Ratio Mathematica Volume 45, 2023 N�̂�*s-Continuous functions in Nano Topological Spaces M. Anto * J. Carolinal † Abstract The aim of this paper is to introduces N�̂�*s-continuous function in nano topological spaces and we also study the relation between N�̂�*s-irresolute functions and N�̂�*s- continuous functions in different closed sets. Keywords: �̂�*s-closed set, �̂�*s-continuous functions, N�̂�*s-irresolute. 2010 AMS subject classification: 54C05‡ * Associate Professor, PG and Research Department of Mathematics, Annai Velankanni College, Tholayavattam 629157, India. e-mail: antorbjm@gmail.com. † Assistant Professor, PG and Research Department of Mathematics, Annai Velankanni College, Tholayavattam 629157, India. e-mail carolinalphonse@gmail.com. ‡ Received on July 21, 2022. Accepted on October 15, 2022. Published on January 30, 2023.doi: 10.23755/rm.v45i0.987. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published under the CC-BY licence agreement. 97 M. Anto and J. Carolinal 1. Introduction Topology which is a branch of mathematics, is formally defined as the study of qualitative properties of certain objects that are invariant under a certain kind of transformation, especially those properties that are invariant under a certain kind of equivalence. The term topology was introduced by a German Mathematician called Johnn Benedict Listing in 1847. The modern topology largely based on the idea of set theory was developed by George Cantor in the later part of 19th century. Since its inception the topic has been growing in different level and in various fields. Nano topology is one of the latest feathers in topology that applies to real life situations. Lellis Thivagar was the main brain behind developing the concept of nano topology. It is constructed in terms of lower and upper approximations and boundary region of a subset of a universe. The term “Nano” can be ascribed to any unit of measure. The concept of continuity plays a very major role in general topology and they are now the research topics of many topologists worldwide. Indeed, a significant theme in general topology concerns the variously modified forms of continuity, separation axioms etc., by utilizing generalized open sets. N. Levine [7] introduced the concept of generalized closed sets in 1970. The concept of �̂�*s –closed sets was introduced by M. Anto [12]. In 2013, M. Lellis Thivagar [6] has introduced nano topological space with respect to a subset X of a universe U, which is defined in terms of lower and upper approximation of X. He has also defined nano-closed sets, nano-interior and nano- closure of a set. He has also introduced, among other, some certain weak form of nano open sets such as nano -open sets, nano semi open sets and nano pre-open sets. The aim of this paper is to introduce a new class of sets on nano topological spaces called N�̂�*s –closed sets. Further, we investigate and discuss the relation of this new sets with existing ones. 2.Preliminaries Definition 2.1 Let U be a non-empty finite set of objects called the universe and R be an equivalence relation on U named as the in-discernibility relation. Then U is divided into disjoint equivalence class, Elements belonging to the same equivalence class are said to be in discernible with one another. The pair (U, R) is said to be the approximation space. Let X ⊆ U. • The lower approximation of X with respect to R is the set of all object which can be for certain classifies 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 equivalence classes determined by X U. • The upper approximation of X with respect to R is the 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) = ⋃𝑥∈𝑈{R(x): R(x) ∩ 𝑋 ≠ ∅}. 98 N�̂�*s-Continuous functions in Nano Topological Spaces • The boundary of the 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) = 𝑈𝑅(X) 𝐿𝑅(X). Definition 2.2 If (UR) is an approximation space and X, Y U, then 1. 𝐿𝑅(X) ⊆ X ⊆ 𝑈 (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. 𝑈𝑅(𝑋𝑌) ⊇ 𝑈𝑅(𝑈𝑅(Y) 6. 𝑈𝑅(𝑋 ⋂𝑌) = 𝑈𝑅(𝑋 𝑈𝑅(𝑌) 7. 