EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 15, No. 4, 2022, 1455-1471 ISSN 1307-5543 – ejpam.com Published by New York Business Global On b-open sets via infra soft topological spaces Radwan Abu-Gdairi1, Mohammed M. Ali Al-Shamiri2,3, S. Saleh4,5, T. M. Al-shami6,7,∗ 1 Department of Mathematics, Faculty of Science, Zarqa University, P.O. Box 13110 Zarqa, Jordan 2 Department of Mathematics, Faculty of Science and Arts, King Khalid University, Muhayl Assir, Saudy Arabia 3 Department of Mathematics and Computer, Faculty of Science, Ibb University,Ibb, Yemen 4 Computer Science Department, Cihan University-Erbil, Kurdistan region, Iraq 5 Department of Mathematics, Hodeidah University, Hodeidah, Yemen 6 Department of Mathematics, Sana’a University, Sana’a, Yemen 7 Future University, Egypt Abstract. This work aims to present the concept of infra soft b-open sets (IS-b-open sets) as a generalized new class of infra open sets (IS-open sets). We first investigate their basic properties and study their behaviours under infra soft homeomorphism maps. Then, we establish some soft operators such as interior, closure, limit and boundary using IS-b-open sets and IS-b-closed sets. The relationships between them are illustrated and discussed. Finally, we display some soft maps (S-map) defined using IS-b-open and IS-b-closed sets and scrutinize their master properties. 2020 Mathematics Subject Classifications: 54A40, 54C08, 54C99 Key Words and Phrases: Infra soft b-open set, infra soft b-interior points, infra soft b-closure points, infra soft b-continuity 1. Introduction Molodtsov [62] proposed the idea of soft set (S-set) as a new mathematical tool to deal with vagueness. He presented some of its applications to some areas. Since the advent of S-set, they have been applied to address some problems and phenomena in different disciplines such as information system [9], economy [14], linear equations [27], computer science [50] and decision-making problems [53]. The main operations and operators via S-set theory such as the difference, intersection, and union between two S-sets, and a complement of an S-set were introduced by Maji et ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v15i4.4497 Email addresses: rgdairi@zu.edu.jo (R. Abu-Gdairi), mal-shamiri@kkul.edu.sa (M.M.A. Al-Shamiri), salem.saleh@cihanuniversity.edu.iq (S. Saleh), tareqalshami83@gmail.com (T.M. Al-shami) https://www.ejpam.com 1455 © 2022 EJPAM All rights reserved. T.M. Al-shami et al. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1455-1471 1456 al. [61]. Then, new operations and operators between S-sets were presented in [27, 44]. Some extensions of S-sets were proposed with the goal of expanding the applications of S-sets such as bipolar S-sets [8] and double-framed S-sets [38]. Recently, topology has been applied to model some real-life issues as showed in [1, 11, 15, 16, 32, 46, 56, 64]. To study topology via S-set theory, Çaǧman et al. [51] and Shabir and Naz [65], in 2011, introduced the concept of soft topology(ST). They followed different techniques for studying ST. This article follows Shabir and Naz’ technique which is defined an ST over a fixed set of universe and a fixed set of parameters. The basic concepts and notions of classical topology have been studied in ST such as caliber