Ratio Mathematica Volume 44, 2022 A New Class of Nano Generalized Closed Sets in Nano Topological Spaces Anbarasi Rodrigo P* Subithra P† Abstract In this paper, we introduce a new class of nano generalized closed sets in nano topological spaces namely nano generalized π›Όβˆ—-closed sets. Then we discuss some of its properties and investigate their relation with many other nano closed sets. Also, we define nano generalized π›Όβˆ—-open set and discuss its relation with other open sets. Finally, we define the properties of nano generalized π›Όβˆ—-interior and nano generalized π›Όβˆ—-closure. Keywords: β„•π‘”π›Όβˆ—- closed sets, β„•π‘”π›Όβˆ—- open sets, β„•π‘”π›Όβˆ—- int, ℕ𝑔𝛼 βˆ—- cl. 2010 AMS subject classification: 54A05‑ *Assistant Professor, Department of Mathematics, St. Mary’s College (Autonomous), (Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli) Thoothukudi-1, TamilNadu, India; anbu.n.u@gmail.com. † Research Scholar, Reg.No. 21212212092003, Department of Mathematics, St. Mary’s College (Autonomous), (Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli) Thoothukudi-1, TamilNadu, India;p.subithra18@gmail.com. ‑ Received on June 4, 2022. Accepted on September 1, 2022. Published on Nov 30, 2022. DOI: 10.23755/rm.v44i0.900. ISSN: 1592-7415. eISSN: 2282-8214. Β©The Authors. This paper is published under the CC-BY licence agreement. 132 mailto:anbu.n.u@gmail.com mailto:p.subithra18@gmail.com P. Anbarasi Rodrigo and P. Subithra 1. Introduction The theory of nano topology proposed by Lellis Thivagar [4] and Carmel Richard is an extension of set theory for the study of intelligent systems characterized by insufficient and incomplete information. The elements of a nano topological space are called the nano open set. The author has defined nano topological space in terms of lower and upper approximations and boundary region. He has defined nano closed sets, nano-interior and nano-closure of a set. He also introduced certain weak forms of nano open sets such as nano 𝛼-open set, nano semi-open sets and nano pre-open sets. Levine [5] introduced the class of 𝑔-closed sets in 1970. K. Bhuvaneswari introduced nano 𝑔- closed [1], nano 𝑔𝑠-closed [3], nano 𝛼𝑔-closed [6], nano 𝑔𝑝-closed [2], nano π‘”π‘Ÿ- closed [12] sets and studied their properties. Nano π‘”βˆ—π‘-closed sets was introduced by Rajendran [10] and investigated. The aim of this paper is to introduce and study the properties of nano π‘”π›Όβˆ—-closed sets and nano π‘”π›Όβˆ—-open sets in nano topological spaces. Finally, we define the properties of nano generalized π›Όβˆ—-interior and nano generalized π›Όβˆ—-closure. 2. Preliminaries Throughout this paper (π•Œ, πœβ„(𝕏)) represent nano topological spaces on which no separation axioms are assumed unless and otherwise mentioned. For a subset π•Š of (π•Œ, πœβ„(𝕏)), 𝑁𝑐𝑙(π•Š) and 𝑁𝑖𝑛𝑑(π•Š) denote the nano closure of π•Š and nano interior of π•Š respectively. We recall the following definitions which are useful in the sequel. Definition 2.1. [4] Let π•Œ be a non-empty finite set of objects called the universe and ℝ be an equivalence relation on π•Œ named as the indiscernibility relation. Elements belonging to the same equivalence class are said to be indiscernible with one another. The pair (π•Œ,ℝ) is said to be the approximation space. Let 𝕏 βŠ†β„• π•Œ. Then 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). 𝕃ℝ(𝕏) = ⋃ {ℝ(π‘₯): ℝ(π‘₯) βŠ†β„• 𝕏} π‘₯βˆˆπ•Œ 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 π•Œβ„(𝕏). π•Œβ„(𝕏) = ⋃ {ℝ(π‘₯): ℝ(π‘₯) ∩ 𝕏 β‰  Ο•} π‘₯βˆˆπ•Œ 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). 