σ)⊆̃(Ψ,∆), if Σ ⊆ ∆ and Ω(η) ⊆ Ψ(η) for all η ∈ Σ. If (Ω,Σ)⊆̃(Ψ,∆) and (Ψ,∆)⊆̃(Ω,Σ), then (Ω,Σ) and (Ψ,∆) are called soft equal. Definition 9. [21] The Cartesian product of (Ω,Σ) and (Ψ,∆), symbolized by (Ω×Ψ,Σ× ∆), is defined as (Ω×Ψ)(η, η′) = Ω(η)×Ψ(η′) for each (η, η′) ∈ Σ×∆. Definition 10. [31] A soft map fτ from C(XΣ) to C(S∆) is a pair of crisp maps f and τ , where f : X → S, τ : Σ → ∆. Let (Ω,M) and (Ψ,N ) be respectively subsets of C(XΣ) and C(S∆). Then the image of (Ω,M) and pre-image of (Ψ,N ) are given by the following. (i) fτ (Ω,M) = (f(Ω),∆) is a soft set in C(V∆) such that f(Ω)(ω) = { ⋃̃ η∈τ−1(ω) ⋂ Mf(Ω(η)) : τ−1(ω) ̸= ∅ ∅ : τ−1(ω) = ∅ for each ω ∈ ∆. T.M. Al-shami, H.A. Othman / Eur. J. Pure Appl. Math, 15 (1) (2022), 261-280 264 (ii) f−1 τ (Ψ,N ) = (f−1(Ψ),Σ) is a soft set in C(XΣ) such that f−1(Ψ)(η) = { f−1(Ψ(τ(η))) : τ(η) ∈ N ∅ : τ(η) ̸∈ N for each η ∈ Σ. Definition 11. [31] We call a soft map fτ : C(XΣ) → C(S∆) injective (resp., surjective, bijective) if f and τ are injective (resp., surjective, bijective). 2.2. Infra soft topological spaces Definition 12. [10] A family ξ of soft sets over X with Σ as a parameters set is said to be an infra soft topology on X if it is closed under finite intersection and Φ is a member of ξ. The triple (X, ξ,Σ) is called an infra soft topological space (briefly, ISTS). We call a member of ξ an infra soft open set and called its complement an infra soft closed set. We call (X, ξ,Σ) stable if all its infra soft open sets are stable. Definition 13. [10] Let (Ω,Σ) be a subset of (X, ξ,Σ). (i) the intersection of all infra soft closed subsets of (X, ξ,Σ) which contains a soft set (Ω,Σ) is called the infra soft closure points of (Ω,Σ). It is denoted by Cl(Ω,Σ). (ii) the union of all infra soft open subsets of (X, ξ,Σ) which are contained in a soft set (Ω,Σ) is called the infra soft interior points of (Ω,Σ). It is denoted by Int(Ω,Σ). It was showed in [10] that Cl(Ω,Σ) and Int(Ω,Σ) need not be infra soft closed and infra soft open, respectively. Through this paper, (Ω,Σ) is called ξ-infra soft open (resp., ξ-infra soft closed) if Int(Ω,Σ) = (Ω,Σ) (resp., Cl(Ω,Σ)) = (Ω,Σ). Proposition 1. [10] Let (Ω,Σ) and (Ψ,Σ) subsets of an ISTS (X, ξ,Σ). Then (i) Cl[(Ω,Σ) ⋃̃ (Ψ,Σ)] = Cl(Ω,Σ) ⋃̃ Cl(Ψ,Σ), and (ii) Int[(Ω,Σ) ⋂̃ (Ψ,Σ)] = Int(Ω,Σ) ⋂̃ Int(Ψ,Σ). Proposition 2. [10] Let (Ω,Σ) be an infra soft open set. Then (Ω,Σ) ⋂̃ Cl(Ψ,Σ)⊆̃Cl[(Ω,Σ) ⋃̃ (Ψ,Σ)] for any subset (Ψ,Σ) of (X, ξ,Σ). Proposition 3. [10] Let (Ω,Σ) be an infra soft closed set. Then Int[(Ω,Σ) ⋃̃ (Ψ,Σ)]⊆̃(Ω,Σ) ⋃̃ Int(Ψ,Σ) for any subset (Ω,Σ) of (X, ξ,Σ). Definition 14. A soft map fτ : (X, ξ,Σ) → (S, π,∆) is said to be an infra soft homeo- morphism if it is bijective, infra soft continuous (i.e,the image of every infra soft open set is infra soft open), and infra soft open (i.e,the image of every infra soft open set is infra soft open). T.M. Al-shami, H.A. Othman / Eur. J. Pure Appl. Math, 15 (1) (2022), 261-280 265 We call a property which is kept by any infra soft homeomorphism an infra soft topo- logical property (in short, IST property). Definition 15. [10] Let Eτ : (X, ξ,Σ) → (S, π,∆) be a soft map and M ̸= ∅ be a subset of X. A soft map Eτ|M : (M, ξM,Σ) → (S, π,∆) which given by Eτ|M(δmη ) = Eτ (δ m η ) for every δmη ∈ M̃ is called a restriction soft map of Eτ on M. Proposition 4. Let {(Xk, ξk,Σk) : k ∈ K} be a family of ISTSs. Then ξ = { ∏ k∈K(ηk,Σk) : (ηk,Σk) ∈ τk} is an infra soft topology on T = ∏ k∈K Xk under a set of parameters B = ∏ k∈K Σk. We call ξ given in proposition above, a product of infra soft topologies, and (T, ξ,B) a product of infra soft spaces. 