Study of Neutrosophic b-continuous maps in Neutrosophic Topological Spaces

University of New Mexico Study of Neutrosophic b-continuous maps in Neutrosophic Topological Spaces Sujeewa Malkanthi1,∗and Panchadcharam Elango2 1Department of Mathematics, Faculty of Science, Eastern University, Sri Lanka; sujeewadissanayak[email protected] 2Department of Mathematics, Faculty of Science, Eastern University, Sri Lanka; [email protected] ∗Correspondence:sujeewadissanayak[email protected] ; Tel.:0762093579 Abstract.Neutrosophic sets were introduced to deal with indeterminancy and persistent issues and they were used to develop the neutrosophic topological spaces. In this paper, the concept of neutrosophic b-continuous maps and strongly neutrosophic b-continuous maps are defined and some important properties are developed. We prove that every neutrosophic strongly b-continuous map is a neutrosophic b-continuous map. We examine the relationship of neutrosophic b-continuous maps with the already defined continuous maps such as neutrosophic semi-continuous maps, neutrosophic pre-continuous maps, neutrosophic α-continuous maps. We prove that if a map is any of these neutrosophic continuous map, then that is a neutrosophic b-continuous map. Further, we prove that the surjective neutrosophic b-continuous image of a neutrosophic b-connected space, and a neutrosophic b-compact space is a neutrosophic b-connected space, and neutrosophic b-compact space respectively. Keywords: Neutrosophic topological spaces; Neutrosophic b-open sets; Neutrosophic b-Continuous map. —————————————————————————————————————————- 1. Introduction To avert obstacles dealing with ambiguous data, fuzzy sets [25], intuitionistic fuzzy sets [2], vague sets [6], rough sets [11], and soft sets [10] were introduced as mathematical tools. All these approaches have their implicit crisis in solving the problems involving indeterminant and inconsistent data due to inadequacy of parametrization tools. The idea of neutrosophic set as an approach for solving issues that cover unreliable, indeterminacy and persistent data was studied by Smarandache [19]. Wang et.al. [24] introduced single valued neutrosophic sets. Peng et al. [12] studied operations of neutrosophic numbers and introduced the idea of neutrosophic numbers. Neutrosophic topological space was introduced by Salama et.al. [14] Author(s), Paper’s title Neutrosophic Sets and Systems, Vol. 97, 2026 in 2012. This was studied by many mathematicians [15, 17, 18]. The idea of neutrosophic soft set was initiated by Maji [8]. Iswaraya et.al. [7] studied the concept of neutrosophic semi-open sets and neutrosophic semi-closed sets. Arokiarani et.al. [1] defined neutrosophic semi-open (resp. pre-open and α-open) functions and investigated their relations. Rao et.al. [22] introduced neutrosophic pre-open sets. P. Evanzalin Ebenanjar et.al. [5] introduced the notion of neutrosophic b-open sets are initiated and their properties are investigated. Debnath et.al. [3] studied on neutrosophic pδ-irresolute functions in neutrosophic topological spaces. Vetrivel et.al. [23] considered the forgotten topological index and the edge forgotten index in the 3-valued logic neutrosophic graph and came up with some important theorem results and applications. Subasree and BasariKodi [13] tested the proofs and examples on heptagonal neutrosophic semi-open sets in heptagonal neutrosophic topological spaces. Smarandache [20] had an overview, examples, trend analysis, research issues, challenges