Ratio Mathematica Volume 48, 2023 Results on Binary Soft Topological Spaces P. G. Patil* Asha G. Adaki† Abstract The Baire space is used in the proof of results in many areas of anal- ysis and geometry, including some of the fundamental theorems of functional analysis. The concept of Baire spaces has been studied ex- tensively in general topology in [5, 8, 9, 10, 11, 15]. Thangaraj and Anjalmose [35] studied Baire spaces in the context of fuzzy theory. In this paper, we discussed the notions of the binary soft nowhere dense, binary soft dense, binary soft Gδ-set, binary soft first and sec- ond category sets, binary soft Baire spaces. Many of their properties are revealed and different characterizations of each are given. In con- clusion, we determined some conditions under which the subspace property of a Baire space is preserved. Keywords: Binary soft set, binary soft nowhere dense set, binary soft dense set, binary soft Gδ-set, binary soft Baire space. 2020 AMS subject classifications: 54A05, 54E52. 1 1 Introduction The idea of soft sets was introduced by Molodtsov [14] in 1999. Soft set theory allows researchers to choose the type of parameters they need, which greatly simplifies decision-making and makes the method more productive in the absence of partial data. Later, Shabir et al. [19] started researching on soft topo- logical spaces. Many researchers continued their work on soft topology, including *Department of Mathematics, Karnatak University, Dharwad - 580 003, India. pg- patil@kud.ac.in. †Department of Mathematics, Karnatak University, Dharwad - 580 003, India. asha.adaki@kud.ac.in 1Received on February 18, 2023. Accepted on July 20, 2023. Published on August 1, 2023. DOI: 10.23755/rm.v42i0.1136. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published under the CC-BY licence agreement. P. G. Patil and Asha G. Adaki Aygunoghu [4], Ahmad [2], Maji [13] and Hussain [12]. In 2016, Acikgöz et al. [1] presented the idea of binary soft set theory on two universal sets and investi- gated various features. Later, Patil et al. [16] introduced the separation axioms in binary soft topological spaces such as n − T0, n − T1 and n − T2 spaces. Baire space is a space through which applications of completeness are made in analysis. Also, it is used in well-known theorems as the closed graph theorem, the open mapping theorem and the uniform boundedness theorem. The Baire category theorem (BCT), which specifies sufficient conditions for a topo- logical space to be a Baire space, is a significant result in general topology and functional analysis. Further, BCT is used to prove Hartogs’s theorem, a funda- mental result in the theory of several complex variables. The Baire space was initially introduced in Bourbaki’s [7] Topologie Gen- erale Chapter lX and was named in honour of Rene Louis Baire. Baire spaces have been extensively studied in classical topology [5, 8, 9, 10, 11, 15]. The first and second category sets were first established by Rene Louis