Ratio Mathematica Volume 44, 2022 Nano Semi* -open sets Reena C * Kanaga M † Abstract In this paper, we introduce a new class of sets called nano semi* -open sets and discuss some of its properties in nano topological space. We also, present nano semi* -interior, nano semi* -closure and study some of its fundamental properties. Keywords: nano semi* -open, nano semi* -closed, nano semi* -interior, nano semi* –closure. 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; reenastephany@gmail.com. † Sec Research Scholar, Reg.No. 21122212092007, Department of Mathematics, St. Mary’s College (Autonomous), (Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli), Thoothukudi-1, TamilNadu, India; kanahaspm@gmail.com. ‡ Received on June 12th, 2022. Accepted on Sep 1 st , 2022. Published on Nov 30th, 2022. doi: 10.23755/rm.v44i0.917. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors.This paper is published under the CC-BY licence agreement. 288 mailto:reenastephany@gmail.com C. Reena and M. Kanaga 1. Introduction The notion of Nano topology was introduced by Lellis Thivagar [8] which was defined in terms of approximations and boundary region of a subset of a universe using an equivalence relation on it and also defined Nano closed sets, Nano-interior and Nano-closure. He also introduced the weak forms of Nano open sets namely Nano - open sets, Nano semi-open sets and Nano pre-open sets. In 2017, the concept of nano semi open sets was introduced by Qays Hatem Imran [3]. In 2014, A. Robert and S. Pious Missier [7] have introduced and studied semi* -open sets in general topology. In this paper we introduce nano semi* -open sets and nano semi * -closed sets in nano topological spaces. We investigate its fundamental properties and find its relation with other Nano sets and study some of its properties. 2. Preliminaries Throughout this chapter (U, (X)) is a nano topological space with respect to X where X ⊆ U, R is an equivalence relation on U, U/R denotes the family of equivalence classes of U by R. Definition 2.1 [8]: 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. Then U is divided into disjoint equivalence classes. Elements belonging to the same equivalence class are said to be discernible 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 . That is (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 possibly defined as X with respect to R and it is denoted by UR(X). That is UR(X)= 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 is denoted by BR(X). That is BR(X) = UR(X) – LR(X). Proposition 2.2 [8] If (U, R) is an approximation space and X, Y ⊆ U, then 1. LR(X) ⊆ X ⊆ UR(X) 2. LR ( ) = UR ( ) = and LR (U) = UR(U) = U 3. UR (X ∪Y) = UR(X) ∪UR(Y) 4. UR (X ∩ Y) ⊆ UR(X) ∩ UR(Y) 5. LR (X ∪Y) ⊇ LR(X) ∪ LR(Y) 6. LR (X ∩ Y) = LR(X) ∩ LR(Y) 7. LR(X) ⊆LR(Y) and UR(X) ⊆UR(Y) whenever X ⊆Y 8. UR(X c ) = [LR(X)] c and LR(X c ) = [UR(X)] c 9. URUR(X) = LR UR(X) = UR(X) 10. LRLR(X) = UR LR(X) = (X) 289 Nano Semi* -open sets Definition 2.3 [8]: Let U be the universe, R be an equivalence relation on U and (X) = {U, , LR(X), UR(X), BR(X)} where X ⊆ U. Then by the proposition 2.2, R(X) satisfies the following axioms: 1. U and (X) 2. The union of the elements of any subcollection of (X) is in (X). 