Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 390 https://internationalpubls.com Exploration of NαIg-Closed and NαIg-Open Sets in Nano Ideal Topological Spaces with Practical Applications of Nano Ideal Topological spaces M. Mary Jansirani1, K. Lakshmanan2, L. Senthil Kumar3, K. Sivaranjani4, R.Kulandaivelu5,S. Santhiya6, A Stephan Antony Raj7, D. Vinodhini8* 1. SRM- Institute of Science and Technology, Trichy- 621105, India 2. St. Joseph University, Chumoukedima, Nagaland- 797 115, India 3. Dr. Mahalingam College of Engineering and Technology, Pollachi, Tamil Nadu- 642 003,India. 4. Sri Eshwar College of Engineering, Coimbatore, India 5. Dr.N.G.P.InstituteofTechnology,Coimbatore,TamilNadu-641 008,India. 6. Sri Krishna College of Engineering and Technology, Coimbatore, Tamil Nadu- 641 008,India. 7. SNS College of Engineering, TamilNadu, India 8. Amrita School of Agricultural Sciences, Amrita Vishwa Vidyapeetham University, Coimbatore, TamilNadu– 642109,India. Correspondence: D. Vinodhini,Email:d_vinodhini@cb.amrita.edu Article History: Received: 11-05-2024 Revised: 20-06-2024 Accepted: 07-07-2024 Abstract: In this study, we explore the concepts of NαIg-closed sets (nano αIg-closed sets) and NαIg-open sets (nano αIg-open sets) in nano ideal topological spaces. We discuss their relationships with other forms of nano ideal sets and illustrate the abstract concepts using appropriate examples. This research delves into the application of NαIg- closed and NαIg-open sets across diverse domains, ranging from network security to healthcare and environmental monitoring. By leveraging these mathematical concepts, we aim to enhance anomaly detection, optimize network operations, and improve decision-making processes across various sectors. This article outlines the methodologies and potential benefits of integrating NαIg-closed and NαIg-open sets in different application domains. Keywords: Ideals, Nanotopology,NαIg-closedsets,NαIg-opensets,nanoαopen. 1. Introduction On topological spaces, Levine [1] introduced the concept of generalized closed sets during 1970. Numerous results in general topology have been developed using this concept. In 1991, Balachandran et. al [2] introduced and examined the notion of generalized continuous functions in topological spaces. The notion of α-open sets was introduced and investigated by Njastad [3]. By using α-open, Mashbour et al. [4] defined and studied the concept of α-closed sets, α- closure of a set, α-continuity . The concept of ideal topological space was introduced by kuratowski [5] in 1966. He also defined the local functions in ideal topological spaces. Furthermore, during the period 1990, Jankovic and Hamlett [6] investigated the properties of ideal topological spaces. In 2014, αIg- closed is introduced in ideal topological spaces. The notion of nano topology was introduced by Lellis Thivagar [7, 8, 9] which was defined in terms of approximations and boundary region of a subset of an universe using an equivalence relation on it. He also established and analyzed the nano forms of weakly open sets such as nanoα- open sets, nano mailto:d_vinodhini@cb.amrita.edu Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 391 https://internationalpubls.com semi-open sets and nano pre-open sets. In 2016, Bhuvaneswari et. al [10], introduced and studied the characteristic of Nano generalized closed sets. The structure of this manuscript is as follows. In section 2, we recall some fundamental definitions and results which are useful to prove our main results. In section 3, we define and study the notion of NαIg- closed sets and NαIg- open sets in nano ideal topological spaces. We also discuss the concept of NαIg-closed sets and discussed the relationships between the other existing nano ideal sets. In section 4, The integration of NαIg-closed and NαIg-open sets in nano ideal topological spaces which presents a novel approach to address complex challenges across multiple domains is discussed. This research explores the diverse applications of these mathematical concepts, highlighting their potential to revolutionize various industries. 