𝐿𝑅( ) 𝐿𝑅(𝑌) and 𝑈𝑅(X) ⊆ 𝑈𝑅(𝑌) whenever X U 8. 𝑈 ((𝑋𝑐)) [𝐿 [(𝑋)]𝑐 and 𝐿 [(𝑋𝑐)] [𝑈 (𝑋)]𝑐 9. 𝑈𝑅(𝑈𝑅((X) = 𝐿 (𝑈𝑅((X) 𝑈𝑅(X) 10. 𝐿𝑅(𝐿𝑅(X) 𝑈𝑅((𝐿𝑅(X) 𝐿𝑅(X) Definition2.3 Let U be the universe, R be an equivalence relation on U and 𝜏𝑅(X) = {U, ∅, 𝐿𝑅(X), 𝑈𝑅(X), 𝐵𝑅(X)} where X ⊆ U. 𝜏𝑅(X)satisfies the following axioms: 1. U and ∈ 𝜏 (X) 2. The union of elements of any sub collection of 𝜏𝑅(X) is in 𝜏𝑅(X). 3. The intersection of the elements of any finite sub collection of 𝜏𝑅(X) is in 𝜏𝑅(X). That is, 𝜏𝑅(X) forms a topology on U is called the nano topology on U with respect to X. We call (U, 𝜏𝑅(X)) is called the nano topological space. Definition2.4 If (U, 𝜏𝑅(X)) is a nano topological space with respect to X where X ⊆ U and if A ⊆ U, then the nano interior of A is defined as the union of all nano open subsets of A and it is de noted by Nint(A). That is, Nint(A) is the largest nano open subset of A. The nano closure of A is defined as the intersection of all nano closed sets containing A and its denoted by Ncl(A). Ncl(A) is the smallest nano closed set containing A. REMARK 2.5 If 𝜏𝑅(X) is the nano topology on U with respect to X, then the set B= {U, 𝐿𝑅(X), 𝑈𝑅(X), 𝐵𝑅(X)} is the basis for 𝜏𝑅(X). Definition 2.6 A function 𝑓: (U, 𝜏𝑅(X)) (V, 𝜎𝑅(Y)) is called • Nano-continuous [2] if 𝑓−1(A) is nano-closed in (X, 𝜏) for every nano-closed set A in (Y, 𝜎). • Nano -continuous [7] if 𝑓−1(A) is nano 𝛼-closed in (X, 𝜏) for every nano-closed set A in (Y, 𝜎). • Nano semi-continuous [4] if 𝑓−1(A) is nano semi closed in (X, 𝜏) for every nano- closed set A in (Y, 𝜎). • Nano regular-continuous if 𝑓−1(A) is nano regular closed in (X, 𝜏) for every nano- closed set A in (Y, 𝜎). 99 M. Anto and J. Carolinal Definition 2.7 If (U, 𝜏𝑅(X)) is a nano topological space if A U., then A is said to be • N𝑔#𝛼-closed if N cl(A) ⊆ V whenever A V and V is Nĝ-open in (U, 𝜏𝑅(X) • N𝑔∗-closed if Ncl(A) ⊆ V whenever A V and V is Ng-open in (U, 𝜏𝑅(X) • N𝑔∗s- closed if Nscl(A) ⊆ V whenever A V and V is Ng-open in (U, 𝜏𝑅(X)). • Nsĝ-closed if Ncl(A) ⊆ U whenever A U and U is Nsg-open in (U, 𝜏𝑅(X)). • Nĝ -closed if N cl(A) ⊆ U whenever A U and U is Nĝ-open in (U, 𝜏𝑅(X)) • N𝛼𝑔∗s-closed if N cl(A) ⊆ U whenever A U and U is N g-open in (U, 𝜏𝑅(X)). • N𝑔𝛼𝑔 - closed if Ncl(A) ⊆ U whenever A U and U is N g-open in (U, 𝜏𝑅(X)) Definition 2.8 If (U, 𝜏𝑅(X)) is a nano topological space if A U., then A is said to be • N𝑔#𝛼-continuous [9] if 𝑓−1(A) is N𝑔#𝛼-closed in (U, 𝜏𝑅(X)) for every nano-closed set A in (V, 𝜎𝑅(Y)) • N𝑔∗-continuous [12] if 𝑓−1(A) is N𝑔∗-closed in (U, 𝜏𝑅(X)) for every nano-closed set A in (V, 𝑅(Y)) • N𝑔∗s-continuous [12] if 𝑓−1(A) is N𝑔∗s-closed in (U, 𝜏𝑅(X)) for every nano-closed set A in (V, (Y)) • Nĝ-continuous [6] if 𝑓−1(A) is Nĝ-closed in (U, 𝜏𝑅(X)) for every nano-closed set A in (V, 𝜎𝑅(Y)) • Nsĝ-continuous [11] if 𝑓−1(A) is Nsĝ-closed in (U, 𝜏𝑅(X)) for every nano-closed set A in (V, 𝜎𝑅(Y)) • Nĝ -continuous [10] if 𝑓−1(A) is Nĝ -closed in (U, 𝜏𝑅(X)) for every nano-closed set A in (V, 𝜎𝑅(Y)) • N𝛼𝑔∗s-continuous [8] if 𝑓−1(A) is N𝛼𝑔∗s-closed in (U, 𝜏𝑅(X)) for every nano-closed set A in (V, 𝜎𝑅(Y)) • N𝑔𝛼g-continuous [5] if 𝑓−1(A) is N𝑔𝛼g-closed in (U, 𝜏𝑅(X)) for every nano-closed set A in (V, 𝜎𝑅(Y)) Definition 2.9[1] A subset A of a nano topological space (U, 𝜏𝑅(X)) is called N𝑔 *s - closed set if Nscl(A) U whenever A U and U is Nĝ-open in (U, 𝜏𝑅(X)). 3. N𝒈 *s-Continuous Functions In this section we define N𝑔 *s-Continuous functions and discuss some of their properties. Definition 3.1 Let (U, 𝜏𝑅(X)) and (V, 𝜎𝑅(Y)) be two nano topological spaces. Then a mapping : (U, 𝜏𝑅(X)) (V, 𝜎𝑅(Y)) is called N𝑔 *s-continuous if 𝑓−1(S) is N𝑔 *s-closed in (U, 𝜏𝑅(X)) for every nano-closed set S in (V, 𝜎𝑅(Y)). Definition 3.2 A function 𝑓: (U, 𝜏𝑅(X)) (V, 𝜎𝑅(Y)) is N𝑔 *s-irresolute if 𝑓−1(S) is N𝑔 *s closed in (U, 𝜏𝑅(X)) for each N𝑔 *s -closed set S in (V, 𝜎𝑅(Y)). Example 3.3. Let (U, 𝜏𝑅(X)) and (V, 𝜎𝑅(Y)) be two nano topological spaces. Let U = {a, b, c, d} with U/R = {{b, c}, {a}, {d}} and X = {a}. Then the nano topology 𝜏𝑅(X) 100 N�̂�*s-Continuous functions in Nano Topological Spaces ={ ,U,{a}}. Then N𝑔 *s-C(U, 𝜏𝑅(X)) = { , U,{b},{c},{a, b},{a, c}, {b, c}, {b, d}, {c, d}, {a, b, c}, {a, b, d}, {a, c, d }, {b, c, d}}. Let V = {a, b, c, d} with V/R = {{a}, {c}, {b, d}} and Y = {a, b, c}. 