3.The intersection of the elements of any finite subcollection of (X) is in (X). That is (X) is a topology on U called the nano topology on U with respect to X. We call ( , (X)) as the nano topological space. The elements of (X) are called as nano-open sets. Definition 2.4 [8]: If (U, (X)) is a nano topological space with respect to X and if A ⊆ U, then (i) nano interior of A is defined as the union of all nano-open sets contained in A and is denoted by NInt (A). That is, NInt(A) is the largest nano-open subset of A. (ii) nano closure of A is defined as the intersection of all nano-closed sets containing A and it is denoted by NCl(A). That is, NCl(A) is the smallest nano-closed set containing A. Definition 2.5 [2]: Let (U, (X)) be a nano topological space. A subset A of (U, (X)) is called nano generalized-closed (briefly Ng- closed) if NCl(A) ⊆V where A ⊆V and V is Nano-open. The complement of nano generalized -closed set is called as nano generalized-open. Definition 2.6 [2]: For every set A ⊆U, the nano generalized closure of A is defined as the intersection of all Ng- closed sets containing A and is denoted by NCl*(A). Definition 2.7 [2]: For every set A ⊆U, the nano generalized interior of A is defined as the union of all Ng- open sets contained in A and is denoted by NInt*(A). Definition 2.8: Let (U, (X)) be a nano topological space and A ⊆U. Then A is said to be (i) nano -open [8] if A ⊆ N Int (NCl (NInt (A))) (ii) nano semi*-open [1] if A ⊆ NCl*(NInt(A)) (iii) nano semi -open [3] if A ⊆ NCl (NInt (NCl (NInt A))) (iv) nano semi pre-open [6] if A ⊆ NCl (NInt (NCl (A))) (v) nano regular-open [8] if A = NInt (NCl (A)) (vi) nano regular *-open [5] if (vii) nano pre *-open [4] if ⊆ (viii) nano pre-open [8] If A ⊆ NInt (NCl (A)) (ix) nano -open [9], if for each x A, there exists a nano open set G such that x G ⊆ Ncl (A) ⊆ A. The complements of the above-mentioned sets are called their respective nano-closed sets. 290 C. Reena and M. Kanaga 3. Nano Semi* -open sets Definition 3.1: A subset A of a nano topological space is called nano semi* - open if there is a nano -open set G in U such that ⊆ ⊆ . The collection of all nano semi* -open sets is denoted by . Example 3.2: Let with U/R= {{ },{b,c,d}} Let Then The nano-closed sets are The nano generalized- closed sets are . The nano generalized-open sets are . Theorem 3.3: For a subset A of a nano topological space the following statements are equivalent: (i)A is nano -open. (ii) ⊆ (iii) Proof: (i) (ii) If A is a nano semi * -open, then there is a nano -open set G in U such that ⊆ ⊆ Now ⊆ G= ⊆ A ⊆ ⊆ . (ii) (iii)By assumption, A⊆ . we have (A)⊆ = Now ⊆ implies that ⊆ Therefore (iii) (i) Take G= Then G is a nano -open set in U such that G⊆ ⊆ Therefore by definition 3.1, A is nano semi * - open. Theorem 3.4: Arbitrary union of nano semi * -open set is nano semi * -open. Proof: Let { } be a collection of nano semi * -open sets in nano topological space U. Then there exists a nano -open set such that ⊆ ⊆ for each Hence ∪ ⊆ ⊆ ⊆ ∪ .Since ∪ is nano -open , by definition 3.1 ∪ is nano semi * -open. Remark 3.5: The intersection of two nano semi * - open sets need not be a nano semi * -open as seen from the following example. Example 3.6: Let , .Let .Then , and . Here the subsets } are nano semi * - open sets, but A is not nano semi * - open. 291 Nano Semi* -open sets Theorem 3.7: If A is nano semi * -open in U and B is nano open in X ,then is nano semi * -open in U. Proof: Since A is nano semi * -open in U,there is an nano -open sets G such that ⊆ ⊆ .Since B is nano open , ⊆ ⊆ ⊆ Since is nano -open,by definition 3.1, is nano semi * -open. Theorem 3.8: Every nano -open set is nano semi * -open. Proof: Let A be a nano -open set in U. Then and hence ⊆ Hence A is nano semi * -open. Remark 3.9: The converse of the above theorem is not true as shown in the following example. Example 3.10: Let , .Let . Then and . Clearly the sets{ and are nano semi * -open but not nano - open. Theorem 3.11: Every nano open set is nano semi * -open. Proof: Let A be any nano open set. Since every nano open set is nano -open and hence by theorem 3.8, A is nano semi * -open. Remark 3.12: The converse of the above theorem is not true as shown in the following example. Example 3.13: Let , .Let . Then and , .Clearly the sets and is nano semi * - open but not nano open. Theorem 3.14: Every nano semi * -open set is nano semi * -open. Proof: Let A be any nano semi * open set. Then there is a nano open set G in U such that ⊆ ⊆ .Since every nano open set is nano -open ,A is nano semi * -open. Remark 3.15: The converse of the above theorem is not true as shown in the following example. Example 3.16: Let Let X = }.Then , . . Clearly the sets , are nano semi * -open but not nano semi * -open. Theorem 3.17: Every nano semi * open set is nano semi -open. 292 C. Reena and M. Kanaga Proof: Let A be any nano semi * open set. Then there is a nano -open set G in U such that ⊆ ⊆ .Since ⊆ ,we have ⊆ ⊆ Hence A is nano semi -open. Remark 3.18: The converse of the above theorem is not true as shown in the following example. Example3.19: Let . Let . }.Clearly the subset and are semi nano -open but not nano semi * -open. Theorem 3.20: Every nano semi * open set is nano semi pre-open. Proof: Let A be any nano semi * -open set. Then there is a nano -open set G such that ⊆ ⊆ .Since every nano -open set is nano pre-open and ⊆ , A is nano semi pre-open. Remark 3.21: The converse of the above theorem is not true as shown in the following example. Example 3.22: Let U={ }, U/R= { }.Let X= . . and ={ , , , . Clearly the subsets , , are nano semi pre-open but not nano semi * -open. Theorem 3.23: Every nano regular open set is nano semi * -open. Proof: Let A be any nano regular open set. Since every nano regular open set is nano open and by theorem 3.11, we have A is nano semi * -open. Remark 3.24: The converse of the above theorem is not true as shown in the following example. Example 3.25: Let . Let . . . = { .Clearly the subset is nano semi * -open but not nano regular open. Theorem 3.26: Every nano regular*-open set is nano semi * -open. Proof: Let A be any nano regular*-open set. Since every nano regular*-open set is nano open and by theorem 3.11, we have A is nano semi * -open. 293 Nano Semi* -open sets Remark 3.27: The converse of the above theorem is not true as shown in the following example. Example 3.28: Let Let X = }.Then }, , , . * .Clearly the subsets , , , are nano semi * -open but not nano regular*-open. Theorem 3.29: Every nano semi * -open set is nano pre *-open. Proof: Let A be any nano semi * -open set. Then there is a nano -open set G in U such that ⊆ ⊆ .Since every nano -open set is nano pre * -open, we have A is nano pre*- open. Remark 3.30: The converse of the above theorem is not true as shown in the following example. Example3.31: Let Let X = }.Then { , } and * , { , Clearly the subsets , { , are nano pre * -open but not nano semi * -open. Remark 3.32: The concept of nano semi * -open sets and nano pre-open sets are independent as shown in the following example. Example 3.33: Let Let X = .Then , , and ={ , , , , Clearly the subset is nano semi * -open but not nano pre- open and the subsets , , are nano pre-open but not nano semi * -open. Remark 3.34: The concept nano semi * -open and nano -open sets are independent as shown in the following example. Example 3.35: Let . Let . . and .Clearly the subsets are nano semi * -open but not nano -open and the subsets are nano -open but not semi * -open. Diagram 3.36: From the above discussions we have the following diagram. 