3. Main properties of infra soft pre-open sets In this section, we define infra soft pre-open and infra soft pre-closed sets which are the core concepts of this article. We characterize them and investigate some of their properties. We show that the class of infra soft pre-open sets forms a supra soft topology and discuss under what conditions this class forms a soft topology. We complete this section by proving that this class is kept under infra soft homeomorphism maps and finite product of soft spaces. Definition 16. A subset (Ω,Σ) of an ISTS (X, ξ,Σ) is said to be infra soft pre-open if (Ω,Σ)⊆̃Int(Cl(Ω,Σ)). Its complement is said to be an infra soft pre-closed set. Proposition 5. If (Ω,Σ) is an infra soft pre-open subset of an ISTS (X, ξ,Σ), then Cl(Ω,Σ) is infra soft semi-open. Proof. Since (Ω,Σ) is an infra soft pre-open set, (Ω,Σ)⊆̃Int(Cl(Ω,Σ)). Therefore, Cl(Ω,Σ) ⊆̃Cl(Int(Cl(Ω,Σ)))⊆̃Cl(Ω,Σ). Thus, Cl(Ω,Σ) = Cl(Int(Cl(Ω,Σ))). Hence, Cl(Ω,Σ) is infra soft semi-open. In the next two results, we present some characterizations for infra soft pre-open and infra soft pre-closed sets. Proposition 6. A subset (Ω,Σ) of an ISTS (X, ξ,Σ) is infra soft pre-open iff there exists an ξ-infra soft open set (Ψ,Σ) such that (Ω,Σ)⊆̃(Ψ,Σ)⊆̃Cl(Ω,Σ). Proof. Necessity : Let (Ω,Σ) be an infra soft pre-open set. Then (Ω,Σ)⊆̃Int(Cl(Ω,Σ)) ⊆̃Cl(Ω,Σ). Putting (Ψ,Σ) = Int(Cl(Ω,Σ)). Then Int(Ψ,Σ) = Int(Int(Cl(Ω,Σ))) = (Ψ,Σ). Therefore, (Ψ,Σ) is an ξ-infra soft open set. Sufficiency : Let (Ψ,Σ) is an ξ-infra soft open set such that (Ω,Σ)⊆̃(Ψ,Σ)⊆̃Cl(Ω,Σ). Then Int(Ω,Σ)⊆̃(Ψ,Σ)⊆̃Int(Cl(Ω,Σ)). Therefore, (Ω,Σ)⊆̃Int(Cl(Ω,Σ)) which means that (Ω,Σ) is an infra soft pre-open set. T.M. Al-shami, H.A. Othman / Eur. J. Pure Appl. Math, 15 (1) (2022), 261-280 266 Proposition 7. A subset (Ω,Σ) of an ISTS (X, ξ,Σ) is infra soft pre-closed iff there exists an ξ-infra soft closed set (Ψ,Σ) such that Int(Ω,Σ)⊆̃(Ψ,Σ)⊆̃(Ω,Σ). Proof. Similar to the proof of Proposition 6. Proposition 8. The class of infra soft pre-open sets is closed under arbitrary unions. Proof. Consider {(Ωj ,Σ) : j ∈ J} as a family of infra soft pre-open sets. Sup- pose that J ̸= ∅. Then (Ωj ,Σ)⊆̃Int(Cl(Ωj ,Σ)) for each j ∈ J . Consequentially, ⋃̃ j∈J (Ωj ,Σ)⊆̃ ⋃̃ j∈JInt(Cl(Ωj ,Σ)) ⊆̃Int(Cl( ⋃̃ j∈J(Ωj ,Σ))). Hence, ⋃̃ j∈J(Ωj ,Σ) is infra soft pre-open. Corollary 1. The class of infra soft pre-closed sets is closed under arbitrary intersections. Corollary 2. The class of infra soft pre-open subsets of an ISTS (X, ξ,Σ) forms a supra soft topology over X. To illustrate that the class of infra soft pre-open sets does not form an infra soft topology, we present the following example. Example 1. Let X = {x1, x2, x3, x4} and Σ = {η1, η2}. Then ξ = {Φ, X̃, (Ω1,Σ), (Ω2,Σ)} is an infra soft topology on X with Σ as a set of parameters, where (Ω1,Σ) = {(η1, {x1}), (η2, {x1})} and (Ω2,Σ) = {(η1, {x2}), (η2, {x2})}. Let (Ω5,Σ) = {(η1, X), (η2, {x1, x3})} and (Ω6,Σ) = {(η1, {x1, x3}), (η2, X)}. Then (Ω5,Σ) and (Ω6,Σ) are infra soft pre-open sets because Int(Cl(Ω5,Σ)) = X̃ and Int(Cl(Ω6,Σ)) = X̃. But (Ω5,Σ) ⋂̃ (Ω6,Σ) is not infra soft pre-open because Int(Cl[(Ω5,Σ) ⋂̃ (Ω6,Σ)]) = {(η1, {x1}), (η2, {x1})}˜̸⊇[(Ω5,Σ) ⋂̃ (Ω6,Σ)]. Proposition 9. The intersection of infra soft open and infra soft pre-open sets is an infra soft pre-open set. Proof. Let (Ω1,Σ) be an infra soft open set and (Ω2,Σ) be an infra soft pre-open set. Then (Ω1,Σ) ⋂̃ (Ω2,Σ)⊆̃(Ω1,Σ) ⋂̃ Int(Cl(Ω2,Σ)) = Int[(Ω1,Σ) ⋂̃ Cl(Ω2,Σ)]; by Proposi- tion 2 we obtain Int[(Ω1,Σ) ⋂̃ Cl(Ω2,Σ)]⊆̃Int(Cl[(Ω1,Σ) ⋂̃ (Ω2,Σ)]. Hence, (Ω1,Σ) ⋂̃ (Ω2,Σ) is an infra soft pre-open set, as required. Corollary 3. The union of infra soft closed and infra soft pre-closed sets is an infra soft pre-closed set. Definition 17. An ISTS (X, ξ,Σ) is said to be infra soft hyperconnected if the intersection of any two non-null ξ-infra soft open sets is non-null. Otherwise, (X, ξ,Σ) is said to be infra soft dishyperconnected. Proposition 10. The intersection of two infra soft pre-open subsets of an