The aim of this paper is to introduce and study the properties the nano generalized pre c- interior of a set such as nano generalized pre c-border and nano generalized pre c-exterior in a nano topological space

Padmavathi, P

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Ratio Mathematica Volume 46, 2023Properties of nano generalized prec-interior in a nano topological space.Padmavathi P*AbstractThe aim of this paper is to introduce and study the properties the nanogeneralized pre c- interior of a set such as nano generalized prec-border and nano generalized pre c-exterior in a nano topologicalspace.Keywords:Nano generalized pre c-border, Nano generalized pre c-exterior.2020 AMS subject classifications: 06F20, 06F15, 20Cxx. 1*Department of Mathematics, Sri G.V.G Visalakshi College for Women (Autonomous), Udu-malpet, Tamilnadu, India. padmasathees74@gmail.com1Received on September 15, 2022. Accepted on March, 2023. Published on March 20, 2023. DOI:10.23755/rm.v46i0.1075. ISSN: 1592-7415. eISSN: 2282-8214. ©P.Padmavathi. This paper ispublished under the CC-BY licence agreement.194P.Padmavathi1 IntroductionThe concept of generalized-semi closed sets to characterize the S-normalityaxiom was introduced by S.P.Arya et.al. The semi-generalized mappings andgeneralized-semi mappings were studied. In 2013 , Govindappa Navalagi investi-gated some of the regularity axioms, normality axioms and continuous functionsthrough gs-open sets and sg-open sets. Also, Govindappa Navalagi continuedthe study of gs-continuous and sg-continuous functions to introduce the new no-tions like generalized semiclosure and generalized semi-interior operators. Lel-lis Thivagar [1] obtained the notion of nano topology and he studied the variousforms of nano sets, their closures and interiors and their homeomorphisms LellisThivagar et al introduced nano topological space with respect to a subset of a Uni-verse which is defined in terms of approximations and boundary region. In thispaper, I have introduced the properties of nano generalized pre c-interior in a nanotopological space.2 PreliminariesDefinition 2.1. [3] Let = be a non empty finite set of objects called the universeand < be an equivalence relation on = named as indiscernibility relation. Then= is divided into disjoint equivalence classes. Elements belonging to the sameequivalence class are said to be indiscernible with one another. The pair (=,<)is said to be approximation space. Let ℵ ⊆ =. Then(i) The lower approximation of ℵ with respect to < is the set of all objects, whichcan be for certain classified as ℵ with respect to < and it is denoted byγ<(ℵ). γ<(ℵ) = =x∈=<(x) : <(x) ⊆ ℵ by γ<(ℵ).where <(ℵ) denotes the equivalence class determined.(ii) The upper approximation of ℵ with respect to < is the set of all objects whichcan be possibly classified as ℵ with respect to < and it is denoted by τ<(ℵ).