𝔹ℝ(𝕏). = π•Œβ„(𝕏) βˆ’ 𝕃ℝ(𝕏). Definition 2.2. [4] Let π•Œ be the universe, ℝ be an equivalence relation on π•Œ and πœβ„(𝕏) = {π•Œ, πœ™, π•Œβ„(𝕏), 𝕃ℝ(𝕏), 𝔹ℝ(𝕏)} where 𝕏 βŠ†β„•π•Œ. Then ℝ(𝕏) satisfies the following axioms: 1) π•Œ and Ο• ∈ πœβ„(𝕏), 2) The union of the elements of any sub collection of πœβ„(𝕏) is in πœβ„(𝕏), 133 A New Class of Nano Generalized Closed Sets in Nano Topological Spaces 3) The intersection of the elements of any finite sub collection of πœβ„(𝕏) is in πœβ„(𝕏). That is, πœβ„(𝕏)is a topology on π•Œ called the nano topology on π•Œ with respect to 𝕏. We call (π•Œ, πœβ„(𝕏)) as the nano topological space(𝑁𝑇𝑆). The elements of πœβ„(𝕏) are called as nano open sets. The complement of nano-open sets is called nano closed sets. Definition 2.3. [4] If (π•Œ, πœβ„(𝕏)) is a 𝑁𝑇𝑆 with respect to 𝕏 and if π•Š βŠ†β„•π•Œ, then β€’ The nano interior of π•Š is defined as the union of all nano open subsets of π•Š and it is denoted by ℕ𝑖𝑛𝑑(π•Š). That is, ℕ𝑖𝑛𝑑(π•Š) is the largest open subset of π•Š. β€’ The nano closure of π•Š is defines as the intersection of all nano closed sets containing π•Š and it is denoted by ℕ𝑐𝑙(π•Š). That is, ℕ𝑐𝑙(π•Š) is the smallest nano closed set containing π•Š. Definition 2.4. A subset π•Š of a 𝑁𝑇𝑆 (π•Œ, πœβ„(𝕏)) is called; 1) Nano pre-open [4] if π•Š βŠ†β„• ℕ𝑖𝑛𝑑(ℕ𝑐𝑙(π•Š)) 2) Nano semi-open [4] if π•Š βŠ†β„• ℕ𝑐𝑙(ℕ𝑖𝑛𝑑(π•Š)) 3) Nano 𝛼-open [4] if π•Š βŠ†β„• ℕ𝑖𝑛𝑑(ℕ𝑐𝑙(ℕ𝑖𝑛𝑑(π•Š))) 4) Nano 𝛽-open [11] if π•Š βŠ†β„• ℕ𝑐𝑙(ℕ𝑖𝑛𝑑(ℕ𝑐𝑙(π•Š))) 5) Nano regular-open [4] if π•Š = ℕ𝑖𝑛𝑑(ℕ𝑐𝑙(π•Š)) The complements of the above-mentioned sets are called their respective closed sets. Definition 2.5. A subset π•Š of a 𝑁𝑇𝑆 (π•Œ, πœβ„(𝕏)) is called; 1) ℕ𝑔-closed [1] if ℕ𝑐𝑙(π•Š) βŠ†β„• 𝔽, whenever π•Š βŠ†β„•π”½ and 𝔽 is nano open in π•Œ. 2) ℕ𝑔𝑠-closed [3] if ℕ𝑠𝑐𝑙(π•Š) βŠ†β„• 𝔽, whenever π•Š βŠ†β„•π”½ and 𝔽 is nano open in π•Œ. 3) ℕ𝛼𝑔-closed [6] if ℕ𝛼𝑐𝑙(π•Š) βŠ†β„• 𝔽, whenever π•Š βŠ†β„•π”½ and 𝔽 is nano open in π•Œ. 4) ℕ𝑔𝑝-closed [2] if ℕ𝑝𝑐𝑙(π•Š) βŠ†β„• 𝔽, whenever π•Š βŠ†β„•π”½ and 𝔽 is nano open in π•Œ. 5) ℕ𝑔𝛽-closed [7] if ℕ𝛽𝑐𝑙(π•Š) βŠ†β„• 𝔽, whenever π•Š βŠ†β„•π”½ and 𝔽 is nano open in π•Œ. 6) β„•π‘”π‘Ÿ-closed [12] if β„•π‘Ÿπ‘π‘™(π•Š) βŠ†β„• 𝔽, whenever π•Š βŠ†β„•π”½ and 𝔽 is nano open in π•Œ. 7) β„•π‘”βˆ—-closed [8] if ℕ𝑐𝑙(π•Š) βŠ†β„• 𝔽, whenever π•Š βŠ†β„•π”½ and 𝔽 is ℕ𝑔-open in π•Œ. 8) β„•π‘”βˆ—π‘ -closed [9] if ℕ𝑠𝑐𝑙(π•Š) βŠ†β„• 𝔽, whenever π•Š βŠ†β„•π”½ and 𝔽 is ℕ𝑔-open in π•Œ. 9) β„•π‘”βˆ—π‘-closed [10] if ℕ𝑝𝑐𝑙(π•Š) βŠ†β„• 𝔽, whenever π•Š βŠ†β„•π”½ and 𝔽 is ℕ𝑔-open in π•Œ. 10) β„•π‘”βˆ—π‘Ÿ-closed [13] if β„•π‘Ÿπ‘π‘™(π•Š) βŠ†β„• 𝔽, whenever π•Š βŠ†β„•π”½ and 𝔽 is ℕ𝑔-open in π•Œ. Theorem 2.6. [1] Every nano open set is ℕ𝑔-open. 3. Nano generalized πœΆβˆ—-closed sets Definition 3.1. A Nano generalized π›Όβˆ—(in short, β„•π‘”π›Όβˆ—) closed set is a subset π•Š of a 𝑁𝑇𝑆(π•Œ, πœβ„(𝕏)) ifℕ𝛼𝑐𝑙(π•Š) βŠ†β„• ℕ𝑖𝑛𝑑 βˆ—(𝔽)whenever π•Š βŠ†β„• 𝔽and𝔽 𝑖𝑠 ℕ𝑔-open in π•Œ. Instance 3.2. Let π•Œ = {p, q, r, s} with π•Œ/ℝ ={{p}, {r}, {q, s}} and 𝕏 = {p, q} βŠ†β„• π•Œ. Then πœπ‘… (𝕏) = {π•Œ, Ο•, {p}, {q, s}, {p, q, s}}. Here ΕŠπ‘”π›Ό βˆ—-closed = {π•Œ, Ο•, {r}, {p, r}, {q, r}, {r, s}, {p, q, r}, {p, r, s}, {q, r, s}}. 