infra soft hy- perconnected space is an infra soft pre-open set. T.M. Al-shami, H.A. Othman / Eur. J. Pure Appl. Math, 15 (1) (2022), 261-280 267 Proof. Let (Ω1,Σ) and (Ω2,Σ) be infra soft pre-open sets. If one of them is the null soft set, then we obtain the desired result. Suppose that (Ω1,Σ) and (Ω2,Σ) are non-null. Ac- cording to Proposition 6 there are two ξ-infra soft open sets (Ψ1,Σ) ̸= Φ and (Ψ2,Σ) ̸= Φ such that (Ω1,Σ)⊆̃(Ψ1,Σ)⊆̃Cl(Ω1,Σ) and (Ω2,Σ)⊆̃(Ψ2,Σ)⊆̃Cl(Ω2,Σ). By hypothesis of infra soft hyperconnectedness, (Ψ1,Σ) ⋂̃ (Ψ2,Σ) is a non-null ξ-infra soft open set. Now, (Ω1,Σ) ⋂̃ (Ω2,Σ)⊆̃(Ψ1,Σ) ⋂̃ (Ψ2,Σ)⊆̃Cl[(Ω1,Σ) ⋂̃ (Ω2,Σ)]. Hence, (Ω1,Σ) ⋂̃ (Ω2,Σ) is an in- fra soft pre-open set. Lemma 1. Let Eτ : (X1, ξ1,Σ1) → (X2, ξ2,Σ2) be an infra soft homeomorphism map. Then for any subset (Ω,Σ1) we have the next two results. (i) Eτ (Int(Ω,Σ1)) = Int(Eτ (Ω,Σ1)). (ii) Eτ (Cl(Ω,Σ1)) = Cl(Eτ (Ω,Σ1)). Proof. To prove (i), let δsη′ ∈ Eτ (Int(Ω,Σ1)). Then there is δxη ∈ Int(Ω,Σ1) such that Eτ (δ x η ) = δsη′ . This means there exists an infra soft open set (Ψ,Σ1) such that δxη ∈ (Ψ,Σ1)⊆̃(Ω,Σ1). Therefore, δsη′ = Eτ (δ x η ) ∈ Eτ (Ψ,Σ1)⊆̃Eτ (Ω,Σ1). This im- plies that δsη′ ∈ Int(Eτ (Ω,Σ1)). Thus, Eτ (Int(Ω,Σ1))⊆̃Int(Eτ (Ω,Σ1)). Conversely, let δsη′ ∈ Int(Eτ (Ω,Σ1)). Then there exists an infra soft open set (Ψ,Σ2) such that δsη′ ∈ (Ψ,Σ2)⊆̃Eτ (Ω,Σ1). Therefore, E−1 τ (δsη′) ∈ E−1 τ (Ψ,Σ2)⊆̃(Ω,Σ1). Automatically, we ob- tain E−1 τ (δsη′) ∈ Int(Ω,Σ1). So that, δ s η′ ∈ Eτ (Int(Ω,Σ1)). Thus, Int(Eτ (Ω,Σ1))⊆̃Eτ (Int(Ω,Σ1)). Hence, the proof is complete. Following similar arguments, one can prove (ii). Proposition 11. The infra soft homeomorphism image of an infra soft pre-open set is an infra soft pre-open set. Proof. Consider Eτ : (X1, ξ1,Σ1) → (X2, ξ2,Σ2) as an infra soft continuous map and let (Ω,Σ1) be an infra soft pre-open subset of (X1, ξ1,Σ1). Then Eτ (Ω,Σ1)⊆̃Eτ (Int(Cl(Ω,Σ1))). It follows from the above lemma that Eτ (Ω,Σ1)⊆̃Int(Cl(Eτ (Ω,Σ1))). Hence, Eτ (Ω,Σ1) is an infra soft pre-open subset of (X2, ξ2,Σ2), as required. Lemma 2. Consider (Ω1,Σ1) and (Ω2,Σ2) as subsets of (X1, ξ1,Σ1) and (X2, ξ2,Σ2), respectively. Then (i) Cl[(Ω1,Σ1)× (Ω2,Σ2)] = Cl(Ω1,Σ1)× Cl(Ω2,Σ2). (ii) Int[(Ω1,Σ1)× (Ω2,Σ2)] = Int(Ω1,Σ1)× Int(Ω2,Σ2). Proof. (i): Let δ (t,s) (η,ϑ) ̸∈ Cl[(Ω1,Σ1)× (Ω2,Σ2)]. Then there is an infra soft open subset (Ψ1,Σ1)×(Ψ2,Σ2) of X̃1×X̃2 containing δ (t,s) (η,ϑ) such that [(Ω1,Σ1)×(Ω2,Σ2)] ⋂̃ [(Ψ1,Σ1)× (Ψ2,Σ2)] = ΦΣ1×Σ2 . This implies that (Ω1,Σ1) ⋂̃ (Ψ1,Σ1) = ΦΣ1 or (Ω2,Σ2) ⋂̃ (Ψ2,Σ2) = T.M. Al-shami, H.A. Othman / Eur. J. Pure Appl. Math, 15 (1) (2022), 261-280 268 ΦΣ2 . Therefore, δtη ̸∈ Cl(Ω1,Σ1) or δsϑ ̸∈ Cl(Ω2,Σ2). Thus, δ (t,s) (η,ϑ) ̸∈ [Cl(Ω1,Σ1) × Cl(Ω2,Σ2)]. Hence, Cl(Ω1,Σ1) × Cl(Ω2,Σ2)⊆̃Cl[(Ω1,Σ1) × (Ω2,Σ2)]. Conversely, let δ (t,s) (η,ϑ) ̸∈ Cl(Ω1,Σ1) × Cl(Ω2,Σ2). Then δxη ̸∈ Cl(Ω1,Σ1) or δsϑ ̸∈ Cl(Ω2,Σ2). Suppose, without loss of generality, that δxη ̸∈ Cl(Ω1,Σ1). Then there is an infra soft open sub- set (Ψ1,Σ1) of (X1, ξ1,Σ1) containing δxη such that (Ω1,Σ1) ⋂̃ (Ψ1,Σ1) = ΦΣ1 . Obvi- ously, (Ψ1,Σ1) × X̃2 is an infra soft open subset of X̃1 × X̃2 containing δ (t,s) (η,ϑ) such that [(Ψ1,Σ1)×X̃2] ⋂̃ [(Ω1,Σ1)×(Ω2,Σ2)] = ΦΣ1×Σ2 . Therefore, δ (t,s) (η,ϑ) ̸∈ Cl[(Ω1,Σ1)×(Ω2,Σ2)]. Thus, Cl[(Ω1,Σ1)× (Ω2,Σ2)]⊆̃Cl(Ω1,Σ1)× Cl(Ω2,Σ2). Hence, the proof is complete. Following similar arguments, one can prove (ii). Proposition 12. The product of infra soft pre-open sets is an infra soft pre-open set. Proof. Let (Ω1,Σ1) and (Ω2,Σ2) be infra soft pre-open subsets of (X1, ξ1,Σ1) and (X2, ξ2,Σ2), respectively. Then (Ω1,Σ1) × (Ω2,Σ2)⊆̃Int(Cl(Ω1,Σ1)) × Int(Cl(Ω2,Σ2)). According to the above lemma, we obtain (Ω1,Σ1)×(Ω2,Σ2)⊆̃Int(Cl[(Ω1,Σ1)×(Ω2,Σ2)]) which means that (Ω1,Σ1)× (Ω2,Σ2) is an infra soft pre-open subset of X̃1 × X̃2. 