τ<(ℵ) = =x∈=<(x) : <(x)⋂ℵ 6= 0.(iii) The boundary region of ℵ with respect to < is the set of all objects which canbe classified neither as ℵ nor as not-X with respect to < and it is denotedby B<(ℵ).B<(ℵ) = τ<(ℵ)− γ<(ℵ).Proposition 2.1. [3] If (=,<) is an approximation space and ℵ, Y ⊆ =,then1. γ<(ℵ) ⊆ ℵ ⊆ τ<(ℵ)2. γ<(φ) = τ<(ℵ) = φ195Properties of nano generalized pre c-interior in a nano topological space3. γ<(U) = τ<(=) = =4. τ<(ℵ ∪ Y ) = τ<(ℵ) ∪ τ<(Y )5. τ<(ℵ ∩ Y ) ⊆ τ<(ℵ) ∩ τ<(Y )6. γ<(ℵ ∪ Y ) ⊇ γ<(ℵ) ∪ γ<(Y )7. γ<(ℵ ∩ Y ) = γ<(ℵ) ∩ γ<(Y )8. γ<(ℵ) ⊆ γ<(Y )andτ<(ℵ) ⊆ τ<(Y ), whenever ℵ ⊆ Y .9. τ<(ℵc) = [γ<(ℵ)]candγ<(ℵc) = [τ<(ℵ)]c10. τ<[τ<(ℵ)] = γ<[τ<(ℵ)] = τ<(ℵ)11. γ<[γ<(ℵ)] = τ<[γ<(ℵ)] = γ<(ℵ)Definition 2.2. [1] Let = be the universe, < be an equivalence relation on = andr<(ℵ) = {=, φ, γ<(ℵ), τ<(ℵ), B<(ℵ)} where ℵ ⊆ =. Then r<(ℵ) satisfies thefollowing axioms.1. = and φ ∈ r<(ℵ).2. The union of all the elements of any sub-collection of r<(ℵ) is in r<(ℵ).3. The intersection of the elements of any finite sub collection of r<(ℵ) is inr<(ℵ).Then r<(ℵ) is a topology on = called the nano topology on = withrespect to ℵ. The elements of r<(ℵ) are called as nano open sets in = and(=, r<(ℵ)) is called as a nano topological space. The complement of thenano open sets are called nano closed sets.Definition 2.3. [1] If (=, r<(ℵ)) is a nano topological space with respect to ℵ,where ℵ ⊆ = and if A ⊆ =, then1. The nano interior of A is defined as the union of all nano open subsetscontained in A and is denoted by Nint(A). That is Nint(A) is the largestnano open subset of A .2. The nano closure of A is defined as the intersection of all nano closed setscontaining A and is denoted by Ncl(A). That is Ncl(A) is the smallestnano closed set containing A.Definition 2.4. [2] A subset A of a nano topological space (=, r<(ℵ)) is calleda nano generalized pre c-closed set (briefly Ngpc−closed set) if Npcl(A) ⊆ Gwhenever A ⊆ G and C is nano c-set.The complement of a Ngpc−closed set is called Ngpc−open set.196P.PadmavathiDefinition 2.5. [2] The Nano generalized pre c-interior of a setA in (=, r<(ℵ)) isdefined as the union of all Ngpc−open sets of U contained in A and it is denotedby Ngpc− int(A). That is Ngpc− int(A) is the largest Ngpc−open subset of A.Definition 2.6. [2] The Nano generalized pre c-closure of a set A in (=, r<(ℵ))is defined as the intersection of all Ngpc−closed sets of U containing A and it isdenoted by Ngpc − cl(A). That is Ngpc − cl(A) is the smallest Ngpc−closedsuperset of A in IU .Remark 2.1. [2]1. A subset A of (=, r<(ℵ)) is Ngpc−open if and ony if Ngpc− int(A) = A.2. A subset A of (=, r<(ℵ)) is Ngpc−closed if and only if Ngpc− cl(A) = A.Theorem 2.1. [2] Let A and B be subsets of (=, r<(ℵ)). Then1. Ngpc− int(=) = = and Ngpc− int(φ) = φ.2. Ngpc− int(A) ⊂ A.3. If B is any Ngpc−open set contained in A, then B ⊂ Ngpc− int(A).4. If A ⊂ B then Ngpc− int(A) ⊆ Ngpc− int(B).5. Ngpc− int(Ngpc− int(A)) = Ngpc− int(A).Theorem 2.2. [2] If A and B are subsets of =, then the following statements aretrue.1. Ngpc− int(A) ∪Ngpc− int(B) ⊂ Ngpc− int(A ∪B).2. Ngpc− int(A ∩B) = Ngpc− int(A) ∩Ngpc− int(B).Theorem 2.3. [2] IfA is a subset of (=, r<(ℵ)), thenNint(A) ⊂ Ngpc−int(A).Theorem 2.4. [2] For the subsets A and B of =, the following statements aretrue.1. =−Ngpc− cl(A) ⊂ Ngpc− cl(=− A).2. If A is Ngpc−closed thenNgpc−cl(A)−Ngpc−cl(B) ⊂ Ngpc−cl(A−B).197Properties of nano generalized pre c-interior in a nano topological space3 Properties of nano generalized pre c-interiorIn this section the nano generalized pre c-border and nano generalized pre c-exterior of a set are defined in terms of nano generalized pre c-interior and someof their properties are derived.Definition 3.1. The nano generalized pre c-border of a set A in (=, r<(ℵ)) isdefined as A−Ngpc− int(A) and it is denoted by Ngpc−Bd(A).Definition 3.2. The nano generalized pre c-exterior of a set A in (=, r<(ℵ)) isdefined as Ngpc− int(=− A) and it is denoted by Ngpc− ext(A).Example 3.1. Let = = {a, b, c, d} with =/< = {{a} , {b} , {c, d}} and ℵ ={b, d}. Then r<(ℵ) = {=, φ, {b} , {c, d} , {b, c, d}} is a nano topology on U withrespect to ℵ. The complement of r<(ℵ) is given by rC(ℵ) = {U, φ, {a} , {a, b} ,{a, c, d}. Ngpc−closed sets are {φ,=, {a} , {c} , {d} ,{a, b} , {a, c} , {a, d} , {a, b, c} , {a, b, d} , {a, c, d}. Ngpc−open sets are φ,=,{b} , {c} , {d} , {b, c} , {b, d} , {a, b, c} , {a, b, d} , {b, c, d}.Here Ngpc − int({a}) = φ, Ngpc − int({b}) = {b}, Ngpc − int({a, c, d}) ={c, d}, Ngpc− int({c, d}) = {c, d} and Ngpc− int({a, b, d}) = {a, b, d}.Then Ngpc−Bd({a}) = {a} , Ngpc−Bd({b}) = φ ,Ngpc−Bd({a, b, d}) = φ and Ngpc−Bd({a, c, d}) = {a}.Ngpc− ext({a}) = {b, c, d}, Ngpc− ext({b}) = {c, d}, Ngpc− ext({a, b}) ={c, d} and Ngpc− ext({a, c, d}) = {b}.Theorem 3.1. For a subset A of = the following statements hold.1. Ngpc−Bd(φ) = Ngpc−Bd(=) = φ.2. Ngpc−Bd(A) ⊂ NBd(A).3. A = Ngpc− int(A) ∪Ngpc−Bd(A).4. Ngpc− int(A) ∩Ngpc−Bd(A) = φ.5. Ngpc− int(A) = A−Ngpc−Bd(A).6. Ngpc− int(Ngpc−Bd(A)) = Ngpc−Bd(Ngpc− int(A)) = φ.7. A is Ngpc−open if and only if Ngpc−Bd(A) = φ.8. Ngpc−Bd(Ngpc−Bd(A)) = Ngpc−Bd(A).Proof. 1. The proof is an immediate consequence of definition (3.1).198P.Padmavathi2. Let x ∈ Ngpc−Bd(A). ⇒ x ∈ A−Ngpc− int(A). By theorem (2.3),Nint(A) ⊂ Ngpc− int(A)⇒ A−Ngpc− int(A) ⊂ A−Nint(A).Hence x ∈ A−Ngpc− int(A)⇒ x ∈ A−Nint(A).⇒ x ∈ NBd(A). Therefore Ngpc−Bd(A) ⊂ NBd(A).3. Ngpc−int(A)∪Ngpc−Bd(A) = Ngpc−int(A)∪(A−Ngpc−int(A)) =A.4. Ngpc−int(A)∩Ngpc−Bd(A) = Ngpc−int(A)∩(A−Ngpc−int(A)) =φ.5. The proof directly follows from definition (3.1).6. Let x ∈ Ngpc− int(Ngpc−Bd(A)). Then x ∈ Ngpc−Bd(A) as Ngpc−Bd(A) ⊂ A.Also x ∈ Ngpc− int(Ngpc−Bd(A)) ⊂ Ngpc− int(A)R.Therefore x ∈ Ngpc − int(A) ∩ Ngpc − Bd(A) which is a contradictionto (d).Hence Ngpc− int(Ngpc−Bd(A)) = φ.7. By