The neutrality’s origin, character, and extent are studied in the Neutrosophic set. The neutrosophic set is an essential issue to research since it opens the door to a wide range of scientific and technological applications. The neutrosophic set can find its spot to research because the universe is filled with indeterminacy. Neutrosophic set is currently being developed to express uncertain, imprecise, partial, and inconsistent data. Truth membership function, indeterminacy membership function, and falsity membership function are used to express a neutrosophic set in order to address uncertainty. The neutrosophic set produces more rational conclusions in a variety of practical problems. The neutrosophic set displays inconsistencies in data and can solve real-world problems. We are directed to do our work in semi-continuous and almost continuous mapping on the basis of the neutrosophic set by observing these. Since we are going to study the properties of semi continuous and almost continuous mapping, we present the meaning of N∽ semi-open set, N∽ semi-closed set, N∽ regularly open set, N∽ regularly closed set, N∽ continuous mapping, N∽ open mapping, N∽ closed mapping, N∽ semi-continuous mapping, N∽ semi-open mapping, N∽ semi-closed mapping. Additionally, we attempt to demonstrate a portion of their properties and furthermore referred to some examples.
𝓝∽ regularly open set𝓝∽ regularly closed set𝓝∽ semi-continuous mapping𝓝∽ almost continuous mappingIntroduction
After Zadeh [1] created fuzzy set theory (FST), FST was used to define the idea of membership value and explain the concept of uncertainty. Many researchers attempted to apply FST to a variety of other fields of science and technology. Atanassov [2] expanded on the concept of fuzzy set theory and introduced the concept of degree of non-membership, as well as proposing intuitionistic fuzzy set theory (IFST). Chang [3] introduced fuzzy topology (FT), and Coker [4] generalized the concept of FT to intuitionistic fuzzy topology (IFT). Rosenfeld [5] introduced the concept of fuzzy groups and Foster [6] proposed the idea of fuzzy topological groups. Azad [7] went through fuzzy semi-continuity (FSC), fuzzy almost continuity (FAC), and fuzzy weakly continuity (FWC). Smarandache [8,9] suggested neutrosophic set theory (NST) by generalizing FST and IFST and valuing indeterminacy as a separate component. Many researchers have attempted to apply NST to a variety of scientific and technological fields. Kandil et al. [10] studied the fuzzy bitopological spaces. Mwchahary et al. [11] did their work in neutrosophic bitopological space. Neutrosophic topology was proposed by Salama et al. [12,13]. The semi-continuous mapping was investigated by Noiri [14] and the term almost continuous mappings were coined by Singal et al. [15]. The idea of fuzzy neutrosophic groups and a topological group of the neutrosophic set was studied by Sumathi et al. [16,17]. NST was used as a tool in a group discussion framework by Abdel-Basset et al. [18]. Abdel-Basset et al. [19] investigated the use of the base-worst technique to solve chain problems using a novel plithogenic model.
Motivations
In this current decade, neutrosophic environments are mainly interested by different fields of researchers. In Mathematics also much theoretical research has been observed in the sense of neutrosophic environment. It will be necessary to carry out more theoretical research to establish a general framework for decision-making and to define patterns for complex network conceiving and practical application. Salama et al. [13] studied neutrosophic closed set and neutrosophic continuous functions. The idea of almost continuous functions is done in 1968 [15] in topology. Similarly, the notion of fuzzy almost contra continuous and fuzzy almost contra α-continuous functions was discussed in [20]. Recently, Al-Omeri et al. [21,22] introduced and studied a number of the definitions of neutrosophic closed sets, neutrosophic mapping, and obtained several preservation properties and some characterizations about neutrosophic of connectedness and neutrosophic connectedness continuity. More recently, in [23–26] authors have given how a new trend of Neutrosophic theory is applicable in the field of Medicine and multimedia with a novel and powerful model. From the literature survey, it is noticed that exactly the properties of neutrosophic semi-continuous and almost continuous mapping are not done. To update this research gap, in this research article, we attempt to investigate the neutrosophic semi-continuous and almost continuous mapping and its properties. Also, we study properties of the neutrosophic semi-open set (NSOS), neutrosophic semi-closed set (NSCoS), neutrosophic regularly open set (NROS), neutrosophic regularly closed set (NRCoS), neutrosophic semi-continuous (NSC), and neutrosophic almost continuous mapping (NACM).