2. Premilinaries Throughout this study (U, 𝑟𝑅(𝑥)) (or U) represent nano topological spaces on which no separation axioms are assumed unless otherwise mentioned. For a subset A of a space (U,𝑟𝑅(𝑥)), Ncl(A) and Nint(A) denote the nano closure of A and the nano interior of A respectively. We recall the following definition which are useful in the sequel. Definition 2.1. [11] Let U be a non-empty finite set of objects called the universe 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) = {𝖴𝑥∈𝑈 {𝑅(𝑥): 𝑅(𝑥) ⊆𝑋}}, where R(x) denotes the equivalence class determined by x. (2) The Upper 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 UR (X). That is , UR(X) = {𝖴𝑥∈𝑈 {𝑅(𝑥): 𝑅(𝑥) ∩ 𝑋 ≠ ∅}} (3) The Boundary region of X with respect to R is the set of all objects, which can be classified as 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) Definition 2.2. Let U be the universe, R be an equivalence relation on U and 𝑟𝑅(𝑥) = {𝑈, ∅, 𝐿𝑅 (X), UR(X), BR(X)} where X ⊆ U. Then 𝑟𝑅(𝑥) satisfies the following axioms: (1) U and ∅∈𝑟𝑅(𝑥) (2) The union of elements of any subcollection of 𝑟𝑅(𝑥) is in 𝑟𝑅(𝑥). (3) The intersection of the elements of any finite subcollection of 𝑟𝑅(𝑥) is in 𝑟𝑅(𝑥). That is , 𝑟𝑅(𝑥) forms a topology on U called the nano topology on U with respect to X. We call { U, 𝑟𝑅(𝑥) } is called the nano topological space. The elements of 𝑟𝑅(𝑥) are called as nano-open sets. The complement of the nano-open sets are called nano-closed sets. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 392 https://internationalpubls.com Definition 2.3. An ideal I on a topological space is a non-empty collection of subsets of X which satisfies (1) A 𝜖 I and B ⊆ A  B 𝜖 I. (2) A 𝜖 I and B 𝜖 I  A 𝖴 B 𝜖 I. Definition 2.4 [13] Let ( X, 𝑟) be a topological space and I be an ideal on X. A subset A of X is said to be αIg-closed if 𝐴∗⊆𝑈 whenever A ⊆𝑈 and U is α-open. Definition 2.5. [12] A nano topological space {U, 𝑟𝑅(𝑥)} with an ideal I on U is called a nano ideal topological space or nano ideal space and denoted as ( U, 𝑟𝑅(𝑥), I ) Definition 2.6. Let { U, 𝑟𝑅(𝑥), I } be a nano ideal topological space. A set operator (A)*N:P(U) →P(U)is called the nano local function of I on U with respect to I on 𝑟𝑅(𝑥) is defined as (A)*N = {x 𝜖U: U ∩ A ∉ I; for every U 𝜖𝑟𝑅(𝑥)} and is denoted by (A)*N, where nano closure operator is defined as Ncl*(A) = A 𝖴(A)*N. Result 2.7. Let (U, 𝑟𝑅(𝑥), I) be a nano ideal topological space and let A and B be subsets of U, then (1) (∅)∗𝑁 = ∅ (2) 𝐴⊂𝐵 → (𝐴)∗𝑁⊂ (𝐵)∗𝑁 (3) For another 𝐽⊇𝐼𝑜𝑛𝑈, (𝐴)∗𝑁(𝐽) ⊂ (𝐴)∗𝑁(𝐼) (4) (𝐴)∗𝑁⊂𝑁𝑐𝑙∗(𝐴) (5) (𝐴)∗𝑁 is a nano closed set. (6) ((𝐴)∗𝑁)∗𝑁⊂ (𝐴)∗𝑁 (7) (𝐴)∗𝑁𝖴 (𝐵)∗𝑁 = (𝐴𝖴𝐵)∗𝑁 (8) (𝐴 ∩ 𝐵)∗𝑁 = (𝐴)∗𝑁 ∩ (𝐵)∗𝑁 (9) For every nano open set V, V∩ (𝑉 ∩ 𝐴)∗𝑁⊂ (𝑉 ∩ 𝐴)∗𝑁 (10) For I ∈𝐼, (𝐴𝖴𝐼)∗𝑁 = (𝐴)∗𝑁 = (𝐴 − 𝐼)∗𝑁 Result 2.8 . Let { U, 𝑟𝑅(𝑥), I } be a nano ideal topological space and A be a subset of U, If A ⊂ (𝐴)∗𝑁, then (𝐴)∗𝑁 = 𝑁𝑐𝑙 (𝐴)∗𝑁 = 𝑁𝑐𝑙(𝐴) = 𝑁𝑐𝑙∗(𝐴). Definition 2.9. Let {U, 𝑟𝑅(𝑥)} be a nano topological space and A ⊆ U. Then A is said to be (1) Nano semi- closed, if Nint (N cl (A)) ⊆ A. (2) Ng-closed , Ncl(A) ⊆G whenever A ⊆ G and G is nano open. (3) N�̂�-closed , Ncl(A) ⊆G whenever A ⊆ G and G is nano semi-open. (4) N𝑔∗-closed , Ncl(A) ⊆G whenever A ⊆ G and G is nano g-open. (5) Nano pre closed if N cl (N int (A)) ⊆ A. (6) Nano α-closed set if N cl (N int (N cl(A)) ⊆ A Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 393 https://internationalpubls.com The complements of the above mentioned sets are called their respective open sets. Definition 2.10. A subset A of a nano ideal space. Let ( U, 𝑟𝑅(𝑥), I ) is said to be (1) ∗𝑁- closed , if (𝐴)∗𝑁⊆𝐴 (2) ∗𝑁- dense, if A ⊆ (𝐴)∗𝑁 (3) N Ig closed, if (𝐴)∗𝑁⊆𝐺 whenever A ⊆𝐺 and G is nano open (4) N Ig* closed, if (𝐴)∗𝑁⊆𝐺 whenever A ⊆𝐺 and G is nano g- open 3. N αIg- CLOSED SETS In this section we define and study the notion of NαIg-closed sets and N αIg-open sets in nano ideal topological spaces. Also we discuss their basic properties and study the relationship between other existing nano closed sets in nano ideal topological spaces. Definition 3.1. A subset A of a nano ideal space ( U, 𝑟𝑅(𝑥), I ) is said to be N αIg-closed if (𝐴)∗𝑁⊆𝐺whenever A ⊆𝐺 and G is nano α open. Example 3.2: Let U = {a, b, c, d}, U/R ={a},{d}, {b, c}} and X = {a, d}. Let the nano ideal space 𝑟𝑅(𝑥) = { U, ∅, {𝑎, 𝑑}} with a nano ideal I = { ∅, {a}}. Then N αIg-closed sets are {U, ∅,{a}, {b, c}, {b,d}, {a, b, c}, {b, c, d}}. Definition 3.3. Let ( U, 𝑟𝑅(𝑥), I ) be a Nano ideal topological space. A subset A of X is said to be N αIg-open if X – A is N αIg-closed. Theorem 3.4. If (U, 𝑟𝑅(𝑥), I ) is any nano ideal space, then the following are equivalent (1) A is N αIg-closed (2) 𝑁𝑐𝑙∗(𝐴) ⊆𝐺 whenever A ⊆𝐺 and G is nano α open in U (3) For all x 𝑁𝑐𝑙∗(𝐴), N αcl({x})  A ≠  (4) 𝑁𝑐𝑙∗(𝐴) - A contains no non empty nano α closed set. (5) (𝐴)∗𝑁-Acontains no nonempty nano α closed set. Proof: (1)  (2) : If A is N αIg-closed, then (𝐴)∗𝑁⊆𝐺whenever A ⊆𝐺 and G is nano α open in X and so 𝑁𝑐𝑙∗(𝐴) = 𝐴⋃(𝐴)∗𝑁⊆𝐺 and G is nano α open in U. This proves (2). (2) (3) : Suppose x 𝑁𝑐𝑙∗(𝐴). If N αcl({x}) A=,then A ⊆ X- N αcl({x}). By (2), 𝑁𝑐𝑙∗(𝐴) ⊆ X- N αcl({x}), which is a contradiction to x 𝑁𝑐𝑙∗(𝐴). This Proves (3) (3)  (4) : Suppose F ⊆𝑁𝑐𝑙∗(𝐴) – A, F is nano α-closed and x  F. Since F ⊆𝑋 − 𝐴 and F is nano α- closed, then A ⊆𝑋 − 𝐹 and hence N αcl({x})  A = . Therefore, 𝑁𝑐𝑙∗(𝐴) – A contains no non empty nano α-closed set. (4)  (5) : Since 𝑁𝑐𝑙∗(𝐴) – A = (A (𝐴)∗𝑁) – A = (A (𝐴)∗𝑁) ∩ 𝐴𝑐 = (A∩𝐴𝑐)((𝐴)∗𝑁∩ 𝐴𝑐)=(𝐴)∗𝑁∩𝐴𝑐=(𝐴)∗𝑁−𝐴. Therefore, (𝐴)∗𝑁−𝐴contains no nonempty nano α closed set. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 394 https://internationalpubls.com Theorem 3.5: Every ∗𝑁 closed set is N αIg-closed but not conversely. Proof: Let A be a ∗𝑁 closed, then (𝐴)∗𝑁⊆ A. Let A ⊆ G and G is Nano α open . This implies (𝐴)∗𝑁⊆ G. Hence A is N αIg-closed. Example 3.6: Let U = { a, b, c, d}, U/R = {{a}, {c}, {b, d}} and X = {a, b}. Let the nano ideal space 𝑟𝑅(𝑥) = { U, ∅, {a}, {a, b, d}, {b, d}} with a nano ideal I = { ∅, {a}, {a, b, d}}. Then N αIg- closed sets are {U, ∅, {a}, {c}, {a, c}, {b, c}, {c, d}, {a, b, c}, {a, b, d}, {a, c, d},{b, c, d}} and ∗𝑁 closed set are {U, ∅, {a}, {c}, {a, c}, {b, c, d}}. It is clear that {a, b, c} is N αIg-closed set but it is not ∗𝑁closed. Theorem 3.7: Every NIg*closed set is N αIg-closed. But not conversely. Proof: Let A ⊆ G and G is Nano α open. Clearly every Nano α open set is Nano semi open. Since A is NIg*closed set, (A*)N⊆ G, which implies that A is an NαIg-closed set. Example 3.8: Let U = {a, b, c, d}, U/R ={a},{d}, {b, c}} and X = {a, d}. Let the nano ideal space 𝑟𝑅(𝑥) = { U, ∅, {𝑎, 𝑑}} with a nano ideal I = { ∅, {a}}. Then NαIg-closed sets are {U, ∅, {a}, {b, c}, {b,d}, {a, b, c}, {b, c, d}} and NIg*closed sets are {U, ∅, {a}, {b, c}, {a, b, c}, {b, c, d}}. It is clear that {b, d} is N αIg-closed but it is not NIg* closed. Theorem 3.9.EveryNαIg-closedsetisNIgclosed. ButConverseisnottrue. Proof: Let A ⊆ G and G is Nano α open. Clearly every nano open set is Nano α open. Since A is N αIg-closed set, (A*)N⊆ G, which implies that A is NIgclosed. Example 3.10. Let