result (2.8), A is Ngpc− open⇔ Ngpc− int(A) = A⇔ A−Ngpc−int(A) = φ⇔ Ngpc−Bd(A) = φ. (by definition (3.1))8. In definition (3.1) let A = Ngpc−Bd(A).ThenNgpc−Bd(Ngpc−Bd(A)) = Ngpc−Bd(A)−Ngpc−int(Ngpc−Bd(A)) = Ngpc−Bd(A)− φ= Ngpc−Bd(A). (Using (6)).Theorem 3.2. For the subsets A and B of = the following statements hold.1. Ngpc− ext(φ) = = and Ngpc− ext(=) = φ.2. Next(A)Ngpc− ext(A).3. If A ⊂ B then Ngpc− ext(B) ⊂ Ngpc− ext(A).4. Ngpc− ext(A) is Ngpc−open.5. Ngpc− ext(A) = =−Ngpc− cl(A).6. A is Ngpc−closed if and only if Ngpc− ext(A) = =− A.7. Ngpc− ext(Ngpc− ext(A)) = Ngpc− int(Ngpc− cl(A))199Properties of nano generalized pre c-interior in a nano topological space8. Ngpc − ext(Ngpc − ext(A)) = Ngpc − ext(Ngpc − int(= − A)) =Ngpc− ext(=−Ngpc− cl(A)).9. Ngpc− ext(A ∪B) ⊂ Ngpc− ext(A) ∪Ngpc− ext(B).10. Ngpc− ext(A ∪B) = Ngpc− ext(A) ∩Ngpc− ext(B).11. Ngpc− ext(A) ∩Ngpc− ext(B) ⊂ Ngpc− ext(A ∩B).Proof. 1. The proof is immediate from definition (3.2).2. Next(A) ⊂ Ngpc− ext(A) follows from theorem (2.3).3. If A ⊂ B then =−B ⊂ =−A. By (iv) of theorem (2.2), Ngpc− int(=−B) ⊂ Ngpc− int(=− A). Hence Ngpc− ext(B) ⊂ Ngpc− ext(A).4. Consider Ngpc− int(Ngpc− ext(A)) = Ngpc− int(Ngpc− int(=−A))= Ngpc − int(= − A) = Ngpc − ext(A). (by (v) of theorem (2.9)) Byremark (2.1),Ngpc− ext(A) is Ngpc−open.5. Ngpc− ext(A) = Ngpc− int(=−A) = =−Ngpc− cl(A). (from (ii) oftheorem (2.4)).6. By remark (2.1), A isNgpc−closed⇔ Ngpc−cl(A) = A⇔ =−Ngpc−cl(A) = =− A−Ngpc− int(=− A)= =− A⇔ Ngpc− ext(A) = =− A.7. In definition let A = Ngpc− ext(A).Then Ngpc− ext(Ngpc− ext(A)) =Ngpc− int(=−Ngpc− ext(A)= Ngpc− int(Ngpc− cl(A)). (Using (5)).8. It follows from definition (3.2) and (5).9. We know that A ⊂ A ∪ B and B ⊂ A ∪ B. From (c) Ngpc − ext(A ∪B) ⊂ Ngpc− ext(B) and Ngpc− ext(A ∪B) ⊂ Ngpc− ext(B). HenceNgpc− ext(A ∪B) ⊂ Ngpc− ext(A) ⊂ Ngpc− ext(B).10. Ngpc− ext(A ∪B) = Ngpc− int(=− (A ∪B)). (by definition (3.2))= Ngpc− int((=− A) ∩ (=−B)).= Ngpc− int(=− A) ∩Ngpc− int(=−B). (by (ii) of theorem(2.4))= Ngpc− ext(A) ∩Ngpc− ext(B).Hence Ngpc− ext(A ∪B) = Ngpc− ext(A) ∩Ngpc− ext(B).200P.Padmavathi11. We know that A ∩ B ⊂ A and A ∩ B ⊂ B. From (c) Ngpc − ext(A) ⊂Ngpc − ext(A ∩ B) and Ngpc − ext(B) ⊂ Ngpc − ext(A ∩ B). HenceNgpc− ext(A) ∩Ngpc− ext(B) ⊂ Ngpc− ext(A ∩B).References[1] M.lellis Thivagar and C. Richard. Note on nano topological spaces (commu-nicated).[2] P.Padmavathi and R. Nithyakala. A note on nano generalized pre c-closedsets. International Journal of Advanced Science and Technology, 29(3s):194– 2021, 2020.[3] Z.Pawlak. Rough sets, Theoretical Aspects of Reasoning about Data. KluwerAcademic Publishers, Boston, 1991.201

Properties of Nano Generalized Pre c-Interior in a Nano Topological Space

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Properties of Nano Generalized Pre c-Interior in a Nano Topological Space

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