Ratio Mathematica Volume 47, 2023 ΥG-operator in grill N-topology Antony George A* Davamani Christober M† Abstract In 2017, Lellis Thivagar et.al [4] introduced a closure operator NτG- cl by using the local function ΦG in grill N-topology. In this article, we introduce a new operator ΥG in the same topological space. We study the properties of this new operator which helps us to derive a few equivalent expressions and a characterizing condition, in terms of ΥG. Then a suitability condition for a grill in N-topological space X is formulated. Also, we discuss the characterizing condition for the discussed suitability condition. In addition, we introduce and study Υ̂G–sets and utilize the ΥG -operator to define a generalized open set and their properties. Keywords: Grill N-topology, N-topology suitable for grill, relatively G-dense, anti-co dense. 2020 AMS subject classifications: 54A05, 54A99, 54C10. 1 *The American College, Madurai, India; email: antonygeorge@americancollege.edu.in. †The American College, Madurai, India; email: christober.md@gmail.com. 1Received on November 29, 2022. Accepted on April 28, 2023. Published on June 30, 2023. DOI: 10.23755/rm.v41i0.967. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published under the CC-BY licence agreement. 385 Antony George A and Davamani Christober M 1 Introduction The grill concept is a powerful supporting tool, like nets and filters, in dealing with many a topological concept quite effectively. The idea of grill on a topolog- ical space was first introduced by Coquet [1]. Later Cattopadhyay and Thorn [2] proved the grills are always unions of ultra-filters. Further Roy and Mukherjee [6] established typical topology associated with grill on a topological space (X,τ). Lellis thivagar et.al [4] initiated the concept of grill in N-topological space and a topology NτG was introduced in terms of an operator ΦG, constructed rather naturally from a grill G on a N-topological space (X,τ). In this paper, we endeavour for an investigation in grill-associated N-topology with a new orientation. we introduce a new operator ΥG, defined in terms of the previously introduced operator ΦG, as a kind of dual of ΦG. We study the prop- erties of this new operator which helps us to derive equivalent expressions for the operator ΥG and a characterizing condition, in terms of ΥG. Also, a suitability condition is formed for a grill in N-topological space. Also, we discuss the char- acterizing condition for the discussed suitability condition. Finally, we introduce and study Υ̂G–sets and utilize the ΥG -operator to define a generalized open set and their properties. 2 Prerequisites In this section we recollect some definitions and results which are beneficial in the sequent. By a space X, we mean a grill N-topological space (X,Nτ,G) with N-topology Nτ and grill G on X on which no separation axioms are assumed unless explicitly stated. Definition 2.1. [4] A non-empty collection G of non empty subsets of a N- topo- logical space (X,Nτ) is called a grill on X if, (i) A ∈ G and A ⊆ B ⊆ X =⇒ B ∈ G, (ii) A,B ⊆ X and A∪B ∈ G =⇒ A ∈ G or B ∈ G. Then a N-topological space (X,Nτ) together with a grill G is called a grill N- topological space and it is denoted by (X,Nτ,G). 