134 P. Anbarasi Rodrigo and P. Subithra Instance 3.3. Let π•Œ = {p, q, r} with π•Œ/ ℝ = {{p}, {p, q}} and 𝕏 = {p} βŠ†β„• π•Œ. Then πœβ„(𝕏) = {π•Œ, Ο•, {p}, {q}, {p, q}}. Here ℕ𝑔𝛼 βˆ—- closed = {Ο•, π•Œ, {r}, {p, r}, {q, r}}. Theorem 3.4. Every nano closed set is β„•π‘”π›Όβˆ—-closed. Proof: Let π•Š be a nano closed set in 𝑁𝑇𝑆 (π•Œ, πœβ„(𝕏)). Then we have, ℕ𝑐𝑙(π•Š) = π•Š. Let 𝔽 be a nano open set in π•Œ such that π•Š βŠ†β„• 𝔽 and𝔽 = ℕ𝑖𝑛𝑑(𝔽). By theorem 2.6, we have 𝔽 is ℕ𝑔-open in π•Œ. Then ℕ𝛼𝑐𝑙(π•Š) βŠ†β„• ℕ𝑐𝑙(π•Š) = π•Š βŠ†β„• 𝔽 = ℕ𝑖𝑛𝑑(𝔽) βŠ†β„• ℕ𝑖𝑛𝑑 βˆ—(𝔽). Thus ℕ𝛼𝑐𝑙(π•Š) βŠ†β„• ℕ𝑖𝑛𝑑 βˆ—(𝔽) whenever π•Š βŠ†β„• 𝔽 and 𝔽 is ℕ𝑔-open in π•Œ. Therefore π•Š is β„•π‘”π›Όβˆ—-closed. Remark 3.5 The invert of the preceding theorem does not hold as witnessed in the succeeding instance. Instance 3.6. Let π•Œ = {p, q, r, s} with π•Œ/ℝ = {{r}, {p, q, s}} and 𝕏 = {q, s}. Then πœβ„(𝕏) = {Ο•, π•Œ, {p, q, s}}. Here {π•Œ, Ο•, {r}, {p, r}, {q, r}, {r, s}, {p, q, r}, {p, r, s}, {q, r, s}} is β„•π‘”π›Όβˆ—- closed set but the set is not nano closed. Theorem 3.7. Every β„•π‘”π›Όβˆ—- closed set is ℕ𝑔 - closed. Proof. Let π•Š be a β„•π‘”π›Όβˆ—- closed set in 𝑁𝑇𝑆(π•Œ, πœβ„(𝕏)) and let 𝔽 be a nano open set in π•Œ such that π•Š βŠ†β„• 𝔽 and 𝔽 = ℕ𝑖𝑛𝑑(𝔽). By theorem 2.6, we have 𝔽 is ℕ𝑔-open in π•Œ. Also, π•Š is β„•π‘”π›Όβˆ—-closed, then ℕ𝛼𝑐𝑙(π•Š) βŠ†β„• ℕ𝑖𝑛𝑑 βˆ—(𝔽). Then, ℕ𝛼𝑐𝑙(π•Š) βŠ†β„• ℕ𝑐𝑙(π•Š) βŠ†β„• ℕ𝑖𝑛𝑑 βˆ—(𝔽) βŠ†β„• 𝔽. Thus ℕ𝑐𝑙(π•Š) βŠ†β„• 𝔽 whenever π•Š βŠ†β„• 𝔽 and 𝔽 is nano open in π•Œ. Hence π•Š is ℕ𝑔-closed. Remark 3.8. The invert of the preceding theorem does not hold as witnessed in the succeeding instance. Instance 3.9. Let π•Œ = {p, q, r, s} with π•Œ/ℝ = {{p, q}, {r, s}} and 𝕏 = {p, q}. Then πœβ„(𝕏) = {π•Œ, Ο•, {p, q}}. Here {π•Œ, Ο•, {r}, {s}, {p, r}, {p, s}, {q, r}, {q, s}, {r, s}, {p, q, r}, {p, q, s}, {p, r, s}, {q, r, s}} is ℕ𝑔 – closed set but the set is not β„•π‘”π›Όβˆ—- closed. Theorem 3.10. Every β„•π‘”π›Όβˆ—- closed set is ℕ𝛼𝑔- closed. Proof. Let π•Š be a β„•π‘”π›Όβˆ—-closed set in 𝑁𝑇𝑆(π•Œ, πœβ„(𝕏)) and let 𝔽 be a nano open set in π•Œ such that π•Š βŠ†β„• 𝔽 and𝔽 = ℕ𝑖𝑛𝑑(𝔽). By theorem 2.6, we have 𝔽 is ℕ𝑔-open in π•Œ. Also, π•Š is β„•π‘”π›Όβˆ—-closed, then ℕ𝛼𝑐𝑙(π•Š) βŠ†β„• ℕ𝑖𝑛𝑑 βˆ—(𝔽). Then ℕ𝛼𝑐𝑙(π•Š) βŠ†β„• ℕ𝑖𝑛𝑑 βˆ—(𝔽) = 𝔽. Thus ℕ𝛼𝑐𝑙(π•Š) βŠ†β„• 𝔽 whenever π•Š βŠ†β„• 𝔽 and 𝔽 is nano open in π•Œ. Hence π•Š is ℕ𝛼𝑔- closed. Remark 3.11. The invert of the preceding theorem does not hold as witnessed in the succeeding instance. Instance 3.12. Let π•Œ = {p, q, r} with π•Œ/ℝ = {{{r}, {p, q}} and 𝕏 = {r}. Then πœβ„(𝕏) = {Ο•, π•Œ, {r}}. Here {Ο•, {p}, {q}, {p, q}, {p, r}, {q, r}} is ℕ𝛼𝑔 - closed but the set is not β„•π‘”π›Όβˆ—- closed. 135 A New Class of Nano Generalized Closed Sets in Nano Topological Spaces Theorem 3.13. Every β„•π‘”π›Όβˆ—-closed set is ℕ𝑔𝑠-closed. Proof. Let π•Š be a β„•π‘”π›Όβˆ—-closed set in 𝑁𝑇𝑆 (π•Œ, πœβ„(𝕏)). Let 𝔽 be a nano open set in π•Œ such that π•Š βŠ†β„• 𝔽 and𝔽 = ℕ𝑖𝑛𝑑(𝔽). By theorem 2.6, 𝔽 is ℕ𝑔-open in π•Œ. Since π•Š is β„•π‘”π›Όβˆ—-closed, ℕ𝛼𝑐𝑙(π•Š) βŠ†β„• ℕ𝑖𝑛𝑑 βˆ—(𝔽). Then ℕ𝑠𝑐𝑙(π•Š) βŠ†β„• ℕ𝛼𝑐𝑙(π•Š) βŠ†β„• ℕ𝑖𝑛𝑑 βˆ—(𝔽) = 𝔽. Thus ℕ𝑠𝑐𝑙(π•Š) βŠ†β„• 𝔽 whenever π•Š βŠ†β„• 𝔽 and 𝔽 is nano open in π•Œ. Hence π•Š is ℕ𝑔𝑠- closed. Remark 3.14. The invert of the former theorem does not holds as witnessed in the succeeding instance. Instance 3.15. Let π•Œ = {p, q, r, s} with π•Œ/ℝ = {{p}, {q}, {r, s}} and 𝕏 = {q, s}. Then πœβ„(𝕏) = {Ο•, π•Œ, {q}, {r, s}, {q, r, s}. Here {Ο•, π•Œ, {p}, {q}, {r}, {s}, {p, q}, {p, r}, {p, s}, {r, s}, {p, q, r}, {p, q, s}, {p, r, s}} is ℕ𝑔𝑠 - closed but it is not β„•π‘”π›Όβˆ—- closed. Theorem 3.16. Every β„•π‘”π›Όβˆ—-closed set is ℕ𝑔𝑝 - closed. Proof: Let π•Š be a β„•π‘”π›Όβˆ—-closed set in 𝑁𝑇𝑆 (π•Œ, πœβ„(𝕏)). Let 𝔽 be a nano open set in π•Œ such that π•Š βŠ†β„• 𝔽 and𝔽 = ℕ𝑖𝑛𝑑(𝔽). By theorem 2.6, we have 𝔽 is ℕ𝑔-open in π•Œ. Since π•Š is β„•π‘”π›Όβˆ—-closed, ℕ𝛼𝑐𝑙(π•Š) βŠ†β„• ℕ𝑖𝑛𝑑 βˆ—(𝔽). Then ℕ𝑝𝑐𝑙(π•Š) βŠ†β„• ℕ𝛼𝑐𝑙(π•Š) βŠ†β„• ℕ𝑖𝑛𝑑 