and future directions on revolutionary topologies. Dey et.al. [4] studied some covering properties in neutrosophic topological spaces using neutrosophic b-open sets. Continuous property is one of the important topological properties in topological spaces. Salama, A.A. and Florentin Smarandache [16] introduced neutrosophic continuous maps. In molecular graph theory or chemical graph theory, the topological index becomes useful for testing the chemical and pharmacological properties of drug molecular structures. The Forgotten Topological index concept in neutrosophic graphs to attain some important results and its properties and application were studied in [23]. In this article, the neutrosophic b-continuous maps and strongly neutrosophic b-continuous maps are defined and their properties are investigated. The properties of neutrosophic b-connectedness and neutrosophic b-compactness are developed using neutrosophic b-continuous maps. 2. Materials and Methods Definition 2.1. [14] Let Xbe a non-empty fixed set. A neutrosophic set Ais an object having the form A={⟨x, µA(x), σA(x), γA(x)⟩:x∈X}where µA(x), σA(x) and γA(x) represents the degree of membership function, the degree of indeterminacy function and the degree of nonmembership function respectively of each element x∈Xto the set A. Definition 2.2. [14] Let Xbe a non-empty set, and let the neutrosophic sets Aand Bbe A={⟨x, µA(x), σA(x), γA(x)⟩:x∈X}and B={⟨x, µB(x), σB(x), γB(x)⟩:x∈X}. Then we may consider two possible definitions for subsets (A⊆B). That is, A⊆Bmay be defined as in the following two different ways (1) A⊆B⇔µA(x)≤µB(x), σA(x)≤σB(x)andγA(x)≥B(x)∀x∈X (2) A⊆B⇔µA(x)≤µB(x), σA(x)≥σB(x)andγA(x)≥B(x)∀x∈X Definition 2.3. [14] Let Xbe a non-empty set and let the neutrosophic sets Aand Bbe A=⟨x, µA(x), σA(x), γA(x)⟩, B =⟨x, µB(x), σB(x), γB(x)⟩. Then, Sujeewa Malkanthi and Panchadcharam Elango, Study of Neutrosophic b-continuous maps in Neutrosophic Topological Spaces Neutrosophic Sets and Systems, Vol. 97, 2026 592 1.A ∩Bmay be defined as in the following in two different ways. (1) A∩B=⟨x, µA(x)∧µB(x), σA(x)∧σB(x) and γA(x)∨γB(x)⟩ (2) A∩B=⟨x, µA(x)∧µB(x), σA(x)∨σB(x) and γA(x)∨γB(x)⟩ 2.A ∪Bmay be defined as in the following in two different ways. (1) A∪B=⟨x, µA(x)∨µB(x), σA(x)∨σB(x) and γA(x)∧γB(x)⟩ (2) A∪B=⟨x, µA(x)∨µB(x), σA(x)∧σB(x) and γA(x)∧γB(x)⟩ Here the notations ∧and ∨means minimum and maximum respectively. Definition 2.4. [14] A neutrosophic topology for a non-empty set Xis a family τof neutrosophic subsets in Xsatisfying the following axioms : (1) 0N,1N∈τ, (2) G1∩G2∈τfor any G1, G2∈τ, (3) ∪Gi∈τfor every {Gi:i∈J} ⊆ τ The pair (X, τ) is called a neutrosophic topological space. The elements of τare called neutrosophic open sets. The complement of a neutrosophic open set is called a neutrosophic closed set. Definition 2.5. [14] Let (X, τ) be neutrosophic topological space and A=⟨x, µA(x), σA(x), γA(x)⟩be a neutrosophic set in X. Then the neutrosophic closure and neutrosophic interior of Aare defined by (1) Ncl(A) = ∩{K:Kis a neutrosophic closed set in Xand A⊆K} (2) Nint(A) = ∪{G:Gis a neutrosophic open set in Xand G⊆A}. It can be also shown that Ncl(A) is neutrosophic closed set and Nint(A) is a neutrosophic open set in X. Further, a) Ais a neutrosophic open set if and only if A=Nint(A) . b) Ais a neutrosophic closed set if and only if A=Ncl(A). Definition 2.6. Let (X, τ) be a neutrosophic topological space and let Ube a neutrosophic set in X. Then, Uis called a (1) neutrosophic α-open set [15] iff U⊆Nint(Ncl(Nint(U))) (2) neutrosophic semi-open set [14] iff U⊆Ncl(Nint(U)) (3) neutrosophic pre-open set [16] iff U⊆Nint(Ncl(U)) Definition 2.7. [5] A neutrosophic set Uin a neutrosophic topological space Xis called a neutrosophic b-open set if U⊆Nint(Ncl(U)) ∪Ncl(Nint(U)). Further, Uis called a neutrosophic b-closed set if U⊇Nint(Ncl(U)) ∩Ncl(Nint(U)) Sujeewa Malkanthi and Panchadcharam Elango, Study of Neutrosophic b-continuous maps in Neutrosophic Topological Spaces Neutrosophic Sets and Systems, Vol. 97, 2026 593 Example 2.1. Let X={a, b}and A=⟨(0.3,0.6,0.4),(0.6,0.3,0.1)⟩, B=⟨(0.2,0.5,0.7),(0.5,0.2,0.4)⟩be neutrosophic sets in X. Then, τ={0N, A, B, 1N}is a neutrosophic topological space on X. Now,A1=⟨(0.4,0.6,0.4),(0.8,0.3,0.4)⟩is a neutrosophic b-open set. Definition 2.8. [16] Let fbe a map from a neutrosophic topological space (X, τ) to a neutrosophic topological space (Y, σ). Then fis called a neutrosophic continuous map if the preimage of each neutrosophic open set in (Y, σ) is a neutrosophic open set in (X, τ). Definition 2.9. [1] Let fbe a map from a neutrosophic topological space (X, τ) to a neutrosophic topological space (Y, σ). Then fis called a neutrosophic semi-continuous map if f−1(B) is a neutrosophic semi-open set in Xfor every neutrosophic open set Bin Y. Definition 2.10. [1] Let fbe a map from a neutrosophic topological space (X, τ) to a neutrosophic topological space (Y, σ). Then fis called a neutrosophic pre-continuous map if f−1(B) is a neutrosophic pre-open set in Xfor every neutrosophic open set Bin Y. Definition 2.11. [1] Let fbe a map from a neutrosophic topological space (X, τ) to a neutrosophic topological space (Y, σ). Then fis called a neutrosophic α-continuous map if f−1(B) is a neutrosophic α-open set in Xfor every neutrosophic open set Bin Y. Definition 2.12. [9] A neutrosophic topological space Xis neutrosophic b-disconnected space if there exist neutrosophic b-open sets Aand Bin X, with A= 0N,B= 0Nsuch that A∪B= 1Nand A∩B= 0N. If Xis not a neutrosophic b-disconnected space, then it is said to be a neutrosophic b-connected space. Definition 2.13. [21] A neutrosophic topological space (X, τ) is said to be a neutrosophic b-compact space if each neutrosophic b-open cover of Xhas a finite sub-cover. 3. Results Lemma 3.1. Every neutrosophic α-open set is a neutrosophic b-open set in a neutrosophic topological space X. Proof. Let Abe a neutrosophic α-open set in a neutrosophic topological space X. Then A⊆ Nint[Ncl(Nint(A))] which implies that, A⊆Nint(Ncl(A)) ⊆Nint(Ncl(A))∪Ncl(Nint(A)). Thus Ais a neutrosophic b-open set. It was shown in [17] that every neutrosophic semi-open set is a neutrosophic b-open set and also every neutrosophic pre-open set is a neutrosophic b-open set. Sujeewa Malkanthi and Panchadcharam Elango, Study of Neutrosophic b-continuous maps in Neutrosophic Topological Spaces Neutrosophic Sets and Systems, Vol. 97, 2026 594 Definition 3.1. Let fbe a map from a neutrosophic topological space (X, τ) to a neutrosophic topological space (Y, σ). Then fis said to be a neutrosophic b-continuous map if f−1(B) is a neutrosophic b-open set in Xfor every neutrosophic open set Bin Y. Example 3.1. Let X={a, b, c}and Y={a, b, c}. Define neutrosophic sets A, B and C as follows: A=⟨(0.5,0.5,0.5),(0.4,0.5,0.5),(0.4,0.5,0.5)⟩,B= ⟨(0.3,0.4,0.4),(0.7,0.5,0.5),(0.3,0.4,0.4)⟩, and C=⟨(0.3,0.4,0.4),(0.3,0.3,0.3),(0.4,0.5,0.5)⟩. Let τ={0N, A, B, 1N}be a neutrosophic topological space on Xand σ={0N, C, 1N}be a neutrosophic