and chain conditions [43], compactness [2, 26, 28, 29, 40, 49], local compactness [47] separation axioms [18, 21, 22, 42, 45, 52], fixed point theorem [7, 19], connectedness [54, 57, 60], mappings [20, 25, 30, 58], bioperators [48], covering properties [34, 35, 59], sum of topologies [31, 36] and generalized open sets [3]. Additionally, STs and supra STs were discussed in ordered settings as given in [24]. Al-shami and Kočinac [33] elucidated the conditions under which the soft operators and classical operators of interior and closure are interchangeable. It should be noted that some classical topological properties were generalized to STs without consideration for the divergences between STs and classical topologies, which causes some incorrect forms of some results; so some articles were conducted to put forward the correct frame of these results via soft structures; see [4–6]. In 2021, Al-shami [13] familiarized the structure of infra soft topologies(ISTs) [13] and showed the motivations for studying this structure. He with his coauthors continued investigating several topological concepts and properties via this structure such as com- pactness [12], homeomorphisms [10], connectedness [17], separation axioms [37, 39], infra soft semi-open (IS-semi-open) [23] and infra soft pre-open (IS-pre-open) sets [41]. In this article, we display the notion of soft b-open sets (S-b-open sets) and applied to initiate new operators and mappings via infra soft structures. The structure of this article is designed as follows. In Sect. 2, we recall the main ideas and findings that make this work self-contained. In Sect. 3, we introduce the notion of infra soft b-open sets(IS-b-open sets) and establish its master characterizations. In Sect. 4, we define new operators and discuss their main properties. In Sect. 5, we explore novel kinds of mappings and demonstrated their features. Ultimately, we give the main contributions of the article and propose some future works. 2. Preliminaries 2.1. Soft set theory Definition 1. [62] A mapping H from a set of parameters O into 2X , where 2X is the power set of X, is called an S-set denoted by (H,O), and it can written as follows (H,O) = {(o,H(o)) : o ∈ O and H(o) ∈ 2X}. C(XO) refers to the class of all S-sets over X with the set of parameters O. T.M. Al-shami et al. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1455-1471 1457 Definition 2. [44] A complement of an S-set (H,O), denoted by (Hc,O), provided that a map Hc : O → 2X is given by Hc(o) = X \ H(o) for each o ∈ O. Definition 3. [61] If H(o) = ∅ (resp., H(o) = X) for all o ∈ O, then (H,O) is called a null S-set (resp., an absolute) S-set over X. Φ and X̃ are the symbols of null S-set and absolute S-set,respectively. Definition 4. [63] (H,O) is called a soft point (S-point) on X if there is o ∈ O such that H(o) = x ∈ X and H(o′) = ∅ for each o′ ̸= o. The symbol of an S-point will be δxo . Definition 5. [44] The intersection of S-sets (H,O) and (F ,∆) on X, symbolized by (H,O)∩̃(F ,∆), is an S-set (G, T ), where T = O ∩∆ ̸= ∅, and a map G : T → 2X is given by G(o) = H(o) ∩ F(o) for each o ∈ T . Definition 6. [61] The union of S-sets (H,O) and (F ,∆) on X, symbolized by (H,O)∪̃(F ,∆), is an S-set (G, T ), where T = O ∪∆ and a map T : O → 2X is given as follows: G(o) =  H(o) : o ∈ O \∆ F(o) : o ∈ ∆ \ O H(o) ∪ F(o) : o ∈ O ∩∆ Definition 7. [55] A S-set (H,O) is a subset of an S-set (F ,∆), symbolized by (H,O)⊆̃(F ,∆), if O ⊆ ∆ and H(o) ⊆ F(o) for all o ∈ O. If (H,O)⊆̃(F ,∆) and (F ,∆)⊆̃(H,O), then (H,O) and (F ,∆) are called soft equal. The definition of soft maps(S-map) in [58] was adjusted as follows. Definition 8. [10] Let f : X → S and ψ : O → ∆ be two maps. A S-map fψ of C(XO) into C(S∆) is a relation such that any S-point in C(XO) is related to one and only one S-point in C(S∆) such that fψ(δ x o ) = δ f(x) ψ(o) for any δxo ∈ C(XO). In addition, f−1 ψ (δyγ) = ⊔ λ∈ψ−1(γ) x∈f−1(y) δxλ for any δyγ ∈ C(S∆). Definition 9. [63] For an S-map fψ : C(XO) → C(S∆), if f and ψ are injective (resp., surjective, bijective), then fψ is called injective (resp., surjective, bijective). 2.2. Infra soft topological spaces Definition 10. [13] A subfamily µ of C(XO) is called an infra soft topology(IST) on X if it contains Φ and it is closed under finite intersection. The triple (X,µ,O) is called an ISTS. The elements of µ are called IS-open sets and their complements are called IS-closed sets. Definition 11. [13] Let (H,O) be a subset of (X,µ,O). T.M. Al-shami et al. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1455-1471 1458 (i) the IS-closure points of (H,O), denoted by cl(H,O), is the intersection of all IS-closed subsets of (X,µ,O) containing (H,O). (ii) the IS-interior points of (H,O), denoted by int(H,O) is the union of all IS-open subsets of (X,µ,O) which are contained in (H,O). Proposition 1. [13] Let (H,O) and (F ,O) subsets of an ISTS (X,µ,O). Then (i) cl[(H,O)∪̃(F ,O)] = cl(H,O)∪̃cl(F ,O), and (ii) int[(H,O)∩̃(F ,O)] = int(H,O)∩̃int(F ,O). Proposition 2. [13] Let (H,O) be an IS-open set. Then (H,O)∩̃cl(F ,O)⊆̃cl[(H,O)∪̃(F ,O)] for any (F ,O) in (X,µ,O). Proposition 3. [13] Let (H,O) be an IS-closed set. Then int[(H,O)∪̃(F ,O)]⊆̃(H,O)∪̃int(F ,O) for any (H,O) in (X,µ,O). Definition 12. [10] A bijective S-map fψ : (X,µ,O) → (S, ν,∆) is said to be an IS- homeomorphism if it is IS-open (i.e,the image of any IS-open set is IS-open), and IS- continuous (i.e,the pre-image of any IS-open set is IS-open). We call a property which is kept by any IS-homeomorphism an IS-topological property. Definition 13. [10] Let fψ : (X,µ,O) → (S, ν,∆) be an S-map and M ≠ ∅ be a subset of X. A S-map fψ|M : (M, µM,O) → (S, ν,∆) which given by fψ|M(δmo ) = fψ(δ m o ) for every δmo ∈ M̃ is called a restriction S-map of fψ on M. Lemma 1. [23, 41] Let fψ : (X1, µ1,O1) → (X2, µ2,O2) be an IS-homeomorphism map. Then for any (H,O1) we have: (i) fψ(int(H,O1)) = int(fψ(H,O1)). (ii) fψ(cl(H,O1)) = cl(fψ(H,O1)). 3. Main properties of infra soft b-open sets Definition 14. A S-set (H,O) in an ISTS (X,µ,O) is said to be IS-b-open if (H,O)⊆̃ int(cl(H,O))∪̃cl(int(H,O)). Its complement is said to be an IS-b-closed set. Proposition 4. Every IS-semi-open (IS-pre-open) set is IS-b-open. Proof. Let (H,O) be an IS-semi-open (resp. IS-pre-open) set. Then, (H,O)⊆̃cl(int(H,O)) (resp. (H,O)⊆̃int(cl(H,O))). Automatically, we obtain (H,O)⊆̃int(cl(H,O))∪̃cl(int(H,O)), which means that (H,O) is IS-b-open. The converse of the above proposition fails as the next example shows. T.M. Al-shami et al. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1455-1471 1459 Example 1. Let X = {x1, x2, x3} and O = {o1, o2}. Then µ = {Φ, X̃, (H1,O), (H2,O)} is an IST on X, where (H1,O) = {(o1, {x1}), (o2, {x2, x3})} and (H2,O) = {(o1, {x3}), (o2, {x1})}. Let (H5,O) = {(o1, {x3}), (o2, {x3})} and (H6,O) = {(o1, {x1, x2}), (o2, {x2, x3})}. Then (H5,O) and (H6,O) are IS-b-open sets because cl(H5,O) = X̃ and cl(int(H6,O)) = (H6,O). But (H5,O) is not IS-semi-open because int(H5,O) = Φ, and (H6,O) is not IS-pre-open because int(cl(H5,O)) = {(o1, {x1}), (o2, {x2, x3})}⊉̃(H6,O). Proposition 5. The unions of IS-b-open sets is IS-b-open. Proof. Consider {(Hj ,O) : j ∈ J} as a family of IS-b-open sets. Suppose J ̸= ∅. Then (Hj ,O)⊆̃int(cl(H,O))∪̃cl(int(H,O)) for each j ∈ J . Thus, ∪̃j∈J(Hj ,O)⊆̃∪̃j∈J [int(cl(H,O)) ∪̃cl(int(H,O))] ⊆̃int(cl(∪̃j∈J(Hj ,O)))∪̃ cl(int(∪̃j∈J(Hj ,O))). Hence, ∪̃j∈J(Hj ,O) is IS- b-open. Corollary 1. The intersections of IS-b-closed sets is IS-b-closed. Proposition 6. If (H1,O) is IS-open and (H2,O) is IS-b-open, then (H1,O)∩̃(H2,O) is IS-b-open. Proof. Let (H1,O) and (H2,O) be as given in the proposition. Then (H1,O)∩̃(H2,O)⊆̃ (H1,O)∩̃[int(cl(H2,O))∪̃cl(int(H2,O))] = [(H1,O)∩̃int(cl(H2,O))] ∪̃[(H1,O)∩̃ cl(int(H2,O))]. It follows from Proposition 2 that (H1,O)∩̃ int(cl(H2,O))⊆̃int(cl[(H1,O)∩̃(H2,O)] and (H1,O)∩̃ cl(int(H2,O))⊆̃cl(int[(H1,O)∩̃(H2,O)] Hence, (H1,O)∩̃(H2,O) is an IS-b-open set. Corollary 2. If (H1,O) is IS-closed and (H2,O) is IS-b-closed, then (H1,O)∩̃(H2,O) is IS-b-closed. Proposition 7. The image of an IS-b-open set under an IS-homeomorphism is IS-b-open. Proof. Consider fψ : (X1, µ1,O1) → (X2, µ2,O2) as an IS-continuous map and let (H,O1) be an IS-b-open subset of (X1, µ1,O1). Then fψ(H,O1)⊆̃fψ[cl(int(H,O1))∪̃int(cl(H,O1))]. It follows from Lemma 1 that fψ(H,O1)⊆̃cl(int(fψ(H,O1)))∪̃int(cl(fψ(H,O1))). Hence, fψ(H,O1) is an IS-b-open subset of (X2, µ2,O2), as required. 4. Infra b-interior, infra b-closure, infra b-limit and infra b-boundary soft points of a soft set Definition 15. Let (H,O) be an S-set in (X,µ,O). Then: (i) the IS-b-interior of (H,O), denoted by bint(H,O), is the union of all IS-b-open sets that are contained in (H,O). T.M. Al-shami et al. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1455-1471 1460 (ii) the IS-b-closure of (H,O), denoted by bcl(H,O), is the intersection of all IS-b-closed sets containing (H,O). Proposition 8. We have the following properties. (i) (H,O) is an IS-b-open subset of (X,µ,O) iff bint(H,O) = (H,O). (ii) (H,O) is an IS-b-closed subset of (X,µ,O) iff bcl(H,O) = (H,O). Proof. It comes from Proposition 5 and Corollary 1. The two characterizations given in the the above proposition are generally false for IS-open and IS-closed sets. Proposition 9. Let (H,O) be a subset of (X,µ,O). (i) δxo ∈ bint(H,O) iff there exists an IS-b-open set (F ,O) such that δxo ∈ (F ,O)⊆̃(H,O). (ii) δxo ∈ bcl(H,O) iff the intersection of any IS-b-open set (F ,O) containing δxo and (H,O) is non-null. Proof. The proof of (i) is obvious, so we prove (ii). Let δxo ∈ bcl(H,O). Then every IS-b-closed set contains (H,O) contains δxo as well. Suppose that there exists an IS-b-open set (F ,O) containing δxo such that (H,O)∩̃(F ,O) = Φ. Therefore, (H,O)⊆̃(Fc,O) which means that δxo ̸∈ bcl(H,O). This is a contradiction. Conversely, suppose that there exists an IS-b-open set (F ,O) containing δxo such that (H,O)∩̃(F ,O) = Φ. Therefore, bcl(H,O)⊆̃(Fc,O) which means that δxo ̸∈ bcl(H,O). Hence, the result holds. Proposition 10. Let (H,O) be a subset of (X,µ,O). Then: (i) (bint(H,O))c = bcl(Hc,O). (ii) (bcl(H,O))c = bint(Hc,O). Proof. (i): (bint(H,O))c = { ∪̃ j∈J (Fj ,O) : (Fj ,O) is an IS-b-open set contained in (H,O)}c = ∩̃ j∈J {(Fc j ,O) : (Fc j ,O) is an IS-b-closed set containing (Hc,O)} = bcl(Hc,O). The proof of (ii) is similar to (i). Proposition 11. Let (F ,O) be an IS-open set and (Λ,O) be an IS-closed set in (X,µ,O). Then: (i) (F ,O)∩̃bcl(H,O)⊆̃bcl((F ,O)∩̃(H,O)). (ii) bint((Λ,O)∪̃(H,O))⊆̃(Λ,O)∪̃bint(H,O). T.M. Al-shami et al. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1455-1471 1461 Proof. (i): Let δxo ∈ (F ,O)∩̃bcl(H,O). Then δxo ∈ (F ,O) and δxo ∈ bcl(H,O). This implies (U ,O)∩̃(H,O) ̸= Φ for every IS-b-open set (U ,O) containing δxo . It fol- lows from Proposition 6 that (F ,O)∩̃(U ,O) is an IS-b-open set containing δxo . There- fore, [(F ,O)∩̃(U ,O)]∩̃(H,O) ̸= Φ. Now, (U ,O)∩̃[(F ,O)∩̃(H,O)] ̸= Φ which means that δxo ∈ bcl((F ,O)∩̃(H,O)). Hence, (F ,O)∩̃bcl(H,O)⊆̃bcl((F ,O)∩̃(H,O)). One can prove (ii) following similar arguments. Theorem 1. Let (H,O) and (F ,O) are in (X,µ,O). Then we have: (i) bint(X̃) = X̃. (ii) bint(H,O)⊆̃(H,O). (iii) If (F ,O)⊆̃(H,O), then bint(F ,O)⊆̃bint(H,O). (iv) bint(bint(H,O)) = bint(H,O). (v) bint(F ,O)∩̃bint(H,O)⊆̃bint((F ,O)∩̃(H,O)). Proof. (i): Since X̃ is IS-b-open, bint(X̃) = X̃. (ii) and (iii) are obvious. (iv): Clearly bint(bint(H,O)) is the largest IS-b-open set contained in bint(H,O); however, bint(H,O) is an IS-b-open set; hence, bint(bint(H,O)) = bint(H,O). (v): It comes from (iii). Theorem 2. Let (H,O) and (F ,O) be subsets of (X,µ,O). Then we have: (i) bcl(Φ) = Φ. (ii) (H,O)⊆̃bcl(H,O). (iii) If (F ,O)⊆̃(H,O), then bcl(F ,O)⊆̃bcl(H,O). (iv) bcl(bcl(H,O))⊆̃bcl(H,O). (v) bcl((F ,O)∪̃(H,O)) = bcl(F ,O)∪̃bcl(H,O). Proof. It can be proved following similar arguments given in the proof of Theorem 1. Definition 16. A S-point δxo is called an IS-b-limit point of a subset (H,O) of (X,µ,O) provided that [(F ,O)\δxo ]∩̃(H,O) ̸= Φ for any IS-b-open set (F ,O) containing δxo . The S-set of all IS-b-limit points of (H,O) is called an infra b-derived S-set. It is denoted by (H,O)bs′. Proposition 12. Consider (F ,O) and (H,O) as S-sets in (X,µ,O). Then (i) Φbs′ = Φ and X̃bs′⊆̃X̃. T.M. Al-shami et al. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1455-1471 1462 (ii) If (F ,O)⊆̃(H,O), then (F ,O)bs′⊆̃(H,O)bs′. (iii) If δxo ∈ (H,O)bs′, then δxo ∈ ((H,O) \ δxo )bs′. (iv) (F ,O)bs′∪̃(H,O)bs′⊆̃((F ,O)∪̃(H,O))bs′. Proof. Straightforward. Theorem 3. Let (H,O) be an S-set in (X,µ,O). Then (i) If (H,O) is an IS-b-closed set, then (H,O)bs′ ⊆ (H,O). (ii) ((H,O)∪̃(H,O)bs′)bs′⊆̃(H,O)∪̃(H,O)bs′. (iii) bcl(H,O) = (H,O)∪̃(H,O)bs′. Proof. (i) Consider (H,O) as an IS-b-closed set such that δxo ̸∈ (H,O). Then δxo ∈ (Hc,O). Now, (Hc,O) is an IS-b-open set such that (Hc,O)∩̃(H,O) = Φ which means that δxo ̸∈ (H,O)bs′. Thus, (H,O)bs′⊆̃(H,O). (ii) Consider δxo ̸∈ (H,O)∪̃(H,O)bs′. Then δxo ̸∈ (H,O) and δxo ̸∈ (H,O)bs′. Therefore, there exists an IS-b-open set (F ,O) such that (F ,O)∩̃(H,O) = Φ (1) This implies that (F ,O)∩̃(H,O)bs′ = Φ (2) It follows from (1) and (2) that (F ,O)∩̃((H,O)∪̃(H,O)bs′) = Φ. Thus, δxo ̸∈ ((H,O)∪̃ (H,O)bs′)bs′. Hence, ((H,O)∪̃(H,O)bs′)bs′⊆̃((H,O)∪̃(H,O)bs′), as