EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 15, No. 1, 2022, 126-134 ISSN 1307-5543 – ejpam.com Published by New York Business Global Nearly Soft β - Open Sets via Soft Ditopological Spaces Radwan Abu- Gdairi1,∗, A. A. Azzam2,3, Ibrahim Noaman4,5 1 Department of Mathematics Faculty of Science Zarqa University, Jordan 2 Department of Mathematics, Faculty of Science and Humanities, Prince Sattam Bin Abdulaziz University, Alkharj 11942, Kingdom of Saudi Arabia 3 Department of Mathematics, Faculty of Science, New Valley University, Elkharga 72511, Egypt 4 Department of Mathematics, Faculty of Science and Arts in Al-Mandaq, AL Baha University, P.O.Box1988, Kingdom of Saudi Arabia 5 Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt Abstract. As a result of the importance of topological space in data analysis and some applica- tions, many researches have used various methods to expand that space, including the concept of ditopology. T. Dizman and et al. presented soft ditopolgical spaces in 2016. We define new types of nearly soft open sets in soft ditopology as soft β - open, soft β - closed, soft preopen, soft semi - open, and some related properties in this paper. Soft β - continuous and soft β - cocontinuous functions were also introduced . Finally, soft β - compact, soft β - stable and soft β - irresolute concepts were discussed, and some of the concepts were studied in this field. 2020 Mathematics Subject Classifications: 54A05, 54A20, 54E55 Key Words and Phrases: Soft set, soft topological space, ditopological space, soft β - open and soft β - closed sets, soft β - continuous, soft β - compact 1. Introduction and Preliminaries In the late twentieth century, Molodtsov[11] introduced the theory of soft set as a generalization of the set theory, which widely used to deal with incomplete, insufficient information for its study and analysis, which similar to the rough set theory. Soft set the- ory and its applications are now advancing rapidly in a variety of fields[5, 7, 8, 13–15, 19]. Maji et al.[21, 22] presented some new definitions of soft sets as well as an application of soft sets in decision making problems. Jose Carlos et al.[6] participated in the development and improvement of soft topology. The idea of a generalization of the topological space by using novel concepts as ideal, grill, filter[3, 9, 16, 24] coming as a result of the importance of topological space and used it to solve some of the measures things that were previously ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v15i1.4249 Email addresses: rgdairi@zu.edu.jo (R. Abu-Gdairi), azzam0911@yahoo.com (A. A. Azzam), noaman20102001@yahoo.com (I.Noaman) http://www.ejpam.com 126 © 2022 EJPAM All rights reserved. R. Abu-Gdairi, A. A. Azzam, I. Noaman / Eur. J. Pure Appl. Math, 15 (1) (2022), 126-134 127 difficult to solve. The mysterious set theory and other uncertain knowledge models have led to new approaches to decision - making as [2, 4]. So, Brown et al.