topological space on Y. Define f: (X, τ)−→ (Y, σ)by f(a) = c, f(b) = a, f(c) = b. Then, fis a neutrosophic b-continuous map. Theorem 3.2. A mapping f: (X, τ)−→ (Y, σ)is neutrosophic b-continuous if and only if the inverse image of every neutrosophic closed set in Yis a neutrosophic b-closed set in X. Proof. Suppose that f: (X, τ)−→ (Y, σ) be a neutrosophic b-continuous map and let Bbe a neutrosophic closed set in Y. Then (Y−B) is neutrosophic open set in Y. Since fis neutrosophic b-continuous map, f−1(Y−B) = X−f−1(B) is a neutrosophic b-open set in X. Hence f−1(B) is a neutrosophic b-closed set in X. Conversely suppose that f−1(B) is a neutrosophic b-closed set in Xfor every neutrosophic closed set Bin Y. Let Ube a neutrosophic open set in Y.Y−Uis neutrosophic closed in Y and by hypothesis, f−1(Y−U) is a neutrosophic closed set in (X, τ) and therefore, f−1(U) is a neutrosophic b-open set in (X, τ). Thus fis a neutrosophic b-continuous map. Proposition 3.1. Every neutrosophic continuous map is a neutrosophic b-continuous map. Proof. Let (X, τ) and (Y, σ) be two neutrosophic topological spaces. Let f: (X, τ)−→ (Y, σ) be a neutrosophic continuous map and let Ube a neutrosophic open set in Y. Since fis neutrosophic continuous, f−1(U) is neutrosophic open set in X. Since every neutrosophic open set is neutrosophic b-open set, f−1(U) is a neutrosophic b-open set in X. Thus fis a neutrosophic b-continuous map. Converse of the above proposition need not be true in general. This can be shown in the following example: Example 3.3. Let X={a, b, c}and Y={a, b, c}. Define Neutrosophic sets A,B and C as follows: A=⟨(0.5,0.5,0.5),(0.4,0.5,0.5),(0.4,0.5,0.5)⟩,B= ⟨(0.3,0.4,0.4),(0.7,0.5,0.5),(0.3,0.4,0.4)⟩, and C=⟨(0.3,0.4,0.4),(0.3,0.3,0.3),(0.4,0.5,0.5)⟩. Sujeewa Malkanthi and Panchadcharam Elango, Study of Neutrosophic b-continuous maps in Neutrosophic Topological Spaces Neutrosophic Sets and Systems, Vol. 97, 2026 595 Let τ={0N, A, B, 1N}be a neutrosophic topological space on Xand let σ={0N, C, 1N}is a neutrosophic topological space on Y. Define f: (X, τ)−→ (Y, σ)by f(a) = c, f(b) = a, f(c) = b. Then, clearly fis a neutrosophic b-continuous map; but not a neutrosophic continuous map. Proposition 3.2. Let fbe a map from a neutrosophic topological space (X, τ)to a neutrosophic topological space (Y, σ). If fis neutrosophic α-continuous map, then fis neutrosophic b-continuous map. Proof. Let Ube a neutrosophic open set in (Y, σ). Since fis neutrosophic α-continuous map, f−1(U) is neutrosophic α-open set in (X, τ). Then f−1(U) is neutrosophic b-open set in (X, τ). Hence fis a neutrosophic b-continuous map. Proposition 3.3. Let fbe a map from a neutrosophic topological space (X, τ)to a neutrosophic topological space (Y, σ). If fis neutrosophic semi-continuous map, then fis neutrosophic b-continuous map. Proof. Let Ube a neutrosophic open set in (Y, σ). Since fis neutrosophic semi-continuous map, f−1(U) is neutrosophic semi-open set in (X, τ). Then f−1(U) is neutrosophic b-open set in (X, τ) by lemma 3.1. Hence fis a neutrosophic b-continuous map. Proposition 3.4. Let fbe a map from a neutrosophic topological space (X, τ)to a neutrosophic topological space (Y, σ). If fis neutrosophic pre-continuous map, then fis neutrosophic b-continuous map. Proof. Let Ube a neutrosophic open set in (Y, σ). Since fis neutrosophic pre-continuous map, f−1(U) is neutrosophic pre-open set in (X, τ). Then, f−1(U) is neutrosophic b-open set in (X, τ). Hence fis a neutrosophic b-continuous map. Sujeewa