Baire in 1899 [6]. Thangaraj and Anjalmose introduced and studied the concept of Baire spaces in fuzzy topology [20]. Riaz and Fatima [18] studied the concept of soft Baire spaces in soft metric spaces. Ameen and Khalaf investigated the soft Baire in- variance using soft semicontinuous, soft somewhat continuous and soft somewhat open functions in [3]. In this article, we introduced a class of binary soft topological space called binary soft Baire space and studied some topological properties that are preserved by proper subspaces such as binary soft nowhere dense, binary soft dense, binary soft first and second category sets and binary soft Baire spaces. We obtained some conditions under which subspace property is preserved. Later, we shall show that binary soft compact, n − T2 space falls in the class of binary soft Baire space. We have referred the paper [1] for the definition of binary soft (briefly BS) set, binary absolute soft set, binary null soft set, union of two BS sets, intersection of two BS sets and [16, 17] for BS topology, binary soft topological space (briefly BSTS), BS subspace and BS relative topology. 2 Binary Soft Nowhere Dense Set Definition 2.1. A BS subset (F, E) of a BSTS (X1, X2, τb, E) is called a BS dense set (resp. BS co-dense set) if (F, E) = ˜̃E (resp. (F, E)⊚ = ˜̃ϕ). Definition 2.2. A BS subset (F, E) of a BSTS (X1, X2, τb, E) is called a BS nowhere dense set if ( (F, E) )⊚ = ˜̃ ϕ. Definition 2.3. A BS subset (F, E) of a BSTS (X1, X2, τb, E) is called a BS Gδ-set if it is countable BS intersection of BS open sets. Results on Binary Soft Topological Spaces Example 2.1. Let X1 = R and X2 = Q be two initial universal sets and E = {e1, e2} be a set of parameters and τb = { ˜̃ E, ˜̃ ϕ, (F1, E), (F2, E), (F3, E)}, where (F1, E) = {(e1, (N, N)), (e2, (Q, Q))}, (F2, E) = {(e1, (Q ′ , ϕ)), (e2, (ϕ, ϕ))}, (F3, E) = {(e1, (N ∪ Q ′ , N)), (e2, (Q, Q))} then BS closed sets are ˜̃E, ˜̃ϕ (F1, E) ′ ={(e1, (R \ N, Q \ N)), (e2, (Q ′ , ϕ))}, (F2, E) ′ ={(e1, (Q, Q)), (e2, (R, Q))}, (F3, E) ′ ={(e1, (Q \ N, Q \ N)), (e2, (Q ′ , ϕ))}. Consider the BS sets (F3, E) = {(e1, (Q ′ ∪ N, N)), (e2, (Q, Q))} then, (F3, E) = ˜̃ E. Therefore, (F3, E) is a BS dense set in (R, Q, τb, E). (G1, E) = {(e1, (Q−, Q−)), (e2, (Q ′ , ϕ))} then, (G1, E)⊚ = ˜̃ ϕ and ( (G1, E) )⊚ = ˜̃ ϕ. Therefore, (G2, E) BS co-dense and BS nowhere dense set (R, Q, τb, E). (G2, E) = {(e1, (Q+ \N, Q+ \N)), (e2, (Q ′ , ϕ))} then, (G2, E)⊚ = ˜̃ ϕ. Therefore (G2, E) is also a BS nowhere dense set (R, Q, τb, E). Theorem 2.1. Let (F, E) be a BS subset of a BSTS (X1, X2, τb, E) then the fol- lowing assertions are interchangeable: 1. (F, E) is BS nowhere dense in (X1, X2, τb, E). 2. ˜̃E − (F, E) is BS dense in (X1, X2, τb, E). 3. For each non-empty BS open set (G, E) in (X1, X2, τb, E), there exist a non-empty BS open set (H, E) in (X1, X2, τb, E) such that (H, E) ˜̃⊆(G, E) and (H, E)˜̃∩(F, E) = ϕ. Proof. (1) ⇒ (2), Let (F, E) is BS nowhere dense set in (X1, X2, τb, E) then ( (F, E) )⊚ = ˜̃ ϕ. Let (W, E) be a BS open subset of (X1, X2, τb, E). Since,( (F, E) )⊚ = ˜̃ ϕ therefore (W, E) intersects ˜̃E − ((F, E)) this holds for all BS open subsets of (X1, X2, τb, E). Therefore, ˜̃ E − ((F, E)) is BS dense set in (X1, X2, τb, E). (2) ⇒ (3), Let ˜̃E − ((F, E)) be a BS dense set in (X1, X2, τb, E) and (G, E) be non-empty BS open subset of (X1, X2, τb, E) then, (G, E)˜̃∩( ˜̃ E−(F, E)) ̸= ϕ. Let (H, E) = (G, E)˜̃∩( ˜̃E−(F, E)) then clearly (H, E)˜̃⊆(G, E) and (H, E)˜̃∩(F, E) = ϕ. (3) ⇒ (1), Suppose (F, E) is not BS nowhere dense set in (X1, X2, τb, E). That P. G. Patil and Asha G. Adaki is ( (F, E) )⊚ ̸= ˜̃ϕ then, any non-empty BS open subset of ( (F, E) )⊚ would in- tersect (F, E), which is a contradiction. Therefore, (F, E) is BS nowhere dense in (X1, X2, τb, E). The following theorem gives the relation between BS dense and BS nowhere dense sets. Theorem 2.2. If (F, E) is a BS open and dense subset of (X1, X2, τb, E) then, (F, E)c is BS nowhere dense set. Proof. Let (F, E) is a BS open and dense subset of (X1, X2, τb, E) then, (F, E) = ˜̃ E, this implies ˜̃E − (F, E) = ˜̃ϕ, ( ˜̃E − (F, E))⊚ = ˜̃ϕ, ((F, E)c)⊚ = ˜̃ϕ. Since, (F, E)c is BS closed subset. Therefore, (F, E)c is BS nowhere dense set. Remark 2.1. The converse of the preceding theorem is generally untrue. It can be demonstrated using the example below. Example 2.2. Let X1 = {a1, a2, a3}, X2 = {b1, b2, b3}, E = {e1, e2} and τb = { ˜̃ E, ˜̃ ϕ, (F1, E), (F2, E), (F3, E), (F4, E), (F5, E), (F6, E), (F7, E), (F8, E)} where (F1, E) = {(e1, ({a1}, {b1})), (e2, ({a2}, {b2}))}, (F2, E) = {(e1, ({a2}, {b2})), (e2, ({a3}, {b1}))}, (F3, E) = {(e1, ({a1, a2}, {b1, b2})), (e2, ({a2, a3}, {b1, b2}))}, (F4, E) = {(e1, ({a1}, {b2})), (e2, ({a2}, {b1}))}, (F5, E) = {(e1, ({a1}, {b1, b2})), (e2, ({a2}, {b1, b2}))}, (F6, E) = {(e1, ({a1}, ϕ)), (e2, ({a2}, ϕ))}, (F7, E) = {(e1, ({a1, a2}, {b2})), (e2, ({a2, a3}, {b1}))}, (F8, E) = {(e1, (ϕ, {b2})), (e2, (ϕ, {b1}))} then binary soft closed sets are ˜̃E, ˜̃ϕ (F1, E) ′ ={(e1, ({a2, a3}, {b2, b3})), (e2, ({a1, a3}, {b1, b3}))}, (F2, E) ′ ={(e1, ({a1, a3}, {b1, b3})), (e2, ({a1, a2}, {b2, b3}))}, (F3, E) ′ ={(e1, ({a3}, {b3})), (e2, ({a1}, {b3}))}, (F4, E) ′ ={(e1, ({a2, a3}, {b1, b3})), (e2, ({a1, a3}, {b2, b3}))}, (F5, E) ′ ={(e1, ({a2, a3}, {b3})), (e2, ({a1, a3}, {b3}))}, (F6, E) ′ ={(e1, ({a2, a3}, {b1, b2, b3})), (e2, ({a1, a3}, {b1, b2, b3}))}, (F7, E) ′ ={(e1, ({a3}, {b1, b3})), (e2, ({a1}, {b2, b3}))}, (F8, E) ′ ={(e1, ({a1, a2, a3}, {b1, b3})), (e2, ({a1, a2, a3}, {b2, b3}))}. Let (H, E) = {(e1, ({a3}, ϕ)), (e2, ({a3}, {b3}))} be a binary soft subset of (X1, X2, τb, E). Clearly, (H, E) = {(e1, ({a2, a3}, {b3})), (e2, ({a1, a3}, {b3}))} then ( (H, E) )⊚ = ˜̃ ϕ. Therefore, (H, E) is BS nowhere dense set in (X1, X2, τb, E) but, (H, E)c = {(e1, ({a1, a2}, {b1, b2, b3})), (e2, ({a1, a2}, {b1, b2}))} is not BS open set. Results on Binary Soft Topological Spaces Theorem 2.3. If (F, E) is a BS closed and nowhere dense subset of (X1, X2, τb, E) then, (F, E)c is BS dense subset of (X1, X2, τb, E). Proof: Let (F, E) is a BS closed and nowhere dense subset of (X1, X2, τb, E) then, ( (F, E) )⊚ = ˜̃ ϕ, this implies (F, E)⊚ = ˜̃ϕ, ˜̃E−(F, E)⊚ = ˜̃E, ( ˜̃E − (F, E)) = ˜̃ E, ((F, E)c) = ˜̃E. Therefore, (F, E)c is BS dense set. Remark 2.2. The previous theorem’s converse is generally not true. The example below can be used to illustrate it. Example 2.3. In the example 2.2 consider, (G, E) = {(e1, ({a1, a2, a3}, {b1, b2, b3})), (e2, ({a1, a2}, {b1, b2, b3}))} be a BS dense subset of (X1, X2, τb, E) but, (G, E)c = {(e1, (ϕ, ϕ)), (e2, ({a3}, ϕ))} is not BS closed set. Theorem 2.4. Let (Y, τby, E) be a BS subspace of (X1, X2, τb, E) and (F, E) be a BS subset of ˜̃Y . If (F, E) is BS nowhere dense set in ˜̃Y , then (F, E) is BS nowhere dense set in (X1, X2, τb, E). Proof: Suppose (F, E) is a BS nowhere dense subset of ˜̃Y , Let (G, E) be a BS open subset of (X1, X2, τb, E) then, there exists a non-empty BS open set (H, E) in ˜̃Y such that (H, E)˜̃⊆(G, E)˜̃∩(Y, E) and (H, E)˜̃∩(F, E) = ˜̃ϕ. Now there exists a BS open set (W, E) in (X1, X2, τb, E) such that (H, E) = (W, E)˜̃∩(Y, E) thus (W, E)˜̃⊆(G, E) and (W, E)˜̃∩(F, E) = ˜̃ϕ. Therefore, (F, E) is BS nowhere dense set in (X1, X2, τb, E). Remark 2.3. The previous theorem’s converse is generally not true. The example below can be used to illustrate it. Example 2.4. In the example 2.2, consider the BS subspace ˜̃ Y = {(e1, ({a3}, {b3})), (e2, ({a3}, {b3}))} with BS subspace topology τby = { ˜̃ Y, ˜̃ ϕ, (G1, E)}, where (G1, E) = {(e1, (ϕ, ϕ)), (e2, ({a3}, ϕ))} and (G1, E) ′ ={(e1, ({a1, a2, a3}, {b1, b2, b3})), (e2, ({a1, a2}, {b1, b2, b3}))}. Let (H, E) = {(e1, ({a3}, ϕ)), (e2, ({a3}, {b3}))} be a binary soft subset of (X1, X2, τb, E). Clearly, (H, E) is a BS nowhere dense set in (X1, X2, τb, E). In BS subspace (Y, τby, E), (H, E) y = ˜̃ Y then ( (H, E) y)⊚y = ( ˜̃ Y )⊚y ̸= ˜̃ϕ. Therefore, (H, E) is not BS nowhere dense set in (˜̃Y, τby, E). Theorem 2.5. Let (˜̃Y, τby, E) is BS open or BS dense subspace of (X1, X2, τb, E) and (F, E) is BS nowhere dense in (X1, X2, τb, E) then, (F, E) is BS nowhere dense in (˜̃Y, τby, E). P. G. Patil and Asha G. Adaki Proof: Let (˜̃Y, τby, E) be a BS open subspace of (X1, X2, τb, E) and (F, E) is BS nowhere dense in (X1, X2, τb, E). Let (G, E) is BS open subset of ( ˜̃ Y, τby, E) then (G, E) is BS open subset of (X1, X2, τb, E). Therefore, there exists a non- empty BS open subset (H, E) of (G, E) such that (H, E)˜̃∩(F, E) = ˜̃ϕ, (H, E) is also BS open subset of (˜̃Y, τby, E). Therefore, (F, E) is also BS nowhere dense in ( ˜̃ Y, τby, E). The proof is similar in the case of (˜̃Y, τby, E) is BS dense subspace of (X1, X2, τb, E). Corolary 2.1. The above theorem need not be true for BS closed subspaces. That is, if (˜̃Y, τby, E) is a BS closed subspace of a BS topological space (X1, X2, τb, E) and (F, E) is BS nowhere dense set in (X1, X2, τb, E) then, (F, E) need not to be BS nowhere dense set in (˜̃Y, τby, E). Example 2.5. Let X1 = {a1, a2, a3}, X2 = {b1, b2, b3}, E = {e1, e2} and τb = { ˜̃ E, ˜̃ ϕ, (F1, E), (F2, E), (F3, E), (F4, E), (F5, E), (F6, E), (F7, E), (F8, E), (F9, E), (F10, E), (F11, E)} where (F1, E) = {(e1, ({a1, a2}, {b1, b2})), (e2, ({a1, a2}, {b1, b2}))}, (F2, E) = {(e1, ({a3}, {b3})), (e2, ({a3}, {b3}))}, (F3, E) = {(e1, ({a2}, {b1})), (e2, ({a1}, {b2}))}, (F4, E) = {(e1, ({a2, a3}, {b1, b3})), (e2, ({a1, a3}, {b2, b3}))}, (F5, E) = {(e1, ({a3}, {b1})), (e2, ({a3}, {b2}))}, (F6, E) = {(e1, ({a1, a2, a3}, {b1, b2})), (e2, ({a1, a2, a3}, {b1, b2}))}, (F7, E) = {(e1, (ϕ, {b1})), (e2, (ϕ, ϕ))}, (F8, E) = {(e1, (ϕ, {b2})), (e2, (ϕ, {b1}))}, (F9, E) = {(e1, ({a3}, {b1, b3})), (e2, ({a3}, {b2, b3}))}, (F10, E) = {(e1, ({a2, a3}, {b1})), (e2, ({a1, a3}, {b2}))}, (F11, E) = {(e1, (ϕ, {b1})), (e2, (ϕ, {b2}))} then binary soft closed sets are ˜̃E, ˜̃ϕ (F1, E) ′ = {(e1, ({a3}, {b3})), (e2, ({a3}, {b3}))}, (F2, E) ′ = {(e1, ({a1, a2}, {b1, b2})), (e2, ({a1, a2}, {b1, b2}, (F3, E) ′ = {(e1, ({a1, a3}, {b2, b3})), (e2, ({a2, a3}, {b1, b3}))}, (F4, E) ′ = {(e1, ({a1}, {b2})), (e2, ({a2}, {b1}))}, (F5, E) ′ = {(e1, ({a1, a2}, {b2, b3})), (e2, ({a1, a2}, {b1, b3}))}, (F6, E) ′ = {(e1, (ϕ, {b3})), (e2, (ϕ, {b3}))}, (F7, E) ′ = {(e1, ({a1, a2, a3}, {b2, b3})), (e2, ({a1, a2, a3}, {b1, b2, b3}))}, (F8, E) ′ = {(e1, ({a1, a2}, {b2})), (e2, ({a1, a2}, {b1}))}, (F9, E) ′ = {(e1, ({a1, a2}, {b1, b2, b3})), (e2, ({a1, a2}, {b1, b2, b3}))}, (F10, E) ′ = {(e1, ({a1}, {b2, b3})), (e2, ({a2}, {b1, b3}))}, (F11, E) ′ = {(e1, ({a1, a2, a3}, {b2, b3})), (e2, ({a1, a2, a3}, {b1, b3}))}. Results on Binary Soft Topological Spaces Let ˜̃Y = {(e1, ({a1, a2}, {b1, b2, b3})), (e2, ({a1, a2}, {b1, b2, b3}))} be a BS closed subset of (X1, X2, τb, E) then the BS subspace topology is τy = { ˜̃ Y, ˜̃ ϕ, (G1, E), (G2, E), (G3, E), (G4, E), (G5, E), (G6, E), (G7, E)} where (G1, E) = {(e1, ({a1, a2}, {b1, b2})), (e2, ({a1, a2}, {b1, b2}))}, (G2, E) = {(e1, (ϕ, {b3})), (e2, (ϕ, {b3}))}, (G3, E) = {(e1, ({a2}, {b1})), (e2, ({a1}, {b2}))}, (G4, E) = {(e1, ({a2}, {b1, b3})), (e2, ({a1}, {b2, b3}))}, (G5, E) = {(e1, (ϕ, {b1})), (e2, (ϕ, {b2}))}, (G6, E) = {(e1, (ϕ, {b1})), (e2, (ϕ, ϕ))}, (G7, E) = {(e1, (ϕ, {b1, b3})), (e2, (ϕ, {b2, b3}))} then BS closed sets are ˜̃Y, ˜̃ϕ (G1, E) ′ = {(e1, (ϕ, {b3})), (e2, (ϕ, {b3}))}, (G2, E) ′ = {(e1, ({a1, a2}, {b1, b2})), (e2, ({a1, a2}, {b1, b2}))}, (G3, E) ′ = {(e1, ({a1}, {b2, b3})), (e2, ({a2}, {b1, b3}))}, (G4, E) ′ = {(e1, ({a1}, {b2})), (e2, ({a2}, {b1}))}, (G5, E) ′ = {(e1, ({a1, a2}, {b2, b3})), (e2, ({a1, a2}, {b1, b3}))}, (G6, E) ′ = {(e1, ({a1, a2}, {b2, b3})), (e2, ({a1, a2}, {b1, b2, b3}))}, (G7, E) ′ = {(e1, ({a1, a2}, {b2})), (e2, ({a1, a2}, {b1}))}. Consider the BS set {(e1, (ϕ, {b3})), (e2, (ϕ, {b3}))} which is BS nowhere dense set in (X1, X2, τb, E), but it is not BS nowhere dense set in the BS closed subspace ( ˜̃ Y, τby, E). 3 Binary Soft Baire Spaces Definition 3.1. A BS subset (F, E) of a BSTS (X1, X2, τb, E) is called BS first category set if it is the BS union of countable family of BS nowhere dense sets. Definition 3.2. A BS subset (F, E) of a BSTS (X1, X2, τb, E) is called BS second category set if it is not BS first category set. Definition 3.3. A BS Baire space is a BSTS such that every non-empty BS open subset is BS second category. Example 3.1. Consider the example 2.1, the BS set (H, E) = {(e1, (Q \ N, Q \ N)), (e2, (Q ′ , ϕ))} is a BS first category set, because (H, E) = (G1, E) ˜̃∪(G2, E). Also, (R, Q, τb, E) is a BS Baire space. Theorem 3.1. The following assertions are interchangeable for a BSTS (X1, X2, τb, E): 1. (X1, X2, τb, E) is a BS baire space. P. G. Patil and Asha G. Adaki 2. The BS intersection of any sequence of BS dense open sets is BS dense in (X1, X2, τb, E). 3. The BS complement of any BS first category set in (X1, X2, τb, E) is BS dense in (X1, X2, τb, E). 4. Every countable BS union of BS closed sets with empty BS interior in (X1, X2, τb, E) has empty BS interior in (X1, X2, τb, E). Proof:(1) ⇒ (2), Suppose (X1, X2, τb, E) is a BS Baire space. Let {(Gi, E)} be a sequence of BS dense open sets in (X1, X2, τb, E). Let us assume that their BS intersection ˜̃∩(Gi, E) is not BS dense in (X1, X2, τb, E), then there exists a BS open set (H, E) such that it doesnot intersect ˜̃∩(Gi, E). That is (H, E) = ˜̃ E − ˜̃∩(Gi, E), (H, E) = ˜̃∪( ˜̃ E − (Gi, E)) where each ˜̃ E − (Gi, E)) is BS nowhere dense set in (X1, X2, τb, E). Therefore, (H, E) is of BS first cate- gory set. Which is contradiction to (X1, X2, τb, E) is a BS baire space. Therefore, ˜̃∩(Gi, E) is a BS dense set in (X1, X2, τb, E). (2) ⇒ (3), Let {(Fi, E)} be a sequence of BS closed nowhere dense subsets of (X1, X2, τb, E), then their BS union ˜̃∪(Fi, E) is a BS first category set in (X1, X2, τb, E), we have to prove ˜̃ E− ˜̃∪(Fi, E) is BS dense in (X1, X2, τb, E). Let us assume that it is not dense in (X1, X2, τb, E). Consider a BS open set (G, E) in (X1, X2, τb, E) such that it does not intersect with ˜̃ E−˜̃∪(Fi, E) = ˜̃∩( ˜̃ E−(Fi, E)), where each ( ˜̃E − (Fi, E)) is BS dense openset in (X1, X2, τb, E) then their binary soft intersection is not BS dense in (X1, X2, τb, E), which is contradiction. There- fore, ˜̃E − ˜̃∪(Fi, E) is BS dense in (X1, X2, τb, E). (3) ⇒ (4), Let {(Fi, E)} be a sequence of BS closed sets with empty interiors. Suppose ˜̃∪(Fi, E) does not has empty BS interior, then ˜̃ E − ˜̃∪(Fi, E) would not be BS dense in (X1, X2, τb, E), which is a contradiction. Therefore, ˜̃∪(Fi, E) has empty BS interior. (4) ⇒ (1), Let {(Fi, E)} be a sequence of BS nowhere dense subsets in (X1, X2, τb, E). Suppose ˜̃∪(Fi, E) is a BS open set, then each cl(Fi, E) has no BS interior point, that is ( (Fi, E) )⊚ = ˜̃ E , then ˜̃∪(Fi, E) = (˜̃∪(Fi, E)), since ˜̃∪(Fi, E) is BS open, this implies it contains all its interior points, therefore ˜̃∪(Fi, E) has BS interior points, which is contradiction to (4), therefore ˜̃∪(Fi, E) is not BS openset. Therefore, BS opensets cannot be written as BS union of BS nowhere dense sets. Therefore, (X1, X2, τb, E) is a BS Baire space. Theorem 3.2. the following assertions are interchangeable for a BSTS (X1, X2, τb, E). Results on Binary Soft Topological Spaces 1. (X1, X2, τb, E) is a BS baire space. 