𝜎𝑅(Y) = { , V, {a, c}, {b, d}}. (𝜎𝑅(Y)) = {∅, V, {a, c}, {b, d}}. N𝑔 *s-C (V, 𝜎𝑅(Y)) = { , U, {b}, {c}, {a, b}, {a, c}, {b, c}, {b, d}, {c, d}, {a, b, c}, {a, b, d}, {a, c, d}, {b, c, d}}. Define 𝑓: U V as f (a) = b, (b) = a, (c) = d, (d) = c. We have 𝑓−1 (a, c) = {b, d}, 𝑓−1 (b, d) = {a, c}. Thus, the inverse image of every N-closed set in V is N𝑔 *s -closed U. Therefore, f is N𝑔 *s –continuous. 4. Main Results Proposition 4.1 Let (U, 𝜏𝑅(X)) and (V, 𝜎𝑅(Y)) be two nano topological spaces. A function : (U, 𝜏𝑅(X)) (V, 𝜎𝑅(Y)) is Nano-continuous then is N𝑔 *s-continuous. Proof: Let A be any nano closed set in (V, 𝜎𝑅(Y)). Since is Nano continuous, 𝑓−1(A) is nano closed in (U, 𝜏𝑅(X)). Since every nano closed set is N𝑔 *s-closed. Therefore 𝑓−1(A) is N𝑔 *s-closed in (U, 𝜏𝑅(X)). Hence is N𝑔 *s-continuous. Remark 4.2 The converse of the above theorem need not be true, as proved by the following example. Example 4.3 Let (U, 𝜏𝑅(X)) be a nano topological space where U = {a, b, c, d} with U/R = {{b}, {c}, {a, d}} and X = {a, b}. 𝜏𝑅(X) = { , U, {b}, {a, d}, {a, b, d}}. (𝜏𝑅(X))𝑐 = { , U, {a, c, d}, {c}, {b, c}}. Nĝ*s C (U, 𝜏𝑅(X)) = { , U, {b}, {c},{a, c}, {a, d}, {b, c}, {c, d}, {a, b, c}, {a, c, d},{b, c, d}}. NSC (U, 𝜏𝑅(X)) = { , U, {b}, {c}, {b, c}, {a, d}, {a, c, d}}. N C (U, 𝜏𝑅(X)) = { , U, {b}, {c}, {b, c}, {a, d}, {a, c, d}}. Let (V, 𝜎𝑅(Y)) be a nano topological space where V = {a, b, c, d} with V/ R = {{b}, {d}, {a, c}} and Y = {a, d}. Let 𝜎𝑅(Y) = { , V, {d}, {a, c}, {a, c, d}}. (𝜎𝑅(Y))𝑐 = { , V, {a, b, c}, {b}, {b, d}}. Define (a) = d, (b) = a, (c) = b, (d) = c. Let 𝑓−1(b) = {c}, 𝑓−1(b, d) = {a, c}, 𝑓−1(a, b, c) = {b, c, d}. Here {b, c, d} is Nĝ*s-closed but not N-closed. Proposition 4.4 Let (U, 𝜏𝑅(X)) and (V, 𝜎𝑅(Y)) be two nano topological spaces. A function 𝑓: (U, 𝜏𝑅(X)) (V, 𝜎𝑅(Y)) is N -continuous then is N𝑔 *s-continuous. Proof Let A be any nano closed set in (V, 𝜎𝑅(Y)). Since is -continuous, 𝑓−1(A) is nano -closed in (U, 𝜏𝑅(X)). Since every nano -closed set is N𝑔 *s-closed. Therefore 𝑓−1(A) is N𝑔 *s-closed in (U, 𝜏𝑅(X)). Hence is N𝑔 *s-continuous. Remark 4.5 The converse of the above theorem need not be true, as proved by the following example. From the example 3.6, the sub set A= {b, c, d} is not nano -closed set in (U, 𝜏𝑅(X)). Hence f is not nano -continuous. Proposition 4.6 Let (U, 𝜏𝑅(X)) and (V, 𝜎𝑅(Y)) be two nano topological spaces. A function : (U, 𝜏𝑅(X)) (V, 𝜎𝑅(Y)) is Nsemi-continuous then is N𝑔 *s-continuous. 101 M. Anto and J. Carolinal Proof Let A be any nano closed set in (V, 𝜎𝑅(Y)). Since is semi-continuous, 𝑓−1(A) is nano semi-closed in (U, 𝜏𝑅(X)). Since every nano semi-closed set is N𝑔 *s-closed. Therefore 𝑓−1(A) is N𝑔 *s-closed in (U, 𝜏𝑅(X)). Hence is N𝑔 *s-continuous. Remark 4.7 The converse of the above theorem need not be true, as proved by the following example. From the example 3.6, the sub set A= {b, c, d} is not nano semi-closed set in (U, 𝜏𝑅(X)). Hence f is not nano semi-continuous. Proposition 4.8 Let (U, 𝜏𝑅(X)) and (V, 𝜎𝑅(Y)) be two nano topological spaces. A function : (U, 𝜏𝑅(X)) (V, 𝜎𝑅(Y)) is N𝑔#𝛼-continuous then is N𝑔 *s-continuous. Proof Let A be any nano closed set in (V, 𝜎𝑅(Y)). Since is N𝑔#𝛼-continuous, 𝑓−1(A) is N𝑔#𝛼-closed in (U, 𝜏𝑅(X)). Since every N𝑔#𝛼-closed set is N𝑔 *s-closed. Therefore 𝑓−1(A) is N𝑔 *s-closed in (U, 𝜏𝑅(X)). Hence is N𝑔 *s-continuous. Remark 4.9 The converse of the above theorem need not be true, as proved by the following example. Example 4.10 Let (U, 𝜏𝑅(X)) be a nano topological space where U = {a, b, c, d} with U/R = {{a}, {c}, {b, d}} and X = {a}. 𝜏𝑅(X) = { , U, {a}}. Nĝ*sC (U, 𝜏𝑅(X)) = { , U, {b},{c}, {d}, {a, b}, {a, d}, {a, c}, {b, c}, {b, d}, {c, d}, {a, b, d}, {a, b, c,},{a, c, d}}. N𝑔#𝛼C (U, 𝜏𝑅(X)) = { , U, {b}, {c}, {d}, {b, c}, {b, d}, {c, d}, {b, c, d}}. Let (V, 𝜎𝑅(Y)) be a nano topological space where V = {a, b, c, d} with V/ R = {{a}, {c}, {b, d}} and Y = {a, b}. 