[12] introduced the concept of ditopological space as a generalization of topological spaces. The concept of ditopological space via the soft set theory with separation axioms of soft ditopological space introduced by Senel in 2016 [23]. Where the idea of ditopological spaces depends on two structures soft topology and soft cotopology. Also, Senel [23] introduced soft di- topological spaces as a soft generalization of ditopology concept, which depends on two structures a soft topology and a soft subspace topology. S. Dost et al. In[12] introduced the concept of β - open and β - closed in ditopological texture spaces. In this paper, we will introduce some of the nearly soft β - open sets, the study of soft β - compactness and soft β - cocompactness. Also, soft β - stable and soft β - irresolute were introduced in soft ditopological spaces and study some of their properties. Through this section, we recall several basic notions related to soft set, soft topologi- cal space, soft cotopological space, and some of the nearly soft open sets through soft topological space, which handled in mentioned in [10–12, 17, 18, 20]. Through this paper, we notice that U refers to an universal set, E is the soft parameters and P (U) is the power set of U . . Definition 1. [11] On universal set U , a pair (f,E) is called a soft set if and only if f is a mapping from E into the power set P (U). To put it another way, the soft set is a parametrized family of subsets of the set U . Every setf(e), e ∈ E in this family can be thought of as the set of e-elements of the soft set (f,E), or as the set of e-approximate elements of the soft set.. Definition 2. [20] If τ is defined as the collection of soft sets over X, then τ is said to be a soft topology on X if it fulfills the following axioms: (1)X,Φ ∈ τ , where Φ(e) = Φ and X(e) = X, ∀e ∈ E. (2) The union of any number of soft sets in τ belongs to τ . (3) The intersection of any two soft sets in τ belongs to τ . The triple (X, τ,E) is referred to as a soft topological space, and the members of τ are referred to as soft open sets. Definition 3. Let (X, τ,E) represent a soft topological space over X and (F,A) represent a soft set over X. (1) The soft interior of (F,A) [18] is the soft set int (F,A) = X̃{(O,A) : (O,A) is the soft open and (O,A)⊆̃(F,A)}. (2) The soft closure of (F,A) [20] is the soft set cl (F,A) = ∩̃{(C,A) : (C,A) is soft closed and (F,A)⊆̃(C,A)}. Definition 4. [12] If κ is the collection of complement soft sets over X, then κ is said to be a soft cotopology on X if it obeys the following axioms: (1) Φ and X̃ ∈ κ. (2) The intersection of any number of soft sets in κ ∈ κ. (3) The union of any two soft sets in κ ∈ κ. The triple (X,κ,E) is referred to as a soft cotopological space, and the members of κ are referred to as soft closed sets. Definition 5. A soft set(F,E) of a soft topological space(X, τ,E) is said to be: (1) Soft β - open [17] if (F,A) ⊆̃ cl(int(cl(F,A))). R. Abu-Gdairi, A. A. Azzam, I. Noaman / Eur. J. Pure Appl. Math, 15 (1) (2022), 126-134 128 (2) Soft preopen [17] if (F,A) ⊆̃ int(cl(F,A)). (3) Soft α - open [20] if (F,A) ⊆̃ int(cl(intl(F,A))). (4) Soft semi- open [10] if (F,A) ⊆̃ cl(int(F,A)). (5) Soft open [20] if its complement is soft closed. Definition 6. [1] A function f : (X, τ) → (Y, σ) is said to be β - irresolute if the preimages of β - open sets are β - open. 2. Soft β - open and soft β - closed sets Definition 7. Let U be a universel set and E be the parameters. A family (τ, κ) of a subsets of ŨE is called a soft ditopology on a soft subspace