required. (iii) It is clear that (H,O)∪̃(H,O)bs′⊆̃bcl(H,O). Conversely, let δxo ∈ bcl(H,O). Then for every IS-b-open set containing δxo we have (H,O)∩̃(F ,O) ̸= Φ. Without loss of generality, let δxo ̸∈ (H,O). Then [(H,O)\δxo ]∩̃(F ,O) ̸= Φ. Consequentially, δxo ∈ (H,O)bs′. Hence, the proof is complete. Definition 17. The IS-b-boundary points of a subset (H,O) of (X,µ,O), denoted by bB(H,O), are all the S-points which belong to the complement of bint(H,O)∪̃bint(Hc,O). Proposition 13. Let (H,O) be an S-set in (X,µ,O). Then: (i) bB(H,O) = bcl(H,O)∩̃bcl((Hc,O)). (ii) bB(H,O) = bcl(H,O) \ bint(H,O). Proof. T.M. Al-shami et al. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1455-1471 1463 (i) bB(H,O) = {δxo ∈ X̃ : δxo ̸∈ bint(H,O) and δxo ̸∈ bint((Hc,O))} = {δxo ∈ X̃ : δxo ̸∈ (bcl(Hc,O))c and δxo ̸∈ (bcl(H,O))c} = {δxo ∈ X̃ : δxo ∈ bcl(Hc,O) and δxo ∈ bcl(H,O)} = bcl(H,O)∩̃bcl(Hc,O) (ii) bB(H,O) = bcl(H,O)∩̃bcl(Hc,O) = bcl(H,O)∩̃(bint(H,O))c = bcl(H,O) \ bint(H,O) Corollary 3. Let (H,O) be a subset of (X,µ,O). Then (i) bB(H,O) = bB(Hc,O) (ii) bcl(H,O) = bint(H,O)∪̃bB(H,O) Proposition 14. Let (H,O) be a subset of (X,µ,O). Then (i) (H,O) is IS-b-open iff bB(H,O)∩̃(H,O) = Φ. (ii) (H,O) is IS-b-closed iff bB(H,O)⊆̃(H,O). Proof. (i) bB(H,O) ∩ (H,O) = bB(H,O) ∩ bint(H,O) = Φ. Conversely, let δxo ∈ (H,O). Then δxo ∈ bint(H,O) or δxo ∈ bB(H,O). Since bB(H,O) ∩ (H,O) = Φ, δxo ∈ bint(H,O). Thus, (H,O) ⊆ bint(H,O) which means that (H,O) = bint(H,O). Hence, (H,O) is IS-b-open. (ii) (H,O) is IS-b-closed⇔ (Hc,O) is IS-b-open⇔ bB(Hc,O)∩(Hc,O) = Φ⇔ bB(H,O)∩ (Hc,O) = Φ ⇔ bB(H,O) ⊆ (H,O). Corollary 4. A subset (H,O) of (X,µ,O) is IS-b-open and IS-b-closed iff bB(H,O) = Φ. 5. Infra soft b-homeomorphism maps Definition 18. fψ : (X,µ,O) → (S, ν,∆) is said to be IS-b-continuous at δxo ∈ X̃ if for any IS-b-open set (F ,∆) containing fψ(δ x o ), there is an IS-b-open set (H,O) containing δxo such that fψ(H,O)⊆̃(F ,∆). fψ is called IS-b-continuous if it is IS-b-continuous at all δxo ∈ X̃. Theorem 4. If fψ : (X,µ,O) → (S, ν,∆) is IS-b-continuous, then the next properties are equivalent. (i) fψ is an IS-b-continuous map; T.M. Al-shami et al. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1455-1471 1464 (ii) The inverse image of each IS-b-closed set is IS-b-closed; (iii) bcl(f−1 ψ (H,∆))⊆̃f−1 ψ (bcl(H,∆)) for each (H,∆)⊆̃S̃; (iv) fψ(bcl(F ,O))⊆̃bcl(fψ(F ,O)) for each (F ,O)⊆̃X̃; (v) f−1 ψ (bint(H,∆))⊆̃bint(f−1 ψ (H,∆)) for each (H,∆)⊆̃S̃. Proof. (i) ⇒ (ii): Let (H,∆) be an IS-b-closed set in (S, ν,∆). Then f−1 ψ (Hc,∆) is an IS-b-open subset of X̃. Obviously, f−1 ψ (Hc,∆) = X̃ − f−1 ψ (H,∆); hence, f−1 ψ (H,∆) is an IS-b-closed subset of X̃. (ii) ⇒ (iii): According to (ii), f−1 ψ (bcl(H,∆)) is an IS-b-closed subset of X̃. Then bcl(f−1 ψ (H,∆))⊆̃bcl(f−1 ψ (bcl(H,∆))) = f−1 ψ (bcl(H,∆)). (iii) ⇒ (vi): According to (iii), bcl(f−1 ψ (fψ(F ,O)))⊆̃f−1 ψ (bcl(fψ(F ,O))). Then fψ(bcl(F ,O))⊆̃fψ(f−1 ψ (bcl(fψ(F ,O))))⊆̃bcl(fψ(F ,O)). (iv) ⇒ (v): According to (iv), fψ(bcl(X̃ − f−1 ψ (H,∆)))⊆̃bcl(fψ(X̃ − f−1 ψ (H,∆))). There- fore, fψ(X̃ − bint(f−1 ψ (H,∆))) = fψ(bcl(X̃ − f−1 ψ (H,∆))) ⊆ bcl(S̃ − (H,∆)) = S̃ − bint(H,∆). Thus X̃−bint(f−1 ψ (H,∆))⊆̃f−1 ψ (S̃ −bint(H,∆)) = f−1 ψ (S̃)−f−1 ψ (bint(H,∆)). Hence f−1 ψ (bint(H,∆))⊆̃bint(f−1 ψ (H,∆)). (v) ⇒ (i): Let (H,∆) be an IS-b-open subset of S̃. According to (v), f−1 ψ (H,∆)⊆̃bint(f−1 ψ (H,∆)). This implies f−1 ψ (H,∆) = bint(f−1 ψ (H,∆)). Hence, fψ is IS-b-continuous. Theorem 5. If fψ : (X,µ,O) → (S, ν,∆) is IS-b-continuous, then the restriction S-map fψ|M : (M, µM,O) → (S, ν,∆) is IS-b-continuous provided that M̃ is an IS-open set. Proof. Consider (H,∆) is an IS-b-open set in (S, ν,∆). By hypothesis, f−1 ψ (H,∆) is IS-b-open. Now, f−1 ψ|M (H,∆) = f−1 ψ (H,∆)∩̃M̃. Since M̃ is an IS-open set, it follows from Proposition 6 that f−1 ψ|M (H,∆) is IS-b-open. Hence, fψ|M is an IS-b-continuous map. Definition 19. If the image of each IS-b-open (resp., IS-b-closed) set under an S-map fψ : (X,µ,O) → (S, ν,∆) is IS-b-open (resp., IS-b-closed), then fψ is called IS-b-open (resp., IS-b-closed). Proposition 15. fψ : (X,µ,O) → (S, ν,∆) is an IS-b-open map iff fψ(bint(H,O)) ⊆̃bint(fψ(H,O)) for each subset of (H,O) of X̃. Proof. ⇒: Let (H,O) be a subset of X̃. Now, fψ(bint(H,O))⊆̃fψ(H,O) and bint(H,O) is an IS-b-open set. By hypothesis, fψ(bint(H,O)) is IS-b-open. Therefore, fψ(bint(H,O))⊆̃ bint(fψ(H,O)). ⇐: Let (Λ,O) be an IS-open subset of X̃. Then fψ(H,O)⊆̃bint(fψ(H,O)). Therefore, fψ(H,O) = bint(fψ(H,O)) which means that fψ is an IS-b-open map. T.M. Al-shami et al. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1455-1471 1465 Proposition 16. fψ : (X,µ,O) → (S, ν,∆) is an IS-b-closed map iff bcl(fψ(H,O)) ⊆̃fψ(bcl(H,O)) for each subset (H,O) of X̃. Proof. ⇒: Let fψ be an IS-b-closed map and (H,O) be an S-set of X̃. By hypothesis, fψ(bcl(H,O)) is IS-b-closed. Since fψ(H,O)⊆̃fψ(bcl(H,O)), bcl(fψ(H,O)) ⊆̃fψ(bcl(H,O)). ⇐: Suppose that (H,O) is an IS-b-closed subset of X̃. By hypothesis, fψ(H,O)⊆̃ bcl(fψ(H,O)) ⊆̃fψ(bcl(H,O)) = fψ(H,O). Therefore, fψ(H,O) is IS-b-closed. Hence, fψ is an IS-b- closed map. Proposition 17. The concepts of IS-b-open and IS-b-closed maps are equivalent under bijectiveness. Proof. It comes from the fact that a bijective soft map fψ : (X,µ,O) → (S, ν,∆) implies fψ(Hc,O) = (fψ(H,O))c. Proposition 18. Let fψ : (X,µ,O) → (S, ν,∆) and Fν : (S, ν,∆) → (V, σ,U) be two S-maps. Then: (i) If fψ and Fν are IS-b-open maps, then Fν ◦ fψ is an IS-b-open map. (ii) If Fν ◦ fψ is an IS-b-open map and fψ is a surjective IS-b-continuous map, then Fν is an IS-b-open map. (iii) If Fν ◦ fψ is an IS-b-open map and Fν is an injective IS-b-continuous map, then fψ is an IS-b-open map. Proof. (i) Straightforward. (ii) Consider (H,∆) as an IS-b-open set in (S, ν,∆). By hypothesis, f−1 ψ (H,∆) is an IS-b-open subset of (X,µ,O). Again, by hypothesis, (Fν ◦ fψ)(f−1 ψ (H,∆)) is an IS-b-open subset of (V, σ,U). Since fψ is surjective, then (Fν ◦ fψ)(f−1 ψ (H,∆)) = Fν(fψ(f −1 ψ (H,∆))) = Fν(H,∆). Hence, Fν is an IS-b-open map. (iii) Consider (H,O) as an IS-b-open subset of (X,µ,O). By hypothesis, (Fν ◦ fψ)(H,O) is an IS-b-open subset of (V, σ,U). Again, by hypothesis, F−1 ν (Fν ◦ fψ(H,O)) is an IS-b-open subset of (S, ν,∆). Since Fν is injective, then F−1 ν (Fν ◦ fψ(H,O)) = (F−1 ν Fν)(fψ(H,O)) = fψ(H,O). Hence, fψ is an IS-b-open map. Definition 20. A bijective S-map fψ : (X,µ,O) → (S, ν,∆) is said to be an IS-b- homeomorphism if it is IS-b-continuous and IS-b-open. The proofs of the following two results are easy and so is omitted. T.M. Al-shami et al. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1455-1471 1466 Proposition 19. Let fψ : (X,µ,O) → (S, ν,∆) and Fν : (S, ν,∆) → (V, σ,U) be IS-b- homeomorphism maps. Then Fν ◦ fψ is an IS-b-homeomorphism map. Proposition 20. If fψ : (X,µ,O) → (S, ν,∆) is a bijective S-map, then the following items are equivalent. (i) fψ is an IS-b-homeomorphism. (ii) fψ and f−1 ψ is IS-b-continuous. (iii) fψ is IS-b-closed and IS-b-continuous. Proposition 21. If fψ : (X,µ,O) → (S, ν,∆) is an IS-b-homeomorphism map, then the following items hold for each (H,O) ∈ S(X)A. (i) fψ(bint(H,O)) = bint(fψ(H,O)). (ii) fψ(bcl(H,O)) = bcl(fψ(H,O)). Proof. (i): According to Proposition 15 (i), we obtain fψ(bint(H,O))⊆̃bint(fψ(H,O)). Conversely, let δsκ ∈ bint(fψ(H,O). Then there is an IS-b-open set (F ,∆) such that δsκ ∈ (F ,∆)⊆̃fψ(H,O). By hypothesis, δxo = f−1 ψ (δsκ) ∈ f−1 ψ (F ,∆)⊆̃(H,O) such that f−1 ψ (F ,∆) is an infra soft b-open set. So that, δxo ∈ bint(H,O) which means that δsκ ∈ fψ(bint(H,O)). One can achieve item (ii) following similar arguments. Theorem 6. The property of an IS-b-dense set is an IS-topological invariant. Proof. Let fψ : (X,µ,O) → (S, ν,∆) be an IS-b-homeomorphism map and consider (H,O) as an IS- b-dense subset of (X,µ,O), i.e. bcl(H,O) = X̃. It comes from Proposition 21 (ii) that bcl(fψ(H,O)) = fψ(bcl(H,O)) = fψ(X̃) = bcl(S̃) = S̃. Thus, fψ(H,O) is an IS-b-dense set in (S, ν,∆), as required. We complete this section by studying the concept of fixed soft points with respect to IS-b-open sets. Definition 21. We say that (X,µ,O) has a b-fixed S-point property provided that for every IS-b-continuous map fψ : (X,µ,O) → (X,µ,O) there exists δso ∈ X such that fψ(δ s o) = δso. Proposition 22. The property of being a b-fixed S-point is preserved under an IS-b- homeomorphism. Proof. Consider (X1, µ1,O1) and (X2, µ2,O2) as two IS-b-homeomorphism. This means that there exists a bijective S-map fψ : (X1, µ1,O1) → (X2, µ2,O2) such that fψ and f−1 ψ are IS-b-continuous. Suppose that (X1, µ1,O1) has the property of b-fixed soft point. That is any IS-b-continuous map fψ : (X1, µ1,O1) → (X1, µ1,O1) has a b-fixed S-point. Now, consider Cψ : (X2, µ2,O2) → (X2, µ2,O2) is IS-b-continuous. It is clear that Cψ ◦ fψ : (X1, µ1,O1) → (X2, µ2,O2) is IS-b-continuous. Therefore, f−1 ψ ◦ Cψ ◦ fψ : (X1, µ1,O1) → (X1, µ1,O1) is IS-b-continuous. Since (X1, µ1,O1) has a b-fixed S-point property, f−1 ψ (hψ(fψ(δ s o))) = δso for some δso ∈ X̃. Thus, fψ(f −1 ψ (hψ(fψ(δ s o)))) = fψ(δ s o). This implies that hψ(fψ(δ s o)) = fψ(δ s o). Hence, fψ(δ s o) is a b-fixed soft point of Cψ which means that (X2, µ2,O2) has a b-fixed S-point property. REFERENCES 1467 6. Concluding remark and further work In this paper, we have formulated the concept of IS-b-open sets and discussed its main properties. Then, we have defined novel operators and mappings between ISTSs depending on the classes of IS-b-open and IS-b-closed sets. We have revealed the relationships between these operators and mappings and investigated their basic features. As we have noted that several topological characterizations still have been valid via the structures of infra topologies, which confirms the importance of infra ST-structures. Our future works will focus on studying further topological concepts and notions via infra ST-structures. 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