βˆ—(𝔽) = 𝔽. Thus ℕ𝑝𝑐𝑙(π•Š) βŠ†β„• 𝔽 whenever π•Š βŠ†β„• 𝔽 and 𝔽 is nano open in π•Œ. Hence π•Š is ℕ𝑔𝑝- closed. Remark 3.17. The invert of the former theorem does not holds as witnessed in the succeeding instance. Instance 3.18. Let π•Œ = {p, q, r, s} with π•Œ/ℝ = {(p}, {r}, {q, s}} and 𝕏 = {r, s}. Then πœβ„(𝕏) = {Ο•, π•Œ, {r}, {q, s}, {q, r, s}}. Here {Ο•, π•Œ, {p}, {q}, {s}, {p, q}, {p, r}, {p, s}, {p, q, r}, {p, q, s} {p, r, s}} is ℕ𝑔𝑝-closed but it is not β„•π‘”π›Όβˆ—- closed. Theorem 3.19. Every β„•π‘”π›Όβˆ—-closed set is ℕ𝑔𝛽-closed. Proof. Let π•Š be a β„•π‘”π›Όβˆ—-closed set in 𝑁𝑇𝑆(π•Œ, πœβ„(𝕏)) and let 𝔽 be a nano open set in π•Œ such that π•Š βŠ†β„• 𝔽 and𝔽 = ℕ𝑖𝑛𝑑(𝔽). By theorem 2.6, we have 𝔽 is ℕ𝑔-open in π•Œ. Since π•Š is β„•π‘”π›Όβˆ—-closed, ℕ𝛼𝑐𝑙(π•Š) βŠ†β„• ℕ𝑖𝑛𝑑 βˆ—(𝔽). Then ℕ𝛽𝑐𝑙(π•Š) βŠ†β„• ℕ𝛼𝑐𝑙(π•Š) βŠ†β„• ℕ𝑖𝑛𝑑 βˆ—(𝔽) = 𝔽. Thus ℕ𝛽𝑐𝑙(π•Š) βŠ†β„• 𝔽 whenever π•Š βŠ†β„• 𝔽 and 𝔽 is nano open in π•Œ. Hence π•Š is ℕ𝑔𝛽- closed. Remark 3.20. The invert of the former theorem does not hold as witnessed in the succeeding instance. Instance 3.21. Let π•Œ = {p, q, r} with π•Œ/ℝ = {{p}, {q, r}} and 𝕏 = {p, q}. Then πœβ„(𝕏) = {Ο•, π•Œ, {p}, {q, r}}. Here {Ο•, π•Œ, {p}, {q}, {r}, {p, q}, {p, r}, {q, r}} is ℕ𝑔𝛽-closed but it is not β„•π‘”π›Όβˆ—-closed. Theorem 3.22. Everyβ„•π‘”π›Όβˆ—- closed set is β„•π‘”π‘Ÿ - closed. 136 P. Anbarasi Rodrigo and P. Subithra Proof. Let π•Š be a β„•π‘”π›Όβˆ—-closed set in 𝑁𝑇𝑆 (π•Œ, πœβ„(𝕏)). Let 𝔽 be a nano open set in π•Œ such that π•Š βŠ†β„• 𝔽 and𝔽 = ℕ𝑖𝑛𝑑(𝔽). By theorem 2.6, we have 𝔽 is ℕ𝑔-open in π•Œ. Since π•Š is β„•π‘”π›Όβˆ—-closed, ℕ𝛼𝑐𝑙(π•Š) βŠ†β„• ℕ𝑖𝑛𝑑 βˆ—(𝔽). Then ℕ𝛼𝑐𝑙(π•Š) βŠ†β„• β„•π‘Ÿπ‘π‘™(π•Š) βŠ†β„• ℕ𝑖𝑛𝑑 βˆ—(𝔽) = 𝔽. Thus β„•π‘Ÿπ‘π‘™(𝐴) βŠ†β„• 𝔽 whenever π•Š βŠ†β„• 𝔽 and 𝔽 is nano open in π•Œ. Hence π•Š is β„•π‘”π‘Ÿ- closed. Remark 3.23. The transpose of the preceding theorem does not hold as witnessed in the succeeding instance. Instance 3.24. Let π•Œ = {p, q, r} with π•Œ/ℝ = {{r}, {p, q}} and 𝕏={r}. Then πœβ„(𝕏) = {Ο•, π•Œ, {r}}. Here {Ο•, {p}, {q}, {p, q}, {p, r}, {q, r}} is β„•π‘”π‘Ÿ-closed which is not β„•π‘”π›Όβˆ—- closed. Theorem 3.25. Every β„•π‘”βˆ—-closed set is β„•π‘”π›Όβˆ—-closed. Proof: Let π•Š be a β„•π‘”βˆ—-closed set in 𝑁𝑇𝑆 (π•Œ, πœβ„(𝕏)). Let 𝔽 be a nano open set in π•Œ such that π•Š βŠ†β„• 𝔽 and𝔽 = ℕ𝑖𝑛𝑑(𝔽). Through theorem 2.6, we have 𝔽 is ℕ𝑔-open in π•Œ. Since π•Š is β„•π‘”βˆ—-closed, ℕ𝑐𝑙(π•Š) βŠ†β„• 𝔽. Then ℕ𝛼𝑐𝑙(π•Š) βŠ†β„• ℕ𝑐𝑙(π•Š) βŠ†β„• 𝔽 = ℕ𝑖𝑛𝑑(𝔽) βŠ†β„• ℕ𝑖𝑛𝑑 βˆ—(𝔽). Thus ℕ𝛼𝑐𝑙(π•Š) βŠ†β„• ℕ𝑖𝑛𝑑 βˆ—(𝔽)whenever π•Š βŠ†β„• 𝔽 and 𝔽 is ℕ𝑔- open in π•Œ. Hence π•Š is β„•π‘”π›Όβˆ—-closed. Remark 3.26. The transpose of the preceding theorem does not hold as witnessed in the succeeding instance. Instance 3.27. Let π•Œ = {p, q, r} with π•Œ/ℝ = {{p}, {q, r}} and 𝕏 = {p}. Then πœβ„(𝕏) = {Ο•, π•Œ, {p}}. Here {Ο•, {q}, {r}, {q, r}} is β„•π‘”π›Όβˆ—-closed which is not β„•π‘”βˆ—-closed. Theorem 3.28. Every β„•π‘”π›Όβˆ—-closed set is β„•π‘”βˆ—π‘ -closed. Proof. Let π•Š be a β„•π‘”π›Όβˆ—-closed set in 𝑁𝑇𝑆 (π•Œ, πœβ„(𝕏)). Then ℕ𝛼𝑐𝑙(π•Š) βŠ†β„• ℕ𝑖𝑛𝑑 βˆ—(𝔽) whenever π•Š βŠ†β„• 𝔽 and 𝔽 is ℕ𝑔-open in π•Œ.Thus ℕ𝑠𝑐𝑙(π•Š) βŠ†β„• ℕ𝛼𝑐𝑙(π•Š) βŠ†β„• ℕ𝑖𝑛𝑑 βˆ—(𝔽) βŠ†β„• 𝔽, we get ℕ𝑠𝑐𝑙(π•Š) βŠ†β„• 𝔽 whenever π•Š βŠ†β„• 𝔽 and 𝔽 is ℕ𝑔-open in π•Œ. Hence π•Š is ℕ𝑔 βˆ—π‘ -closed. Remark 3.29. The transpose of the preceding theorem does not hold as witnessed in the succeeding instance. Instance 3.30. Let π•Œ = {p, q, r, s} with π•Œ/ℝ = {{p}, {r}, {q, s}} and 𝕏 = {q, r}. Then πœβ„(𝕏) = {Ο•, π•Œ, {r}, {q, s}, {q, r, s}}. Here {Ο•, π•Œ, {p}, {r}, {p, q}, {p, r}, {p, s}, {q, s}, {p, q, r}, {p, q, s}, {p, r, s}} is β„•π‘”βˆ—π‘ -closed which is not β„•π‘”π›Όβˆ—-closed. Theorem 3.31. Every β„•π‘”π›Όβˆ—-closed set is β„•π‘”βˆ—π‘-closed. Proof: Let π•Š be a β„•π‘”π›Όβˆ—-closed set in 𝑁𝑇𝑆 (π•Œ, πœβ„(𝕏)). Then ℕ𝛼𝑐𝑙(π•Š) βŠ†β„• ℕ𝑖𝑛𝑑 βˆ—(𝔽) whenever π•Š βŠ†β„• 𝔽 and 𝔽 is ℕ𝑔-open in π•Œ.Thus ℕ𝑝𝑐𝑙(π•Š) βŠ†β„• ℕ𝛼𝑐𝑙(π•Š) βŠ†β„• ℕ𝑖𝑛𝑑 βˆ—(𝔽) βŠ†β„• 𝔽, we get ℕ𝑝𝑐𝑙(π•Š) βŠ†β„• 𝔽 whenever π•Š βŠ†β„• 𝔽 and 𝔽 is ℕ𝑔-open in π•Œ. Hence π•Š is ℕ𝑔 βˆ—π‘-closed. 