𝜎𝑅(Y) = { , V, {a}, {b, d}, {a, b, d}, (𝜎𝑅(Y))= { , V, {c}, {a, c}, {b, c, d}}. Define (a) = b, (b) = a, (c) = c, (d) = d. Let 𝑓−1(c) = {c}. 𝑓−1(a, c) ={b, c}, 𝑓−1(b, c, d) ={a, c, d}. Here {a, c, d} is Nĝ*s closed but not N𝑔#𝛼-closed. Proposition 4.11 Let (U, 𝜏𝑅(X)) and (V, 𝜎𝑅(Y)) be two nano topological spaces. A function : (U, 𝜏𝑅(X)) (V, 𝜎𝑅(Y)) is N𝑔 𝛼-continuous then is N𝑔 *s-continuous. Proof Let A be any nano closed set in (V, 𝜎𝑅(Y)). Since is N𝑔 𝛼-continuous, 𝑓−1(A) is N𝑔 𝛼-closed in (U, 𝜏𝑅(X)). Since every N𝑔 𝛼-closed set is N𝑔 *s-closed. Therefore 𝑓−1(A) is N𝑔 *s-closed in (U, 𝜏𝑅(X)). Hence is N𝑔 *s-continuous. Remark 4.12 The converse of the above theorem need not be true, as proved by the following example. Example 4.13 Let (U, 𝜏𝑅(X)) be a nano topological space where U = {a, b, c, d} with U/R = {{a}, {b}, {c, d}} and X = {b, d}. 𝜏𝑅(X) = { , U, {b}, {c, d}, {b, c, d}}. Nĝ*sC (U, 𝜏𝑅(X)) = { , U, {a}, {b}, {a, b}, {a, d}, {a, c}, {c, d}, {a, b, d}, {a, b, c,}, {a, c, d}}. N𝑔 𝛼C (U, 𝜏𝑅(X)) = { , U, {a}, {a, b}, {a, c}, {a, d}, {c, d}, {a, b, c}, {a, b, d}, {a, c, d}}. Let (V, 𝜎𝑅(Y)) be a nano topological space where V = {a, b, c, d} with V/ R = {{a}, {c}, {b, d}} and Y = {a, b, d}. 𝜎𝑅(Y) = { , V, {a, b, d}}, (𝜎𝑅(Y)) = { , V, {c}}. 102 N�̂�*s-Continuous functions in Nano Topological Spaces Define (a) = b, (b) = c, (c) = d, (d) = a. Let 𝑓−1(c) = {b}. Here {b} is Nĝ*s-closed but not N𝑔 𝛼-closed. Proposition 4.14 Let (U, 𝜏𝑅(X)) and (V, 𝜎𝑅(Y)) be two nano topological spaces. A function : (U, 𝜏𝑅(X)) (V, 𝜎𝑅(Y)) is Ng*-continuous then is Nĝ*s -continuous. Proof Let A be any nano closed set in (V, 𝜎𝑅(Y)). Since is Ng*-continuous, 𝑓−1(A) is Ng*-closed in (U, 𝜏𝑅(X)). Since every Ng*-closed set is N𝑔 *s-closed. Therefore 𝑓−1(A) is N𝑔 *s-closed in (U, 𝜏𝑅(X)). Hence is N𝑔 *s-continuous. Remark 4.15 The converse of the above theorem need not be true, as proved by the following example. Example 4.16 Let (U, 𝜏𝑅(X)) be a nano topological space where U = {a, b, c, d} with U/R = {{a}, {c}, {b, d}} and X = {b, c}. 𝜏𝑅(X) = { , U, {c}, {b, d}, {b, c, d}}. (𝜏𝑅(X))𝑐 = { , U, {a, b, d}, {a}, {a, c}}. Nĝ*sC (U, 𝜏𝑅(X)) = { , U, {a}, {c}, {a, b}, {a, c}, {a, d}, {b, d}, {a, b, d}, {a, b, c,}, {a, c, d}}. Ng*C (U, 𝜏𝑅(X)) = { , U, {a}, {a, b}, {a, c}, {a, d}, {b, d}, {a, b, c}, {a, b, d}, {a, c, d}}. Let (V, 𝜎𝑅(Y)) be a nano topological space where V = {a, b, c, d} with V/ R = {{b}, {c}, {a, d}} and Y = {a, c}. 𝜎𝑅(Y)= { , V, {c}, {a, d}, {a, c, d}}. (𝜎𝑅(Y))𝑐= { , V, {a, b, d}, {b}, {b, c}}. Define (a) = c, (b) = d, (c) = b, (d) = a. Let 𝑓−1(a, b, d) = {a, b, c}, 𝑓−1(b, c) = {a, c}, 𝑓−1(b) = {c}. Here {c} is Nĝ*s-closed but not Ng*-closed. Proposition 4.17 Let (U, 𝜏𝑅(X)) and (V, 𝜎𝑅(Y)) be two nano topological spaces. A function : (U, 𝜏𝑅(X)) (V, 𝜎𝑅(Y)) is Ng*s-continuous then is Nĝ*s -continuous. Proof Let A be any nano closed set in (V, 𝜎𝑅(Y)). Since is Ng*s-continuous, 𝑓−1(A) is Ng*s-closed in (U, 𝜏𝑅(X)). Since every Ng*s-closed set is N𝑔 *s-closed. Therefore 𝑓−1(A) is N𝑔 *s-closed in (U, 𝜏𝑅(X)). Hence is N𝑔 *s-continuous. Remark 4.18 The converse of the above theorem need not be true, as proved by the following example. Example 4.19 Let (U, 𝜏𝑅(X)) be a nano topological space where U = {a, b, c, d} with U/R = {{a}, {b}, {c, d}} and X = {a, c}. 𝜏𝑅(X) = { , U, {a}, {c, d}, {a, c, d}}. (𝜏𝑅(X))𝑐 = { , U, {b, c, d}, {b}, {a, b}}. Nĝ*sC (U, 𝜏𝑅(X)) = { , U, {a}, {b}, {a, b}, {b, c}, {b, d}, {c, d}, {a, b, d}, {a, b, c,}, {b, c, d}}. Ng*sC (U, 𝜏𝑅(X)) = { , U, {a}, {a, b}, {b, c}, {b, d}, {c, d}, {a, b, c}, {a, b, d}, {b, c, d}}. Let (V, 𝜎𝑅(Y)) be a nano topological space where V = {a, b, c, d} with V/ R = {{a}, {c}, {b, d}} and Y = {a, b}. 𝜎𝑅(Y) = { , V, {a}, {b, d}, {a, b, d}}. (𝜎𝑅(Y))𝑐= { , V, {b, c, d}, {c}, {a, c}}. Define (a) = d, (b) = c, (c) = b, (d) = a. Let 𝑓−1(b, c, d) = {a, b, c}, 𝑓−1(a, c) = {b, d}, 𝑓−1(c) = {b}. Here {b} is Nĝ*s-closed but not Ng*s-closed. 