ŨE, where τ is a soft topology, κ is a soft cotopology and the space (ŨE , τ, κ) is called soft ditopological space. If we take (τ, κ) = Ω, then (ŨE ,Ω) is said to be soft ditopological space. Definition 8. Let (ŨE , τ, κ) be a soft ditopological space over ŨE and f be a soft set over ŨE such that f = {(e,A) : e ∈ E,A ∈ P (U) and (e,A) = F : E → P (U)}. Definition 9. Let ŨE ∈ S. The power soft set of ŨE is defined by P (ŨE) = {fi⊆̃ŨE : i ∈ I} and its cardinality is defined by |P (ŨE)| = 2 ∑ e∈E |ŨE(e)| where |ŨE(e)| is the cardinality ŨE(e) Example 1. Let U = {u1, u2}, E = {e1, e2} and ŨE = {(e1, {u1, u2}), (e2, {u1, u2})} then the soft sets are: f1 = {(e1, {u1})}, f2 = {(e1, {u2})}, f3 = {(e1, {u1, u2})}, f4 = {(e2, {u1})}, f5 = {(e2, {u2})}, f6 = {(e2, {u1, u2})}, f7 = {(e1, {u1}), (e2, {u1})}, f8 = {(e1, {u1}), (e2, {u2})}, f9 = {(e1, {u2}), (e2, {u1})}, f10 = {(e1, {u2}), (e2, {u2})}, f11 = {(e1, {u1}), (e2, {u1, u2})}, f12 = {(e1, {u2}), (e2, {u1, u2})}, f13 = {(e1, {u1, u2}), (e2, {u1})}, f14 = {(e1, {u1, u2}), (e2, {u2})}, f15 = ŨE, f16 = Φ. And we get the complement the soft sets are: f c1 = {(e1, {u2})}, f c2 = {(e1, {u1})}, f c3 = {(e1,Φ)}, f c4 = {(e2, {u2})}, f c5 = {(e2, {u1})}, f c6 = {(e2,Φ)}, f c7 = {(e1, {u2}), (e2, {u2})}, f c8 = {(e1, {u2}), (e2, {u1})}, f c9 = {(e1, {u1}), (e2, {u2})}, f c10 = {(e1, {u1}), (e2, {u1})}, f c11 = {(e1, {u2}), (e2,Φ)}, f c12 = {(e1, {u1}), (e2,Φ)}, f c13 = {(e1,Φ), (e2, {u2})}, f c14 = {(e1,Φ), (e2, {u2})}, f c15 = Φ, f c16 = ŨE. Also we get a soft ditopological space (τ, κ) = {Φ, ŨE , {(e1, {u2})}, {(e1, {u1})}} on ŨE, such that τ = {ŨE ,Φ, {(e1, {u2})}} and κ = {Φ, ŨE , {(e1, {u1})}. Definition 10. A soft β interior of a soft set f is denoted by sβ - int (f) which is defined by. sβ - int (f) = ∪̃{h : h is a soft β - open and h⊆̃f}. A soft β closure of a soft set f is denoted by sβ - cl (f) which is defined by. sβ - cl (f) = ∩̃{k : k is a soft β - closed and f⊆̃k}. Definition 11. Let (ŨE , τ, κ) be a soft ditopological space and f ∈ P (ŨE) then: (1) f is a soft β - open if f⊆̃cl(int(cl(f))). R. Abu-Gdairi, A. A. Azzam, I. Noaman / Eur. J. Pure Appl. Math, 15 (1) (2022), 126-134 129 (2) f is a soft β - closed if int(cl(int(f)))⊆̃f . (3) f is a soft α - open if f⊆̃int(cl(int(f))). (4) f is a soft preopen if f⊆̃int(cl(f)). (5) f is a soft preclosed if cl(int(f))⊆̃f . (6) f is a soft semi - open if f⊆̃cl(int(f)). (7) f is a soft β - open if the complement of f is a soft β - closed. Remark 1. In a soft ditopological space it is easy to see the set of all soft β - open con- tains each of a soft semi - open, soft preopen and soft α - open, as shown in the following diagram but the converse need not be true in general as Example 2 . Example 2. Let (ŨE , τ, κ) be a soft ditopological space, U = {u1, u2}, E = {e1, e2} such that ŨE = {(e1, {u1, u2}), (e2, {u1, u2})}, τ = {ŨE ,Φ, {(e1, {u2})}, {(e2, {u1, u2})}, {(e1, {u1, u2}), (e2, {u1)}, {(e1, {u2}), (e2, {u1})}, κ = {Φ, ŨE , {(e1, {u1}), (e2,Φ)}, {(e1,Φ), (e2, {u2})}, {(e1, {u1}), (e2, {u2})}, we notice that the soft set {(e1,Φ), (e2, {u1})} in soft ditopological space is soft preopen set and not soft α - open set. Also it is soft β - open and not soft semi - open set. Theorem 1. If h is a soft closed and f is a soft β - open then f ∪̃ h