Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 358 https://internationalpubls.com On Generalized Nano ℕ�̌�- Closed and Nano ℕα̂- Open Sets in Nano- Topological Spaces Abdulaziz .S. Hameed1, Layla Hindi2, Nabila I. Aziz3, Faeyda Yaseen Taha4 1Ministry of Education-General directorate for breeding Baghdad Third Karkh azizsaad201357@gmail.com 2Department of Mathematics, Faculty of Computer Science and Mathematics, University of Kufa, Al-Najaf 54001, Iraq laylah.algharrawi@uokufa.edu.iq 3Department of Physics, College of Education -Tuzkhurmatu , Tikrit University nabila.be@tu.edu.iq 4Samarra University, College of Education, Department of Chemistry faedayaseen@ uosamarra. edu. iq Article History: Received: 25-04-2024 Revised: 10-06-2024 Accepted: 21-06-2024 Abstract: This work aims to define a new class of sets in nano topological spaces called Nano (Nα) ̌- closed and (Nα) ̂-open sets, and to prove its verifiable properties and theorems . Subject Classification: 54A05, 54A 10 Keywords: nano topology, (Nα) ̌- closed set and Nano (Nα) ̂- open Set. 1. Introduction In 2021, Ą regular closed sets in nano topological spaces were presented by Narmatha S., Harshitha S., and others [3]. Within micro topological spaces, πgβ-closed sets are studied by Rajasekaran I. and others [4]. In nano topological spaces, ng∗α− closed sets were first presented by Rajendran V. and colleagues [5]..Crossley and Hildebrand [7] conducted research on semi-closure in 1971. Dunham [14] provided a definition of the closure operator C* notion along with various attributes. Operator of regular closed sets was first defined by S. Bhattacharya [6] in 2011. Soft W -int. and soft W -Cl. in Soft topological spaces are studied by Savita R. [8]. Soft g* closed in Soft topological spaces are studied by Kalavathi, A. and Krishnan, G.[2]. Regular generalized* in topological spaces, closure regular generalized* Closure Regular Generalized*, and regular generalized closed sets are new classes of operators introduced by Siham I. Aziz and Nabila I. Aziz [9, 10, 11] and, respectively. 2014 saw the introduction of Nano closure and Nano Interior operator in Nano topological spaces by Thivagar M. Lellis and Carmel Rechard [12], a novel class of operators Open and Closed Nano operators Ɲ ̂ (Ᾱ) and Ɲ ̌ (Ᾱ) introduced by ABDULAZIZ .S.[ 1 ]. The aim of this work is to investigate and characterize a new class of operators in nano topological spaces called Nano ℕα̌- closed and ℕα̂- open sets , and to establish their verifiable characteristics and theorems. 