294 C. Reena and M. Kanaga 4. Nano semi * - closed sets Definition 4.1: The complement of nano semi * -open set is called as nano semi * - closed. The collection of all nano semi * -open sets is denoted by . Example 4.2: Let with Let . Then The nano-closed sets are {U, }. The nano generalized – closed sets are { }. The nano generalized open sets are { . . Theorem 4.3: Arbitrary intersection of nano semi * -closed sets is nano semi * - closed. Proof: Let be a collection of nano semi * -closed sets in U. Since each is nano semi * - closed, is a nano semi * -open. Since ∪ and hence by thm 3.4, is nano semi * -open. Hence is nano semi * -closed. Remark 4.4: Union of two nano semi * -closed sets need not be nano semi * -closed as shown in the following example. Example 4.5: Consider , U/R = { }. . Then , , and . The sets and are nano semi * -closed but their union ∪ is not nano semi * -closed. Theorem 4.6: In any nano topological space. (i)Every nano -closed set is nano semi * -closed. (ii) Every nano-closed set is nano semi * -closed. (iii) Every nano semi * -closed set is nano semi * -closed. (iv) Every nano semi * -closed set is nano semi -closed. (v) Every nano semi * - closed set is nano semi pre-closed. (vi) Every nano regular closed set is nano semi * -closed. 295 Nano Semi* -open sets (vii) Every nano regular*- closed set is nano semi * -closed. (viii) Every nano semi * - closed set is nano pre *-closed. Proof: (i)Let A be any nano -closed set in U, then U\A is nano -open. By theorem 3.8, U\A is nano semi * -open. Hence A is nano semi * -closed. (ii) Let A be any nano- closed set in U. Then U\A is nano open. By theorem 3.11, U\A is nano semi * -open. Hence A is nano semi * -closed. (iii)Let A be any nano semi * -closed set in U, then U\A is nano semi * -open. By theorem 3.14, U\A is nano semi * -open. Hence A is nano semi * -closed.(iv) Let A be a nano semi * -closed set in U. Then U\A is nano semi * - open. By theorem 3.16, U\A is nano semi -open. Hence A is nano semi -closed. (v) Let A be a nano semi * -closed set in U, then U\A is nano semi * -open. By theorem 3.20, U\A is nano semi pre-open. Hence A is nano semi pre-closed. (vi) Let A be a nano regular closed set in U. Then U\A is nano regular open. By theorem 3.23, U\A is nano semi * -open. Hence A is nano semi * -closed. (vii) Let A be a nano regular*-closed set in U. Then U\A is nano regular*-open. By theorem 3.26, U\A is nano semi * - open .Hence A is nano semi * -closed .(viii) Let A be a nano semi * -closed set in U. Then U\A is nano semi * -open set. By theorem 3.28, U\A is nano pre * -open. Hence A is nano pre * -closed. Remark 4.7: The converse of each of the statements in theorem 4.6 is not true as shown in the following examples. Example 4.8: Let with U/R={ .Let .Then and , and .Clearly the subsets and are nano semi * -closed but not nano closed . Example 4.9: Let with U/R = { }. Let X = { .Then and .Clearly the subsets nano semi * -closed but not nano-closed. Example 4.10: Let with U/R = { }. Let X = . Then and . ={ }. Clearly the subsets are nano semi * -closed but not nano semi * -closed. Example 4.11:Let , .Then and .Clearly the subsets are nano semi - closed but not nano semi * -closed . Example 4.12: Let with Let . Then and . 296 C. Reena and M. Kanaga . Clearly the subsets {b}, {c} are nano semi pre- closed but not nano semi * -closed. Example 4.13: Let with Let . Then , }. . Clearly the subset is nano semi * -closed but not nano regular-closed. Example 4.14: Let withU/R Let . Then , . Clearly the subset is nano semi * - closed but not nano regular*- closed. Example 4.15: Let with Let . Then , . , . Clearly the subsets are nano pre * - closed but not nano semi * -closed. Remark 4.16: The concept of nano semi * -closed sets and nano pre-closed sets are independent as shown in the following example. Example 4.17: Let Let X = . Then ={ , . ={ , . , ,{c,d}, , , . Clearly the subset is nano semi * -closed but not nano pre closed and the subsets , ,{c,d}, , , , , are nano pre- closed but not nano semi * -closed. Remark 4.18: The concept of nano semi * -closed sets and nano - closed sets are independent as shown in the following example. Example 4.19: Let U= ,U/R = { }. Let X= { }. = {U, {d}, }. Then = , .Clearly the subsets are nano semi * -closed but not nano - closed and the subsets are nano -closed but not semi * -closed. 5.Nano semi* - interior and nano semi* -closure Definition 5.1: The nano semi* - interior of A is defined as the union of all nano semi* - open sets contained in A. It is denoted by s* Int(A). Definition 5.2: Let A be a subset of U. A point u in U is called a nano semi* -interior point of A if A contains a nano semi* -open set containing u. 297 Nano Semi* -open sets Theorem 5.3: If A is any subset of a nano topological space ,then (i) s* Int(A) is the largest nano semi* -open set contained in A. (ii)A is nano semi* -open if and only if s* Int(A)=A. Proof: (i) Being the union of all nano semi* -open subsets of A, by theorem 3.4, s* Int(A) is nano semi* -open and contains every nano semi* -open subsets of A. (ii) A is nano semi* -open implies s* Int(A)=A is obvious from definition 5.1. On the other hand, suppose s* Int(A)=A. Hence by (i) s* Int(A) is nano semi* - open and hence A is nano semi* -open. Theorem 5.4: In any nano topological space (U, (X), if A and B are subsets of U, then the following results hold: i) s* Int ( )= ii) s* Int(U)=U iii) s* Int(A)⊆A iv) A⊆B s* Int(A)⊆s* Int(B) v) s* Int(s* Int(A)) = s* Int(A). vi) Int(A)⊆s* Int(A)⊆s Int(A)⊆A vii) s* Int(A∪ )⊇s* Int(A)∪s* Int(B) viii) s* Int(A B) ⊆s* Int(A) s* Int(B) Proof: (i), (ii), (iii) and (iv) follows from definition 5.1. By theorem 5.3(i), s* Int(A) is nano semi* -open and by theorem 5.3(ii), s* Int(s* Int(A)) = s* Int(A). Thus (v) proved. (vi) follows from theorem 3.11 and 3.17. (vii)Since A⊆A∪B, from statement (iv) we have s* Int(A) ⊆ s* Int(A∪B). Similarly, s* Int(B) ⊆ s* Int(A∪B). Then s* Int(A∪ ) ⊇ s* Int(A) ∪ s* Int(B). (viii) Since ⊆ ,from statement (iv) we have s* Int( ) ⊆ s* Int(A).Similarly s* Int( ) ⊆ s* Int(B). Therefore s* Int( ) ⊆ s* Int(A) Int(B). Remark 5.5: In Theorem 5.4(vi), each of the inclusions may be strict and equality may also hold. This can be seen from the following examples: Example 5.6: Let , , .Then and , Let .Then . Let Then int (B)= , , . Here Let Then , , . Let ,Then , . Here . Let .Then , , . Here 298 C. Reena and M. Kanaga Remark 5.7: In Theorem 5.5(vii) and (viii), each of the inclusions may be strict and equality may also hold. This can be seen from the following examples: Example 5.8: Let , ,X= . Then and = { , { . Let B = { .Then ∪ and , , ∪ . Therefore ∪ ∪ LetC , . Then ∪ and , , ∪ Therefore ∪ ∪ Let , .Then and , , . Therefore Let G , Then and , , . Therefore Definition 5.9: If A is a subset of a nano topological space U, the Nano semi * -closure of A is defined as the intersection of all nano semi * -closed sets in U containing A. It is denoted by . Theorem 5.10: If A is any