386 ΥG-operator in grill N-topology Remark 2.2. [4] In (X,Nτ), the following are true. (i) The grill G = P(A)−{∅} is maximal grill in any N-topological space (X,Nτ) (ii) The grill G = {X} is the minimal grill in any N-topological space (X,Nτ). Definition 2.3. [4] Let (X,Nτ,G) be a grill N-topological space and for each A ⊆ X, the operator ΦG(A,Nτ) = {x ∈ X | A ∩ U ∈ G,∀ U ∈ Nτ(x)} is called the local function associated with the grill G and the N-topology Nτ. It is denoted as ΦG(A). For any point x of a N-topological space (X,Nτ), Nτ(x) means the collection of all Nτ-open sets containing x. Theorem 2.4. [4] Let (X,Nτ) be a N-topological space. Then the following are true. (i) If G is any grill on X, then ΦG is an increasing function in the sense that A ⊆ B implies ΦG(A,Nτ) ⊆ ΦG(B,Nτ). (ii) If G1 and G2 are two grills on X with G1 ⊆ G2, then ΦG1 (A,Nτ) ⊆ ΦG2 (A,Nτ) for all A ⊆ X. (iii) For any grill G on X and if A /∈ G, then ΦG(A,Nτ) = ∅. Theorem 2.5. [4] Let (X,Nτ,G) be a grill N-toplogical space. Then for all A,B ⊆ X. (i) ΦG(A∪B) ⊇ ΦG(A) ∪ ΦG(B) (ii) ΦG(ΦG(A)) ⊆ ΦG(A) = Nτ-cl(ΦG(A)) ⊆ Nτ-cl(A). Theorem 2.6. [4] If G is a grill on a N-toplogical space (X,Nτ) with Nτ −{∅} ⊆ G, then for all U ∈ Nτ, U ⊆ ΦG(U). Lemma 2.7. [4] For any grill G on a N-topological space (X,Nτ) and any A,B ⊆ X, ΦG(A) − ΦG(B) = ΦG(A−B) − ΦG(B). Corolary 2.8. [4] Let G is a grill on a N-topological space (X,Nτ) and suppose A,B ⊆ X with B /∈ G. Then ΦG(A∪B) = ΦG(A) = ΦG(A−B). Proposition 2.9. [4] Corresponding to a grill G on a N-topological space (X,Nτ), the operator NτG-cl : P(X) → P(X) defined by NτG-cl(A) = A ∪ ΦG(A), for all A ⊆ X, satisfies Kuratowski’s closure axioms and also there exists a unique topology NτG = {U ⊆ X | NτG-cl(Uc) = Uc} which is finer than Nτ. 387 Antony George A and Davamani Christober M Proposition 2.10. [4] In a grill N-topological space (X,Nτ,G), Nτ ⊆ β(G,Nτ) ⊆ NτG and in particular if G = P(X) −{∅} then Nτ = β(G,Nτ) = NτG. Proposition 2.11. [4] In a grill N-topological space (X,Nτ,G) and A ⊆ X such that A ⊆ ΦG(A) , then Nτ-cl(A) = NτG-cl(A) = Nτ-cl(ΦG(A)) = ΦG(A). Definition 2.12. [9] In (X,Nτ), S ⊆ X then S is Nτ-dense if Nτ-cl(S) = X. 3 ΥG-operator via grills In this section, we suggest a new operator is called ΥG(A,Nτ) (upsilon) in grill N-topological space, and take up some basic associated results. Throughout this section (X,Nτ,G) denotes a grill N-topological space. Definition 3.1. Let G be a grill on a N-topological space (X,Nτ). We define a map ΥG : P(X) → P(X), given by ΥG(A,Nτ) = X − ΦG(X − A) for any A ⊆ X. We shall simply write ΥG(A), assuming that the grill G under consideration is understood. Remark 3.2. It follows from theorem 2.5 (ii) that ΥG is open in (X,Nτ) for any subset A of X. Thus ΥG treated as a mapping from P(X) to Nτ. Remark 3.3. In view of Theorem 2.4(ii) it turns out that for two grills G1 and G2 on X, G1 ⊆ G2 =⇒ ΥG1 (A) ⊇ ΥG2 (A). But for a given grill G on X, ΥG is increasing in the sense that whenever A ⊆ B ⊆ X, then ΥG(A) ⊆ ΥG(B). This is again an immediate consequence of theorem 2.4 (i); however it may so happen that ΥG(A) ⊆ ΥG(B) even if A * B. The following is an example to justify our contention. Example 3.4. Let N = 3 and X = {s,t,u} and consider τ1O(X) = {∅,X,{t,u}}, τ2O(X) = {∅,X,{s}} and