MethodologiesDefinition [8]
A neutrosophic set (NS) AN on X can be expressed as AN={<x∈X,TAN(x),IAN(x),FAN(x)>}, where T,I,F:X⟶]-0, 1+[. Note that 0≼TAN(x)+IAN(x)+FAN(x)≼3.
For a family {A}α of NSs of NTS (X,TXN), ⋓N∽Cl(Aα)≼N∽Cl(⋓(Aα)). In case B is a finite set, ⋓N∽Cl(Aα)≼N∽Cl(⋓(Aα)). Also, ⋓N∽Int(Aα)≼N∽Int(⋓(Aα)), where a subfamily B of (X,TXN) is said to be subbase for (X,TXN) if the collection of all intersections of members of B forms a base for (X,TXN).
3.8 Lemma
For a NS A of NTS (X,TXN), (a) 1−N∽Int(A)=N∽Cl(1−A), and (b) 1−N∽Cl(A)=N∽Int(1−A).
Prove is Straightforward.
3.9 Theorem
The statements below are equivalent:
A is a NSCoS,
Ac is a NSOS,
N∽Int(N∽Cl(A))≼A, and
N∽Cl(N∽Int(Ac))≽Ac.
Proof:
(i) and (ii) are equivalent follows from Lemma 3.8, since for a NS A of NTS (X,TXN) such that 1−N∽Int(A)=N∽Cl(1−A) and 1−N∽Cl(A)=N∽Int(1−A).
(i)⇒(iii). By definition ∃ a NCoS B such that N∽Int(B)≼A≼B and hence N∽Int(B)≼A≼N∽Cl(A)≼B. Since N∽Int(B) is the greatest NOS contained in B, we have N∽Int(N∽Cl(B))≼N∽Int(B)≼A.
(iii)⇒(i) follows by taking B=N∽Cl(A).
(ii)⇔(iv) can similarly be proved.
3.10 Theorem
Arbitrary union of NSOSs is a NSOS, and
Arbitrary intersection of NSCoSs is a NSCoS.
Proof:
(i) Let {Aα} be a collection of NSOSs of NTS (X,TXN). Then ∃ a Bα∈TXN such that Bα≼Aα≼N∽Cl(Bα), for each α. Thus, ⋒Bα≼⋓Aα≼⋓N∽Cl(Bα)≼N∽Cl(⋓(Bα)) [Lemma 3.7], and ⋓Bα∈TXN, this shows that ⋓Bα is a NSOS.
(ii) Let {Aα} be a collection of NSCoSs of NTS (X,TXN). Then ∃ a Bα∈TXN such that N∽Int(Bα)≼Aα≼Bα, for each α. Thus, N∽Int(⋒(Bα))≼⋒N∽Int(Bα)≼⋒Aα≼⋒Bα [Lemma 3.7], and ⋓Bα∈TXN, this shows that ⋒Bα is a NSCoS.
3.11 Remark
It is clear that every neutrosophic open set (NOS) (neutrosophic closed set (NCoS)) is a NSOS (NSCoS). The converse is false, it is seen in Example3.12. It also shows that the intersection (union) of any two NSOSs (NSCoSs) need not be a NSOS (NSCoS). Even the intersection (union) of a NSOS (NSCoS) with a NOS (NCoS) may fail to be a NSOS (NSCoS). It should be noted that the ordinary topological setting the intersection of a NSOS with an NOS is a NSOS.
Further, the closure of NOS is a NSOS and the interior of NCoS is a NSCoS.
3.12 Example
Let X={a,b} and A,B be neutrosophic subsets of X such that
A={<a(0.6,0.3,0.2)>,<b(0.5,0.2,0.3)>}
B={<a(0.5,0.4,0.3)>,<b(0.4,0.2,0.3)>}
Then, TXN={1XN,0XN,A,B,A⋓B,A⋒B} is a NTS on X.
Let P={<a(0.8,0.2,0.1)>,<b(0.7,0.2,0.3)>} be any neutrosophic set XN, then N∽Int(P)=⋓{G:Gisopenset,G≼P}=A⋓B=A and N∽Cl(P)=⋒{K≽P:KisclosedsetinTXN}=1XN. Therefore, P is a NSOS which is not a NOS and also by Theorem 3.9, Pc is a NSCoS which is not an NCoS.
3.13 Theorem
If (X,TXN) and (Y,TYN) are NTSs and X is product related to Y. Then the product A×B of a NSOS A of X and a NSOS B of Y is NSOS of the neutrosophic product space X×Y.