U = {a, b, c, d}, U/R ={a},{d}, {b, c}} and X = {a, d}. Let the nano ideal space 𝑟𝑅(𝑥) = { U, ∅, {𝑎, 𝑑}} with a nano ideal I = { ∅, {a}}. Then NαIg-closed sets are {U, ∅,{a}, {b, c}, {b,d}, {a, b, c}, {b, c, d}} and NIgclosed sets are {U, ∅, {a}, {b}, {c}, {a, b}, {a,c}, {b, c}, {b, d}, {c, d}, {a, b, c}, {a, b, d}, {a, c, d}, {b, c, d}}. It is clear that {b} is NIgclosed but it is not NIg- closed. Theorem 3.11. Let ( U, 𝑟𝑅(𝑥), I ) be an nano ideal space. For every A  I, A is N αIg-closed set. Proof: Let A ⊆ G and G is Nano α open. Since (A*)N = ∅ for every A  I, then (A*)N⊆ A. This implies (A*)N⊆ G. Hence for every A  I, A is an N αIg-closed set. Theorem 3.12. If A and B are NαIg-closed sets in ( U, 𝑟𝑅(𝑥), I ), then A  B is also an N αIg- closed set. Proof: Let A  B  Gwhere G is a Nano -open set in X. Then, A  G and B  G. By hypothesis, A and B are two N αIg-closed set. This implies (A*)N G and (B*)N G. Hence (A*)N (B*)N G. By Result 2.6 (vii), ((A  B)*)N = (A*)N (B*)N G. Therefore, A  B is an N αIg-closed set. Remark 3.13. The intersection of N αIg-closed sets in ( U, 𝑟𝑅(𝑥), I) need not be a N αIg-closed set. This can be proved from the example given below. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 395 https://internationalpubls.com Example 3.14. Let U = {a, b, c, d}, U/R ={{a},{d}, {b, c}} and X = {a, d}. Let the nano ideal space 𝑟𝑅(𝑥) = { U, ∅, {𝑎, 𝑑}} with a nano ideal I = { ∅, {a}}. Then N αIg-closed sets are {U, ∅, {a}, {b, c}, {b,d}, {a, b, c}, {b, c, d}}. If A = {b, c} and B = {b, d}, then their intersection A B = {b} is not N αIg-closed. Theorem 3. 15. If ( U, 𝑟𝑅(𝑥), I ) is a nano ideal space, then (A*)N is always a N αIg-closed set for every subset A of X. Proof. Let (A*)N G, where G is Nano α open. Since, ((A*)N)*  (A*)N , we have ((A*)N)*  G whenever (A*)N G and G is Nano α open. Hence (A*)N is a N αIg-closed set 4. Application: 4.1 Network Anomaly Detection: The research would enhance the potential of anomaly detection inside computer networks. Subsequently, this method empowers the security system through the early detection of various threats by utilizing notions of NαIg-closed set and NαIg-open set. Thus, security analyses could review all network traffic, catching even the smallest irregularities in network behavior. 4.2 Social Network Analysis: Additionally, the research could contain applications in the field of social network analysis. In this regard, NαIg-closed set and NαIg-open set demonstrate the potential to identify the most influencing nodes and communities within social networks. Therefore, specific information flow patterns, social dynamics, and community structures could be analyzed and introduce some strategic decisions. 4.3 Biological Network Analysis: The use of NαIg-closed and NαIg-open sets for biological network analysis also allows understanding some gene-protein interactions and biological pathways. The conducting of analysis of complex biological systems through these mathematical concepts will help determine the complex biological processes and ways to affect them in case of drug development. 4.4 Transportation Network Optimization: The combination of NαIg-closed and NαIg-open sets in transportation network optimization implies their use for planning routes and managing traffic. The optimization of transportation networks through these mathematical concepts will allow improvements in the performance of multiple systems and reducing congestion to achieve better overall performance . 4.5 Pattern Recognition: Pattern recognition uses NαIg-closed and NαIg-open sets to improve image processing, natural language processing, and other important machine learning tasks. By utilizing the defined mathematics, machine learning practitioners can come up with superior-pattern-recognition algorithms that are widely applicable. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 396 https://internationalpubls.com 4.6 IoT Security: NαIg-closed and NαIg-open sets help in improving security in IoT devices and networks. The mathematics of the two important concepts can help in abnormality detection and therefore providing stake holders with severe stability to prevent possible losses and data compromise. 4.7 Financial Market Analysis: The integration of NαIg-closed and NαIg-open sets in financial market analysis enables the detection of abnormal trading patterns and assists in risk management and fraud detection. By analyzing financial market data using these mathematical concepts, stakeholders can make informed decisions and mitigate financial risks. 4.8 Environmental Monitoring: NαIg-closed and NαIg-open sets play a crucial role in environmental monitoring by detecting anomalies in ecosystems and contributing to conservation efforts. By leveraging these mathematical concepts, researchers can monitor environmental data effectively and implement sustainable solutions to address environmental challenges. 4.9 Healthcare Systems Analysis: In healthcare systems analysis, NαIg-closed and NαIg-open sets are utilized for analyzing healthcare data and improving patient outcomes. By applying these mathematical concepts, stakeholders can identify irregularities in patient records, optimize healthcare delivery, and enhance patient care. 4.10 Supply Chain Management: Applying NαIg-closed and NαIg-open sets in supply chain management enables optimization of supply chain networks and enhances resilience and responsiveness. By leveraging these mathematical concepts, stakeholders can identify inefficiencies, vulnerabilities, and potential disruptions in supply chains, leading to improved supply chain performance. Conclusion: The exploration of NαIg-closed and NαIg-open sets across various domains showcases their versatility and potential impact in addressing complex challenges. By integrating these mathematical concepts into diverse applications, researchers and practitioners can unlock new insights, optimize operations, and drive innovation across multiple industries. Further research and experimentation in this area are essential to fully realize the potential of NαIg-closed and NαIg-open sets in multi- domain applications. Acknowledgement: We would like to express our sincere gratitude to all those who contributed to the publication of this paper. Our sincere gratitude goes out to Amrita University and other organizations for providing us with resources and support. We would also like to thank our colleagues for their feedback and support throughout the research process. Without their support, we would not have been able to complete this study. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 397 https://internationalpubls.com References [1] Levine,N.(1970).Generalizedclosedsetsintopology.RendicontidelCircoloMatematicodiPalermo,19,89-96. [2] Balachandran,K.,P.Sundaram andH.Maki(1991a).Ongeneralizedcontinuousmapsintopologicalspaces.Mem.Fac.Sci.KochiUniv.Ser.A,Math.,12,5- 13. [3] Njȧstad,O.(1965). Onsomeclassesofnearlyopensets. Pacificjournalofmathematics, 15(3),961-970. [4] Mashhour, A. S., Hasanein, I. A., & El-Deeb, S. N. (1983). α-continuousandα openmappings. ActaMathematicaHungarica, 41(3-4), 213-218. [5] Kuratowski,K.(1966).Topologyvol1(PWN,WarsawAcademicPress,NewYork).Russiantransl:Mir,Moscow. [6] Janković,D.,&Hamlett,T.R.(1990).Newtopologiesfromoldviaideals.Theamericanmathematicalmonthly, 97(4),295- 310. [7] Thivagar,M.L.,&Richard,C.(2014).Noteonnanotopologicalspaces.Communicated,1(2.2),2-3. 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