Malkanthi and Panchadcharam Elango, Study of Neutrosophic b-continuous maps in Neutrosophic Topological Spaces Neutrosophic Sets and Systems, Vol. 97, 2026 596 Figure 1. An overview of neutrosophic semi-continuous, b-continuous, precontinuous, and α-continuous mappings. Theorem 3.4. Let (X, τ),(Y, σ)and (Z, η)be three neutrosophic topological spaces and let f: (X, τ)−→ (Y, σ)and g: (Y, σ)−→ (Z, η)be two mappings. If fis a neutrosophic b-continuous map and gis a neutrosophic continuous map, then g◦fis a neutrosophic bcontinuous map. Proof. Let Ube a neutrosophic open set in (Z, η). Since gis neutrosophic continuous map, g−1(U) is neutrosophic open set in (Y, σ). Now as fis neutrosophic b-continuous map, we get f−1(g−1(U)) is neutrosophic b-open set in (X, τ). So (g◦f)−1(U) is neutrosophic b-open set in (X, τ). Thus g◦fis a neutrosophic b-continuous map. If fis neutrosophic α-continuous map or neutrosophic pre-continuous map or neutrosophic semi-continuous map, Then also g◦fis a neutrosophic b-continuous map by proposition 3.3,3.4,3.5 and theorem 3.4. Definition 3.2. A map f: (X, τ)−→ (Y, σ) is said to be strongly neutrosophic b-continuous if f−1(B) is a neutrosophic b-open set in Xfor every neutrosophic b-open set in Y. Example 3.5. Let X={a, b, c}and Y={a, b, c}. Define neutrosophic sets A, B and C as follows : A=⟨(0.5,0.5,0.5),(0.4,0.5,0.5),(0.4,0.5,0.5)⟩,B= ⟨(0.3,0.4,0.4),(0.7,0.5,0.5),(0.3,0.4,0.4)⟩, and C=⟨(0.3,0.4,0.4),(0.3,0.3,0.3),(0.4,0.5,0.5)⟩. Let τ={0N, A, B, 1N}be a neutrosophic topological space on Xand σ={0N, C, 1Nbe a neutrosophic topological space on Y. Define f: (X, τ)−→ (Y, σ)by f(a) = c, f(b) = a, f(c) = b. Consider D=⟨(0.3,0.6,0.4),(0.6,0.3,0.1),(0.5,0.2,0.4)⟩be a neutrosophic b-open set in Y. Since f−1(D)is a neutrosophic b-open set in Y,fis strongly neutrosophic b-continuous map. Sujeewa Malkanthi and Panchadcharam Elango, Study of Neutrosophic b-continuous maps in Neutrosophic Topological Spaces Neutrosophic Sets and Systems, Vol. 97, 2026 597 Proposition 3.5. Every neutrosophic strongly b-continuous map is a neutrosophic bcontinuous map. Proof. Let (X, τ) and (Y, σ) be two neutrosophic topological spaces. Let f: (X, τ)−→ (Y, σ) be a neutrosophic b-continuous map and Ube a neutrosophic open set in Y. Then by theorem [21], Uis a neutrosophic b-open set in Y. Since fis strongly neutrosophic b-continuous, f−1(U) is neutrosophic b-open set in X. Thus fis a neutrosophic b-continuous map. Proposition 3.6. Let (X, τ),(Y, σ)and (Z, η)be three neutrosophic topological spaces. Let f: (X, τ)−→ (Y, σ)and g: (Y, σ)−→ (Z, η)be strongly neutrosophic b-continuous maps. Then the composition g◦fis also a strongly neutrosophic b-continuous map. Proof. Let Ube a neutrosophic b-open set in (Z, η). Since gis strongly neutrosophic bcontinuous map, g−1(U) is neutrosophic b-open set in (Y, σ). Now as fis also strongly neutrosophic b-continuous map, we get f−1(g−1(U)) is neutrosophic b-open set in (X, τ). So (g◦f)−1(U) is neutrosophic b-open set in (X, τ). Thus g◦fis strongly neutrosophic bcontinuous map. Theorem 3.6. Let (X, τ),(Y, σ)and (Z, η)be three neutrosophic topological spaces and let f: (X, τ)−→ (Y, σ)be a strongly neutrosophic b-continuous map and g: (Y, σ)−→ (Z, η)be a neutrosophic b-continuous map, then g◦fis a neutrosophic b-continuous map. Proof. Let Ube a neutrosophic open set in Z. Since gis neutrosophic b-continuous map, g−1(U) is neutrosophic b-open set in Y. Now as fis strongly neutrosophic b-continuous map, we get f−1(g−1(U)) is neutrosophic b-open set in X. So (g◦f)−1(U) is neutrosophic b-open set in X. Thus g◦fis neutrosophic b-continuous map. Theorem 3.7. The image of a neutrosophic b-connected space under a surjective neutrosophic b-continuous map is a neutrosophic b-connected space. Proof. Let f: (X, τ)→(Y, σ) be a neutrosophic b-continuous map from a neutrosophic topological space Xonto a neutrosophic topological space Yand Xbe a neutrosophic b-connected space. Suppose Y is neutrosophic b-disconnected. Then there exist two neutrosophic b-open sets Band Csuch that B∪C= 1Nand B∩C= 0N. Thus f−1(B)∪f−1(C) = f−1(B∪C) = f−1(1N)=1Nand f−1(B)∩f−1(C) = f−1(B∩C) = f−1(0N)=0Nwhich is a contradiction. Hence Yis a neutrosophic b-connected space. Sujeewa Malkanthi and Panchadcharam Elango, Study of Neutrosophic b-continuous maps in Neutrosophic Topological Spaces Neutrosophic Sets and Systems, Vol. 97, 2026 598 Theorem 3.8. The image of neutrosophic b-compact space under a surjective neutrosophic b-continuous map is neutrosophic compact space. Proof. Let f: (X, τ)→(Y, σ) be a neutrosophic b-continuous map from a neutrosophic topological space Xonto a neutrosophic topological space Y. Let {Gλ:λ∈∆}be an neutrosophic open cover for Y. Then {f−1(Gλ) : λ∈∆}is a neutrosophic b-open cover for X. Since Xis neutrosophic b-compact, this neutrosophic bopen cover has a finite sub cover {f−1(Gλ) : λ= 1, ..., n}. Since fis onto, {Gλ:λ=1,...,n}be a finite cub cover for Y. Thus Yis neutrosophic compact space. Theorem 3.9. Let (X, τ)and (Y, σ)be two neutrosophic topological spaces and let f:X→Y be a neutrosophic strongly b-continuous map. If the neutrosophic set Ais neutrosophic bcompact in (X, τ)then f(A)is neutrosophic b-compact in (Y, σ). Proof. Let B={Gλ:λ∈∆}be an neutrosophic open cover of f(A), where Gλ∈(Y, σ) for each λ∈∆. Then f(A)⊆ ∪λ∈∆⇒f−1(f(A)) ⊆f−1(∪λ∈∆Gλ)⇒f−1(f(A)) ⊆ ∪λ∈∆f−1(Gλ)⇒A⊆ ∪λ∈∆f−1(Gλ). Since Gλis b-open neutrosophic set in Y, so f−1(Gλ) is b-open neutrosophic set in Xas fis neutrosophic strongly b-continuous. Therefore C={f−1Gλ:λ∈∆}is an neutrosophic b-open cover of A. Since Ais neutrosophic bcompact, so Chas a finite neutrosophic b-open sub cover {f−1Gλ:λ= 1, ..., n}. Therefore A⊆ ∪n i=1f−1(Gλ)⇒f(A)⊆f(∪n i=1f−1G(λ)) ⇒f(A)⊆ ∪n i=1f(f−1(Gλ)) ⇒f(A)⊆ ∪n i=1Gλ. Thus the neutrosophic b-open cover Bof f(A) has a finite neutrosophic b-open sub cover. Therefore f(A) is neutrosophic b-compact in (Y, σ). Remark 3.1. Every neutrosophic b-compact space is a neutrosophic semi-compact space,neutrosophic pre-compact space and neutrosophic α-compact space. 4. Conclusions Here we studied some notions of neutrosophic b-open sets and further we discussed some new properties related to neutrosophic b-open sets. we discussed some relationship of neutrosophic b-open sets with neutrosophic semi-open,neutrosophic pre-open and neutrosophic α-open sets. We defined and investigated some properties of neutrosophic b-continuous and strongly neutrosophic b-continuous maps. Finally, we developed some properties of neutrosophic b-connectedness and neutrosophic b-compactness using neutrosophic b-continuous maps. Acknowledgments:The authors are highly thankful to the editor and referees for the valuable comments and suggestions for improving the quality of the paper. Conflicts of Interest:The authors declare that there is no conflict of interest in the research. Sujeewa Malkanthi and Panchadcharam Elango, Study of Neutrosophic b-continuous maps in Neutrosophic Topological Spaces Neutrosophic Sets and Systems, Vol. 97, 2026 599