Exploration of Neutrosophic b-semi-open and b-semi-closed Sets

University of New Mexico Exploration of Neutrosophic b-semi-open and b-semi-closed Sets Sudeep Dey1and Gautam Chandra Ray2,∗ 1Department of Mathematics, Science College, Kokrajhar, Assam, India; sudeep.dey[email protected] 2Department of Mathematics, Central Institute of Technology Kokrajhar, Assam, India; [email protected] ∗Correspondence: [email protected] Abstract.In this write-up, we introduce and develop the concepts of neutrosophic b-semi-open sets and neutrosophic b-semi-closed sets, examining their properties and behaviours under various topological operations. Additionally, we explore their interactions with other established classes of open sets within neutrosophic topological spaces. The notions of neutrosophic b-semi-interior and neutrosophic b-semi-closure are also presented, and their numerous properties are thoroughly investigated. Keywords: Neutrosophic b-semi-open set ; Neutrosophic b-semi-closed set ; Neutrosophic b-semi-interior ; Neutrosophic b-semi-closure.) —————————————————————————————————————————- 1. Introduction The term “fuzzy set” was coined by L.A. Zadeh [25] in 1965, and a more generalized version known as the intuitionistic fuzzy set was introduced by K. Atanassov [1] in 1986. Following these developments, Florentin Smarandache [18, 19] further extended the concept, giving rise to the neutrosophic set. A neutrosophic set is characterized by three membership functions: truth-membership, falsity-membership, and indeterminacy-membership functions. Notably, all three neutrosophic components remain unbiased to one another. The introduction of neutrosophy sparked global interest, leading researchers [20–22, 24] to contribute significantly to its advancement. Neutrosophic theory offers a more general and suitable approach to addressing real-life problems. Numerous practical-based works [2, 7, 9] have been carried out in a neutrosophic environment. In 2012, Salama & Alblowi [22] proposed the concept of a neutrosophic topological space, a generalization of intuitionistic fuzzy topological space developed by D.Coker [4] in 1997. Subsequently, various concepts related to neutrosophic topological spaces were developed by S.Dey, G.C.Ray, Exploration of NBSO and NBSC Sets Neutrosophic Sets and Systems, Vol. 97, 2026 different researchers [6, 12, 16, 17, 21, 23]. These included the introduction of various types of open and closed sets [3,5,10,11,13–15] connected to neutrosophic topological spaces. In 2018, Ebenanjar et al. [8], described the concept of neutrosophic b-open sets and investigated some properties. In this article, we present a new type of open set called neutrosophic b-semi-open set and examine some of its fundamental properties. Additionally, we define and study neutrosophic b-semi-interior and neutrosophic b-semi-closure of a neutrosophic set, exploring some properties associated with these concepts. 