103 M. Anto and J. Carolinal Proposition 4.20 Let (U, 𝜏𝑅(X)) and (V, σR′(Y)) be two nano topological spaces. A function : (U, 𝜏𝑅(X)) (V, 𝜎𝑅(Y)) is Nsĝ-continuous then is Nĝ*s -continuous. Proof Let A be any nano closed set in (V, σR′(Y)). Since is Nsĝ-continuous, 𝑓−1(A) is Nsĝ -closed in (U, 𝜏𝑅(X)). Since every Nsĝ-closed set is N𝑔 *s-closed. Therefore 𝑓−1(A) is N𝑔 *s closed in (U, 𝜏𝑅(X)). Hence is N𝑔 *s-continuous. Remark 4.21 The converse of the above theorem need not be true, as proved by the following example. Let (U, 𝜏𝑅(X)) be a nano topological space where U = {a, b, c, d} with U/R = {{b}, {c}, {a, d}} and X = {a, b}. (U) = { , U, {b}, {a, d}, {a, b, d}}. (𝜏𝑅(X))𝑐 = { , U, {a, c, d}, {c}, {b, c}}. Nĝ*sC (U, 𝜏𝑅(X)) = { , U, {b}, {c}, {a, c}, {a, d}, {b, c}, {c, d}, {a, c, d}, {a, b, c,}, {b, c, d}}. NsĝC (U, 𝜏𝑅(X)) = { , U, {c}, {b, c}, {a, c, d}}. Let (V, 𝜎𝑅(Y)) be a nano topological space where V = {a, b, c, d} with V/ R = {{a}, {b}, {c, d}} and Y = {b, c}. 𝜎𝑅(Y) = { , V, {b}, {c, d}, {b, c, d}}. (𝜎𝑅(Y))𝑐= { , V, {a, c, d}, {a}, {a, b}}. Define (a) = b, (b) = a, (c) = d, (d) = c. Let 𝑓−1(a, c, d) = {b, c, d}, 𝑓−1(a, b) = {a, b}, 𝑓−1(a) = {b}. Here {b} is Nsĝ-closed but not Ng*s-closed. Proposition 4.22 Let (U, 𝜏𝑅(X)) and (V, 𝜎𝑅(Y)) be two nano topological spaces. A function : (U, 𝜏𝑅(X)) (V, 𝜎𝑅(Y)) is Ng𝛼𝑔-continuous then is Nĝ*s -continuous. Proof Let A be any nano closed set in (V, 𝜎𝑅(Y)). Since is Ng𝛼𝑔-continuous, 𝑓−1(A) is Ng𝛼𝑔-closed in (U, 𝜏𝑅(X)). Since every Ng𝛼𝑔-closed set is N𝑔 *s-closed. Therefore 𝑓−1(A) is N𝑔 *s-closed in (U, 𝜏𝑅(X)). Hence is N𝑔 *s-continuous. Remark 4.23 The converse of the above theorem need not be true, as proved by the following example. Example 4.24 Let (U, 𝜏𝑅(X)) be a nano topological space where U = {a, b, c, d} with U/R = {{b}, {c}, {a, d}} and X = {a, b}. 𝜏𝑅(X) = { , U, {b}, {a, d}, {a, b, d}}. (𝜏𝑅(X))𝑐 = { , U, {a, c, d}, {c}, {b, c}}. Nĝ*sC (U, 𝜏𝑅(X)) = { , U, {b}, {c}, {a, c}, {a, d}, {b, c}, {c, d}, {a, c, d}, {a, b, c,},{b, c, d}}. NsĝC (U, 𝜏𝑅(X)) = { , U, {c}, {b, c}, {a, c, d}}. Let (V, 𝜎𝑅(Y)) be a nano topological space where V = {a, b, c, d} with V/ R' = {{a}, {b}, {c, d}} and Y = {b, c}. 𝜎𝑅(Y) = { , V, {b}, {c, d}, {b, c, d}}. (𝜎𝑅(Y))𝑐= { , V, {a, c, d}, {a}, {a, b}}. Define (a) = b, (b) = a, (c) = d, (d) = c. Let 𝑓−1(a, c, d) = {b, c, d}, 𝑓−1(a, b) = {a, b}, 𝑓−1(a) = {b}. Here {b} is Nsĝ-closed but not Ng*s-closed. Proposition 4.25 Let (U, 𝜏𝑅(X)) and (V, 𝜎𝑅(Y)) be two nano topological spaces. A function : (U, 𝜏𝑅(X)) (V, 𝜎𝑅(Y)) is N𝛼𝑔∗𝑠-continuous then is Nĝ*s -continuous. Proof Let A be any nano closed set in (V, 𝜎𝑅(Y)). Since is N𝛼𝑔∗𝑠-continuous, 𝑓−1(A) is N𝛼𝑔∗𝑠-closed in (U, 𝜏𝑅(X)). Since every N𝛼𝑔∗𝑠-closed set is N𝑔 *s-closed. Therefore 𝑓−1(A) is N𝑔 *s-closed in (U, 𝜏𝑅(X)). Hence is N𝑔 *s-continuous. 104 N�̂�*s-Continuous functions in Nano Topological Spaces Remark 4.26 The converse of the above theorem need not be true, as proved by the following example. Example 4.27 Let (U, 𝜏𝑅(X)) be a nano topological space where U = {a, b, c, d} with U/R = {{b}, {c}, {a, d}} and X = {b, c}. 𝜏𝑅(X) = { , U, {b, c}}. (𝜏𝑅(X))𝑐 = { , U, {a, d}} Nĝ*sC(U, 𝜏𝑅(X)) = { , U, {a},{d},{a, b}, {a, d}, {b, c}, {b, d}, {c, d}, {a, b, d}, {a, c, d}, {a, b, c,},{b, c, d}}. N𝛼𝑔∗𝑠C (U, 𝜏𝑅(X)) = { , U, {a}, {d}, {a, d}}. Let (V, σR′(Y)) be a nano topological space where V = {a, b, c, d} with V/ R' = {{b}, {c}, {a, d}} and Y = {a, c}. 𝜎𝑅(Y) = { , V, {c}, {a, d}, {a, c, d}}. (𝜎𝑅(Y))𝑐= { , V, {a, b, d}, {b}, {b, c}}. Define (a) = b, (b) = c, (c) = d, (d) = a. Let 𝑓−1(a, b, d) = {a, c, d}, 𝑓−1(b, c) = {a, b}, 𝑓−1(b) = {a}. Here {a, b} is N𝑔 *s-closed but not Ng*s-closed. Proposition 4.28 Composition of two Nĝ*s-continuous function need not be Nĝ*s-continuous. Let (U, 𝜏𝑅(X)) be a nano topological space where U = {a, b, c, d} with U/R = {{b}, {c}, {a, d}} and X = {a, b}. 