is a soft β - open. Proof: Since f ⊆̃ cl(int(cl(f))), (h ∪̃ f) ⊆̃ h ∪̃ cl(int(cl(f))) = cl(int(cl(h))) ∪̃ cl(int(cl(f))) ⊆̃ cl(int(cl((h)∪̃ (f)))). This show that f ∪̃ h is soft β - open. The class of all soft β - open ( resp. soft β - closed, soft preopen, soft semi - open, soft α - open, soft α - closed and soft preclosed )in ditopological spaces (ŨE , τ, κ) denoted by SβO ( resp. SβC , SPO, SSO, SαO, SαC and SPC). Theorem 2. Let (ŨE , τ, κ) be a soft ditopological space we have: (1) If f ∈ SPO, f ⊆̃ h ⊆̃ scl(f) then h ∈ SβO. (2) If f ∈ SPC, sint(f) ⊆̃ h ⊆̃f then h ∈ SβC. Proof: (1) Since f is soft preopen ⇒ f⊆̃int(cl(f))⊆̃h⊆̃∩̃{f : f is soft closed } ⊆̃cl(int(cl(f))) ⇒ h ∈ SβO . (2) Since f is soft preclosed ⇒ cl(int(f))⊆̃f , sint(f)⊆̃h⊆̃f ⇒ h ∈ SβC. Lemma 1. Let (ŨE , τ, κ) be a soft ditopological space, then (1) τ ⊆̃ SPO ⊆̃ SβO and κ ⊆̃ SPC ⊆̃ SβC. (2) SPO and SβO are closed under arbitrary unions. (3) SPC and SβC are closed under arbitrary intersections. Proof: (1) Since the element of τ is a soft open then, τ ⊆̃ SPO and SPO ⊆̃ SβO, that is R. Abu-Gdairi, A. A. Azzam, I. Noaman / Eur. J. Pure Appl. Math, 15 (1) (2022), 126-134 130 τ ⊆̃ SPO ⊆̃ SβO. Similary, the element of κ is a soft closed then, κ ⊆̃ SPC and SPC ⊆̃ SβC, that is κ ⊆̃ SPC ⊆̃ SβC. (2) and (3)are obvious. Lemma 2. Let (ŨE , τ, κ) be a soft ditopological space and f is a soft set on ŨE then : (1) f ∈ SβO ⇔ f = sβ - int(f). (2) f ∈ SβC ⇔ f = sβ - cl(f). Proof: (1) Let f = sβ - int(f). Since sβ - int(f) = ∪̃{h : h is a soft β - open and h⊆̃f} this show that f ∈ {h : h is a soft β - open and h⊆̃f} hance f is a soft β - open . Conversely let f ∈ SβO, since f⊆̃f , f ∈ {h : h is a soft β - open and h⊆̃f} further, h⊆̃f ∀ f , since f = ∪̃{h : h is a soft β - open and h⊆̃f}. (2) Similar (1) Lemma 3. Let (ŨE ,Ω) be a soft ditopological space the following hold for soft β - closure. (1) sβ - cl (Φ) = Φ. (2) If f ⊆̃ h⇒ sβ - cl(f) ⊆̃ sβ - cl(h). Definition 12. A soft ditopological space (ŨE ,Ω) is called . (1) Soft β - compact if every cover of ŨE by soft β - open sets has a finite subcover. (2) Soft β - cocompact if every cocover of Φ by soft β - closed sets has a finite subcocover. Proposition 1. Let (ŨE ,Ω) be a soft ditopological space and (ŨE ,Ω c) is a complement of soft ditopological space. Then h ∈ SβC ⇐⇒ hc ∈ SβO, h ∈ ŨE. Proposition 2. Let Ωc be a complement soft ditopology on ŨE. Then (ŨE ,Ω c) is soft β - compact if and only if it is soft β - cocompact. Proof: Let Ω be soft β - compact and f = {fi : i ∈ J} ∈ SβC with ∩̃f = Φ. that G = {f ci : i ∈ J} ∈ SβO, Moreover ∪̃G = ∪̃{f ci : i ∈ J} = {∩̃fi : i ∈ J}c = Φc = ŨE. Similary, if Ω is soft β - compact then it is soft β -cocompact. Definition 13. Let (τ, κ) be a soft ditopology on ŨE. (1) (τ, κ) will be called sβ - stable if every sβ - closed set h ∈ Ω \ {ŨE} is sβ - compact in ŨE. (2) (τ, κ) will be called sβ - costable if every sβ - open set f ∈ Ω \Φ is sβ - cocompact in ŨE. Example 3. Let (τ, κ) be a soft ditopological space on ŨE such that U = {u1, u2, u3}, E = {e1, e2, e3}, ŨE = {(e1, {u1, u2}), (e2, {u2, u3})}, τ = {Φ, ŨE} and κ = {Φ, {(e1, {u1}), (e2, {u2})}}. Firstly, we notice that, the only soft β - open are Φ, ŨE in soft ditopolgical space (ŨE , τ, κ), that is it is soft β - compact. Also, the soft