2. (G, E)⊚ = ˜̃ϕ for every BS first category set (G, E) in (X1, X2, τb, E). 3. (H, E) = ˜̃E for every BS residual set (H, E) in (X1, X2, τb, E). Proof: (1) ⇒ (2), Let (X1, X2, τb, E) be a BS baire space, let (G, E) be a BS first category set then (G, E) = ˜̃∪(Gi, E) where (Gi, E)’s are BS nowhere dense sets then (G, E)⊚ = ( ˜̃∪(Gi, E) )⊚ = ˜̃ ϕ. Therefore, (G, E)⊚ = ˜̃ϕ (2) ⇒ (3), Let (H, E) be a BS residual set then (H, E)′ is BS first category set i.e. ˜̃ E−(H, E) is BS first category set, then ( ˜̃E−(H, E))⊚ = ˜̃ϕ, then ˜̃E−(H, E) = ˜̃ϕ. Therefore, (H, E) = ˜̃E. (3) ⇒ (1), Let (F, E) be a BS first category set then ˜̃E − (F, E) is BS resid- ual set then ( ˜̃E − (F, E)) = ˜̃E, then ˜̃E − (F, E)⊚ = ˜̃E, (F, E)⊚ = ˜̃ϕ there- fore ( ˜̃∪(Fi, E) )⊚ = ˜̃ ϕ where (Fi, E)’s are BS nowhere dense sets. Therefore, (X1, X2, τb, E) is a BS Baire space. Theorem 3.3. Every BS open or BS dense subspace of a BS Baire space is a BS Baire space. Proof: Let (X1, X2, τb, E) be a BS baire space, let ( ˜̃ Y, τby, E) be a BS open subspace of (X1, X2, τb, E), consider (Fi, E) be a sequence of BS nowhere dense sets in (˜̃Y, τby, E), by theorem 2.4 (Fi, E) is also a sequence of BS nowhere dense sets in (X1, X2, τb, E). Since (X1, X2, τb, E) is a BS baire space then ˜̃∪(Fi, E) is also BS nowhere dense set in (X1, X2, τb, E). Let us assume that ˜̃∪(Fi, E) is not BS nowhere dense set in (˜̃Y, τby, E), then ( (˜̃∪(Fi, E)) y)⊚y ̸= ˜̃ϕ then there exists a BS open set (G, E) in (˜̃Y, τby, E) such that (G, E) ˜̃∩(˜̃∪(Fi, E)) ̸= ˜̃ ϕ, since ( ˜̃ Y, τby, E) is a BS open subspace of (X1, X2, τb, E). Thus (G, E) is a BS open set in (X1, X2, τb, E) then (G, E)˜̃∩(˜̃∪(Fi, E)) ̸= ˜̃ ϕ, which is contradiction that (X1, X2, τb, E) is a binary soft baire space, therefore ( (˜̃∪(Fi, E)) y)⊚y = ˜̃ ϕ, thus ˜̃∪(Fi, E) is BS nowhere dense set in ( ˜̃ Y, τby, E). Therefore ( ˜̃ Y, τby, E) is also a BS Baire space. The proof is similar for a BS dense subspace. Example 3.2. Consider the example 2.5, let ˜̃Y = {(e1, ({a1, a2}, {b1, b2})), (e2, ({a1, a2}, {b1, b2}))} be a binary soft open P. G. Patil and Asha G. Adaki subset of a BS Baire space (X1, X2, τb, E) then the BS subspace topology is τby = { ˜̃ Y, ˜̃ ϕ, (G1, E), (G2, E), (G3, E)} where (G1, E) = {(e1, ({a2}, {b1})), (e2, ({a1}, {b2}))}, (G2, E) = {(e1, (ϕ, {b1})), (e2, (ϕ, {b2}))}, (G3, E) = {(e1, (ϕ, {b1})), (e2, (ϕ, ϕ))} then BS closed sets are ˜̃Y, ˜̃ϕ, (G1, E) ′ = {(e1, ({a1}, {b2})), (e2, ({a2}, {b1}))}, (G2, E) ′ = {(e1, ({a1, a2}, {b2})), (e2, ({a1, a2}, {b1}))}, (G3, E) ′ = {(e1, ({a1, a2}, {b1})), (e2, ({a1, a2}, {b1, b2}))}. Since, (X1, X2, τb, E) is a BS Baire space and ( ˜̃ Y, τby, E) is a BS open subspace of (X1, X2, τb, E), then ( ˜̃ Y, τby, E) is also BS Baire space. Corolary 3.1. A BS compact and BS n − T2 space is BS