τ3O(X) = {∅,X,{s},{t,u}}. Then 3τO(X)={∅,X,{s}, {t,u}} is a tri topology and consider the grill G = {{s},{u},{s,u},{s,t},X}. Thus (X, 3τ,G) is a grill tri topological space on X. Now, ΦG({s}) = {s} and ΦG({t}) = ∅. Then ΥG({t,u}) = X − ΦG({s}) = {t,u} and ΥG({s,u}) = X − ΦG({t}) = X. Thus ΥG({t,u}) ⊆ ΥG({s,u}) although {t,u} * {s,u}. Theorem 3.5. Let (X,Nτ,G) be a grill N-topological space. Then the following statements are true: 388 ΥG-operator in grill N-topology (i) If S ∈ NτG then S ⊆ ΥG(S). (ii) If S,T ⊆ X then ΥG(S ∩T) = ΥG(S) ∩ ΥG(T). (iii) If S ⊆ X and S /∈ G, then ΥG(S) = X − ΦG(X). (iv) If S,T ⊆ X with T /∈ G, then ΥG(S) = ΥG(S −T) = ΥG(S ∪T). (v) If S,T ⊆ X with (S −T) ∪ (T −S) /∈ G, then ΥG(S) = ΥG(T). Proof. (i) In fact, S ∈ NτG =⇒ ΦG(X −S) ⊆ X −S by result 2.9, S ⊆ X − ΦG(X −S) = ΥG(S). (ii) ΥG(S ∩T) = X − ΦG(X − (S ∩T)) = X − ΦG[(X −S) ∪ ΦG(X −T)] = [X − ΦG[(X −S)] ∩ ΦG[(X −T)] = ΥG(S) ∩ ΥG(T). (iii) ΥG(S) = X −ΦG(X −S) = X − [ΦG(X −S)−ΦG(S)] = X − [ΦG(X)− ΦG(S)] = X − ΦG(X). (iv) ΥG(S − T) = X − ΦG((X − S) ∪ T) = X − [ΦG(X − S) ∪ ΦG(T)] = X − ΦG(X −S) = X − ΥG(S). (v) Let (S−T)∪(T −S) /∈ G so that S−T,T −S /∈ G. Then by using corollary 2.8 we Have ΥG(S) = ΥG((T−(T−S))∪(S−T) = ΥG(T−(T−S)) = ΥG(T). Remark 3.6. From (ii) of the above theorem we see that the operator ΥG is dis- tributive over finite intersection. That is not necessarily true for finite union which is shown below. Example 3.7. Let N = 2 and X = {1, 2, 3} and consider τ1O(X)={∅,X,{1, 2}}, τ2O(X)={∅,X}. Then 2τO(X)={∅,X,{1, 2}}. Consider the grill G = {{1},{1, 2}, {2},{1, 3},{2, 3},X}. Now ΦG({1, 3}) = {1, 2, 3} =X= ΦG{2, 3} and ΦG({3}) = Φ. Then ΥG({1})=X − ΦG({2, 3}) = ∅, ΥG({2}) = X − ΦG({1, 3}) = ∅ and ΥG({1, 2}) = X − ΥG({3}) = X. Thus ΥG({1}) ∪ ΥG({2}) 6= ΥG({1, 2}). Next we derive two equivalent expressions for ΥG(A) where A ⊆ (X,Nτ). Theorem 3.8. In (X,Nτ,G), Let A ⊆ X. Then the following statements are true: (i) ΥG(A) = {x ∈ X : ∃V ∈ Nτ(x) such that V −A /∈ G}. (ii) Let ΥG(A) = ∪{V ∈ Nτ : V −A /∈ G}. 389 Antony George A and Davamani Christober M Proof. (i) x ∈ ΥG(A) iff x /∈ ΦG(X−A) ⇐⇒ there exist V ∈ Nτ(x) such that V −A = V ∩ (X −A) /∈ G ⇐⇒ x ∈ R.H.S. (ii) Let A# = ∪{V ∈ Nτ : V −A /∈ G}. Now x ∈ A# then there exists V ∈ Nτ with x ∈ V such that V − A /∈ G which implies that there exist V ∈ Nτ(x) such that V − A /∈ G. Thus by (i), x ∈ ΥG(A). From this it is clear that ΥG(A) ⊆ A#. Remark 3.9. Let G = P(x)−{∅}, then by theorem 3.8 (ii) ΥG(A) = ∪{U ∈ Nτ : U −A = ∅} = ∪{U ∈ Nτ : U ⊆ A} = Nτ–int(A), for any space (X,Nτ). Corolary 3.10. Let (X,Nτ,G) be a grill N-topological space and A ⊆ X .Then A∩ ΥG(A) = NτG-intA Proof. We have, x ∈ A ∩ ΥG(A) =⇒ x ∈ A and x ∈ ΥG(A) =⇒ x ∈ A and there exist V ∈ Nτ(x) such that V − A /∈ G (by theorem 3.8 (i)) which implies V − (V − A) is a NτG-open neighbourhood of x such that V − (V − A) ⊆ A =⇒ x ∈ A. Again x ∈ NτG-intA implies there exist a NτG-open neighbourhood U −B of x, where U ∈ Nτ and B /∈ G, such that x ∈ U −B ⊆ A =⇒ U − A ⊆ B and U − A /∈ G. So by theorem 3.8 (i) x ∈ ΥG(A). Thus x ∈ A∩ ΥG(A) = NτG-intA. Theorem 3.11. Let (X,Nτ,G) be a grill N-topological space and if A ∈ Nτ then ΥG(A) = ∪{S ∈ Nτ : S∆A /∈ G}. Proof. Let A# = ∪{S ∈ Nτ : S∆A /∈ G}. Then by Theorem 3.8 (ii), A# ⊆ ΥG(A). Now, x ∈ ΥG(A) which implies there exist S ∈ Nτ(x) such that S−A /∈ G (by theorem 3.8 (i)). Let V = S ∪ A ∈ Nτ. Then V ∆A = S − A /∈ G and x ∈ V ∈ Nτ. Thus x ∈ A#. From the result so far, we arrive at the following simple and alternative de- scription of the topology NτG in terms of our introduced operator. Theorem 3.12. If (X,Nτ,G) is a grill N-topological space then NτG = {S ⊆ X,S ⊆ ΥG(S)}. Proof. T = {S ⊆ X : S ⊆ ΥG(S)}. In fact, ∅ ⊆ ΥG(∅) =⇒ ∅ ∈ T . ΥG(X) = X − ΦG(X −X) = X − ΦG({∅}) = X ∈ T . Now A1,A2 ∈ T then A1 ⊆ ΥG(A1) and A2 ⊆ ΥG(A2) which implies A1 ∩A2 ⊆ ΥG(A1)∩ΥG(A2) = 390 ΥG-operator in grill N-topology ΥG(A1 ∩ A2)(by theorem 3.5(ii)). Again {Ai : i ∈ Λ} ∈ T which implies Ai ⊆ ΥG(Ai) for each i ∈ Λ. This implies Ai ⊆ ΥG(∪i∈ΛAi) for each i ∈ Λ (by remark 3.3) hence ∪i∈ΛAi ⊆ ΥG(∪i∈ΛAi) this implies that ∪i∈ΛAi ∈ T . We will show that T = NτG. Indeed, V ∈ NτG, then V ∈ ΥG(V ) (by theorem 3.5 (i)). This implies that V ∈ T . Conversely, A ∈ T =⇒ A ⊆ ΥG(A) this implies A = A∩ΥG(A) = NτG-intA (Remark 3.9) which implies that A ∈ NτG. 4 Suitable for a grill N-topology In this segment, we intend to do some investigations in respect of NτG, along with certain applications, under the assumption of such a suitability conditions imposed on the concerned grills. Throughout this section (X,Nτ,G) denotes a grill N-topological space. Definition 4.1. Let (X,Nτ,G) be a grill N-topological space and Nτ is called suitable for the grill G if A− ΦG(A) /∈ G, for all A ⊆ X. Example 4.2. Let X = {1, 2, 3}. For N = 3, consider τ1 = {∅,X,{1}},τ2 = {∅,X,{2}},τ3 = {∅,X,{1, 2}} and 3τ = {∅,X,{1},{2},{1, 2},X}. Let G = {{1},{1, 2},{1, 3},X}. For every A ⊆ X, A−ΦG(A) /∈ G. Hence 3τ is suitable for G. Theorem 4.3. In a grill N -topological space (X,Nτ,G), the following are equivalent: (i) Nτ is suitable for the grill G. (ii) For any NτG-closed subset A of X, A− ΦG(A) /∈ G. (iii) For any A ⊆ X and each x ∈ A, there corresponds some U ∈ Nτ(x) with U ∩A /∈ G, it follows that A /∈ G. (iv) A ⊆ X and A∩ ΦG(A) = Φ =⇒ A /∈ G. Proof. (i) =⇒ (ii) It is obvious. 391 Antony George A and Davamani Christober M (ii) =⇒ (iii) Let A ⊆ X and suppose for every x ∈ A there exist U ∈ Nτ(x) such that U∩A /∈ G. Then x /∈ ΦG(A) so that A∩ΦG(A) = Φ. Now as A∪ΦG(A) is NτG-closed, by (ii) we have (A ∪ ΦG(A)) − ΦG(A ∪ ΦG(A)) /∈ G.That is (A∪ΦG(A))−ΦG(A)∪ΦG(ΦG(A)) /∈ G =⇒ (A∪ΦG(A))−ΦG(A) /∈ G (by theorem 2.5 (i)) that is A /∈ G. (iii) =⇒ (iv) If A ⊆ X and A∩ΦG(A) = ∅ then A ⊆ X −ΦG(A). Let x ∈ A. Then x /∈ ΦG(A) implies there exist U ∈ Nτ(x) such that U ∩A /∈ G. Then by (iii), A /∈ G. (iv) =⇒ (i) Let A ⊆ X. We first claim that (A−ΦG(A))∩ΦG(A−ΦG(A)) = ∅. In fact x ∈ (A− ΦG(A)) ∩ ΦG(A− ΦG(A)) =⇒ x ∈ A− ΦG(A) =⇒ x ∈ A and x /∈ ΦG(A) implies there exist U ∈ Nτ(x) such that U ∩ A /∈ G. Now U ∩ (A − ΦG(A)) ⊆ U ∩ A /∈ G =⇒ x /∈ ΦG(A − ΦG(A)), which is contradiction. Hence by (iv), A− ΦG(A) /∈ G. Now we derive, in term of the ΥG, a characterizing condition for a N-topology Nτ to be suitable for a grill G on a N-topological space X. Theorem 4.4. Let G be a grill on a N-topological space X then Nτ is suitable for G iff ΥG(A) −A /∈ G for any A ⊆ X. Proof. Let Nτ be suitable for G and A ⊆ X, We first observe that x ∈ ΥG(A)−A iff ΥG(A) and x /∈ A iff there exists U ∈ Nτ(x) such that x ