Ratio Mathematica On δ-open sets in ideal nano topological spaces N. Sekar ∗ R. Asokan † I. Rajasekaran ‡ Abstract Aim of this paper, the new notions of introduce δ-open sets in ideal nano topological spaces and investigate some of their properties. A comparison between these types of ideal nano continuity will be dis- cussed. Finally, we introduce application examples in ideal nano topological spaces. Keywords: nano-open (resp. n-open), semi-n-open (resp. ns-open) semi-nI-open, pre-nI-open and strong β-nI-open and nIδ-open. 2020 AMS subject classifications: 54B05, 54C60, 54E55. 1 ∗Research Scholar, Department of Mathematics, School of Mathematics, Madurai Kamaraj University, Madurai, Tamil Nadu, India; sekar.skrss@gmail.com. †Department of Mathematics, School of Mathematics, Madurai Kamaraj University, Madurai, Tamil Nadu, India; rasoka mku@yahoo.co.in. ‡Department of Mathematics, Tirunelveli Dakshina Mara Nadar Sangam College, T. Kallikulam-627 113, Tirunelveli District, Tamil Nadu, India; sekarmelakkal@gmail.com. 1Received on September 15, 2022. Accepted on December 15, 2022. Published on January 30, 2023. DOI: 10.23755/rm.v45i0.1013. ISSN: 1592-7415. eISSN: 2282-8214. c©The Authors. This paper is published under the CC-BY licence agreement. Volume 45, 2023 185 N. Sekar, R. Asokan and I. Rajasekaran 1 Introduction An ideal I (16) on a space (X,τ) is a non-empty collection of subsets of X which satisfies the following conditions. 1. A∈ I and B ⊂A imply B ∈ I and 2. A∈ I and B ∈ I imply A∪B ∈ I. Given a space (X,τ) with an ideal I on X if ℘(X) is the set of all subsets of X, a set operator (.)? : ℘(X) → ℘(X), called a local function of A with respect to τ and I is defined as follows: for A ⊂ X, A?(I,τ) = {x ∈ X : U ∩A /∈ I for every U ∈ τ(x)} where τ(x) = {U ∈ τ : x ∈ U} (2). The closure operator defined by cl?(A) = A∪A?(I,τ) (15) is a Kuratowski closure operator which generates a topology τ?(I,τ) called the ?-topology which is finer then τ. We will simply write A? for A?(I,τ) and τ? for τ?(I,τ). If I is an ideal on X, then (X,τ,I) is called an ideal topological space or an ideal space. Rajasekaran and Nethaji, introduced pre-nI-open sets and α-nI-open sets in the concept of ideal nano topological spaces. In this paper, we introduce the notions of δ-open sets in ideal nano topologi- cal spaces and investigate some of their properties. A comparison between these types of ideal nano continuity will be discussed. Finally, we introduce application examples in ideal nano topological spaces. 2 Preliminaries Definition 2.1. (9) Let U be a non-empty finite set of objects called the universe and R be an equivalence relation on U named as the indiscernibility relation. Elements belonging to the same equivalence class are said to be indiscernible with one another. The pair (U, R) is said to be the approximation space. Let X ⊆U. 1. The lower approximation of X with respect to R is the set of all objects, which can be for certain classified as X with respect to R and it is denoted by LR(X). That is, LR(X) = ⋃ x∈U{R(x) : R(x) ⊆ X}, where R(x) denotes the equiva- lence class determined by x. 2. The upper approximation of X with respect to R is the set of all objects, which can be possibly classified as X with respect to R and it is denoted by UR(X). That is, UR(X)= ⋃ x∈U{R(x) :R(x)∩X 6=φ}. 3. The boundary region of X with respect to R is the set of all objects, which can be classified neither as X nor as not - X with respect to R and it is denoted by BR(X). That is, BR(X)=UR(X)−LR(X). 186 On δ-open sets in ideal nano topological spaces Definition 2.2. (3) Let U be the universe, R be an equivalence relation on U and τR(X) = {U,φ,LR(X),UR(X),BR(X)} where X ⊆ U. Then R(X) satisfies the following axioms: 1. U and φ∈ τR(X), 2. The union of the elements of any sub collection of τR(X) is in τR(X), 3. The intersection of the elements of any finite subcollection of τR(X) is in τR(X). Thus τR(X) is a topology on U called the nano topology with respect to X and (U,τR(X)) is called the nano topological space. The elements of τR(X) are called nano-open sets (briefly n-open sets). The complement of a n-open set is called n-closed. In the rest of the paper, we denote a nano topological space by (U,N), where N = τR(X). The nano-interior and nano-closure of a subset O of U are denoted by In(O) and Cn(O), respectively. A nano topological space (U,N) with an ideal I on U is called (6) an ideal nano topological space and is denoted by (U,N ,I). Gn(x)= {Gn |x∈Gn,Gn ∈ N}, denotes (6) the family of nano open sets containing x. In future an ideal nano topological spaces (U,N ,I) is referred as a space. Definition 2.3. (6) Let (U,N ,I) be a space with an ideal I on U. Let (.)?n be a set operator from ℘(U) to ℘(U) (℘(U) is the set of all subsets of U). For a subset O ⊆ U, O?n(I,N) = {x ∈ U : Gn ∩ O /∈ I, for every Gn ∈ Gn(x)} is called the nano local function (briefly, n-local function) of A with respect to I and N . We will simply write O?n for O ? n(I,N). Theorem 2.1. (6) Let (U,N ,I) be a space and O and B be subsets of U. Then 1. O ⊆B ⇒O?n ⊆B ? n, 2. O?n =Cn(O ? n)⊆Cn(O) (O ? n is a n-closed subset of Cn(O)), 3. (O?n) ? n ⊆O ? n, 4. (O∪B)?n =O ? n ∪B ? n, 5. V ∈N ⇒V ∩O?n =V ∩ (V ∩O) ? n ⊆ (V ∩O) ? n, 6. J ∈ I ⇒ (O∪J)?n =O ? n =(O−J) ? n. Theorem 2.2. (6) Let (U,N ,I) be a space with an ideal I and O ⊆ O?n, then O?n =Cn(O ? n)=Cn(O). Definition 2.4. (8) A subset A of a space (U,N ,I) is n?-dense in itself (resp. n?-perfect and n?-closed) if O ⊆O?n (resp. O =O ? n, O ? n ⊆O). The complement of a n?-closed set is said to be n?-open. 187 N. Sekar, R. Asokan and I. Rajasekaran Definition 2.5. (5) A subset O of U in a nano topological space (U,N) is called nano-codense (briefly n-codense) if U −O is n-dense. Theorem 2.3. (6) Let (U,N ,I) be an ideal nano space. Then is I is n-codense ⇐⇒ O ⊆O? for every n-open set O. Definition 2.6. (6) Let (U,N ,I) be a space. The set operator C?n called a nano ?-closure is defined by C?n(O)=O∪O ? n for O ⊆U. It can be easily observed that C?n(O)⊆Cn(O). Theorem 2.4. (7) In a space (U,N ,I), if O and B are subsets of U, then the following results are true for the set operator n-cl?. 1. O ⊆C?n(O), 2. C?n(φ)=φ and C ? n(U)=U, 3. IfO ⊂B, then C?n(O)⊆C ? n(B), 4. C?n(O)∪C ? n(B)=C ? n(O∪B). 5. C?n(C ? n(O))=C ? n(O). Definition 2.7. A subset O of a nano space (U,N), is called a 1. nano pre-open (resp. np-open) set (3) if O ⊆ In(Cn(O)). 2. nano semi-open (resp. ns-open) set (3) if O ⊆Cn(In(O)). 3. nano ε-open (resp. nε-open) set (13) if In(Cn(O))⊆Cn(In(O)). 4. nano nowhere dense (resp. n-nowhere dense) (4)if In(Cn(O))=φ. Definition 2.8. A subset O of an ideal nano space (U,N ,I), is called a 1. nano pre-I-open (resp. pre-nI-open) (10) if O ⊆ In(C ? n(O)). 2. nano semi-I-open (resp. semi-nI-open) (10) if O ⊆C?n(In(O)). 3. nano α-I-open (resp. α-nI-open) (10) if O ⊆ In(C ? n(In(O))). 4. strongly nano β-I-open (resp. Sβ-nI-open) (11) if O ⊆C?n((In(C ? n(O))). Theorem 2.5. (10) In a nano space (U,N ,I), if O is α-nI-open, then O is semi- nI-open. Definition 2.9. (1) A function f : (U,N ,I)−→ (V,N ′) is said to be 1. α-nI-continuous if f−1(H) is α-nI-open set in (U,N ,I) for every n-open set H in (V,N ′). 2. pre-nI-continuous if f−1(H) is pre-nI-open set in (U,N ,I) for every n-open set H in (V,N ′). 3. semi-nI-continuous if f−1(H) is semi-nI-open set in (U,N ,I) for every n-open set H in (V,N ′). 188 On δ-open sets in ideal nano topological spaces 3 On δ-open sets in ideal nano space Definition 3.1. A subset O of an ideal nano space (U,N ,I), is called a nano Iδ-open (resp. nIδ-open) set if In(C ? n(O))⊆C ? n(In(O)). Example 3.1. Let U = {A1,A2,A3,A4} with U/R = {{A2},{A4},{A1,A3}} and X = {A3,A4}. Then the nano topology N = {φ,{A4},{A1,A3},{A1,A3,A4},U} and I = {φ,{A3}}. Clear that {φ,{A2},{A3},{A4},{A1,A3},{A2,A3},{A2,A4}, {A1,A2,A3},{A1,A3,A4},U} is nIδ-open. Proposition 3.1. Let (U,N ,I) be an ideal nano space. Then a subset of U is semi-nI-open ⇐⇒ if it is both nIδ-open and Sβ-nI-open. Proof. Necessity. Let O be a semi-nI-open, then we have O ⊆ C?n(In(O)) ⊆ C?n(In(C ? n(O))). This show that O is Sβ-nI-open. Moreover, In(C ? n(O)) ⊆ C ? n(O) ⊆ C ? n(C ? n(In(O))) = C ? n(In(O)). Therefore O is nIδ-open. Sufficiency. Let O be nIδ-open and Sβ-nI-open, then we have In(C?n(O))⊆ C?n(In(O)). Thus we obtain that C ? n(In(C ? n(O)))⊆C ? n(C ? n(In(O)))=C ? n(In(O)). Since O is Sβ-nI-open, we have O ⊆ C?n(In(C ? n(O))) ⊆ C ? n(In(O)) and O ⊆ C?n(In(O)). Hence O is semi-nI-open. Proposition 3.2. Let (U,N ,I) be an ideal nano space. Then a subset of U is α-nI-open ⇐⇒ if it is both nIδ-open and pre-nI-open. Proof. Necessity. Let O be a α-nI-open, since every α-nI-open set is semi-nI-open (10), by Proposition 3.1 O is a nIδ-open set. Now we prove that O ⊆ In(C ? n(O)). Since O is α-nI-open set, we have O ⊆ In(C ? n(In(O))) ⊆ In(C ? n(O)). Hence O is a pre-nI-open set. Sufficiency. Let O be a nIδ-open and pre-nI-open. Then we have In(C?n(O))⊆ C?n(In(O)) and hence In(C ? n(O)) ⊆ In(C ? n(In(O))). Since O is pre-nI-open, we have O ⊆ In(C ? n(O)). Therefore we obtain that O ⊆ In(C ? n(In(O))) and hence O ia α-nI-open. Remark 3.1. In (U,N ,I) ideal nano space. 1. Sβ-nI-open and nIδ-open are independent. 2. pre-nI-open and nIδ-open are independent. Example 3.2. In Example 3.1, 189 N. Sekar, R. Asokan and I. Rajasekaran 1. the set {A1} is Sβ-nI-open set but not nIδ-open. 2. the set {A2} is not Sβ-nI-open but nIδ-open. 3. the set {A1,A4} is pre-nI-open but not nIδ-open. 4. the set {A2,A3} is not pre-nI-open but nIδ-open. Proposition 3.3. Let O, P be subsets of an ideal nano space (U,N ,I). If O ⊆ P ⊆C?n(O) and O is nIδ-open, then P is nIδ-open. Proof. Suppose that O ⊆P ⊆C?n(O) and O is nIδ-open. Then, since O is nIδ-open, we have In(C ? n(O)) ⊆ C ? n(In(O)). Since, O ⊆ P , C ? n(In(O)) ⊆ C ? n(I(P)) and In(C ? n(O)) ⊆ C ? n(In(P)). Since P ⊆ C ? n(O), we have C ? n(P) ⊆ C ? n(C ? n(O)) = C?n(O) and In(C ? n(P)) ⊆ In(C ? n(O)). Therefore, we obtain that In(C ? n(P)) ⊆ C?n(In(P)). This show that P is a nIδ-open. Proposition 3.4. let O, P and Q be subsets of an ideal nano space (U,N ,I). If O is nIδ-open, then O = P ∪Q, where P is α-nI-open, In(C ? n(Q)) = φ and P ∩Q=φ. Proof. Suppose that O is nIδ-open. Then we have In(C ? n(O)) ⊆ C ? n(In(O)) and In(C ? n(O))⊆ In(C ? n(InO))). Now we have O = ( In(C ? n(O)) ∩ O ) ∪ ( O − In(C ? n(O)) ) . Now, we set P = In(C ? n(O))∩O and Q=O− In(C ? n(O)). We first show that P is α-nI-open, that is, P ⊆ In(C ? n(In(P)). Now we have In(C ? n(In(P)) = In(C ? n(In(In(C ? n(O))∩O))) = In(C ? n(In(C ? n(O))∩ In(O))) = In(C ? n(In(O))). Since O is nIδ-open, In(C ? n(In(O)))⊇ In(C ? n(O))⊇P and thus P is α-nI-open. Next we show that In(C ? n(Q))=φ. Since N ⊆N ?, C?n(K)⊆Cn(K) for any subset K of U. Therefore, we have In(C ? n(Q))= In(C ? n(O∩(U−In(C ? n(O)))))⊆ In(C ? n(O))∩In(C ? n(U−In(C ? n(O))))⊆ In(C ? n(O))∩In(Cn(U−In(C ? n(O))))⊆ In(C ? n(O))∩ (U − In(C ? n(O)))=φ. It is obvious that P ∩Q=(In(C ? n(O))∩O)∩ (O− In(C ? n(O))) =φ. Remark 3.2. In an ideal nano space (U,N ,I), nε-open and nIδ-open sets are independent. Example 3.3. In Example 3.1, 1. the set {A1,A2} is nε-open but not nIδ-open. 2. the set {A3} is not nε-open but nIδ-open. 190 On δ-open sets in ideal nano topological spaces Proposition 3.5. Let (U,N ,I) be an ideal nano space and O ⊆ U. If O is both nIδ-open and ns-closed, then O is a nε-open. Proof. Since O is ns-closed, In(Cn(O)) ⊆ O and hence In(Cn(O)) = In(O). Thus, In(Cn(O))⊆ In(O)⊆ In(C ? n(O))⊆C ? n(In(O))= In(O)∪(In(O)) ? n. (In(O)) ? n ⊆ Cn(In(O)) and hence we obtain In(Cn(O)) ⊆ Cn(In(O)). This show that O is nε-open. Proposition 3.6. Let (U,N ,I) be an ideal nano space. Let I = {φ} or I = H, where H is the ideal of n-nowhere dense. Then a subset O of U is nIδ-open ⇐⇒ O is a nε-open. Proof. 1. Let I = {φ}. Then for every subset O of U, O?n = Cn(O) and C ? n(O) = O∪O?n =O∪Cn(O)=Cn(O). Therefore, the statement holds obviously. 2. Let I =H, we have O?n =Cn(In(Cn(O))). First, let O be a nε-open. Then In(Cn(O)) ⊆ Cn(In(O)) and In(C ? n(O)) = In(O ? n∪O)⊆ In(Cn(O)∪O)= In(Cn(O))⊆Cn(In(O))=Cn(In(Cn(In(O))))= (In(O)) ? n ⊆C ? n(In(O)). Therefore, O is nIδ-open set. Next, let O be a nIδ-open set. Then In(C ? n(O))⊆ C?n(In(O)). We have In(Cn(O))= In(Cn(In(Cn(O))))= In(O ? n)⊆ In(O ? n ∪ O)= In(C ? n(O))⊆C ? n(In(O))= (In(O)) ? n∪In(O)=Cn(In(Cn(In(O))))∪ In(O)=Cn(In(O)). Therefore, nε-open. 4 On semi-δ-nI-continuous Definition 4.1. A function f : (U,N ,I)−→ (V,N ′) is called a 1. semi-δ-nI-continuous if every H ∈N ′, f−1(H)∈nIδ-open. 2. strong β-nI-continuous (resp. Sβ-nI-continuous) if every H ∈N ′, f−1(H)∈ Sβ-nI-open. Theorem 4.1. For a function f : (U,N ,I)−→ (V,N ′), the following properties are equivalent. 1. f is semi-nI-continuous. 2. f is Sβ-nI-continuous and semi-δ-nI-continuous. Proof. The proof is obvious by Proposition 3.1. 191 N. Sekar, R. Asokan and I. Rajasekaran Theorem 4.2. For a function f : (U,N ,I)−→ (V,N ′), the following properties are equivalent. 1. f is α-nI-continuous. 2. f is pre-nI-continuous and semi-nI-continuous. 3. f is pre-nI-continuous and semi-δ-nI-continuous. Proof. The proof is obvious by Proposition 3.1 and Proposition 3.2. 5 Conclusion Because of the spaces is stripped of the geometric form and its is used to mea- sure thing that are difficult to measure, such as intelligence, beauty and goodness. In this paper, different type of ideal nano continuity and ideal nano closed sets are introduced and studied. Also, we introduce an applications example in ideal nano topology. Some applications on them are given in some real life branches such as medicine and physics. References [1] V. Inthumathi, R. Abinprakash and M. Parveen Banu, Some weaker form of continuous and irresolute mapping in nano ideal topological spaces, Journal of new results in science, 8(1)(2019), 14-25. [2] K. Kuratowski, Topology, Vol I. Academic Press (New York) 1966. [3] M. Lellis Thivagar and Carmel Richard, On nano forms of weakly open sets, International Journal of Mathematics and Statistics Invention,1(1)(2013), 31-37. [4] M. Lellis Thivagar, Saeid Safari and V. Sutha Devi, On new class of contra continuity in nano topology, Available on researchgate in https://www.researchgate.net/ publication /315892547. [5] O. Nethaji, R. Asokan and I. Rajasekaran, New generalized classes of an ideal nano topological spaces, Bull. Int. Math. Virtual Inst., 9(3)(2019), 543- 552. [6] M. Parimala, T. Noiri and S. Jafari, New types of nano topological spaces via nano ideals (to appear). [7] M. Parimala and S. Jafari, On some new notions in nano ideal topological spaces, International Balkan Journal of Mathematics(IBJM), 1(3)(2018), 85- 92. 192 On δ-open sets in ideal nano topological spaces [8] M. Parimala, S. Jafari and S. Murali, Nano ideal generalized closed sets in nano ideal topological spaces, Annales Univ. Sci. Budapest., 60(2017), 3-11. [9] Z. Pawlak, Rough sets, International journal of computer and Information Sciences, 11(5)(1982), 341-356. [10] I. Rajasekaran and O. Nethaji, Simple forms of nano open sets in an ideal nano topological spaces, Journal of New Theory, 24(2018), 35-43. [11] I. Rajasekaran, Weak forms of strongly nano open sets in ideal nano topo- logical spaces, Asia Mathematika, 5(2)(2021), 96-102. [12] I. Rajasekaran and O. Nethaji, Unified approach of several sets in ideal nan- otopological spaces, Asia Mathematika, 3(1)(2019), 70-78. [13] I. Rajasekaran, M. Meharin and O. Nethaji, On new classes of some nano open sets, International Journal of Pure and Applied Mathematical Sciences, 10(2)(2017, 147-155. [14] A. Revathy and G. Ilango, On nano β-open sets, Int. Jr. of Engineering, Contemporary Mathematics and Sciences, 1(2)2015, 1-6. [15] R. Vaidyanathaswamy, The localization theory in set topology, Proc. Indian Acad. Sci., 20(1945), 51-61. [16] R. Vaidyanathaswamy, Set topology, Chelsea Publishing Company, New York, 1946. 193