regular. Theorem 3.4. If (X1, X2, τb, E) is a BS compact and BS n − T2 space then (X1, X2, τb, E) is a BS baire space. Proof: Let (X1, X2, τb, E) is a BS compact and BS n−T2 space, let {(Fi, E)} be a countable collection of closed sets of (X1, X2, τb, E) having empty BS inte- riors, to prove ˜̃∪(Fi, E) has empty BS interior, that is ˜̃∪(Fi, E) does not contain any BS open set. Consider a BS openset (G1, E) of (X1, X2, τb, E), then we must find a pair of points (x0, y0) ∈ (G1, E) but does not lie in any of the sets (Fi, E). Let us first consider the set (F1, E), by hypothesis (F1, E) does not contain (G1, E), that is there exist a point (x1, y1) ∈ (G1, E) but (x1, y1) /∈ s (F1, E), since (X1, X2, τb, E) is a BS compact and BS n − T2 space then by corollary 3.1 it is a BS n-regular space, thus for any (x1, y1) ∈ (G1, E) there exist a BS open set (H1, E) such that (x1, y1) ∈ (H1, E)˜̃⊆(H1, E)˜̃⊆(G1, E) also for any (x1, y1) ∈ X1 × X2 and (x1, y1) /∈ s (F1, E) then, there is a BS open set (H1, E) such that (x1, y1) ∈ (H1, E) and (H1, E)˜̃∩(F1, E) = ˜̃ ϕ [16], (H1, E) ˜̃⊆(G1, E). There fore given any nonempty binary soft open set Gn, E, we choose a point (xn, yn) ∈ (Gn, E) and it does not lie in the BS closed set (Fi, E) for all i, then we choose a BS open set (Hn, E) such that (Hn, E)˜̃∩(Fn, E) = ˜̃ ϕ and (Hn, E) ˜̃⊆(Gn, E). Since (X1, X2, τb, E) is a BS compact space then consider {(Hn, E)} be a family of BS closed sets, by finite intersection property of BS closed sets [16], we have ˜̃∩(Hi, E) ̸= ˜̃ ϕ, let (x0, y0) ∈ ˜̃∩(Hi, E), that is (x0, y0) ∈ (Hi, E) for all i, this implies (x0, y0) ∈ (Gi, E) also (x0, y0) /∈ s (Fi, E) for all i, thus (x0, y0) /∈ s ˜̃∪(Fi, E), that is ˜̃∪(Fi, E) is also has empty BS interior, There fore (X1, X2, τb, E) is a BS Baire space. Remark 3.1. The converse of the preceding theorem is generally untrue. It can be demonstrated using the example below. Results on Binary Soft Topological Spaces Example 3.3. Let X1 = {a1, a2}, X2 = {b1, b2}, E = {i, ii} and τb = { ˜̃ E, ˜̃ ϕ, (G1, E), (G2, E), (G3, E), (G4, E), (G5, E), (G6, E), (G7, E)} where (G1, E) = {(i, ({a1, a2}, {b1})), (ii, ({a1, a2}, {b1}))}, (G2, E) = {(i, ({a2}, {b1, b2})), (ii, ({a2}, {b1, b2}))}, (G3, E) = {(i, ({a1, a2}, {b2})), (ii, ({a1, a2}, {b2}))}, (G4, E) = {(i, ({a2}, {b1})), (ii, ({a2}, {b1}))}, (G5, E) = {(i, ({a2}, {b2})), (ii, ({a2}, {b2}))}, (G6, E) = {(i, ({a1, a2}, ϕ)), (ii, ({a1, a2}, ϕ))}, (G7, E) = {(i, ({a2}, ϕ)), (ii, ({a2}, ϕ))}. Then, (X1, X2, τb, E) is a BS Baire space and BS compact space as it is a finite space. But, it is not n−T2 space because, (a1, b1) ̸= (a2, b2) ∈ X1 ×X2 and there does not exist BS disjoint open sets (G, E) and (F, E) such that (a1, b1) ∈ (G, E) and (a2, b2) ∈ (F, E). 4 Discussion and Conclusion This paper contributes to the area of Baire spaces in the BS topological spaces. We defined BS nowhere dense, BS dense, BS first, second category sets and obtained their properties. 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