Proof:
Let P≼A≼N∽Cl(P) and Q≼B≼N∽Cl(Q), where P∈TXN and Q∈TYN. Then P×Q≼A×B≼N∽Cl(P)×N∽Cl(Q). For NSs P’s of X and Q’s of Y, we have
inf{P,Q}=min{infP,infQ},
inf{P×1XN}=(infP)×1XN, and
inf{1XN×Q}=1XN×(infQ).
It is sufficient to prove N∽Cl(A×B)≽N∽Cl(A)×N∽Cl(B). Let P∈TNX and Q∈TNY.
Then
N∽Cl(A×B)=inf{(P×Q)c|(P×Q)c≽A×B}=inf{(Pc×1XN)⋓(1XN×Qc)|(Pc×1XN)⋓(1XN×Qc)≽A×B}=inf{(Pc×1XN)⋓(1XN×Qc)|Pc≽AorQc≽B}=min[inf{(Pc×1XN)⋓(1XN×Qc)|Pc≽A},inf{(Pc×1XN)⋓(1XN×Qc)|Qc≽B}]
(i) The intersection of any two NROSs is a NROS, and
(ii) The union of any two NRCoSs is a NRCoS.
Proof:
(i) Let A1 and A2 be any two NROSs of NTS (X,TXN). Since A1⋒A2 is NOS [from Remark 3.17], we have A1⋒A2≼N∽Int(N∽Cl(A1⋒A2)). Now, N∽Int(N∽Cl(A1⋒A2))≼N∽Int(N∽Cl(A1))=A1 and N∽Int(N∽Cl(A1⋒A2))≼N∽Int(N∽Cl(A2))=A2 implies that N∽Int(N∽Cl(A1⋒A2))≼A1⋒A2. Hence the theorem.
(ii) Let A1 and A2 be any two NROSs of NTS (X,TXN). Since A1⋓A2 is NOS [from Remark 3.17], we have A1⋓A2≽N∽Cl(N∽Int(A1⋓A2)). Now, N∽Cl(N∽Int(A1⋓A2))≽N∽Cl(N∽Int(A1))=A1 and N∽Cl(N∽Int(A1⋓A2))≽N∽Cl(N∽Int(A2))=A2 implies that A1⋓A2≼N∽Cl(N∽Int(A1⋓A2)). Hence the theorem.
3.22 Theorem
(i) The closure of a NOS is NRCoS, and
(ii) The interior of a NCoS is NROS.
Proof:
(i) Let A be a NOS of NTS (X,TXN), clearly, N∽Int(N∽Cl(A))≼N∽Cl(A)⇒N∽Cl(N∽Int(N∽Cl(A)))≼N∽Cl(A). Now, A is NOS implies that A≼N∽Int(N∽Cl(A)) and hence N∽Cl(A)≼N∽Cl(N∽Int(N∽Cl(A))). Thus, N∽Cl(A) is NRCoS.
(ii) Let A be a NCoS of a NTS (X,TXN), clearly, N∽Cl(N∽Int(A))≽N∽Int(A)⇒N∽Int(N∽Cl(N∽Int(A)))≽N∽Int(A). Now, A is NCoS implies that A≽N∽Cl(N∽Int(A)) and hence N∽Int(A)≽N∽Int(N∽Cl(N∽Int(A))). Thus, N∽Int(A) is NROS.
3.23 Definition
Let ϕ:(X,TXN)⟶(Y,TYN) be a mapping from NTS (X,TXN) to another NTS (X,TYN), then ϕ is called a N∽ continuous mapping (NCM), if ϕ−1(A)∈TXN for each A∈TYN; or equivalently ϕ−1(B) is a NCoS of X for each NCoS B of Y.
3.24 Definition
Let ϕ:(X,TXN)⟶(Y,TXN) be a mapping from NTS (X,TXN) to another NTS (Y,TYN), then ϕ is said to be a N∽ open mapping (NOM), if ϕ(A)∈TYN for each A∈TXN.
3.25 Definition
Let ϕ:(X,TXN)⟶(Y,TYN) be a mapping from NTS (X,TXN) to another NTS (Y,TYN), then ϕ is said to be a N∽ closed mapping (NCoM) if ϕ(B) is a NCoS of Y for each NCoS B of X.
3.26 Definition
Let ϕ:(X,TXN)⟶(Y,TYN) be a mapping from NTS (X,TXN) to another NTS (X,TYN), then ϕ is said to be a N∽ semi-continuous mapping (NSCM), if ϕ−1(A) is a neutrosophic semi-open set of X, for each A∈TYN.