subset of a nano topological space (U, ,then (i) is the smallest nano semi* -closed set in U containing A. (ii)A is nano semi* -closed if and only if Proof: (i) Since is the intersection of all nano semi* -closed subsets of U containing A, by theorem 4.3, it is nano semi* -closed and it is contained in every nano semi* -closed set containing A and hence it is the smallest nano semi* -closed set in U containing A. (ii)If A is nano semi* -closed, then is obvious. Conversely, let , By (i) is nano semi* -closed and hence A is nano semi* -closed. Theorem 5.11: In any nano topological space (U, (X), if A and B are subsets of U, then the following results hold: (i)s* cl( )= (ii)s* cl(U)=U (iii)A⊆s* cl(A) (iv)A⊆B s* cl(A)⊆s* cl(B) (v)s* (s* (A))= s* (A). (vi)A⊆s cl(A)⊆s* cl(A)⊆ cl(A) (vii)s* cl(A∪ )⊇s* cl(A) ∪ s* cl(B) (viii)s* cl(A B)⊆s* cl(A) s* cl(B) 299 Nano Semi* -open sets Proof: (i), (ii), (iii) and (iv) follows from definition 5.7. From theorem 5.10(i) s* cl(A) is the nano semi* -closed and from theorem5.10(ii) s* (s* (A))= s* (A).This proves (v). (vi) follows from theorem 4.6 and 4.9. (vii) Since A⊆A∪B, from statement (iv) we have s*Ncl(A) ⊆ s* Ncl(A∪B). Similarly, s* cl(B) ⊆ s* cl(A∪ B). Then s* cl(A∪B) ⊇s* cl(A) ∪ s* cl(B) (viii) Since ⊆ ,from statement (iv) we have s* cl( ) ⊆ s* cl(A).Similarly s* cl( ) ⊆ s* cl(B). Therefore s* cl( ) ⊆ s* Int(A) s* cl(B). Remark 5.12: In Theorem 5.11(vi), each of the inclusions may be strict and equality may also hold. This can be seen from the following examples: Example 5.13: Let , , , Then , and , { . Let .Then s* cl(A)= s cl(A)= cl(A)=U. Let .Then s* cl(B) , s cl(B) , Here cl(A). Let .Thens* cl(C) , , Here cl(C). Let }. Then s* cl(D) , s cl( , cl(A)= Here cl(D). Let E .Then s* cl(E) , s cl(E , cl(E)= Here cl(E). 6. Conclusions In this article, we have introduced nano semi* -open sets and nano semi * -closed sets in nano topological spaces and studied their characterizations with other nano open sets. A diagramatic explanation gives a clear explanation of this article. References [1] J. Arul Jesti, K. Heartlin, A New Class of Nearly Open Sets in Nano topological Spaces, International Journal of Advanced Science and Technology, Vol.29, no.9s, pp.4784-4793, 2020. 300 C. Reena and M. Kanaga [2] K. Bhuvaneswari, K. MythiliGnanapriya, on nano generalized closed sets in nano topological space. International Journal of Scientific and Research Publications, Volume 4, Issues 5, ISSN 2250- 3153, May 2014. [3] Qays Hatem Imran, On nano semi alpha open sets, Journal of Science and Arts, Year 17, No. 2(39), pp. 235-244, 2017. [4] C. Reena, R. Raxi and S. Buvaneswari, Nano pre*- open sets, proceeding of the international conference on Analysis and Applied Mathematics-2022, organized by Ayya Nadar Janaki Ammal College (Autonomous), Sivakasi. [5] C. Reena, B. Santhalakshmi and S.M Janu Priyadharshini , Nano regular*- open sets, Proceeding of the international conference on Analysis and Applied Mathematics-2022, organized by Ayya Nadar Janaki Ammal College (Autonomous), Sivakasi. [6] A. Revathy, G. Ilango, On nano - open sets. Int.J.ENG. Contemp. Math sci 1(2),1- 6(2015). [7] A. Robert and S. Pious Missier, A new class of sets weaker than -open sets, International Journal of Mathematics and soft Computing, Vol.4, No.2., 197 - 206. 2014. [8] M.L. Thivagar, C. Richard, On nano forms of weekly open sets. Int. J. Math. Stat: Inven. 1(1), 31 – 37(2013). [9] A. Vadivel, P. Dhanasekaran, G. Saravanakumar, M. Angayarkanni, Generalizations of Nano - closed sets in nano topological spaces. 301