h = {(e1, {u1}), (e2, {u2})} is soft closed and soft β - closed, so it is not soft compact and not soft β - compact. If follows that (τ, κ) is not sβ - stable. Secondly, we show that the space may be sβ - compact but not soft β - costable. R. Abu-Gdairi, A. A. Azzam, I. Noaman / Eur. J. Pure Appl. Math, 15 (1) (2022), 126-134 131 Let τ = {(e1, {u1}), (e2, {u2})}, κ = {Φ, ŨE}, the soft ditopology (τ, κ) is not sβ - compact since it is not soft compact. On the other hand (τ, κ) is sβ - stable since every sβ - closed set is closed and the only closed sets ŨE and Φ which is sβ - compact. Thirdly, also we can choose τ and κ such that the soft ditopological space is sβ - costable but not sβ - compact. Definition 14. A soft ditopological space is called Sβ - dicompact if it is Sβ - compact, Sβ - cocompact, Sβ - stable and Sβ - costable. Proposition 3. Let (τ, κ) be a soft ditopology on ŨE: (1) Soft β - compact =⇒ strongly soft compact =⇒ soft compact. (2) Soft β - cocompact =⇒ strongly soft cocompact =⇒ soft cocompact. Proof: It is obvious, since every soft open set is soft preopen and every soft closed set is soft preclosed. Proposition 4. For a soft ditopological space: (1) Soft β - stable =⇒ soft strongly stable =⇒ soft stable. (2) Soft β - costable =⇒ strongly soft costable =⇒ soft costable. Moreover, the converse is not true in general, as the following example: Proposition 5. Let Ω be a complemented soft ditopology on (ŨE) c. Then (ŨE ,Ω c) is soft β - compact if and only if it is soft β - cocompact. Proof: Let (ŨE ,Ω) be a soft β - compact and let K = {κi | i ∈ J} be a family of soft β - closed sets with ∩̃K = Φ. Obvious G = {κi | i ∈ J}c is a family of soft β open sets. Moreover , ∪̃G = ∪̃{κi | i ∈ J}c = ŨE, and so we have J 8 ⊆ J finite with ∪̃{κi | i ∈ J 8}c = ŨE . That is ∩̃{κi | i ∈ J 8 = Φ, and so (ŨE ,Ω) is soft β - cocompact. Similarly, if (ŨE ,Ω) is soft β - compact, then it is soft β - compact. Definition 15. A soft ditopological space will be called soft β - dicompact if it is soft β - compact, soft β - cocompact, soft β - stable and soft β - costable. Example 4. (1) Let (τ, κ) be a soft ditopological space on ŨE such that U = {u1, u2, u3}, E = {e1, e2}, ŨE ∈ S , ŨE = {(e1, {u1, u2}), (e2, {u2, u3})}, τ = {ŨE ,Φ}, {(e1, {u1, u2}), (e2, {u3})} and κ = {Φ, ŨE}. Since the only soft β - open sets are ŨE ,Φ in soft ditopology (ŨE , τ, κ), we have that it is soft β - compact. (2) Let τ = {ŨE ,Φ} and κ = {Φ, ŨE , {(e1, {u1, u2}), (e2, {u3})}, then the soft ditopology (ŨE , τ, κ) is soft β - cocompact but not soft β - compact. This example show that in general soft β - compact and soft β - cocompact are independent. Definition 16. Let Ω1 = (τ1, κ1) and Ω2 = (τ2, κ2) are two soft ditopological spaces on ŨE.Then Ω2 is called coarser than Ω1 (denoted by Ω2 ⊆̃ Ω1 if f ∈ τ1 whenever f ∈ τ2 and h ∈ κ1 whenever f ∈ κ2. Theorem 3. If (ŨE ,Ω1) and (ŨE ,Ω2) are two soft ditopological spaces. Then (ŨE ,Ω1∩̃ Ω2) is a soft ditopological space. R. Abu-Gdairi, A. A. Azzam, I. Noaman / Eur. J. Pure Appl. Math, 15 (1) (2022), 126-134 132 Proof: Since Ω1 = (τ1, κ1) and Ω2 = (τ2, κ2) are two a soft ditopological space on ŨE then (ŨE , τ1) and (ŨE , τ2) are two soft topological space ⇒ (ŨE , (τ1∩̃τ2)) is a soft topological space (1). Also, (ŨE , κ1) and (ŨE , κ2) are two a soft cotopological space ⇒ (ŨE , (κ1∩̃κ2)) is a soft ctopological space (2). From (1) and (2), we get (ŨE ,Ω1) and (ŨE ,Ω2) are two soft ditopological spaces. 3. Soft β - continuous mappings Definition 17. Let (ŨE ,Ω1) and (ṼE ,Ω2) be two soft ditopological spaces. A soft function (ϕ,ψ) : (ŨE ,Ω1) → (ṼE ,Ω2) where ϕ : (ŨE , τ1) → (ṼE , , τ2) and ψ : (ŨE , κ1) → (ṼE , κ2) then, a mapping (ϕ,ψ) is called continuous function at a soft point xp ∈ ŨE if ϕ : (ŨE , τ1) → (ṼE , τ2) is continuous function at xp, and ψ : (ŨE , κ1) → (ṼE , κ2) is continuous function at xp. Definition 18. A soft function Γ = (ϕ,ψ) : (ŨE ,Ω1) → (ṼE ,Ω2) is soft continuous if and only if the inverse image of soft open in Ω2 is soft open in Ω1. Definition 19. The soft function (ϕ,ψ) : (ŨE , τ1, κ1) → (ŨE , τ2, κ2) is called : (1) Soft β - continuous if ϕ−1(f) ∈ SβO(ŨE) ∀ f ∈ τ2. (2) Soft β - cocontinuous if ψ−1(h) ∈ SβC(ŨE) ∀ h ∈ κ2. (3) Soft β - bicontinuous if it is both soft β - continuous and soft β - cocontinuous. (4) Soft semi - continuous if ϕ−1(f) ∈ SSO(ŨE) ∀ f ∈ τ2. (5) Soft semi - cocontinuous if ψ−1(h) ∈ SSC(ŨE) ∀ h ∈ κ2. (6) Soft semi - bicontinuous if it semi - continuous and semi - cocontinuous. Example 5. Let (ŨE ,Ω1), (ŨE ,Ω2) be two soft ditopiogical spaces, such that U = {u1, u2, u3}, E = {e1, e2}, ϕ : (ŨE , τ1) → (ŨE , τ2) and ψ : (ŨE , κ1) → (ŨE , κ2), τ1 = {Φ, ŨE , {(e1, {u1}), (e2, {u1})}, {(e1, {u2}), (e2, {u2})}, {(e1, {u1, u2}), (e2, {u1, u2})}, κ1 = {Φ, ŨE , {(e1, {u1}), (e2, {u2})}, {(e1, {u1}), (e2, {u2})}}, and τ2 = {Φ, ŨE , {(e1, {u1}), (e2, {u1})}, {(e1, {u1, u2}), (e2, {u1, u2})}, κ2 = {Φ, ŨE , {(e1, {u2}), (e2, {u2})}}, if we defined the mapping as ϕ(u1) = u1, ϕ(u2) = u3, ϕ(u3) = u2 and ψ(u1) = u1, ψ(u2) = u3, ψ(u3) = u2, then ϕ is a soft β - continuous and ψ is a soft β - cocontinuous, Consequently Ω is a soft β - bicontinuous. Definition 20. A soft function Γ = (ϕ,ψ) : (ŨE , τ1, κ1) → (ŨE , τ2, κ2) is called : (1) Soft β - irresolute if ϕ−1(f) is sβo(ŨE) ∀f is sβo(ŨE) and ψ−1(f) is sβc(ŨE) ∀f is sβc(ŨE). (2) Strongly soft β - irresolute if ϕ−1(f) is sso(ŨE) ∀f is sβo(ŨE) and ψ−1(f) is ssc(ŨE) ∀f is sβc(ŨE). Proposition 6. Let a soft function Γ1 : (ŨE , τ1, κ1) → (ṼE , τ2, κ2) and Γ2 : (ṼE , τ2, κ2) → (W̃E , τ3, κ3) are both soft β irresolute. Then Γ1 ◦ Γ2 : (ŨE , τ1, κ1) → (W̃E , τ3, κ3) is soft β irresolute. REFERENCES 133 Definition 21. Let a soft function (ϕ,ψ) : (ŨE , τ1, κ1) → (ŨE , τ2, κ2), then: (1) ϕ is called soft β - open if the image of each soft β open in τ1 is soft β - open in τ2. (2) ψ is called soft β - closed if the image of each soft β - closed in κ1 is soft β - closed in κ2. 4. Conclusion In recent decades, many applications of topology have merged in different fields. There- fore we have had to expand the topological space in many ways as a result of its contri- bution to solving some issues. So, in this paper, we generalized some of the concepts via soft ditopology, and some properties are obtained. Acknowledgements This research is funded by the Deanship of Research in Zarqa University, Jordan. References [1] M.E. AbdEl-Monsef, R.A. Mahmoud, and A.A. Nasef. Some forms of strongly µ- functions, µ ∈ (α-irresolute, open, closed). Kyungpook Math. J., 36, ,pp.:143–150, 