3.27 Definition
Let ϕ:(X,TXN)⟶(Y,TYN) be a mapping from NTS (X,TXN) to another NTS (X,TYN), then ϕ is said to be a N∽ semi-open mapping (NSOM), if ϕ(A) is a NSOS for each A∈TXN.
3.28 Definition
Let ϕ:(X,TXN)⟶(Y,TYN) be a mapping from NTS (X,TXN) to another NTS (X,TYN), then ϕ is said to be a N∽ semi-closed mapping (NSCoM), if ϕ(B) is a NSCoS for each NCoS B of X.
3.29 Remark
From Remark 3.11, a NCM (NOM, NCoM) is also a NSCM (NSOM, NSCoM). But the converse is not true.
3.30 Example
Let X={a,b}, Y={x,y}, and
A={<a(0.6,0.3,0.2)>,<b(0.5,0.2,0.3)>}
B={<x(0.5,0.4,0.3)>,<y(0.4,0.2,0.3)>},
C={<x(0.8,0.2,0.1)>,<y(0.7,0.2,0.3)>}.
Then TXN={0XN,1XN,A} and TYN={0XN,1XN,B,C} are NTSs on X and Y.
Let ϕ:(X,TXN)⟶(Y,TYN) be a mapping defined as ϕ(a)=y,ϕ(b)=x.
Then ϕ:(X,TXN)⟶(Y,TYN) is NSCM but not NCM.
3.31 Theorem
Let X1, X2, Y1 and Y2 be NTSs such that X1 is product related to X2. Then, the product ϕ1×ϕ2:X1×X2⟶Y1×Y2 of NSCMs ϕ1:X1⟶Y1 and ϕ2:X2⟶Y2 is NSCM.
Proof:
Let A≡⋓(Aα×Bβ), where Aα’s and Bβ’s are NOSs of Y1 and Y2, respectively, be a NOS of Y1×Y2. By using Lemma 3.3(i) and Lemma 3.5, we have
(ϕ1×ϕ2)−1(A)=⋓[ϕ1−1(Aα)×ϕ2−1(Aβ)].
That (ϕ1×ϕ2)−1(A) is a NSOS follows from Theorem 3.13 and Theorem 3.10(i).
3.32 Theorem
Let X, X1 and X2 be NTSs and pi:X1×X2⟶Xi(i=1,2) be the projection of X1×X2 onto Xi. Then, if ϕ:X⟶X1×X2 is a NSCM, piϕ is also NSCM.
Proof:
For a NOS A of Xi, we have (piϕ)−1(A)=ϕ−1(pi−1(A)). That pi is a NCM and ϕ is a NSCM imply that (piϕ)−1(A) is a NSOS of X.
3.33 Theorem
Let ϕ:X⟶Y be a mapping from NTS X to another NTS Y. Then if the graph ψ:X⟶X×Y of ϕ is NSCM, then ϕ is also NSCM.
Proof:
From Lemma 3.6, ϕ−1(A)=1XN⋒ϕ−1(A)=ψ−1(1XN×A), for each NOS A of Y. Since ψ is a NSCM and 1XN×A is a NOS X×Y, ϕ−1(A) is a NSOS of X and hence ϕ is a NSCM.
3.34 Remark
The converse of Theorem 3.33 is not true.
3.35 Definition
A mapping ϕ:(X,TXN)⟶(Y,TYN) from NTS X to another NTS Y is said to be a N∽ almost continuous mapping (NACM), if ϕ−1(A)∈TXN for each neutrosophic regularly open set A of Y.
3.36 Theorem
Let ϕ:(X,TXN)⟶(Y,TYN) be a mapping. Then the statements below are equivalent:
ϕ is a NACM,
ϕ−1(F) is a NCoS, for each NRCoS F of Y,
ϕ−1(A)≼N∽Int(ϕ−1(N∽Int(N∽Cl(A))), for each NOS A of Y,
N∽Cl(ϕ−1(N∽Cl(N∽Int(F))))≼ϕ−1(F), for each NCoS F of Y.
Proof:
Consider that ϕ−1(Ac)=(ϕ−1(A))c, for any NS A of Y, (a) ⇔ (b) follows from Theorem 3.16.