4. Infra pre-interior, infra pre-closure, infra pre-limit and infra pre-boundary soft points of a soft set The goal of this part is to introduce the concepts of infra soft pre-interior and infra soft pre-closure, infra soft pre-limit and infra soft pre-boundary soft points of a soft set. We explore their essential properties and explain the interrelationships between them with the aid of illustrative examples. Definition 18. Let (Ω,Σ) be a subset of (X, ξ,Σ). Then: (i) the infra soft pre-interior of (Ω,Σ), denoted by pInt(Ω,Σ), is the union of all infra soft pre-open sets that are contained in (Ω,Σ). (ii) the infra soft pre-closure of (Ω,Σ), denoted by pCl(Ω,Σ), is the intersection of all infra soft pre-closed sets containing (Ω,Σ). Proposition 13. We have the following properties. (i) (Ω,Σ) is an infra soft pre-open subset of (X, ξ,Σ) iff pInt(Ω,Σ) = (Ω,Σ). (ii) (Ω,Σ) is an infra soft pre-closed subset of (X, ξ,Σ) iff pCl(Ω,Σ) = (Ω,Σ). Proof. It comes from Proposition 8 and Corollary 1. Note that the the above two properties are not valid for infra soft open and infra soft closed sets. T.M. Al-shami, H.A. Othman / Eur. J. Pure Appl. Math, 15 (1) (2022), 261-280 269 Proposition 14. Let (Ω,Σ) be a subset of (X, ξ,Σ). (i) δxη ∈ pInt(Ω,Σ) iff there is an infra soft pre-open set (Ψ,Σ) such that δxη ∈ (Ψ,Σ)⊆̃(Ω,Σ). (ii) δxη ∈ pCl(Ω,Σ) iff the intersection of any infra soft pre-open set (Ψ,Σ) containing δxη and (Ω,Σ) is non-null. Proof. The proof of (i) is obvious, so we prove (ii). Let δxη ∈ pCl(Ω,Σ). Then every infra soft pre-closed set contains (Ω,Σ) contains δxη as well. Suppose that there exists an infra soft pre-open set (Ψ,Σ) containing δxη such that (Ω,Σ) ⋂̃ (Ψ,Σ) = Φ. Therefore, (Ω,Σ)⊆̃(Ψc,Σ) which means that δxη ̸∈ pCl(Ω,Σ). This is a contradiction. Conversely, suppose that there exists an infra soft pre-open set (Ψ,Σ) containing δxη such that (Ω,Σ) ⋂̃ (Ψ,Σ) = Φ. Therefore, pCl(Ω,Σ)⊆̃(Ψc,Σ) which means that δxη ̸∈ pCl(Ω,Σ). Hence, we obtain the desired result. Proposition 15. Let (Ω,Σ) be a subset of (X, ξ,Σ). Then: (i) (pInt(Ω,Σ))c = pCl(Ωc,Σ). (ii) (pCl(Ω,Σ))c = pInt(Ωc,Σ). Proof. (i): (pInt(Ω,Σ))c = { ⋃̃ j∈J (Ψj ,Σ) : (Ψj ,Σ) is an infra soft pre-open set contained in (Ω,Σ)}c = ⋂̃ j∈J {(Ψc j ,Σ) : (Ψc j ,Σ) is an infra soft pre-closed set containing (Ωc,Σ)} = pCl(Ωc,Σ). The proof of (ii) is similar to (i). Proposition 16. Let (Ψ,Σ) be an infra soft open set and (Λ,Σ) be an infra soft closed set in (X, ξ,Σ). Then: (i) (Ψ,Σ) ⋂̃ pCl(Ω,Σ)⊆̃pCl((Ψ,Σ) ⋂̃ (Ω,Σ)). (ii) pInt((Λ,Σ) ⋃̃ (Ω,Σ))⊆̃(Λ,Σ) ⋃̃ pInt(Ω,Σ). Proof. (i): Let δxη ∈ (Ψ,Σ) ⋂̃ pCl(Ω,Σ). Then δxη ∈ (Ψ,Σ) and δxη ∈ pCl(Ω,Σ). This implies that (Γ,Σ) ⋂̃ (Ω,Σ) ̸= Φ for every infra soft pre-open set (Γ,Σ) containing δxη . It follows from Proposition 9 that (Ψ,Σ) ⋂̃ (Γ,Σ) is an infra soft pre-open set containing δxη . Therefore, [(Ψ,Σ) ⋂̃ (Γ,Σ)] ⋂̃ (Ω,Σ) ̸= Φ. Now, (Γ,Σ) ⋂̃ [(Ψ,Σ) ⋂̃ (Ω,Σ)] ̸= Φ which means that δxη ∈ pCl((Ψ,Σ) ⋂̃ (Ω,Σ)). Hence, (Ψ,Σ) ⋂̃ pCl(Ω,Σ)⊆̃pCl((Ψ,Σ) ⋂̃ (Ω,Σ)). One can prove (ii) following similar arguments. Theorem 1. Let (Ω,Σ) and (Ψ,Σ) be subsets of (X, ξ,Σ). Then we have the following properties. (i) pInt(X̃) = X̃. T.M. Al-shami, H.A. Othman / Eur. J. Pure Appl. Math, 15 (1) (2022), 261-280 270 (ii) pInt(Ω,Σ)⊆̃(Ω,Σ). (iii) If (Ψ,Σ)⊆̃(Ω,Σ), then pInt(Ψ,Σ)⊆̃pInt(Ω,Σ). (iv) pInt(pInt(Ω,Σ)) = pInt(Ω,Σ). (v) pInt(Ψ,Σ) ⋂̃ pInt(Ω,Σ)⊆̃pInt((Ψ,Σ) ⋂̃ (Ω,Σ)). Proof. (i): Since X̃ is infra soft pre-open, pInt(X̃) = X̃. (ii) and (iii) are obvious. (iv): It is clear that pInt(pInt(Ω,Σ)) is the largest infra soft pre-open set contained in pInt(Ω,Σ); however, pInt(Ω,Σ) is an infra soft pre-open set; hence, pInt(pInt(Ω,Σ)) = pInt(Ω,Σ). (v): It comes from (iii). Theorem 2. Let (Ω,Σ) and (Ψ,Σ) be subsets of (X, ξ,Σ). Then we have the following properties. (i) pCl(Φ) = Φ. (ii) (Ω,Σ)⊆̃pCl(Ω,Σ). (iii) If (Ψ,Σ)⊆̃(Ω,Σ), then