2. Preliminaries Definition 2.1 [2] : Suppose ϑ be the world, ψ ⊆ ϑ, and Π be an equivalence relation on ϑ. With regard to Ψ, τ ϕ(Ψ) ={ ϑ, ∅, LR(Ψ), ϑ Φ(Ψ), BΦ(Ψ)} and (ϑ, τ Φ(Ψ) ) define the Nano topology on U. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 359 https://internationalpubls.com Definition 2.2 [ 1 ]: Assume A ⊆ ϑ and (U, τ R(Ψ) ) be a Nano topological space. Next, we established 1- ℵ(A) = ∩ {G : A⊆ G, G ∈ N O(ϑ, Ψ)} and 2-N ̌ (A) = ∪ {G : G⊆ A, G ∈ NC (ϑ, Ψ)}. Definition 2.3 [2]: Assuming Ψ, Ψ ⊆ ϑ, let (U, τ R(Ψ) ) be a Nano topological space. If A is not equal to ϑ, then: The union of all of A's open subsets is A's nanointerior, and it is represented by Nint(A). Ncl(A) represents the Nano closure of A, which is the intersection of all Nano closed subsets containing A. Definition 2.4 [13] : When M ⊆ Nint (NCl (Nint M )), a subset M of (ϑ, τ R(Ψ) ) is referred to as a nano α open set (briefly, N𝛼-o-s.). In (U, τ R(Ψ) ), the complement of a N𝛼-o-s. is referred to as a nano 𝛼-closed set (briefly, N𝛼-c-s.). N𝛼-o-( ϑ, Ψ) (resp. N𝛼-c-( ϑ, Ψ)) represents the family of all N𝛼-o-s. (resp. N𝛼-c-s.) of U. 3 - On Generalized Nano ℕ�̌�- closed and Nano ℵ�̂�- open Sets in Nano- Topological Spaces Definition 3.1Assume that A ⊆ U and that (U, τ R(Ψ)) is a Nano topological space. If A ⊆ ℵ (( N) ̌ (ℵ(A))), then a subset A is referred to as an open set (Nα) ̂.The closed set Nano (ℵδ) - is the complement of the open set Nano (Nα) ̂ and is defined as [A ⊇ ( N) ̌ (ℵ( (( N) ̌ (A)))] Example 3.2: Assuming U/R = {{r, p}, { q}} and Ψ = {r, q}, let U = {r, μ, q}. Assuming τR(ψ)={∅,U, { q},{r, p}}, we get τcR(Ψ) = {∅,U, { q},{r, p}}. (Nα) -o(x)={U, ϕ,{r},{μ },{q},{r, },{r,q},{μ,q}}. Theorem 3.3 All subsets of U R(Ψ) are not (Nα) ̂ open sets if τ R(Ψ) is not highly disconnected. Proof Case 1 in the event that τ R(Ψ) ={ U, Ø, U R(Ψ) }. To begin with, assume A = U R(Ψ) ⇒ℵ(A) = U R(Ψ) ⇒ ( N) ̌ (ℵ(U R(Ψ))=∅⇒ℵ( N) ̌(ℵ(∅))=∅. If A⊉ ∅ ∅ ⇒ A ⊈ ℵ((N) ̌(ℵ(A)), then A∉ (Nα) ̂. 2-Take A⊆ Uc R(Ψ)⇒ N̂(A)=U· ( N)· (N̂(U)=U· N̂((N)Œ(N̂(U))=U ⇒ A ⊆N̂( (N) ̌(N̂(A)). 