(a) ⇒ (c). Since A is a NOS of Y, A≼N∽Int(Cl(A)) and hence ϕ−1(A)≼ϕ−1(N∽Int(N∽Cl(A))). From Theorem 3.22(ii), N∽Int(N∽Cl(A)) is a NROS of Y, hence ϕ−1(N∽Int(N∽Cl(A))) is a NOS of X. Thus, ϕ−1(A)≼ϕ−1(N∽Int(N∽Cl(A)))=N∽Int(ϕ−1(N∽Int(N∽Cl(A)))).
(c) ⇒ (a). Let A be a NROS of Y, then we have ϕ−1(A)≼N∽Int(ϕ−1(N∽Int(N∽Cl(A))))=N∽Int(ϕ−1(A)). Thus, have ϕ−1(A)=N∽Int(ϕ−1(A)). This shows that ϕ−1(A) is a NOS of X.
(b) ⇔ (d) similarly can be proved.
3.37 Remark
Clearly, a NCM is NACM. But the converse needs not be true.
3.38 Example
Let X={a,b}, Y={x,y}, and
A={<a(0.6,0.5,0.3)>,<b(0.4,0.5,0.5)>}
B={<a(0.2,0.5,0.7)>,<b(0.4,0.5,0.5)>},
C={<x(0.6,0.5,0.3)>,<y(0.4,0.5,0.5)>},
D={<x(0.2,0.5,0.7)>,<y(0.4,0.5,0.5)>},
E={<x(0.2,0.5,0.5)>,<y(0.3,0.5,0.7)>}.
Then TXN={0XN,1XN,A,B} and TYN={0XN,1XN,C,D,E} are NTSs on X and Y.
Now, let ϕ:(X,TXN)⟶(Y,TYN) be a mapping defined as ϕ(a)=y,ϕ(b)=x and clearly ϕ is NACM.
Here, 0XN,1XN,C,D are open sets in TYN but ϕ−1(E) is not open set in TXN and hence NACM is not NCM.
3.39 Theorem
N∽ semi-continuity and N∽ almost continuity are independent notions.
3.40 Definition
A NTS (X,TXN) is said to be a N∽ semi-regularly space (NSRS) iff the collection of all NROSs of X forms a base for NT TXN.
3.41 Theorem
Let ϕ:(X,TXN)⟶(Y,TYN) be a mapping from NTS X to a NSRS Y. Then ϕ is NACM iff ϕ is NCM.
Proof:
From Remark 3.37, it suffices to prove that if ϕ is NACM then it is NCM. Let A∈TNY, then A=⋓Aα, where Aα’s are NROSs of Y. Now, from Lemma 3.3(i), 3.7 and Theorem 3.36(c), we get
Let X, X1 and X2 be NTSs and pi:X1×X2⟶Xi(i=1,2) be the projection of X1×X2 onto Xi. Then if ϕ:X⟶X1×X2 is a NACM, piϕ is also a NACM.
Proof:
Since pi is NCM Definition 3.23, for any NS A of Xi, we have (i) N∽Cl(pi−1(A))≼pi−1(N∽Cl(A)) and (ii) N∽Int(pi−1(A))≽pi−1(N∽Int(A)). Again, since (i) each pi is a NOM, and (ii) for any NS A of Xi (a) A≼pi−1pi(A), and (b) pi−1pi(A)≼A, we have pi(N∽Int(pi−1(A)))≼pipi−1(A)≼A and hence pi(N∽Int(pi−1(A)))≼N∽Int(A). Thus, N∽Int(pi−1(A))≼pi−1pi(N∽Int(pi−1(A)))≼(pi−1(N∽Int(A)) establishes that N∽Int(pi−1(A))≼pi−1(N∽Int(A)). Now, for any NOS A of Xi,
The truth membership function, indeterminacy membership function, and falsity membership function are all employed in the Neutrosophic Set to overcome uncertainty. First, we developed the definitions of N∽ semi-open set, N∽ semi-closed, N∽ regularly open set, N∽ regularly closed set, N∽ continuous mapping, N∽ open mapping, N∽ closed mapping, N∽ semi-continuous mapping, N∽ semi-open mapping, N∽ semi-closed mapping, set in order to propose the definition of N∽ almost continuous mapping. Some properties of N∽ almost continuous mapping have been demonstrated. We expect that our study may spark some new ideas for the construction of the neutrosophic almost continuous mapping. It will be necessary to carry out more theoretical research to establish a general framework for decision-making and to define patterns for complex network conceiving and practical application. In the future, we would like to extend our work to study some properties in the neutrosophic semi and almost topological group with the help of the neutrosophic semi and almost continuous mapping.
Funding Statement: The authors received no specific funding for this study.
Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.
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