𝜏𝑅(X) = { , U, {b}, {a, b, d}, {a, d}}. (𝜏𝑅(X))𝑐 = { , U, {a, c, d}, {c}, {b, c}} Nĝ*sC(U, 𝜏𝑅(X)) = { , U, {b},{c},{a, c}, {a, d}, {b, c}, {c, d}, {a, b, c}, {a, c, d}, {b, c, d}}. Define (a) = d, (b) = a, (c) = b, (d) = c. Let 𝑓−1(a, b, c) = {b, c, d}, 𝑓−1(b, d) = {a, c}, 𝑓−1(b) = {c}. Here {a, b} is N𝑔 *s-but not Ng*s-closed. Let (V, 𝜎𝑅(Y)) be a nano topological space Let V = {a, b, c, d} with V/ R = {{b}, {d}, {a, c}} and Y = {a, d}. 𝜎𝑅(Y) = { , V, {d}, {a, c}, {a, c, d}}. (𝜎𝑅(Y))𝑐= { , V, {a, b, c}, {b}, {b, d}}. Nĝ*sC(U, 𝜏𝑅(X)) = { , U, {b},{d},{a, b}, {a, c}, {b, c}, {b, d}, {a, b, c}, {a, b, d}, {b, c, d}}. Let (W, (Z)) be a nano topological space Let W = {a, b, c, d} with W/ R = {{a}, {c}, {b, d}} and Z = {b, c}. (Z) = { , W, {c}, {b, d}, {b, c, d}}. ((Z) )𝑐 = { , V, {a, b, d}, {a}, {a, c}}. Define (a) = b, (b) = a, (c) = d, (d) = c. Let 𝑔−1(a, b, d) = {a, b, c}, 𝑔−1(a, c) = {b, d}, 𝑔−1(a) = {b}. Now, ∶ (U, 𝜏𝑅(X)) (W, (Z)) by ( (a) = c, ( (b) = b, ( c) = a, ( (d) = d, 𝑔−1 (𝑓−1(a)) = a, 𝑔−1(𝑓−1(a, c)) =(a, c), 𝑔−1(𝑓−1(a, b, d)) =(a, b, d) is not N𝑔 *s-closed in U but {a, b, d} is closed in Z. Therefore, is not N𝑔 *s-continuous. Proposition 4.29 Let f (U, 𝜏𝑅(X)) (V, 𝜎𝑅(Y)) be N𝑔 *s be a function. Then following are equivalent. (i) f is N𝑔 *s -continuous. (ii) 𝑓−1(A) is N𝑔 *s -open for each open set A in Y. Proof: (i) (ii) Suppose that f is a N𝑔 *s -continuous. Let A be N-open in U. Then Ac is N-closed in V. Since is N𝑔 *s -continuous, we have 𝑓−1 (Ac) is N𝑔 *s -closed in U. But 𝑓−1(Ac) = [𝑓−1(A)]c. Hence f−1 (A) is Nĝ*s-open in U. 105 M. Anto and J. Carolinal (ii) (i) Suppose that 𝑓−1 (A) is N𝑔 *s -open for each N-open set A in V. Let S be N- closed in V. Then Sc is nano open in V. By assumption, 𝑓−1 (Sc) is N𝑔 *s -open in U and hence 𝑓−1(S) is N𝑔 *s -closed in U. Hence f is Nĝ*s-continuous. Proposition 4.30 Let 𝑓 ∶ (U, 𝜏𝑅(X)) (V, 𝜎𝑅(Y)) be function (i) f is N𝑔 *s-continuous. (ii) For each u in U and for each N-open set B containing f (u), there is a N𝑔 *s -open set A containing u such that (A) B. (iii) (N𝑔 *s cl(A)) Ncl( (A)) for each subset A of U. (iv) N𝑔 *scl 𝑓−1(B)) 𝑓−1(Ncl(f(B)) for each subset B of V. Then (i) (ii) (iii) (i) Proof: (i) (ii) Let u U and B be an open set containing f (u). Then, by (i) f -1(B) is N𝑔 *s -open set of U containing u. If A = 𝑓−1(B), then (A) = (𝑓−1(B)) B. (ii) (iii) Let A be a subset of a space U and f(u) Ncl(f(A)). Then there exists N- open set B of V containing (u) such that B f (A) = . Now, by (ii), there is a N𝑔 *s - open set G containing u such that (u) f (G) B. Hence (A) f (G) = . That is, (A G) = . i.e., A G) = . Therefore, u N𝑔 *scl (A) and also (u) N𝑔 *scl 𝑓 (A). Therefore (N𝑔 *s cl(A) Ncl( (A)) (iii) (iv) Let B be a subset of V such that A = 𝑓−1(B). By (iii), (N𝑔 *scl(A) Ncl( (A) for each subset A of U. Therefore, (N𝑔 *scl 𝑓−1(B)) Ncl ( (𝑓−1 (B))). i.e., (N𝑔 *s cl 𝑓−1(B)) Ncl(B). i.e., N𝑔 *scl 𝑓−1 (B) ⊆ 𝑓−1(Ncl(B)). Lemma 4.31 A subset A of a nano topological space (U, 𝜏𝑅(X)) is N𝑔 *s -open iff F Nsint (A) whenever F A and F is Nĝ-closed. Proposition 4.32 Let B be a N𝑔 *s open (or N𝑔 *s -closed) subset of (V, 𝜏𝑅′(Y)) (satisfying Nsint(B) =Nint(B). Then 𝑓−1 (B) is N𝑔 *s -open (or N𝑔 *s -closed) in (U, 𝜏𝑅(X)). If 𝑓 ∶ (U, 𝜏𝑅(X)) (V, 𝜏𝑅′(Y)) is Nĝ*s-continuous and if image of a Nĝ-closed set in U under is Nĝ closed set in V. Proof. Let B be a N𝑔 *s -open set in V. Let F 𝑓−1 (B) where F is a Nĝ-closed set in U. Then f(F) B holds. By our assumption, f(F) is Nĝ-closed set in V and B be a N𝑔 *s - open set in V. Therefore, by lemma 3.7 f(F) Nsint(B) holds. Again, by our assumption, (F) Nint(B) and hence F ⊆ 𝑓−1 (Nint(B)) holds. Since f is N𝑔 *s - continuous and Nint (B) is N open in V, f -1(int (B)) is N𝑔 *s -open in U So, by lemma 3.7, F Nsint (𝑓−1 (Nint (B))) holds. i.e., F Nsint (𝑓−1 (Nint (B))) Nsint (𝑓−1 (B)) holds. Therefore 𝑓−1 (B) is N𝑔 *s -open. By taking complements, we can show that if B is N𝑔 *s -closed in V, then 𝑓−1 (B) is N𝑔 *s -closed in U. Proposition 4.33 Let 𝑓 ∶ (U, 𝜏𝑅(X)) (V, 𝜎𝑅(Y)) be a function. Then the following are equivalent. (i) is Nĝ*s-irresolute. (ii) 𝑓−1(B) is N𝑔 *s -open for each N𝑔 *s -open set B in V. 106 N�̂�*s-Continuous functions in Nano Topological Spaces Proof: (i) (ii) Suppose that f is N𝑔 *s -irresolute. Let B be N𝑔 *s -open in V. Then is N𝑔 *s -closed in V. Since is N𝑔 *s -irresolute, we have 𝑓−1(𝐵𝑐) is N𝑔 *s -closed in U. But −1 (𝐵𝑐) = [𝑓−1 (B)]c. Therefore 𝑓−1(B) is Nĝ*s-open in U. (ii) (i) N𝑔 *s Suppose that 𝑓−1 (B) is N𝑔 *s -open for each is N𝑔 *s -open set B in V. Let H be is N𝑔 *s closed in V. Then 𝐻𝑐 is N𝑔 *s -open in V. Therefore 𝑓−1 (𝐻𝑐) is N𝑔 *s -open in U. Therefore 𝑓−1 (H) is N𝑔 *s -closed in U. Therefore, f is N𝑔 *s -irresolute. Proposition 4.34 If a function 𝑓 ∶ (U, 𝜏𝑅(X)) (V, 𝜎𝑅(Y)) is N𝑔 *s -irresolute, then is N𝑔 *s -continuous. Proof: Let B be a N-closed set of V. But every N-closed set is N𝑔 *s -closed. Therefore, B is a N𝑔 *s -closed set of V. Since is N𝑔 *s -irresolute, 𝑓−1(B) is N𝑔 *s - closed in U. Hence, by definition 3.2 is N𝑔 *s -continuous. Remark 4.35 The converse of Proposition 3.10 need not be true as seen from the following example. Example 4.36 Let (U, 𝜏𝑅(X)) and (V, 𝜎𝑅(Y)) be a nano topological spaces where U = V = {a, b, c, d} with U/R = {{b}, {c}, {a, d}} and X = {a, c}. Then the nano topology 𝜏𝑅(X) = { , U, {c}, {a, c, d}, {a, d}}. (𝜏𝑅(X)) = { , U, {a, b, d}, {b}, {b, c}}. Then N𝑔 *s C (U, 𝜏𝑅(X)) = { , U, {b}, {c}, {a, b}, {a, d}, {b, c}, {b, d},{a, b, d}, {b, c, d}}. Let V/ R = {{a}, {b, c}, {d}} and Y = {a, c}. Then the nano topology 𝜏𝑅(X) = { , V, {a}, {a, b, c}, {b, c}}. (𝜎𝑅(Y))𝑐= { , U, {b, c, d}, {d}, {a, d}}. Then N𝑔 *sC (U, 𝜏𝑅(X)) = { , U, {a}, {d}, {b, d},{c, d}, {a, d}, {b, c, d}, {a, b, d}, {a, c, d}}. Define 𝑓 ∶ U V as (a) = c, (b) = d, (c) = a, (d) = b. We have 𝑓−1(a) = {c}, f -1(d) = {b}, 𝑓−1(b, d) = {a, d}, 𝑓−1(c, d) = {a, b}, 𝑓−1(a, d) = {b, c}, 𝑓−1(a, b, d) = {b, c, d}, 𝑓−1(a, c, d) = {a, b, c}, 𝑓−1(b, c, d) = {a, b, d}. 𝑓−1(B) is Nĝ*s-closed for each nano-closed set B in V. Hence f: U V is N𝑔 *s continuous. But, 𝑓−1(a, c, d) = {a, b, c} is not N𝑔 *s -closed. Therefore 𝑓 ∶ U V is not N𝑔 *s -irresolute. Proposition 4.37 If a function 𝑓 ∶ (U, 𝜏𝑅(X)) (V, 𝜎𝑅(Y)) is N𝑔 *s -irresolute, then for every subset A of U. Then (N𝑔 *scl(A)) Nscl( (A)). Proof: Let A U. We know that every Ns-closed set is N𝑔 *s-closed set in V. Therefore, we have Nscl( (A)) is N𝑔 *s -closed in V. Since is N𝑔 *s -irresolute, then 𝑓−1(Nscl( (A))) is N𝑔 *s-closed in U. Also A 𝑓−1 ( (A)) ⊆ 𝑓−1 (Nscl ( (A))). Since 𝑓−1(Nscl( (A))) is N𝑔 *s -closed, we have N𝑔 *scl(A) 𝑓−1 (Nscl(f(A))). Therefore 𝑓( N𝑔 *scl(A)) {𝑓−1 (Nscl( (A)))} Nscl( (A)). Proposition 4.38 If a function : (U, 𝜏𝑅(X)) (V, 𝜎𝑅(Y)) is bijective, N𝑔 *s - continuous, Nscl (A) = Ncl(A) for all subsets B in V and if image of a Nĝ-open set is Nĝ-open under , then is N𝑔 *s -irresolute. 107 M. Anto and J. Carolinal Proof: Let B be N𝑔 *s -closed set of V. Let 𝑓−1 ((B)) A where A is Nĝ-open in U. Then 𝑓−1( (B)) (A). Since f is surjective, B (A). Since (A) is Nĝ-open and since B is N𝑔 *s -closed in V, we have Nscl(B) (A). By our assumption, Ncl(B) (A). Since is injective, 𝑓−1(Ncl(B)) A. Since is N𝑔 *s -continuous and since Ncl(B) is N-closed in V, 𝑓−1(Ncl(B)) is Nĝ*s-closed in U. Therefore 𝑓−1((B)) is N𝑔 *s - closed in U and hence is N𝑔 *s - irresolute. Definition 4.39 A function 𝑓 ∶ (U, 𝜏𝑅(X)) (V, 𝜎𝑅(Y)) is called a N𝑔 *s -closed map if (A) is N𝑔 *s -closed in V for every N-closed set A of U Definition 4.40 A function 𝑓 ∶ (U, 𝜏𝑅(X)) (V, 𝜎𝑅(Y)) is called a Nĝ*s-open map if (A) is Nĝ*s-open in V for every N-open set A of U Proposition 4.41 If 𝑓 ∶ (U, 𝜏𝑅(X)) (V, 𝜎𝑅(Y)) is N𝑔 *s -irresolute and A is a N𝑔 *s closed subset of U, then (A) is N𝑔 *s -closed in V. Proof: Let (A) B and B is Nĝ-open in V. Then 𝑓−1( (A)) 𝑓−1 (B). i.e., A ⊆ 𝑓−1 (B). Since f is Nĝ-irresolute, 𝑓−1(B) is Nĝ-open in U. Since A is N𝑔 *s -closed, Ncl(A) 𝑓−1 (B). So, (Ncl(A)) (𝑓−1 (B)). i.e., (Ncl(A)) U. Since is N𝑔 *s -closed and Ncl(A) is N closed in U, f(Ncl(A)) is N𝑔 *s -closed in V. Therefore Nscl ( (Ncl (A)) B. Since (A) ⊆ 𝑓 (Ncl (B)), we have Nscl ( (A)) Nscl ( (Ncl (B)) B. Therefore (A) is N𝑔 *s -closed in V. Proposition 4.42 If 𝑓 ∶ (U, 𝜏𝑅(X)) (V, 𝜎𝑅(Y)) is N-closed and: V W