Ratio Mathematica Volume 46, 2023 Properties of nano generalized pre c-interior in a nano topological space. Padmavathi P* Abstract 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. 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.com 1Received 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 is published under the CC-BY licence agreement. 194 P.Padmavathi 1 Introduction The concept of generalized-semi closed sets to characterize the S-normality axiom was introduced by S.P.Arya et.al. The semi-generalized mappings and generalized-semi mappings were studied. In 2013 , Govindappa Navalagi investi- gated some of the regularity axioms, normality axioms and continuous functions through gs-open sets and sg-open sets. Also, Govindappa Navalagi continued the 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 various forms of nano sets, their closures and interiors and their homeomorphisms Lellis Thivagar 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 this paper, I have introduced the properties of nano generalized pre c-interior in a nano topological space. 2 Preliminaries Definition 2.1. [3] Let = be a non empty finite set of objects called the universe and < be an equivalence relation on = named as indiscernibility relation. Then = is divided into disjoint equivalence classes. Elements belonging to the same equivalence 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, which can be for certain classified as ℵ with respect to < and it is denoted by γ<(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 which can 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 can be classified neither as ℵ nor as not-X with respect to < and it is denoted by B<(ℵ).B<(ℵ) = τ<(ℵ)−γ<(ℵ). Proposition 2.1. [3] If (=,<) is an approximation space and ℵ,Y ⊆=,then 1. γ<(ℵ) ⊆ℵ⊆ τ<(ℵ) 2. γ<(φ) = τ<(ℵ) = φ 195 Properties of nano generalized pre c-interior in a nano topological space 3. 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)]c 10. 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 = and r<(ℵ) = {=,φ,γ<(ℵ),τ<(ℵ),B<(ℵ)} where ℵ ⊆ =. Then r<(ℵ) satisfies the following 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 in r<(ℵ).Then r<(ℵ) is a topology on = called the nano topology on = with respect 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 the nano 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 ⊆=, then 1. The nano interior of A is defined as the union of all nano open subsets contained in A and is denoted by Nint(A). That is Nint(A) is the largest nano open subset of A . 2. The nano closure of A is defined as the intersection of all nano closed sets containing A and is denoted by Ncl(A). That is Ncl(A) is the smallest nano closed set containing A. Definition 2.4. [2] A subset A of a nano topological space (=,r<(ℵ)) is called a nano generalized pre c-closed set (briefly Ngpc−closed set) if Npcl(A) ⊆ G whenever A ⊆ G and C is nano c-set. The complement of a Ngpc−closed set is called Ngpc−open set. 196 P.Padmavathi Definition 2.5. [2] The Nano generalized pre c-interior of a set A in (=,r<(ℵ)) is defined as the union of all Ngpc−open sets of U contained in A and it is denoted by 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 is denoted by Ngpc − cl(A). That is Ngpc − cl(A) is the smallest Ngpc−closed superset 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)). Then 1. 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 are true. 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] If A is a subset of (=,r<(ℵ)), then Nint(A) ⊂ Ngpc−int(A). Theorem 2.4. [2] For the subsets A and B of =, the following statements are true. 1. =−Ngpc− cl(A) ⊂ Ngpc− cl(=−A). 2. If A is Ngpc−closed then Ngpc−cl(A)−Ngpc−cl(B) ⊂ Ngpc−cl(A− B). 197 Properties of nano generalized pre c-interior in a nano topological space 3 Properties of nano generalized pre c-interior In 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 some of their properties are derived. Definition 3.1. The nano generalized pre c-border of a set A in (=,r<(ℵ)) is defined 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)) is defined 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 with respect 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). 198 P.Padmavathi 2. 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 contradiction to (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). Then Ngpc−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)) 199 Properties of nano generalized pre c-interior in a nano topological space 8. 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)) By remark (2.1), Ngpc−ext(A) is Ngpc−open. 5. Ngpc−ext(A) = Ngpc− int(=−A) = =−Ngpc−cl(A). (from (ii) of theorem (2.4)). 6. By remark (2.1), A is Ngpc−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). Hence Ngpc−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). 200 P.Padmavathi 11. 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). Hence Ngpc−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-closed sets. International Journal of Advanced Science and Technology, 29(3s):194 – 2021, 2020. [3] Z.Pawlak. Rough sets, Theoretical Aspects of Reasoning about Data. Kluwer Academic Publishers, Boston, 1991. 201