137 A New Class of Nano Generalized Closed Sets in Nano Topological Spaces Remark 3.32. The polar statement of the preceding theorem does not hold as witnessed in the succeeding instance. Instance 3.33. Let π•Œ = {p, q, r, s} with π•Œ/ℝ = {{p, q}, {r, s}} and 𝕏 = {q, r, s}. Then πœβ„(𝕏) = {Ο•, π•Œ, {p, q}, {r, s}}. Here {Ο•, π•Œ, {p}, {q}, {r}, {s}, {p, q}, {p, r}, {p, s}, {q, r}, {q, s}, {r, s}, {p, q, r}, {p, q, s}, {p, r, s}, {q, r, s}} is β„•π‘”βˆ—π‘-closed but it is not β„•π‘”π›Όβˆ—-closed. Theorem 3.34. Every β„•π‘”βˆ—π‘Ÿ-closed set is β„•π‘”π›Όβˆ—-closed. Proof: Let π•Š be a β„•π‘”βˆ—π‘Ÿ-closed set in 𝑁𝑇𝑆(π•Œ, πœβ„(𝕏)) and let 𝔽 be a nano open set in π•Œ such that π•Š βŠ†β„• 𝔽 and𝔽 = ℕ𝑖𝑛𝑑(𝔽). Through theorem 2.6, we have 𝔽 is ℕ𝑔-open in π•Œ. Since π•Š is β„•π‘”βˆ—π‘Ÿ-closed, β„•π‘Ÿπ‘π‘™(𝐴) βŠ†β„• 𝔽. Then ℕ𝛼𝑐𝑙(π•Š) βŠ†β„• β„•π‘Ÿπ‘π‘™(π•Š) βŠ†β„• 𝔽 = ℕ𝑖𝑛𝑑(𝔽) βŠ†β„• ℕ𝑖𝑛𝑑 βˆ—(𝔽). Thus ℕ𝛼𝑐𝑙(π•Š) βŠ†β„• ℕ𝑖𝑛𝑑 βˆ—(𝔽)whenever π•Š βŠ†β„• 𝔽 and 𝔽 is ℕ𝑔- open in π•Œ. Hence π•Š is β„•π‘”π›Όβˆ—-closed. Remark 3.35. The polar statement of the preceding theorem does not hold as witnessed in the succeeding instance. Instance 3.36. Let π•Œ = {p, q, r, s} with π•Œ/ℝ = {{r}, {q, s}} and 𝕏 = {p, r}. Then πœβ„(𝕏) = {Ο•, π•Œ, {r}}. Here {Ο•, π•Œ, {p}, {q}, {s}, {p, q}, {p, s}, {q, s}, {p, q, s}} is β„•π‘”π›Όβˆ—- closed but it is not β„•π‘”βˆ—π‘Ÿ-closed. Remark 3.37. The concepts of nano semi closed and β„•π‘”π›Όβˆ—-closed are independent as witnessed in the succeeding instance. Instance 3.38. Let π•Œ = {p, q, r, s} with π•Œ/ℝ = {{p}, {r}, {q, s}} and 𝕏 = {p, q} βŠ† π•Œ. Then πœβ„(𝕏) = {π•Œ, Ο•, {p}, {q, s}, {p, q, s}}. The set {Ο•, π•Œ, {p}, {r}, {p, r}, {q, s}, {q, r, s}} is nano semi closed yet not β„•π‘”π›Όβˆ—-closed. The set {Ο•, π•Œ, {r}, {p, r}, {q, r}, {r, s}, {p, q, r}, {p, r, s}, {q, r, s}} is β„•π‘”π›Όβˆ—-closed but the set is not nano semi closed. Remark 3.39. β„•π‘”π›Όβˆ—-closed set lies between β„•π‘”βˆ—-closed set and ℕ𝑔-closed set. That is, β„•π‘”βˆ—-closedβŠ†β„• ℕ𝑔𝛼 βˆ—-closedβŠ†β„• ℕ𝑔-closed. Remark 3.40. The diagram that follows exhibit the relation between β„•π‘”π›Όβˆ—-closed sets and other closed sets. β„•π‘”π›Όβˆ—-closed ℕ𝑔-closed ℕ𝛼𝑔-closed ℕ𝑔𝑠-closed ℕ𝑔𝑝-closed ℕ𝑔𝛽-closed β„•π‘”π‘Ÿ-closed β„•π‘”βˆ—π‘ -closed β„•π‘”βˆ—π‘-closed β„•π‘”βˆ—-closed β„•π‘”βˆ—π‘Ÿ-closed 138 P. Anbarasi Rodrigo and P. Subithra Theorem 3.41. If 𝔾 and ℍ are β„•π‘”π›Όβˆ—-closed sets in 𝑁𝑇𝑆(π•Œ, πœβ„(𝕏)), then 𝔾 βˆͺ ℍ is a β„•π‘”π›Όβˆ—-closed set. Proof: Let 𝔾and ℍ be β„•π‘”π›Όβˆ—-closed sets in a 𝑁𝑇𝑆(π•Œ, πœβ„(𝕏)) and let 𝔽 be any ℕ𝑔-open set in π•Œ containing 𝔾 and ℍ. Then 𝔾 βˆͺ ℍ βŠ†β„• 𝔽. Then 𝔾 βŠ†β„• 𝔽 and ℍ βŠ†β„• 𝔽. Since 𝔾 and ℍ are β„•π‘”π›Όβˆ—-closed sets, ℕ𝛼𝑐𝑙(𝔾) βŠ†β„• ℕ𝑖𝑛𝑑 βˆ—(𝔽)and ℕ𝛼𝑐𝑙(ℍ) βŠ†β„• ℕ𝑖𝑛𝑑 βˆ—(𝔽). Now, ℕ𝛼𝑐𝑙(𝔾 βˆͺ ℍ) = ℕ𝛼𝑐𝑙(𝔾) βˆͺ ℕ𝛼𝑐𝑙(𝐻) βŠ†β„• ℕ𝑖𝑛𝑑 βˆ—(𝔽). Thus, ℕ𝛼𝑐𝑙(𝔾 βˆͺ ℍ) βŠ†β„• ℕ𝑖𝑛𝑑 βˆ—(𝔽) whenever 𝔾 βˆͺ ℍ βŠ†β„• 𝔽 and 𝔽 is ℕ𝑔-open in π•Œ. Hence 𝔾 βˆͺ ℍ is a β„•π‘”π›Όβˆ—-closed. Theorem 3.42. If 𝔾 and ℍ are β„•π‘”π›Όβˆ—-closed sets in 𝑁𝑇𝑆 (π•Œ, πœβ„(𝕏)), then 𝔾 ∩ ℍ is a β„•π‘”π›Όβˆ—-closed set. Proof: Let 𝔾 and ℍ be β„•π‘”π›Όβˆ—-closed sets in a 𝑁𝑇𝑆(π•Œ, πœβ„(𝕏)) and let 𝔽 be a ℕ𝑔-open set in π•Œ such that 𝔾 βŠ†β„• 𝔽 and ℍ βŠ†β„• 𝔽. Then 𝔾 ∩ ℍ βŠ†β„• 𝔽. Since 𝔾 and ℍ are ℕ𝑔𝛼 βˆ—- closed sets, ℕ𝛼𝑐𝑙(𝔾) βŠ†β„• ℕ𝑖𝑛𝑑 βˆ—(𝔽)and ℕ𝛼𝑐𝑙(ℍ) βŠ†β„• ℕ𝑖𝑛𝑑 βˆ—(𝔽). Now,ℕ𝛼𝑐𝑙(𝔾 ∩ ℍ) βŠ†β„• ℕ𝛼𝑐𝑙(𝔾) ∩ ℕ𝛼𝑐𝑙(ℍ) βŠ†β„• ℕ𝑖𝑛𝑑 βˆ—(𝔽). Thus ℕ𝛼𝑐𝑙(𝔾 ∩ ℍ) βŠ†β„• ℕ𝑖𝑛𝑑 βˆ—(𝔽) whenever 𝔾 ∩ ℍ βŠ†β„• 𝔽 and 𝔽 is ℕ𝑔-open in π•Œ. Hence 𝔾 ∩ ℍ is a ℕ𝑔𝛼 βˆ—-closed. Corollary 3.43. If 𝔾 is β„•π‘”π›Όβˆ—-closed and ℍ is nano closed in π•Œ, then 𝔾 ∩ ℍ is β„•π‘”π›Όβˆ—- closed. Proof: Let ℍ be nano closed in π•Œ. Then by theorem 3.4, ℍ is β„•π‘”π›Όβˆ—-closed. 𝔾 is also β„•π‘”π›Όβˆ—-closed. By theorem 3.42, 𝔾 ∩ ℍ is β„•π‘”π›Όβˆ—-closed. Corollary 3.44. If 𝔾 is β„•π‘”π›Όβˆ—-closed and ℍ is nano open in π•Œ, then 𝔾\ℍ is β„•π‘”π›Όβˆ—- closed. Proof: Let 𝔾\ℍ = 𝔾 ∩ (π•Œ\ℍ). Since ℍ is nano open in π•Œ, π•Œ\ℍ is nano closed in π•Œ. Since 𝔾 is β„•π‘”π›Όβˆ—-closed and π•Œ\ℍ is nano closed in π•Œ, by corollary 3.43, 𝔾 ∩ (π•Œ\ℍ)is β„•π‘”π›Όβˆ—-closed. Hence 𝔾\ℍ is β„•π‘”π›Όβˆ—-closed. 4. Nano generalized πœΆβˆ—-open sets Definition 4.1. A subset π•Š of a 𝑁𝑇𝑆(π•Œ, πœβ„(𝕏)) is called nano generalized 𝛼 βˆ—(in short, β„•π‘”π›Όβˆ—) open set if its complement is β„•π‘”π›Όβˆ—-closed. Instance 4.2. Let π•Œ = {p, q, r, s} with π•Œ/ℝ = {{q}, {r}, {p, s}} and 𝕏 = {p, r}. Then πœβ„(𝕏) = {π•Œ, Ο•, {r}, {p, s}, {p, r, s}}. Here {Ο•, π•Œ, {p}, {r}, {s}, {p, r}, {p, s}, {r, s}, {p, r, s}} is β„•π‘”π›Όβˆ—-open sets. Theorem 4.3. Every nano open set is β„•π‘”π›Όβˆ—-open but the invert may not be true. Theorem 4.4. Every β„•π‘”π›Όβˆ—-open set is ℕ𝑔-open but the invert may not be true. Theorem 4.5. Every β„•π‘”π›Όβˆ—-open set is ℕ𝛼𝑔-open but the invert may not be true. Theorem 4.6. Every β„•π‘”π›Όβˆ—-open set is ℕ𝑔𝑠-open but the invert may not be true. 139 A New Class of Nano Generalized Closed Sets in Nano Topological Spaces Theorem 4.7. Every β„•π‘”π›Όβˆ—-open set is ℕ𝑔𝑝-open but the invert may not be true. Theorem 4.8. Every β„•π‘”π›Όβˆ—-open set is ℕ𝑔𝛽-open but the invert may not be true. Theorem 4.9. Every β„•π‘”π›Όβˆ—-open set is β„•π‘”π‘Ÿ-open but the invert may not be true. Theorem 4.10. Every β„•π‘”βˆ—-open set is β„•π‘”π›Όβˆ—-open but the invert may not be true. Theorem 4.11. Every β„•π‘”π›Όβˆ—-open set is β„•π‘”βˆ—π‘ -open but the invert may not be true. Theorem 4.12. Every β„•π‘”π›Όβˆ—-open set is β„•π‘”βˆ—π‘-open but the invert may not be true. Theorem 4.13. Every β„•π‘”βˆ—π‘Ÿ-open set is β„•π‘”π›Όβˆ—-open but the invert may not be true. Theorem 4.14. If 𝔾 and ℍ are β„•π‘”π›Όβˆ—-open sets in 𝑁𝑇𝑆, then 𝔾 βˆͺ ℍ is a β„•π‘”π›Όβˆ—-open set. Theorem 4.15. If 𝔾 and ℍ are β„•π‘”π›Όβˆ—-open sets in 𝑁𝑇𝑆, then 𝔾 ∩ ℍ is a β„•π‘”π›Όβˆ—-open set. Corollary 4.16. If 𝔾 is β„•π‘”π›Όβˆ—-open and ℍ is nano open in π•Œ, then 𝔾 ∩ ℍ is β„•π‘”π›Όβˆ—-open. Corollary 4.17. If 𝔾 is β„•π‘”π›Όβˆ—-open and ℍ is nano closed in π•Œ, then 𝔾\ℍ is β„•π‘”π›Όβˆ—-open. 5. β„•π’ˆπœΆβˆ—-interior and β„•π’ˆπœΆβˆ—-closure Definition 5.1. Let π•Œ be a 𝑁𝑇𝑆 and let any point π‘Ž ∈ π•Œ. A subset π•Š of π•Œ is called the β„•π‘”π›Όβˆ—-nbhd of π‘Ž if there exists a β„•π‘”π›Όβˆ—-open set 𝕂 such that π‘Ž ∈ 𝕂 βŠ†β„• π•Š. Definition 5.2. Let π•Š be a subset of the 𝑁𝑇𝑆 (π•Œ, πœβ„(𝕏)). A point π‘Ž ∈ π•Šis called ℕ𝑔𝛼 βˆ—- interior point of π•Š if π•Š is aβ„•π‘”π›Όβˆ—-nbhd of π‘Ž. The set which contains all β„•π‘”π›Όβˆ—-interior points of π•Š is called β„•π‘”π›Όβˆ—-interior of π•Š and symbolized as β„•π‘”π›Όβˆ—-int(π•Š). Definition 5.3. Let π•Š be a subset of the 𝑁𝑇𝑆 (π•Œ, πœβ„(𝕏)). Then the intersection of all