is N𝑔 *s - closed, then o is N𝑔 *s -closed. Proof: Let A be a N-closed set of U. Since is N-closed, (A) is N-closed in V. Since is N𝑔 *s -closed, (A)) is N𝑔 *s -closed in W. Hence o : U W is N𝑔 *s -closed. Proposition 4.43 Let 𝑓 ∶ (U, 𝜏𝑅(X)) (V, 𝜎𝑅(Y)) is N-closed and : (V, 𝜎𝑅(Y)) (W, (Z)) be two maps such that 𝑔o : (U, 𝜏𝑅(X)) (W, 𝜇𝑅(Z)) is N𝑔 *s -open map, if is N continuous and surjective. Proof: Let B be a N-open V. Since f is N-continuous, 𝑓−1 (B) is N-open in U. Since 𝑓−1(B) is N-open in U, o (𝑓−1 (B))) is N𝑔 *s -open in W. i.e., (B) is N𝑔 *s -open in W. Therefore, is a N𝑔 *s -open map. Proposition 4.44 For any bijection 𝑓 (U, 𝜏𝑅(X)) (V, 𝜎𝑅(Y)), the following are equivalent: (i) 𝑓−1 : (V, 𝜎𝑅(Y)) U is Nĝ*s-continuous. (ii) is N𝑔 *s -open. (iii) is N𝑔 *s -closed. 108 N�̂�*s-Continuous functions in Nano Topological Spaces Proof: (i) (ii) Let A be N-open in U. Then U-A is N-closed in U. Since 𝑓−1 is N𝑔 *s - continuous, (𝑓−1)-1 (U-A) = (U – F) = V – (F) is N𝑔 *s -closed in V. Then (F) is N𝑔 *s -open in V. Hence is N𝑔 *s -open. (ii) (iii) Let A be N-closed in U. Then U-A is N-open in U. Since 𝑓 is N𝑔 *s - open, (U – F) = V – (F) is N𝑔 *s -open in V. Then (F) is N𝑔 *s -closed in V. Hence is N𝑔 *s closed. (iii) (i) Let A be N-closed in U. Since : (U, 𝜏𝑅(X)) (V, 𝜎𝑅(Y)) is N𝑔 *s - closed. (A) is N𝑔 *s closed in V. i.e., (𝑓−1)-1(A) N𝑔 *s -closed in V. Therefore 𝑓−1is N𝑔 *s -continuous. 5. Conclusion In this paper, we introduced and studied the concepts of Nĝ*s-continuous and N𝑔 *s - irresolute in nano topological spaces and we compare it with other nano-continuous and irresolute function and proved that composition of two Nĝ*s-continuous functions need not be a Nĝ*s-continuous functions. We also investigate some of its properties and give suitable examples for the reverse which is not true. In future this work will be extended with some real-life applications. References [1] M. Anto and J. Carolinal, “N𝑔 *s-Continuous functions in nano topological spaces” (communicated) in SEAJMMS. [2] M. Lellis Thivagar. M and Carmel Richard, ‘On Nano Continuity’ Mathematical theory of modeling Vol.3, No.7, 2013. [3] Pious Missier, M. Anto, ĝ*s-closed sets in Topological spaces, International Journal of modern Engineering Research, (IJMER),2249- 6645/vol.4/ISS.11/Nov.2014/32 [4] P. Sathishmohan, V. Rajendran, A. Devika and R. Vani on ‘Nano Semi- continuity and nano pre-continuity’, International Journal of Applied Research 2017;3(2):76-79. [5] Qays Hatem Imaran, Murtadha M Abdulkadhim and Mustafa. H. Hadi. ‘On nano generalised alpha generalized closed sets’ in nano topological spaces, General Mathematics notes: Vol 34, issue 2. [6] R. Lalitha, Dr. A. Francina Shalini, ‘On nano generalized^ continuous and irresolute functions in nano topological spaces. International journal of engineering science and computing, vol.7 No.5, may 2017. 109 M. Anto and J. Carolinal [7] R. Thanga Nachiyar, K. Bhuvaneswari,” Nano generalized A-continuous and Nano Ageneralized continuous functions in Nano Topological spaces”, International Journal of Mathematics Trends and Technology-volume 14 number 2 oct2014 [8] S. SathyaPriya, G. Pradeepa, Y. Granaanathan ‘Nano 𝛼𝑔∗𝑠-Continuous function is nano topological spaces’ international conference on new Achievements in multi- disciplinary 26thand 27th september2019,978-93-89107-39-5. [9] V. Kolilavani, S. Visagapriya, “A new class on N𝑔#𝛼-quotient mappings in nano topological space” Journal of scientific research 12(3), 269-277 (2020). [10] V. Rajendran, P. Sathishmohan and M. Malarvizhi ‘On N𝑔 𝛼-continuous function in nano topological spaces, Malaya journal of mathematics vol.5, No.1, 355- 360,2019. [11] V. Rajendran, P. Sathishmohan and R. Mangayarkarasi,’ Ns𝑔 -continuous function in nano topological spaces’, Turkish journal of Computer and Mathematics Education, vol.12, No.10, 2021. [12] V. Rajendran, P. Sathishmohan and R. Nithya Kala, ‘On new class of continuous functions in Nano topological spaces’, Malaya journal of mathematics vol.6, No.2, 385-389, 2018 110