1996. [2] M. Abo-Elhamayel, T.M. Al-shami, and M.E. El-Shafei. On soft topological ordered spaces. Journal of King Saud University – Science, 31:556–566, 2019. [3] Ahmad Al-Omari and Shyamapada Moda. Filter on generalized topological spaces. Scientia Magna, 9, no.1,pp.:62–71, 2013. [4] T. M. AL-Shami and Mohammed E. EL-Shafei. T -soft equality relation. Turkish Journal of Mathematics, 44:1427–1441, 2020. [5] Jose Carlos R. Alcantud. An operational characterization of soft topologies by crisp topologies. Mathematics, 9(14):1656, 2021. [6] Jose Carlos R. Alcantud, Tareq M. Al-shami, and A. A. Azzam. Caliber and chain conditions in soft topologies. Mathematics, 9, no.19:1–15, 2021. [7] T. M. Alsham and A. A. Azzam. Infra soft semiopen sets and infra soft semicontin- uous. Journal of Function Spaces, Volume2021:Article ID 5716876, 2021. [8] T. M. Alshami. On soft separation axioms and their applications on decision making problem. Mathematical Problems in Engineering, Volume2021:Article ID 8876978, 2021. [9] A. A. Azzam. Grill nano topological spaces with grill nano generalized closed sets. Journal of the Egyption Mathematical Socciety, 25, no.2:164–166, 2017. REFERENCES 134 [10] B.Chen. Soft semi-open sets and related properties in soft topological spaces. Applied Mathematics and Information Sciences, 7, no.1:287–294, 2013. [11] D.Molodtsov. Soft set theory-first result. Computers and Mathematics with Applica- tion, 37, no.4–5:19–31, 1999. [12] S. Dost, L. M. Brown, and R. Erturk. β - open and β - closed sets in ditopology texture spaces. Filomat, pages 11–26, 2010. [13] Senel G. A new approach to hausdorff space theory via the soft sets. Mathematical Problems in Engineering, 9:1–6, 2016. [14] Senel G. Soft topology generated by l-soft sets. Journal of New Theory, 4(24):88–100, 2018. [15] Senel G.l, Jeong-Gon Lee, and Kul Hur. Distance and similarity measures for octa- hedron sets and their application to mcgdm problems. Mathematics, 8:1690, 2020. [16] E. Hater and S. Jafari. On some new classes of sets and a new decomposition of continuity via grills. Journal of advanced mathematical studies, 3, no.1,pp.:33–40, 2010. [17] I.Arockiarani and A.Arokialancy. Generalized soft gβ - closed sets and soft gsβ - closed sets in soft topological spaces. nternational Journal of Mathematical Archive, 4, no.2:1–7, 2013. [18] I.Zorlutuna, W. K. Min, M. Akdag, and S. Atmaca. Remarks on soft topological spaces. Annals of Fuzzy Mathematics and Informatics, 3, no.2,pp.:171–185, 2011. [19] M. Matejdes. Methodological remarks on soft topology. Soft computing, 25(5):4149–4156, 2021. [20] M.Shabir and M.Naz. On soft topological spaces. Computers and Mathematical with Applications, 61, no.7,pp.:1786–1799, 2011. [21] P.K.Maji, R.Biswas, and A.R.Roy. An application of soft sets in a decision making problem. Computers and Mathematics with Application, 44, no.8-9:1083–2002, 2002. [22] P.K.Maji, R.Biswas, and A.R.Roy. Soft set theory. Computers and Mathematics with Application, 45, no.4-5:555–562, 2003. [23] Guzide Senel. The theory of soft ditopological spaces. International Journal of Computer Applications, 150, no.4, September 2016. [24] T.Nori and N. Rajesh. Generalized closed sets with respect to an ideal in bitopological spaces. Acta Math. Haunger, 2009.