3. If [U R(Ψ) and Uc R(Ψ)] intersect at A, then ℵ(A)= U ⇒ ( N) ̌ (ℵ(U)=U ⇒ ℵ((N) ̌(ℵ(U))=U ⇒ A ⊆ℵ( (N) ̌(ℵ(A)). Case 2: If 𝜏 R(Ψ) ={ U, Ø, L R(Ψ), B R(Ψ),U R(Ψ) }. First, suppose that A⊆ L R(Ψ) ⇒ℵ(A)= L R(Ψ)⇒ ( N) ̌ (ℵ(L R(Ψ) )=∅ ⇒ℵ( N) ̌(ℵ(∅))=∉. A⊈ ∅⍒⇒ A⊈ ℵ((N) ̌(ℵ(A)). Afterwards, A∉ (Nα) ̂ 2. In the event when A ⊆ B R(Ψ) ⇒ ℵ(A)= B R(Ψ) ⇒ ( N) ̌ (ℵ(B R(Ψ) )=∅ ⇒ℵ( N) ̌(ℵ(∅))=∪. A⊈_∅_⇒ A⊈_ℵ((N) ̌(ℵ(A)). Afterwards, A∉ (Nα) ̂ 3. At the intersection of A with [L R(Ψ) and B R(Ψ)], ℵ(A)= U R(Ψ) and (N) ̌ (ℵ(U R(Ψ)) = ∅ ⇒ℵ( N) ̌(ℵ(∅)) =∅. A⊈ ∅⍒⇒ A⊈ ℵ((N) ̌(ℵ(A)). Afterwards, A∉ (Nα) 4-If A⊆ Uc R(Ψ) ⇒ N̂(A)= U ⇒ ( N) ̌ (N̂(U)=U ⇒ N̂( (N) ̌(N̂(U))=U ⇒ A ⊆N̂( (N) ̌(N̂(A)). 5. In the event when A intersects [L R(Ψ) and Uc R(Ψ)] ⇒ ℵ(A)= U ⇒ ( N) ̌ (ℵ(U)=U ⇒ ℵ( (N) ̌(ℵ(U))=U ⇒ A ⊆ℵ( (N) ̌(ℵ(A)). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 360 https://internationalpubls.com 6-If [B R(Ψ) and Uc R(Ψ)] intersect A, then N̂(A)= U ⇒ ( ℵ) ̌ (N̂(U)=U ⇒ ℵ( (ℵ) ̌(N̂(U))=U ⇒ A ⊆ℵ( (N) ̌(ℵ(A)). Theorem 3.4: All subset of Uc R(Ψ) is ℕα̂ open set. Proof: Case 1- If τ R(Ψ) ={ U, Ø, U R(Ψ) }. If A⊆ Uc R(Ψ) ⇒ ℕ̂(A)=U ⇒ ℕ̌ (ℕ̂(U) = U ⇒ ℕ̂ (ℕ̌(ℕ̂(U))=U ⇒ A ⊆ ℕ̂ (ℕ̌(ℕ̂(A)). Then A∈ ℕα̂. Case 2 : when τ R(Ψ) ={ U, Ø, L R(Ψ), B R(Ψ) ,U R(Ψ) }. If A⊆ Uc R(Ψ) ⇒ ℕ̂(A)=U ⇒ ℕ̌ (ℕ̂(U) = U ⇒ ℕ̂ (ℕ̌(ℕ̂(U))=U ⇒ A ⊆ ℕ̂ (ℕ̌(ℕ̂(A)). Then A∈ ℕα̂. Theorem 3. 5: All subset of U which intersect [U R(Ψ) and Uc R(Ψ) ] is ℕα̂ open set. Proof: Case 1- If τ R(Ψ) ={ U, Ø, U R(Ψ) } Let A ⊆ U Such that A intersect [ U R(Ψ) and Uc R(Ψ) ] ℕ̂(A)=U ⇒ ℵ̌ (ℕ̂(U) = U ⇒ ℵ̂ (ℕ̌(ℵ̂(U))=U ⇒ A ⊆ ℵ̂ (ℕ̌(ℵ̂(A)). Then A∈ ℕα̂ open set. Case 2: 𝜏 R(Ψ) ={ U, Ø, L R(Ψ), B R(Ψ) ,U R(Ψ) }. 1- Let A⊆ U such that A intersect [L R(Ψ) and Uc R(Ψ)]. ℕ̂(A)=U ⇒ ℕ̌ (ℵ(U) = U ⇒ ℵ̂ (ℕ̌(ℵ̂(U))=U ⇒ A ⊆ ℵ̂ (ℕ̌(ℵ̂(A)). Then A∈ ℕα̂ open set. 2- Let A⊆ U such that A intersect [B R(Ψ) and Uc R(Ψ)]. ℕ̂(A)=U ⇒ ℕ̌ (ℕ̂(U) = U ⇒ ℵ̂ (ℕ̌(ℵ̂(U))=U ⇒ A ⊆ ℵ̂ (ℕ̌(ℵ̂(A)). Then A∈ ℕα̂ open set. 3- Let A⊆ U such that A intersect [L R(Ψ) and B R(Ψ) and Uc R(Ψ)]. ℕ̂(A)=U ⇒ ℕ̌ (ℕ̂(U) = U ⇒ ℵ̂ (ℕ̌(ℕ̂(U))=U ⇒ A ⊆ ℵ̂ (ℕ̌(ℕ̂(A)). Then A∈ ℕα̂ open set. Theorem 3.6: When the Nano space U is extremely disconnected then all subset of U is ℕα̂ −open set. Proof: 𝜏 R(Ψ) ={ U, Ø, L R(Ψ), B R(Ψ) }. Assuming that A ⊆ L R(Ψ) ⇒ℵ(A)= L R(Ψ)⇒ ( N) ̌ (ℵ(L R(Ψ)) = L R(Ψ)⇒ ℵ ( N) ̌ (ℵ(L R(Ψ) )= L R(Ψ)A ⊆ ℵ((N) ̌(ℵ(A)) if A⊆ L R(Ψ) ⇒ A. After that, A∈(Nα) ̂ open set. In the event that A ⊆ B R(Ψ) ⇒ ℵ(A)= B R(Ψ) ⇒ ( N) ̌ (ℵ(B R(Ψ) )= B R(Ψ) ⇒ ℵ ( N) ̌ (ℵ(B R(Ψ))= B R(Ψ) ⇒\ A⊆ B R(Ψ) ⇒ A ⊆ B R(Ψ) ⇒ B R(Ψ) ⇒ A ⊆ ℵ((N) ̌(ℵ(A)). A∈(Nα) ̂ is thus an open set.When A crosses across [L R(Ψ) and B R(Ψ)].N (A)=U ⇒ (N) (N̂(U) = U· N̂((N)(N̂(U))=U· A ⊆N̂((N)(ℵ(A)). A∈(Nα) ̂ is thus an open set. 4.Conclusion The purpose of this study is to define a novel class called ( ℕα̌- closed and ℕα̂- open) Sets in Nano- Topological Spaces and to demonstrate its verifiable characteristics and theorems. The (𝛽 , b, regular, and semi) sets can be included in the future generalization of the new concept. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 361 https://internationalpubls.com References [1] ABDULAZIZ .S. HAMEED, , Nabila I. Aziz and Siham I. Aziz , Inside of Nano topological spaces a novel generalized Open and Closed Nano operators, under review [2] Kalavathi, A. and Krishnan, G., Soft g* closed and soft g* open sets in soft topological spaces. Journal of Interdisciplinary Mathematics, 19(1), 65-82, https://doi.org/10.1080/09720502.2015.1103110, (2016). [3] Narmatha S. , Harshitha S. , Kaaviya P. , Harshini R. , Indhuja S. .(2021). "On Generalizd Α Regular-Closed Set in Nano Topoloical Spaces".Jour. Nat. Volatiles & Essent. Oils. No.8. Vol.5,PP. 5057 – 5061. [4] Rajasekaran I. , Nethaji O. and Sajan Joseph M.". (2018). On nano πgβ-closed sets." Global Journal of Pure and Applied Mathematics No.1.Vol. 14 , PP. 181-187. [5] Rajendran V. , Sathish Mohan P. and Chitra M.(2020)." on ng∗α− closed sets in nano topological spaces." journal of critical reviews. No.13.Vol.7, PP. 4121-4127. [6] S. Bhattacharya. On generalized regular closed sets. Contemp. Math. Sciences, 6(3),pp. 145-152, (2011). [7] S. G. Crossley and S. K. Hildebrand, ''Semi-closure'', Teψas J. Sci., 22 ,99–112, (1971) . [8] Mustafa, M.A., Kadham, S.M., Abbass, N.K. et al. A novel fuzzy M-transform technique for sustainable ground water level prediction. Appl Geomat 16, 9–15 (2024). https://doi.org/10.1007/s12518-022-00486-4) [9] Siham I. Aziz and Nabila I. Aziz, On some generalized recent operators in topological Spaces . Tikrit Journal of Pure Science, Vol. 27 (4),( 2022) [10] Kadham, S.M. Acute interstitial pneumonia image enhancement using fuzzy partial transforms. Appl Geomat 16, 35–39 (2024). https://doi.org/10.1007/s12518-023-00509-8 [11] Siham I. Aziz and Nabila I . Aziz ,New generalized Operator in Topological Spaces . V. International Scientific Congress of Pure, Applied and Technological Sciences,(2022). [12] Thivagar M. lellis , Carmel Rechard ,Note on Nano topological space, Communicated, (2014). [13] Thivagar, M.L., Richard, C., On nano forms of weakly open sets, Int. J. Math Statistics Invention, 1(1), 31-37, 2013 . [14] W. Dunham, ''A new closure operator for non-T1topologies'', Kyungpook, Math.J". 22 ,55-602009, (1982).