pCl(Ψ,Σ)⊆̃pCl(Ω,Σ). (iv) pCl(pCl(Ω,Σ))⊆̃pCl(Ω,Σ). (v) pCl((Ψ,Σ) ⋃̃ (Ω,Σ)) = pCl(Ψ,Σ) ⋃̃ pCl(Ω,Σ). Proof. It can be proved following similar arguments given in the proof of Theorem 1. The next example shows that the inclusion relations given in the above two theorems are proper. Example 2. Let X = {x1, x2} and Σ = {η1, η2}. Then ξ = {Φ, X̃, (Ωj ,Σ) : j = 1, 2, 3} is an infra soft topology on X over X with Σ as a set of parameters, where (Ω1,Σ) = {(η1, {x1}), (η2, ∅)}; (Ω2,Σ) = {(η1, ∅), (η2, {x1})} and (Ω3,Σ) = {(η1, X), (η2, {x2})}. Let (Ψ1,Σ) = {(η1, {x2}), (η2, {x1})}. Then pInt(Ψ1,Σ) = {(η1, ∅), (η2, {x1})}⊂̃(Ψ1,Σ) and pCl(Ψ1,Σ) = {(η1, {x2}), (η2, X)}⊃̃(Ψ1,Σ). Also, consider (Ψ2,Σ) = {(η1, {x2}), (η2, ∅)}. Then pCl((Ψ1,Σ) ⋃̃ (Ψ2,Σ)) = {(η1, {x2}), (η2, X)}⊇̃pCl(Ψ1,Σ) ⋂̃ pCl(Ψ2,Σ) = {(η1, {x2}), (η2, {x2})}. Definition 19. A soft point δxη is said to be an infra soft pre-limit point of a subset (Ω,Σ) of (X, ξ,Σ) provided that [(Ψ,Σ)\δxη ] ⋂̃ (Ω,Σ) ̸= Φ for every infra soft pre-open set (Ψ,Σ) containing δxη . The soft set of all infra soft pre-limit points of (Ω,Σ) is said to be an infra pre-derived soft set. It is denoted by (Ω,Σ)ps′. T.M. Al-shami, H.A. Othman / Eur. J. Pure Appl. Math, 15 (1) (2022), 261-280 271 Proposition 17. Consider (Ψ,Σ) and (Ω,Σ) as subsets of (X, ξ,Σ). Then (i) Φps′ = Φ and X̃ps′⊆̃X̃. (ii) If (Ψ,Σ)⊆̃(Ω,Σ), then (Ψ,Σ)ps′⊆̃(Ω,Σ)ps′. (iii) If δxη ∈ (Ω,Σ)ps′, then δxη ∈ ((Ω,Σ) \ δxη )ps′. (iv) (Ψ,Σ)ps′ ⋃̃ (Ω,Σ)ps′⊆̃((Ψ,Σ) ⋃̃ (Ω,Σ))ps′. Proof. Straightforward. Theorem 3. Let (Ω,Σ) be a subset of (X, ξ,Σ). Then (i) If (Ω,Σ) is an infra soft pre-closed set, then (Ω,Σ)ps′ ⊆ (Ω,Σ). (ii) ((Ω,Σ) ⋃̃ (Ω,Σ)ps′)ps′⊆̃(Ω,Σ) ⋃̃ (Ω,Σ)ps′. (iii) pCl(Ω,Σ) = (Ω,Σ) ⋃̃ (Ω,Σ)ps′. Proof. (i) Consider (Ω,Σ) as an infra soft pre-closed set such that δxη ̸∈ (Ω,Σ). Then δxη ∈ (Ωc,Σ). Now, (Ωc,Σ) is an infra soft pre-open set such that (Ωc,Σ) ⋂̃ (Ω,Σ) = Φ which means that δxη ̸∈ (Ω,Σ)ps′. Thus, (Ω,Σ)ps′⊆̃(Ω,Σ). (ii) Consider δxη ̸∈ (Ω,Σ) ⋃̃ (Ω,Σ)ps′. Then δxη ̸∈ (Ω,Σ) and δxη ̸∈ (Ω,Σ)ps′. Therefore, there exists an infra soft pre-open set (Ψ,Σ) such that (Ψ,Σ) ⋂̃ (Ω,Σ) = Φ (1) This implies that (Ψ,Σ) ⋂̃ (Ω,Σ)ps′ = Φ (2) It follows from (1) and (2) that (Ψ,Σ) ⋂̃ ((Ω,Σ) ⋃̃ (Ω,Σ)ps′) = Φ. Thus, δxη ̸∈ ((Ω,Σ) ⋃̃ (Ω,Σ)ps′)ps′. Hence, ((Ω,Σ) ⋃̃ (Ω,Σ)ps′)ps′⊆̃((Ω,Σ) ⋃̃ (Ω,Σ)ps′), as required. (iii) It is clear that (Ω,Σ) ⋃̃ (Ω,Σ)ps′⊆̃pCl(Ω,Σ). Conversely, let δxη ∈ pCl(Ω,Σ). Then for every infra soft pre-open set containing δxη we have (Ω,Σ) ⋂̃ (Ψ,Σ) ̸= Φ. Without loss of generality, let δxη ̸∈ (Ω,Σ). Then [(Ω,Σ)\δxη ] ⋂̃ (Ψ,Σ) ̸= Φ. Consequentially, δxη ∈ (Ω,Σ)ps′. Hence, the proof is complete. Definition 20. The infra soft pre-boundary points of a subset (Ω,Σ) of (X, ξ,Σ), denoted by pB(Ω,Σ), are all the soft points which belong to the complement of pInt(Ω,Σ) ⋃̃ pInt(Ωc,Σ). T.M. Al-shami, H.A. Othman / Eur. J. Pure Appl. Math, 15 (1) (2022), 261-280 272 Proposition 18. Let (Ω,Σ) be a subset of (X, ξ,Σ). Then: (i) pB(Ω,Σ) = pCl(Ω,Σ) ⋂̃ pCl((Ωc,Σ)). (ii) pB(Ω,Σ) = pCl(Ω,Σ) \ pInt(Ω,Σ). Proof. (i) pB(Ω,Σ) = {δxη ∈ X̃ : δxη ̸∈ pInt(Ω,Σ) and δxη ̸∈ pInt((Ωc,Σ))} = {δxη ∈ X̃ : δxη ̸∈ (pCl(Ωc,Σ))c and δxη ̸∈ (pCl(Ω,Σ))c} = {δxη ∈ X̃ : δxη ∈ pCl(Ωc,Σ) and δxη ∈ pCl(Ω,Σ)} = pCl(Ω,Σ) ⋂̃ pCl(Ωc,Σ) (ii) pB(Ω,Σ) = pCl(Ω,Σ) ⋂̃ pCl(Ωc,Σ) = pCl(Ω,Σ) ⋂̃ (pInt(Ω,Σ))c = pCl(Ω,Σ) \ pInt(Ω,Σ) Corollary 4. Let (Ω,Σ) be a subset of (X, ξ,Σ). Then (i) pB(Ω,Σ) = pB(Ωc,Σ) (ii) pCl(Ω,Σ) = pInt(Ω,Σ) ⋃̃ pB(Ω,Σ) Proposition 19. Let (Ω,Σ) be a subset of (X, ξ,Σ). Then (i) (Ω,Σ) is infra soft pre-open iff pB(Ω,Σ) ⋂̃ (Ω,Σ) = Φ. (ii) (Ω,Σ) is infra soft pre-closed iff pB(Ω,Σ)⊆̃(Ω,Σ). Proof. (i) pB(Ω,Σ) ⋂ (Ω,Σ) = pB(Ω,Σ) ⋂ pInt(Ω,Σ) = Φ. Conversely, let δxη ∈ (Ω,Σ). Then δxη ∈ pInt(Ω,Σ) or δxη ∈ pB(Ω,Σ). Since pB(Ω,Σ) ⋂ (Ω,Σ) = Φ, δxη ∈ pInt(Ω,Σ). Thus, (Ω,Σ) ⊆ pInt(Ω,Σ) which means that (Ω,Σ) = pInt(Ω,Σ). Hence, (Ω,Σ) is infra soft pre-open. (ii) (Ω,Σ) is infra soft pre-closed ⇔ (Ωc,Σ) is infra soft pre-open ⇔ pB(Ωc,Σ) ⋂ (Ωc,Σ) = Φ ⇔ pB(Ω,Σ) ⋂ (Ωc,Σ) = Φ ⇔ pB(Ω,Σ) ⊆ (Ω,Σ). Corollary 5. A subset (Ω,Σ) of (X, ξ,Σ) is infra soft pre-open and infra soft pre-closed iff pB(Ω,Σ) = Φ. T.M. Al-shami, H.A. Othman / Eur. J. Pure Appl. Math, 15 (1) (2022), 261-280 273 5. Infra soft pre-homeomorphism maps We devote this section to introducing new types of soft maps called infra soft pre- continuous, infra soft pre-open, infra soft pre-closed and infra soft pre-homeomorphism maps. We study their characterizations and establish main properties. Definition 21. A soft map Eτ : (X, ξ,Σ) → (S, π,∆) is said to be infra soft pre- continuous at δxη ∈ X̃ if for any infra soft pre-open set (Ψ,∆) containing Eτ (δ x η ), there is an infra soft pre-open set (Ω,Σ) containing δxη suchthat Eτ (Ω,Σ)⊆̃(Ψ,∆). If Eτ is infra soft pre-continuous at all soft points of the domain, then it is called infra soft pre-continuous. Theorem 4. Let Eτ : (X, ξ,Σ) → (S, π,∆) be an infra soft pre-continuous map. Then we have the following five equivalent statements: (i) Eτ is an infra soft pre-continuous map; (ii) The pre-image of each infra soft pre-closed set is infra soft pre-closed; (iii) pCl(E−1 τ (Ω,∆))⊆̃E−1 τ (pCl(Ω,∆)) for each (Ω,∆)⊆̃S̃; (iv) Eτ (pCl(Ψ,Σ))⊆̃pCl(Eτ (Ψ,Σ)) for each (Ψ,Σ)⊆̃X̃; (v) E−1 τ (pInt(Ω,∆))⊆̃pInt(E−1 τ (Ω,∆)) for each (Ω,∆)⊆̃S̃. Proof. (i) ⇒ (ii): Let (Ω,∆) be an infra soft pre-closed set in (S, π,∆). Then E−1 τ (Ωc,∆) is an infrasoft pre-open subset of X̃. Obviously, E−1 τ (Ωc,∆) = X̃−E−1 τ (Ω,∆); hence, E−1 τ (Ω,∆) is an infra soft pre-closed subset of X̃. (ii) ⇒ (iii): According to (ii), E−1 τ (pCl(Ω,∆)) is an infra soft pre-closed subset of X̃. Then pCl(E−1 τ (Ω,∆))⊆̃pCl(E−1 τ (pCl(Ω,∆))) = E−1 τ (pCl(Ω,∆)). (iii) ⇒ (vi): According to (iii), pCl(E−1 τ (Eτ (Ψ,Σ)))⊆̃E−1 τ (pCl(Eτ (Ψ,Σ))). Then Eτ (pCl(Ψ,Σ))⊆̃Eτ (E −1 τ (pCl(Eτ (Ψ,Σ))))⊆̃pCl(Eτ (Ψ,Σ)). (iv) ⇒ (v): According to (iv), Eτ (pCl(X̃−E−1 τ (Ω,∆)))⊆̃pCl(Eτ (X̃−E−1 τ (Ω,∆))). There- fore, Eτ (X̃ − pInt(E−1 τ (Ω,∆))) = Eτ (pCl(X̃ − E−1 τ (Ω,∆))) ⊆ pCl(S̃ − (Ω,∆)) = S̃ − pInt(Ω,∆). Thus X̃−pInt(E−1 τ (Ω,∆))⊆̃E−1 τ (S̃−pInt(Ω,∆)) = E−1 τ (S̃)−E−1 τ (pInt(Ω,∆)). Hence E−1 τ (pInt(Ω,∆))⊆̃pInt(E−1 τ (Ω,∆)). (v) ⇒ (i): Let (Ω,∆) be an infra soft open subset of S̃. According to (v), E−1 τ (Ω,∆)⊆̃pInt(E−1 τ (Ω,∆)). This implies that E−1 τ (Ω,∆) = pInt(E−1 τ (Ω,∆)). Hence, Eτ is infra soft pre-continuous. Theorem 5. If Eτ : (X, ξ,Σ) → (S, π,∆) is infra soft pre-continuous, then the restriction soft map Eτ|M : (M, ξM,Σ) → (S, π,∆) is infra soft pre-continuous provided that M̃ is an infra soft open set. Proof. Consider (Ω,∆) is an infra soft pre-open set in (S, π,∆). By hypothesis, E−1 τ (Ω,∆) is infra soft pre-open. Now, E−1 τ|M (Ω,∆) = E−1 τ (Ω,∆) ⋂̃ M̃. Since M̃ is an T.M. Al-shami, H.A. Othman / Eur. J. Pure Appl. Math, 15 (1) (2022), 261-280 274 infra soft open set, it follows from Proposition 9 that E−1 τ|M (Ω,∆) is infra soft pre-open. Hence, Eτ|M is an infra soft pre-continuous map. Proposition 20. Let Eτ : (X, ξ,Σ) → (S, π,∆) and Fν : (S, π,∆) → (V, σ,Γ) be infra soft pre-continuous. Then Fν ◦ Eτ is infra soft pre-continuous. Proof. Straightforward. Definition 22. A soft map Eτ : (X, ξ,Σ) → (S, π,∆) is said to be infra soft pre-open (resp., infra soft pre-closed) if the image of each infra soft pre-open (resp., infra soft pre-closed) set is infra soft pre-open (resp., infra soft pre-closed). Proposition 21. Eτ : (X, ξ,Σ) → (S, π,∆) is an infra soft pre-open map iff Eτ (pInt(Ω,Σ)) ⊆̃pInt(Eτ (Ω,Σ)) for each subset of (Ω,Σ) of X̃. Proof. ⇒: Let (Ω,Σ) be a subset of X̃. Now, Eτ (pInt(Ω,Σ))⊆̃Eτ (Ω,Σ) and pInt(Ω,Σ) is an infra soft pre-open set. By hypothesis, Eτ (pInt(Ω,Σ)) is infra soft pre-open. There- fore, Eτ (pInt(Ω,Σ))⊆̃pInt(Eτ (Ω,Σ)). ⇐: Let (Λ,Σ) be an infra soft open subset of X̃. Then Eτ (Ω,Σ)⊆̃pInt(Eτ (Ω,Σ)). There- fore, Eτ (Ω,Σ) = pInt(Eτ (Ω,Σ)) which means that Eτ is an infra soft pre-open map. Proposition 22. Eτ : (X, ξ,Σ) → (S, π,∆) is an infra soft pre-closed map iff pCl(Eτ (Ω,Σ)) ⊆̃Eτ (pCl(Ω,Σ)) for each subset (Ω,Σ) of X̃. Proof. ⇒: Let Eτ be an infra soft pre-closed map and (Ω,Σ) be a subset of X̃. By hypothesis, Eτ (pCl(Ω,Σ)) is infra soft pre-closed. Since Eτ (Ω,Σ)⊆̃Eτ (pCl(Ω,Σ)), pCl(Eτ (Ω,Σ)) ⊆̃Eτ (pCl(Ω,Σ)). ⇐: Suppose that (Ω,Σ) is an infra soft pre-closed subset of X̃. By hypothesis, Eτ (Ω,Σ)⊆̃ pCl(Eτ (Ω,Σ))⊆̃Eτ (pCl(Ω,Σ)) = Eτ (Ω,Σ). Therefore, Eτ (Ω,Σ) is infra soft pre-closed. Hence, Eτ is an infra soft pre-closed map. Proposition 23. The concepts of infra soft pre-open and infra soft pre-closed maps are equivalent under bijectiveness. Proof. It comes from the fact that a bijective soft map Eτ : (X, ξ,Σ) → (S, π,∆) implies that Eτ (Ω c,Σ) = (Eτ (Ω,Σ)) c. Proposition 24. Let Eτ : (X, ξ,Σ) → (S, π,∆) and Fν : (S, π,∆) → (V, σ,Γ) be two soft maps. Then: (i) If Eτ and Fν are infra soft pre-open maps, then Fν ◦Eτ is an infra soft pre-open map. (ii) If Fν ◦Eτ is an infra soft pre-open map and Eτ is a surjective infra soft pre-continuous map, then Fν is an infra soft pre-open map. T.M. Al-shami, H.A. Othman / Eur. J. Pure Appl. Math, 15 (1) (2022), 261-280 275 (iii) If Fν◦Eτ is an infra soft pre-open map and Fν is an injective infra soft pre-continuous map, then Eτ is an infra soft pre-open map. Proof. (i) Straightforward. (ii) Consider (Ω,∆) as an infra soft pre-open subset of (S, π,∆). By hypothesis, E−1 τ (Ω,∆) is an infra soft pre-open subset of (X, ξ,Σ). Again, by hypothesis, (Fν◦Eτ )(E −1 τ (Ω,∆)) is an infra soft pre-open subset of (V, σ,Γ). Since Eτ is surjective, then (Fν ◦ Eτ )(E −1 τ (Ω,∆)) = Fν(Eτ (E −1 τ (Ω,∆))) = Fν(Ω,∆). Hence, Fν is an infra soft pre- open map. (iii) Consider (Ω,Σ) as an infra soft pre-open subset of (X, ξ,Σ). By hypothesis, (Fν ◦ Eτ )(Ω,Σ) is an infra soft pre-open subset of (V, σ,Γ). Again, by hypothesis, F−1 ν (Fν◦ Eτ (Ω,Σ)) is an infra soft pre-open subset of (S, π,∆). Since Fν is injective, then F−1 ν (Fν ◦ Eτ (Ω,Σ)) = (F−1 ν Fν)(Eτ (Ω,Σ)) = Eτ (Ω,Σ). Hence, Eτ is an infra soft pre-open map. In a similar way, one can prove the next proposition. Proposition 25. Let Eτ : (X, ξ,Σ) → (S, π,∆) and Fν : (S, π,∆) → (V, σ,Γ) be two infra soft maps. Then the following statements hold. (i) If Eτ and Fν are infra soft pre-closed maps, then Fν ◦ Eτ is an infra soft pre-closed map. (ii) If Fν ◦ Eτ is an infra soft pre-closed map and Eτ is a surjective infra soft pre- continuous map, then Fν is an infra soft pre-closed map. (iii) If Fν ◦ Eτ is an infra soft pre-closed map and Fν is an injective infra soft pre- continuous map, then Eτ is an infra soft pre-closed map. Definition 23. A bijective soft map Eτ : (X, ξ,Σ) → (S, π,∆) is said to be an infra soft pre-homeomorphism if it is infra soft pre-continuous and infra soft pre-open. We cancel the proofs of the next two results because they are easy. Proposition 26. Let Eτ : (X, ξ,Σ) → (S, π,∆) and Fν : (S, π,∆) → (V, σ,Γ) be infra soft pre-homeomorphism maps. Then Fν ◦ Eτ is an infra soft pre-homeomorphism map. Proposition 27. If Eτ : (X, ξ,Σ) → (S, π,∆) is a bijective soft map, then the following statements are equivalent. (i) Eτ is an infra soft pre-homeomorphism. (ii) Eτ and E−1 τ is infra soft pre-continuous. (iii) Eτ is infra soft pre-closed and infra soft pre-continuous. T.M. Al-shami, H.A. Othman / Eur. J. Pure Appl. Math, 15 (1) (2022), 261-280 276 Proposition 28. If Eτ : (X, ξ,Σ) → (S, π,∆) is an infra soft pre-homeomorphism map, then the following statements hold for each (Ω,Σ) ∈ S(X)A. (i) Eτ (pInt(Ω,Σ)) = pInt(Eτ (Ω,Σ)). (ii) Eτ (pCl(Ω,Σ)) = pCl(Eτ (Ω,Σ)). Proof. (i): According to Proposition 21 (i), we obtain Eτ (pInt(Ω,Σ))⊆̃pInt(Eτ (Ω,Σ)). Conversely, let δsκ ∈ pInt(Eτ (Ω,Σ). Then there is an infra soft pre-open set (Ψ,∆) such that δsκ ∈ (Ψ,∆)⊆̃Eτ (Ω,Σ). By hypothesis, δxη = E−1 τ (δsκ) ∈ E−1 τ (Ψ,∆)⊆̃(Ω,Σ) such that E−1 τ (Ψ,∆) is an infra soft pre-open set. So that, δxη ∈ pInt(Ω,Σ) which means that δsκ ∈ Eτ (pInt(Ω,Σ)). One can achieve item (ii) following similar arguments. Theorem 6. The property of an infra soft pre-dense set is an infra soft topological in- variant. Proof. Let Eτ : (X, ξ,Σ) → (S, π,∆) be an infra soft pre-homeomorphism map and consider (Ω,Σ) as an infra soft pre-dense subset of (X, ξ,Σ), i.e. pCl(Ω,Σ) = X̃. It comes from Proposition 28 (ii) that pCl(Eτ (Ω,Σ)) = Eτ (pCl(Ω,Σ)) = Eτ (X̃) = pCl(S̃) = S̃. Thus, Eτ (Ω,Σ) is an infra soft pre-dense set in (S, π,∆), as required. We complete this section by studying the concept of fixed soft points with respect to infra soft pre-open sets. Definition 24. We say that (X, ξ,Σ) has a pre-fixed soft point property provided that for every infra soft pre-continuous map Eτ : (X, ξ,Σ) → (X, ξ,Σ) there exists δsη ∈ X such that Eτ (δ s η) = δsη. Proposition 29. The property of being a pre-fixed soft point is preserved under an infra soft pre-homeomorphism. Proof. Consider (X1, ξ1,Σ1) and (X2, ξ2,Σ2) as two infra soft pre-homeomorphism. This means that there exists a bijective soft map Eτ : (X1, ξ1,Σ1) → (X2, ξ2,Σ2) suchthat Eτ and E−1 τ are infra soft pre-continuous. Suppose that (X1, ξ1,Σ1) has the property of pre-fixed soft point. That is any infra soft pre-continuous map Eτ : (X1, ξ1,Σ1) → (X1, ξ1,Σ1) has a pre-fixed soft point. Now, consider Cτ : (X2, ξ2,Σ2) → (X2, ξ2,Σ2) is infra soft pre-continuous. It is clear that Cτ ◦Eτ : (X1, ξ1,Σ1) → (X2, ξ2,Σ2) is infra soft pre-continuous. Therefore, E−1 τ ◦ Cτ ◦ Eτ : (X1, ξ1,Σ1) → (X1, ξ1,Σ1) is infra soft pre- continuous. Since (X1, ξ1,Σ1) has a pre-fixed soft point property, E−1 τ (hτ (Eτ (δ s η))) = δsη for some δsη ∈ X̃. Thus, Eτ (E −1 τ (hτ (Eτ (δ s η)))) = Eτ (δ s η). This implies that hτ (Eτ (δ s η)) = Eτ (δ s η). Hence, Eτ (δ s η) is a pre-fixed soft point of Cτ which means that (X2, ξ2,Σ2) has a pre-fixed soft point property. REFERENCES 277 6. Concluding remark and further work In this paper, we contribute to the area of infra soft topologies. We have generalized infra soft open and infra soft closed sets by introducing the concepts of infra soft pre- open and infra soft pre-closed sets. Then, we have applied them to define new kinds of soft operators and soft maps. To validate and illustrate the obtained findings and relationships, we have constructed some examples. As we have noted, most of soft topological properties of initiated concepts are kept via infra soft topologies. This means the absence of some topology’s stipulations does not effected in the behaviours and properties of topological concepts which considers an advantage of studying infra soft topological spaces. However, there is a few properties of some topological concept are partially losing such as the those given in Proposition 6 and Proposition 21. In the upcoming works, we will apply infra soft pre-open sets to introduce the some topological concepts like separation axioms, compactness and connectedness. Also, we will present the concepts and results given in this paper using new generalizations of infra soft open sets such as infra soft α-open and infra soft b-open sets. Furthermore, we shall define new rough set models using infra soft pre-open sets to improve the accuracy measures of sets following a similar technique what was given in [8]. Conflict of interest The authors declare that there is no conflict of interest regarding the publication of this paper. 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