β„•π‘”π›Όβˆ—-closed sets containing π•Š is called β„•π‘”π›Όβˆ—-closure of π•Š. That is, β„•π‘”π›Όβˆ— βˆ’ 𝑐𝑙(π•Š) =∩ {ℝ: ℝ 𝑖𝑠 β„•π‘”π›Όβˆ— βˆ’ π‘π‘™π‘œπ‘ π‘’π‘‘ 𝑠𝑒𝑑𝑠 π‘Žπ‘›π‘‘ π•Š βŠ†β„• ℝ}. Theorem 5.4. If π•Š be a subset of π•Œ, then β„•π‘”π›Όβˆ— βˆ’ 𝑖𝑛𝑑(π•Š) =βˆͺ {ℝ: ℝ 𝑖𝑠 β„•π‘”π›Όβˆ— βˆ’ π‘œπ‘π‘’π‘› 𝑠𝑒𝑑 π‘Žπ‘›π‘‘ ℝ βŠ†β„• π•Š}. Proof: Let π•Š be a subset of π•Œ. π‘₯ ∈ β„•π‘”π›Όβˆ— βˆ’ 𝑖𝑛𝑑(π•Š) ⇔ π‘₯ is a β„•π‘”π›Όβˆ—-interior point of π•Š β‡”π•Š is a β„•π‘”π›Όβˆ—-nbhd of the point π‘₯ ⇔There exists β„•π‘”π›Όβˆ—-open set ℝ such that π‘₯ ∈ ℝ βŠ†β„• π•Š ⇔π‘₯ ∈βˆͺ {ℝ: ℝ 𝑖𝑠 β„•π‘”π›Όβˆ— βˆ’ π‘œπ‘π‘’π‘› 𝑠𝑒𝑑 π‘Žπ‘›π‘‘ ℝ βŠ†β„• π•Š} Hence π‘π‘”π›Όβˆ— βˆ’ 𝑖𝑛𝑑(π•Š) =βˆͺ {ℝ: ℝ 𝑖𝑠 β„•π‘”π›Όβˆ— βˆ’ π‘œπ‘π‘’π‘› 𝑠𝑒𝑑 π‘Žπ‘›π‘‘ ℝ βŠ†β„• π•Š}. 140 P. Anbarasi Rodrigo and P. Subithra Theorem 5.5. Let ℝ and π•Š be subsets of π•Œ. Then a) β„•gπ›Όβˆ— βˆ’ 𝑖𝑛𝑑(π•Œ) = π•Œ and β„•gπ›Όβˆ— βˆ’ 𝑖𝑛𝑑(πœ™) = πœ™ b) β„•gπ›Όβˆ— βˆ’ 𝑖𝑛𝑑(π•Š) βŠ†β„• π•Š c) If ℝ contains any β„•gπ›Όβˆ—-open set π•Š, then π•Š βŠ†β„• ℕ𝑔𝛼 βˆ— βˆ’ 𝑖𝑛𝑑(ℝ) d) If β„βŠ†β„•π•Š, then ℕ𝑔𝛼 βˆ— βˆ’ 𝑖𝑛𝑑(ℝ) βŠ†β„• ℕ𝑔𝛼 βˆ— βˆ’ 𝑖𝑛𝑑(π•Š) Proof: a) Since π•Œ and Ο• are β„•π‘”π›Όβˆ—-open sets, by theorem 5.4, β„•π‘”π›Όβˆ— βˆ’ 𝑖𝑛𝑑(π•Œ) =βˆͺ {ℝ: ℝ 𝑖𝑠 β„•π‘”π›Όβˆ— βˆ’ π‘œπ‘π‘’π‘› 𝑠𝑒𝑑 π‘Žπ‘›π‘‘ ℝ βŠ†β„• π•Œ}. β‡’ π‘π‘”π›Όβˆ— βˆ’ 𝑖𝑛𝑑(π•Œ) =βˆͺ {π•Š: π•Š 𝑖𝑠 π‘Ž β„•π‘”π›Όβˆ— βˆ’ π‘œπ‘π‘’π‘› 𝑠𝑒𝑑} β‡’ π‘π‘”π›Όβˆ— βˆ’ 𝑖𝑛𝑑(π•Œ) = π•Œ Since Ο• is the only β„•π‘”π›Όβˆ—-open set contained in Ο•, β„•π‘”π›Όβˆ— βˆ’ 𝑖𝑛𝑑(πœ™) = πœ™. b) Let π‘₯ ∈ β„•π‘”π›Όβˆ— βˆ’ 𝑖𝑛𝑑(π•Š) β‡’π‘₯ is a β„•π‘”π›Όβˆ—-interior point of π•Š. β‡’π•Š is a β„•π‘”π›Όβˆ—-nbhd of π‘₯. β‡’π‘₯ ∈ π•Š Thus β„•gπ›Όβˆ— βˆ’ 𝑖𝑛𝑑(π•Š) βŠ†β„• π•Š. c) Let π•Š be any β„•π‘”π›Όβˆ—-open set such that π•Š βŠ†β„• ℝ and let π‘₯ ∈ π•Š. Since π•Š is a β„•π‘”π›Όβˆ—-open set contained in ℝ, π‘₯ is a β„•π‘”π›Όβˆ—-interior point of ℝ. That is, π‘₯ ∈ β„•π‘”π›Όβˆ— βˆ’ 𝑖𝑛𝑑(ℝ) Hence π•Š βŠ†β„• ℕ𝑔𝛼 βˆ— βˆ’ 𝑖𝑛𝑑(ℝ). d) Let ℝ and π•Š be subsets of π•Œ such that β„βŠ†β„•π•Š. Let π‘₯ ∈ β„•π‘”π›Όβˆ— βˆ’ 𝑖𝑛𝑑(ℝ). Then π‘₯ is aβ„•π‘”π›Όβˆ—-interior point of ℝ and so ℝ is aβ„•π‘”π›Όβˆ—-nbhd of π‘₯ contained in π•Š. Therefore π‘₯ is aβ„•π‘”π›Όβˆ—-interior point of π•Š. Thus π‘₯ ∈ β„•π‘”π›Όβˆ— βˆ’ 𝑖𝑛𝑑(π•Š). Hence β„•π‘”π›Όβˆ— βˆ’ 𝑖𝑛𝑑(ℝ) βŠ†β„• ℕ𝑔𝛼 βˆ— βˆ’ 𝑖𝑛𝑑(π•Š). Theorem 5.6. If π•Š is β„•π‘”π›Όβˆ—-open then β„•π‘”π›Όβˆ— βˆ’ 𝑖𝑛𝑑(π•Š) = π•Š. Proof: Let π•Š be a β„•π‘”π›Όβˆ—-open in π•Œ. We know that β„•π‘”π›Όβˆ— βˆ’ 𝑖𝑛𝑑(π•Š) βŠ†β„• π•Š. Also, π•Š is β„•π‘”π›Όβˆ—-open set contained in π•Š. By theorem 5.5 c), π•Š βŠ†β„• ℕ𝑔𝛼 βˆ— βˆ’ 𝑖𝑛𝑑(π•Š). Hence β„•π‘”π›Όβˆ— βˆ’ 𝑖𝑛𝑑(π•Š) = π•Š. Theorem 5.7. If ℝ and π•Š are subsets of π•Œ, then β„•π‘”π›Όβˆ— βˆ’ 𝑖𝑛𝑑(ℝ) βˆͺ β„•π‘”π›Όβˆ— βˆ’ 𝑖𝑛𝑑(π•Š) βŠ†β„• ℕ𝑔𝛼 βˆ— βˆ’ 𝑖𝑛𝑑(ℝ βˆͺ π•Š). Proof: We know that ℝ βŠ†β„• ℝ βˆͺ π•Š and π•Š βŠ†β„• ℝ βˆͺ π•Š. Then by theorem 5.5 d), β„•π‘”π›Όβˆ— βˆ’ 𝑖𝑛𝑑(ℝ) βŠ†β„• ℕ𝑔𝛼 βˆ— βˆ’ 𝑖𝑛𝑑(ℝ βˆͺ π•Š) and β„•π‘”π›Όβˆ— βˆ’ 𝑖𝑛𝑑(π•Š) βŠ†β„• ℕ𝑔𝛼 βˆ— βˆ’ 𝑖𝑛𝑑(ℝ βˆͺ π•Š). Thus β„•π‘”π›Όβˆ— βˆ’ 𝑖𝑛𝑑(ℝ) βˆͺ β„•π‘”π›Όβˆ— βˆ’ 𝑖𝑛𝑑(π•Š) βŠ†β„• ℕ𝑔𝛼 βˆ— βˆ’ 𝑖𝑛𝑑(ℝ βˆͺ π•Š). Theorem 5.8. If ℝ and π•Š subsets of π•Œ, then β„•π‘”π›Όβˆ— βˆ’ 𝑖𝑛𝑑(ℝ ∩ π•Š) = β„•π‘”π›Όβˆ— βˆ’ 𝑖𝑛𝑑(ℝ) ∩ β„•π‘”π›Όβˆ— βˆ’ 𝑖𝑛𝑑(π•Š). Proof: We know that ℝ ∩ π•Š βŠ†β„• ℝ and ℝ ∩ π•Š βŠ†β„• π•Š. By theorem 5.5 d), β„•π‘”π›Όβˆ— βˆ’ 𝑖𝑛𝑑(ℝ ∩ π•Š) βŠ†β„• ℕ𝑔𝛼 βˆ— βˆ’ 𝑖𝑛𝑑(ℝ) and β„•π‘”π›Όβˆ— βˆ’ 𝑖𝑛𝑑(ℝ ∩ π•Š) βŠ†β„• ℕ𝑔𝛼 βˆ— βˆ’ 𝑖𝑛𝑑(π•Š). 141 A New Class of Nano Generalized Closed Sets in Nano Topological Spaces Then β„•π‘”π›Όβˆ— βˆ’ 𝑖𝑛𝑑(ℝ ∩ π•Š) βŠ†β„• ℕ𝑔𝛼 βˆ— βˆ’ 𝑖𝑛𝑑(ℝ) ∩ β„•π‘”π›Όβˆ— βˆ’ 𝑖𝑛𝑑(π•Š)(1) Next, let π‘₯ ∈ β„•π‘”π›Όβˆ— βˆ’ 𝑖𝑛𝑑(ℝ) ∩ β„•π‘”π›Όβˆ— βˆ’ 𝑖𝑛𝑑(π•Š) Then π‘₯ ∈ β„•π‘”π›Όβˆ— βˆ’ 𝑖𝑛𝑑(ℝ) and π‘₯ ∈ β„•π‘”π›Όβˆ— βˆ’ 𝑖𝑛𝑑(π•Š) Hence π‘₯ is a β„•π‘”π›Όβˆ—-interior point of both sets ℝ and π•Š. It follows that ℝ and π•Š is a β„•π‘”π›Όβˆ—-nbhd of π‘₯. Thus π‘₯ ∈ β„•π‘”π›Όβˆ— βˆ’ 𝑖𝑛𝑑(ℝ ∩ π•Š) Hence β„•π‘”π›Όβˆ— βˆ’ 𝑖𝑛𝑑(ℝ) ∩ β„•π‘”π›Όβˆ— βˆ’ 𝑖𝑛𝑑(π•Š) βŠ†β„• ℕ𝑔𝛼 βˆ— βˆ’ 𝑖𝑛𝑑(ℝ ∩ π•Š)(2) From (1) and (2), we get β„•π‘”π›Όβˆ— βˆ’ 𝑖𝑛𝑑(ℝ ∩ π•Š) = β„•π‘”π›Όβˆ— βˆ’ 𝑖𝑛𝑑(ℝ) ∩ β„•π‘”π›Όβˆ— βˆ’ 𝑖𝑛𝑑(π•Š). Theorem 5.9. Let π•Š be a subset of π•Œ, then a) β„•π‘”π›Όβˆ— βˆ’ 𝑐𝑙(π•Š) is β„•π‘”π›Όβˆ—-closed in π•Œ and β„•π‘”π›Όβˆ— βˆ’ 𝑐𝑙(π•Š) is the smallest β„•π‘”π›Όβˆ—- closed set in π•Œ containing π•Š. b) π•Š is β„•π‘”π›Όβˆ—-closed if and only if β„•π‘”π›Όβˆ— βˆ’ 𝑐𝑙(π•Š) = π•Š. Proof: a) Since the intersection of all β„•π‘”π›Όβˆ—-closed subsets of π•Œ containing π•Š is β„•π‘”π›Όβˆ— βˆ’ 𝑐𝑙(π•Š), β„•π‘”π›Όβˆ— βˆ’ 𝑐𝑙(π•Š) is β„•π‘”π›Όβˆ—-closed. β„•π‘”π›Όβˆ— βˆ’ 𝑐𝑙(π•Š) is contained in every β„•π‘”π›Όβˆ—-closed set containing π•Š. Hence, the smallest ΕŠπ‘”π›Όβˆ—-closed set in π•Œ containing π•Š is β„•π‘”π›Όβˆ— βˆ’ 𝑐𝑙(π•Š). b) Suppose π•Š is β„•π‘”π›Όβˆ—-closed. By the definition of β„•π‘”π›Όβˆ—-closure, β„•π‘”π›Όβˆ— βˆ’ 𝑐𝑙(π•Š) = π•Š Conversely, Suppose β„•π‘”π›Όβˆ— βˆ’ 𝑐𝑙(π•Š) = π•Š. By theorem 3.42, β„•π‘”π›Όβˆ— βˆ’ 𝑐𝑙(π•Š)is the β„•π‘”π›Όβˆ—- closed set. Therefore, π•Š is β„•π‘”π›Όβˆ—-closed. Theorem 5.10 Let ℝ and π•Š be subsets of π•Œ, then a) β„•π‘”π›Όβˆ— βˆ’ 𝑐𝑙(πœ™) = πœ™ b) β„•π‘”π›Όβˆ— βˆ’ 𝑐𝑙(π•Œ) = π•Œ c) π•Š βŠ†β„• ℕ𝑔𝛼 βˆ— βˆ’ 𝑐𝑙(π•Š) d) If ℝ βŠ†β„• π•Š then ℕ𝑔𝛼 βˆ— βˆ’ 𝑐𝑙(ℝ) βŠ†β„• ℕ𝑔𝛼 βˆ— βˆ’ 𝑐𝑙(π•Š) e) β„•π‘”π›Όβˆ— βˆ’ 𝑐𝑙(ℝ βˆͺ π•Š) = β„•π‘”π›Όβˆ— βˆ’ 𝑐𝑙(ℝ) βˆͺ β„•π‘”π›Όβˆ— βˆ’ 𝑐𝑙(π•Š) f) β„•π‘”π›Όβˆ— βˆ’ 𝑐𝑙(ℝ ∩ π•Š) βŠ†β„• ℕ𝑔𝛼 βˆ— βˆ’ 𝑐𝑙(ℝ) ∩ β„•π‘”π›Όβˆ— βˆ’ 𝑐𝑙(π•Š) Proof: a), b), c), d) follows from the definition of β„•π‘”π›Όβˆ—-closure. e) We know that ℝ βŠ†β„• ℝ βˆͺ π•Š and π•Š βŠ†β„• ℝ βˆͺ π•Š. By d), β„•π‘”π›Όβˆ— βˆ’ 𝑐𝑙(ℝ) βŠ†β„• ℕ𝑔𝛼 βˆ— βˆ’ 𝑐𝑙(ℝ βˆͺ π•Š),β„•π‘”π›Όβˆ— βˆ’ 𝑐𝑙(π•Š) βŠ†β„• ℕ𝑔𝛼 βˆ— βˆ’ 𝑐𝑙(ℝ βˆͺ π•Š). Then β„•π‘”π›Όβˆ— βˆ’ 𝑐𝑙(ℝ) βˆͺ β„•π‘”π›Όβˆ— βˆ’ 𝑐𝑙(π•Š) βŠ†β„• ℕ𝑔𝛼 βˆ— βˆ’ 𝑐𝑙(ℝ βˆͺ π•Š)(1) Next, we prove β„•π‘”π›Όβˆ— βˆ’ 𝑐𝑙(ℝ βˆͺ π•Š) βŠ†β„• ℕ𝑔𝛼 βˆ— βˆ’ 𝑐𝑙(ℝ) βˆͺ β„•π‘”π›Όβˆ— βˆ’ 𝑐𝑙(π•Š) Let π‘₯ βˆ‰ β„•π‘”π›Όβˆ— βˆ’ 𝑐𝑙(ℝ) βˆͺ β„•π‘”π›Όβˆ— βˆ’ 𝑐𝑙(π•Š) β‡’π‘₯ βˆ‰ β„•π‘”π›Όβˆ— βˆ’ 𝑐𝑙(ℝ) and π‘₯ βˆ‰ β„•π‘”π›Όβˆ— βˆ’ 𝑐𝑙(π•Š) By definition of β„•π‘”π›Όβˆ— βˆ’ 𝑐𝑙, β„•π‘”π›Όβˆ— βˆ’ 𝑐𝑙(ℝ) =∩ {𝐹𝑖 : ℝ βŠ†β„• 𝐹𝑖 , 𝐹𝑖 𝑖𝑠 ℕ𝑔𝛼 βˆ— βˆ’ π‘π‘™π‘œπ‘ π‘’π‘‘} and β„•π‘”π›Όβˆ— βˆ’ 𝑐𝑙(π•Š) =∩ {𝐹𝑖 : π•Š βŠ†β„• 𝐹𝑖 , 𝐹𝑖 𝑖𝑠 ℕ𝑔𝛼 βˆ— βˆ’ π‘π‘™π‘œπ‘ π‘’π‘‘}. Then π‘₯ βˆ‰ 𝐹𝑖 for some i. Since ℝ βŠ†β„• 𝐹𝑖 and π•Š βŠ†β„• 𝐹𝑖, ℝ βˆͺ π•Š βŠ† 𝐹𝑖. Therefore π‘₯ βˆ‰ β„•π‘”π›Όβˆ— βˆ’ 𝑐𝑙(ℝ βˆͺ π•Š) Hence β„•π‘”π›Όβˆ— βˆ’ 𝑐𝑙(ℝ βˆͺ π•Š) βŠ†β„• ℕ𝑔𝛼 βˆ— βˆ’ 𝑐𝑙(ℝ) βˆͺ β„•π‘”π›Όβˆ— βˆ’ 𝑐𝑙(π•Š)(2) From (1) and (2) we have, β„•π‘”π›Όβˆ— βˆ’ 𝑐𝑙(ℝ βˆͺ π•Š) = β„•π‘”π›Όβˆ— βˆ’ 𝑐𝑙(ℝ) βˆͺ β„•π‘”π›Όβˆ— βˆ’ 𝑐𝑙(π•Š). f) We know that ℝ ∩ π•Š βŠ†β„• ℝ and ℝ ∩ π•Š βŠ†β„• π•Š. 142 P. Anbarasi Rodrigo and P. Subithra By d), β„•π‘”π›Όβˆ— βˆ’ 𝑐𝑙(ℝ ∩ π•Š) βŠ†β„• ℕ𝑔𝛼 βˆ— βˆ’ 𝑐𝑙(ℝ) and β„•π‘”π›Όβˆ— βˆ’ 𝑐𝑙(ℝ ∩ π•Š) βŠ†β„• ℕ𝑔𝛼 βˆ— βˆ’ 𝑐𝑙(π•Š) Then β„•π‘”π›Όβˆ— βˆ’ 𝑐𝑙(ℝ ∩ π•Š) βŠ†β„• ℕ𝑔𝛼 βˆ— βˆ’ 𝑐𝑙(ℝ) ∩ β„•π‘”π›Όβˆ— βˆ’ 𝑐𝑙(π•Š). 6. Conclusions In this paper, we have introduced nano generalized π›Όβˆ—-closed sets and discussed some of its properties. Then we investigated its relation with many other nano closed sets. Further nano generalized π›Όβˆ—-open sets are defined and its properties and relations with other nano open sets are studied. 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