2. Preliminaries 2.1. Definition: [18] Let Xbe the universe of discourse. A neutrosophic set Aover Xis defined as A= {⟨x, TA(x),IA(x),FA(x)⟩:x∈X}, where the functions TA,IA,FAare real standard or nonstandard subsets of ]−0,1+[, i.e., TA:X→]−0,1+[,IA:X→]−0,1+[,FA:X→]−0,1+[ and −0≤ TA(x) + IA(x) + FA(x)≤3+. The neutrosophic set Ais characterized by the truth-membership function TA, indeterminacy-membership function IA, falsity-membership function FA. 2.2. Definition: [24] Let Xbe the universe of discourse. A single-valued neutrosophic set Aover Xis defined as A={⟨x, TA(x),IA(x),FA(x)⟩:x∈X}, where TA,IA,FAare functions from Xto [0,1] and 0≤ TA(x) + IA(x) + FA(x)≤3. The set of all single valued neutrosophic sets over Xis denoted by N(X). Throughout this article, a neutrosophic set (NS, for short) will mean a single valued neutrosophic set. 2.3. Definition: [12] Let A, B ∈ N(X). Then (i) (Inclusion): If TA(x)≤ TB(x),IA(x)≥ IB(x),FA(x)≥ FB(x) for all x∈Xthen Ais said to be a neutrosophic subset of Band which is denoted by A⊆B. (ii) (Equality): If A⊆Band B⊆Athen A=B. (iii) (Intersection): The intersection of Aand B, denoted by A∩B, is defined as A∩B= {⟨x, TA(x)∧ TB(x),IA(x)∨ IB(x),FA(x)∨ FB(x)⟩:x∈X}. (iv) (Union): The union of Aand B, denoted by A∪B, is defined as A∪B={⟨x, TA(x)∨ TB(x),IA(x)∧ IB(x),FA(x)∧ FB(x)⟩:x∈X}. (v) (Complement): The complement of the NS A, denoted by Ac, is defined as Ac= {⟨x, FA(x),1− IA(x),TA(x)⟩:x∈X} S.Dey, G.C.Ray, Exploration of NBSO and NBSC Sets Neutrosophic Sets and Systems, Vol. 97, 2026 382 (vi) (Universal Set): If TA(x) = 1,IA(x) = 0,FA(x) = 0 for all x∈Xthen Ais said to be neutrosophic universal set and which is denoted by ˜ X. (vii) (Empty Set): If TA(x) = 0,IA(x) = 1,FA(x) = 1 for all x∈Xthen Ais said to be neutrosophic empty set and which is denoted by ˜ ∅. 2.4. Definition: [22] Let {Ai:i∈△} ⊆ N(X), where △is an index set. Then (i) ∪i∈△Ai={⟨x, ∨i∈△TAi(x),∧i∈△IAi(x),∧i∈△FAi(x)⟩:x∈X}. (ii) ∩i∈△Ai={⟨x, ∧i∈△TAi(x),∨i∈△IAi(x),∨i∈△FAi(x)⟩:x∈X}. 2.5. Definition: [12] Let τ⊆ N(X). Then τis called a neutrosophic topology on Xif (i) ˜ ∅and ˜ Xbelong to τ. (ii) Arbitrary union of neutrosophic sets in τis in τ. (iii) Intersection of any two neutrosophic sets in τis in τ. If τis a neutrosophic topology on Xthen the pair (X, τ) is called a neutrosophic topological space (NTS, for short) over X. The members of τare called neutrosophic τ-open sets (or neutrosophic open sets or open sets, for short) in X. If for a neutrosophic set A,Ac∈τthen Ais said to be a neutrosophic τ-closed set (or neutrosophic closed set or closed set, for short) in X. 2.6. Definition: [12] Let (X, τ) be a NTS and A∈ N(X). Then the neutrosophic (i) interior of A, denoted by int(A), is defined as int(A) = ∪{G:G∈τand G⊆A}. (ii) closure of A, denoted by cl(A), is defined as cl(A) = ∩{G:Gis a neutrosophic closed set and G⊇A}. 2.7. Definition: [12] Let (X, τ) be a NTS and A, B ∈ N (X). Then (i) [cl(A)]c=int(Ac) (ii) [cl(A)]c=int(Ac) (iii) cl(A∪B) = cl(A)∪cl(B) (iv) cl(A∩B)⊆cl(A)∩cl(B) (v) int(A∪B)⊇int(A)∪int(B) (vi) int(A∩B) = int(A)∩int(B) S.Dey, G.C.Ray, Exploration of NBSO and NBSC Sets Neutrosophic Sets and Systems, Vol. 97, 2026 383 2.8. Definition: [8] Let (X, τ) be a NTS and Gbe a NS over X. Then Gis called a (i) neutrosophic b-open (NBO, for short) set iff G⊆[int(cl(G))] ∪[cl(int(G))]. (ii) neutrosophic b-closed (NBC, for short) set iff G⊇[int(cl(G))] ∩[cl(int(G))]. 2.9. Theorem: [8] Let (X, τ) be an NTS and Gbe a NS over X. Then (i) Gis an NBO set iff Gcis an NBC set. (i) Gis an NBC set iff Gcis an NBO set. 2.10. Definition: [8] Let (X, τ) be an NTS and A∈ N(X). Then the neutrosophic (i) b-interior of A, denoted by bint(A), is defined as bint(A) = ∪{G:Gis an NBO set in Xand G⊆A}. (ii) b-closure of A, denoted by bcl(A), is defined as bcl(A) = ∩{G:Gis an NBC set in X and G⊇A}. 2.11. Definition: [10] Let (X, τ) and (Y, σ) be two NTSs and A∈τ,B∈σ. Then (X, τ) is called neutrosophic product related to (Y, σ) if for any NSs C∈ N(X) and D∈ N(Y) whenever C⊈Acand D⊈Bc⇒C×D⊆Acט Y∪˜ X×Bc, there exist A1∈τ,B1∈σsuch that C⊆Ac 1or D⊆Bc 1 and Ac 1ט Y∪˜ X×Bc 1=Acט Y∪˜ X×Bc. 2.12. Definition: [10] Let (X, τ) and (Y, σ) be two NTSs such that Xis neutrosophic product related to Y. Then for the NSs A∈ N (X) and B∈ N(Y), we have (i) cl(A×B) = cl(A)×cl(B). (ii) int(A×B) = int(A)×int(B). 3. Main Results 3.1. Definition: Let (X, τ) be an NTS. A non-empty NS A∈ N(X) is called a (i) neutrosophic b-semi-open set(NBSO set, for short) in Xiff there exists an NBO set G in Xsuch that G⊆A⊆cl(G). (ii) neutrosophic b-semi-closed set(NBSC set, for short) in Xiff there exists an NBC set Gin Xsuch that int(G)⊆A⊆G. S.Dey, G.C.Ray, Exploration of NBSO and NBSC Sets Neutrosophic Sets and Systems, Vol. 97, 2026 384 The collection of all the neutrosophic b-semi-open sets of the NTS (X, τ) will be denoted by NBSO(X). 3.2. Example: Let X={a, b},τ={˜ ∅,˜ X}. Obviously (X, τ) is an NTS. Let us consider the NSs A= {⟨a, 0.9,0,0⟩,⟨b, 0,1,1⟩}, B ={⟨a, 1,0,0⟩,⟨b, 0.7,0.6,0.4⟩}, C ={⟨a, 0,1,1⟩,⟨b, 0.4,0.4,0.7⟩} and D={⟨a, 0,1,0.9⟩,⟨b, 1,0,0⟩} over X. Now cl(A) = ˜ X⇒int(cl(A)) = ˜ Xwhich gives A⊆int(cl(A)), i.e. A⊆int(cl(A)) ∪cl(int(A)). Therefore Ais an NBO set. Clearly A⊆B⊆cl(A). Therefore Bis an NBSO set in X. Clearly Dis an NBC set as Ac=D. Now int(D) = ˜ ∅and C⊆D. Therefore int(D)⊆C⊆Dand so, Cis an NBSC set in X. 3.3. Proposition: (i) Every NBO set in an NTS is an NBSO set. (ii) Every NBC set in an NTS is an NBSC set. Proof: (i) Let (X, τ) be an NTS and Abe an NBO set in X. Since A⊆A⊆cl(A). Therefore Ais an NBSO set. (ii) Let (X, τ) be an NTS and Abe an NBC set in X. Then Acis an NBO set and so, Ac is an NBSO set [by (i)]. Therefore, there exists an NBO set Bsuch that B⊆Ac⊆cl(B)⇒ [cl(B)]c⊆A⊆Bc⇒int(Bc)⊆A⊆Bc⇒Ais an NBSC set as Bcis an NBC set. 3.4. Proposition: (i) Every neutrosophic open set in an NTS is an NBSO set. (ii) Every neutrosophic closed set in an NTS is an NBSC set. Proof: (i) Let (X, τ) be an NTS and A∈τ. Then A=int(A). Now A⊆cl(A)⇒int(A)⊆ int(cl(A)) ⇒A⊆int(cl(A)) ⇒A⊆int(cl(A)) ∪cl(int(A)) ⇒Ais an NBO set. Thus for A∈τ, there exists an NBO set Asuch that A⊆A⊆cl(A). Therefore Ais an NBSO set. Hence proved. (ii) Let (X, τ) be an NTS and Abe a neutrosophic closed set. Then Acis a neutrosophic open set and so, Acis an NBSO set [by (i)]. Then there exists an NBO set Bsuch that B⊆Ac⊆cl(B)⇒[cl(B)]c⊆A⊆Bc⇒int(Bc)⊆A⊆Bc⇒Ais an NBSC set as Bcis an NBC set. Hence proved. 3.5. Remark: Converses of the propositions 3.4(i) and 3.4(ii) are not true. We establish by the following example. S.Dey, G.C.Ray, Exploration of NBSO and NBSC Sets Neutrosophic Sets and Systems, Vol. 97, 2026 385 Let X={a, b},τ={˜ ∅,˜ X}. Obviously (X, τ) is an NTS. Let us consider the NSs A= {⟨a, 1,0,0⟩,⟨b, 0,1,1⟩} and B={⟨a, 0,1,1⟩,⟨b, 1,0,0⟩} over X. Now int(cl(A)) = int(˜ X) = ˜ X which gives A⊆int(cl(A)), i.e. A⊆int(cl(A)) ∪cl(int(A)). Therefore Ais an NBO set and so by 3.3(i), Ais an NBSO set. Clearly Ais not a neutrosophic open set. Thus an NBSO set may not be a neutrosophic open set. Again Bis not a neutrosophic closed set as Bc=Ais not a neutrosophic open set. As A is an NBO set, so by 2.9, Ac=Bis an NBC set and therefore by 3.3(ii), Bis an NBSC set. Thus an NBSC set may not be a neutrosophic closed set. 3.6. Proposition: Let (X, τ) be an NTS and G∈ N(X). Then Gis an NBSO set iff Gcis an NBSC set. Proof: Necessary part: Gis an NBSO set ⇒there exists an NBO set Hsuch that H⊆ G⊆cl(H)⇒[cl(H)]c⊆Gc⊆Hc⇒int(Hc)⊆Gc⊆Hc⇒Gcis an NBSC set as Hcis an NBC set. Sufficient part: Gcis an NBSC set ⇒there exists an NBC set Hsuch that int(H)⊆Gc⊆ H⇒Hc⊆G⊆[int(H)]c⇒Hc⊆G⊆cl(Hc)⇒Gis an NBSO set as Hcis an NBO set. 3.7. Proposition: In an NTS, union of an arbitrary collection of NBSO sets is an NBSO set. Proof: Let (X, τ) be an NTS and {Gλ:λ∈△}be an arbitrary collection of NBSO sets in X, where △is an index set. Since Gλis an NBSO set, so there exists an NBO set Hλfor each Gλ,λ∈△such that Hλ⊆Gλ⊆cl(Hλ). Now Hλ⊆Gλ⊆cl(Hλ) for each λ∈△⇒ ∪λ∈△Hλ⊆ ∪λ∈△Gλ⊆ ∪λ∈△cl(Hλ)⇒ ∪λ∈△Hλ⊆ ∪λ∈△Gλ⊆cl(∪λ∈△Hλ). Since arbitrary union of NBO sets is an NBO set, so ∪λ∈△Hλis an NBO set. Thus there exists an NBO set ∪λ∈△Hλsuch that ∪λ∈△Hλ⊆ ∪λ∈△Gλ⊆cl(∪λ∈△Hλ). Therefore ∪λ∈△Gλis an NBSO set. Hence proved. 3.8. Proposition: (i) In an NTS, union of a neutrosophic open set and an NBSO set is an NBSO set. (ii) In an NTS, union of an NBO set and an NBSO set is an NBSO set. Proof: Very obvious. 3.9. Proposition: In an NTS, intersection of an arbitrary collection of NBSC sets is an NBSC set. Proof: Let (X, τ) be an NTS and {Gλ:λ∈△}be an arbitrary collection of NBSC sets in X, where △is an index set. Then Gc λis an NBSO set for each λ∈△⇒ ∪λ∈△Gc λis an NBSO set [by 3.7] ⇒(∩λ∈△Gλ)cis an NBSO set ⇒ ∩λ∈△Gλis an NBSC set[by 3.6]. Hence proved. S.Dey, G.C.Ray, Exploration of NBSO and NBSC Sets Neutrosophic Sets and Systems, Vol. 97, 2026 386 3.10. Proposition: (i) In an NTS, intersection of a neutrosophic closed set and an NBSC set is an NBSC set. (ii) In an NTS, intersection of an NBC set and an NBSC set is an NBSC set. Proof: Very obvious. 3.11. Proposition: Let (X, τ) be an NTS and G∈ N(X). Then Gis an NBSO set iff G⊆cl(bint(G)). Proof: Necessary part: Since Gis an NBSO set, so there exists an NBO set Hsuch that H⊆G⊆cl(H). Obviously H⊆bint(G) which implies cl(H)⊆cl(bint(G)). Therefore G⊆cl(bint(G)). Sufficient part: Given G⊆cl(bint(G)). Since bint(G)⊆G, so bint(G)⊆G⊆cl(bint(G)). As bint(G) is an NBO set, so Gis an NBSO set. 3.12. Proposition: Let (X, τ) be an NTS and G∈ N(X). Then Gis an NBSC set iff G⊇int(bcl(G)). Proof: Gis an NBSC set ⇔Gcis an NBSO set ⇔Gc⊆cl(bint(Gc))[by 3.11] ⇔Gc⊆ cl[(bcl(G))c]⇔Gc⊆[int(bcl(A))]c⇔G⊇int(bcl(A)). 3.13. Proposition: Let Gbe an NBSO set in an NTS (X, τ). If K∈ N(X) is such that bint(G)⊆K⊆bcl(G) then Kis an NBSO set in X. Proof: Since Gis an NBSO set, so there exists an NBO set Hsuch that H⊆G⊆cl(H). Obviously H⊆bint(G)⊆G⊆bcl(G). Again G⊆cl(H)⇒bcl(G)⊆bcl(cl(H)) ⊆cl(cl(H)) = cl(H). Thus, H⊆bint(G)⊆K⊆bcl(G)⊆cl(H). Therefore, H⊆K⊆cl(H), which ensures that Kis an NBSO set in X. 3.14. Proposition: Let Gbe an NBSC set in an NTS (X, τ). If K∈ N(X) is such that bint(G)⊆K⊆bcl(G) then Kis an NBSC set in X. Proof: Gis an NBSC set ⇔Gcis an NBSO set. Now bint(G)⊆K⊆bcl(G)⇔[bint(G)]c⊇ Kc⊇[bcl(G)]c⇔bint(Gc)⊆Kc⊆bcl(Gc)⇔Kcis an NBSO set [by 3.13] ⇔Kis an NBSC set in X. S.Dey, G.C.Ray, Exploration of NBSO and NBSC Sets Neutrosophic Sets and Systems, Vol. 97, 2026 387 3.15. Definition: Let (X, τ) be an NTS and A∈ N(X). Then the neutrosophic b-semi-interior of A, denoted by NBSint(A), is defined as NBSint(A) = ∪{G:Gis an NBSO set in Xand G⊆A}. 3.16. Proposition: Let (X, τ) be an NTS and A, B ∈ N (X). Then the following hold. (i) NBSint(A) is an NBSO set. (ii) NBSint(A)⊆A (iii) Ais an NBSO set iff A=NBSint(A). (iv) NBSint(˜ ∅) = ˜ ∅ (v) NBSint(˜ X) = ˜ X (vi) NBSint(NBSint(A)) = NBSint(A) Proof: (i) Since NBSint(A) is the union of NBSO sets, so by 3.7, NBSint(A) is an NBSO set. (ii) Since NBSint(A) is the union of all NBSO sets contained in A, so NBSint(A)⊆A. (iii) Suppose that Ais an NBSO set. Since A⊆Aand Ais an NBSO set, so A⊆ NBSint(A). Again NBSint(A)⊆A[by (ii)]. Therefore A=NBSint(A). Conversely if A=NBSint(A) then Ais an NBSO set as NBSint(A) is NBSO set[by (i)]. Hence proved. (iv) Since every neutrosophic open set is an NBSO set, so ˜ ∅is an NBSO set and therefore by (iii), NBSint(˜ ∅) = ˜ ∅. (v) Since every neutrosophic open set is an NBSO set, so ˜ Xis an NBSO set and therefore by (iii), NBSint(˜ X) = ˜ X. (vi) Since by (i), NBSint(A) is an NBSO set, so by (iii), NBSint(NBSint(A)) = NBSint(A). 3.17. Proposition: Let (X, τ) be an NTS and A, B ∈ N (X). Then the following hold. (i) A⊆B⇒NBSint(A)⊆NBSint(B) (ii) NBSint(A∪B)⊇NBSint(A)∪NBSint(B). (iii) NBSint(A∩B)⊆NBSint(A)∩NBSint(B). Proof: (i) Since A⊆Band NBSint(A)⊆A, so NBSint(A)⊆B. Since NBSint(A) is an NBSO set such that NBSint(A)⊆Band since NBSint(B) is the largest NBSO set contained in B, so NBSint(A)⊆NBSint(B). S.Dey, G.C.Ray, Exploration of NBSO and NBSC Sets Neutrosophic Sets and Systems, Vol. 97, 2026 388 (ii) A⊆A∪B⇒NBSint(A)⊆NBSint(A∪B). Similarly NBSint(B)⊆NBSint(A∪B). Therefore NBSint(A∪B)⊇NBSint(A)∪NBSint(B). (iii) A∩B⊆A⇒NBSint(A∩B)⊆NBSint(A). Similarly NBSint(A∩B)⊆NBSint(B). Therefore NBSint(A∩B)⊆NBSint(A)∩NBSint(B). 3.18. Definition: Let (X, τ) be an NTS and A∈ N (X). Then the neutrosophic b-semi-closure of A, denoted by NBScl(A), is defined as NBScl(A) = ∩{G:Gis an NBSC set in Xand G⊇A}. 3.19. Proposition: Let (X, τ) be an NTS and A∈ N (X). Then the following hold. (i) NBScl(A) is an NBSC set. (ii) A⊆NBScl(A). (iii) Ais an NBSC set iff A=NBScl(A). (iv) NBScl(˜ ∅) = ˜ ∅ (v) NBScl(˜ X) = ˜ X (vi) NBScl(NBScl(A)) = NBScl(A) Proof: (i) As NBScl(A) is the intersection of NBSC sets, so by 3.9, NBScl(A) is an NBSC set. (ii) As NBScl(A) is the intersection of all NBSC sets containing A, so A⊆NBScl(A). (iii) Suppose that Ais an NBSC set. Since A⊇Aand Ais an NBSC set, so NBScl(A)⊆A. Again A⊆NBScl(A) [by (ii)]. Therefore A=NBScl(A). Conversely if A=NBScl(A) then Ais an NBSC set as NBScl(A) is an NBSC set[by (i)]. Hence proved. (iv) Since every neutrosophic closed set is an NBSC set, so ˜ ∅is an NBSC set and therefore by (iii), NBScl(˜ ∅) = ˜ ∅. (v) Since every neutrosophic closed set is an NBSC set, so ˜ Xis an NBSC set and therefore by (iii), NBScl(˜ X) = ˜ X. (vi) Since by (i), NBScl(A) is an NBSC set, so by (iii), NBScl(NBScl(A)) = NBScl(A). 3.20. Proposition: Let (X, τ) be an NTS and A, B ∈ N (X). Then the following hold. (i) A⊆B⇒NBScl(A)⊆NBScl(B) (ii) NBScl(A∪B)⊇NBScl(A)∪NBScl(B). (iii) NBScl(A∩B)⊆NBScl(A)∩NBScl(B). Proof: S.Dey, G.C.Ray, Exploration of NBSO and NBSC Sets Neutrosophic Sets and Systems, Vol. 97, 2026 389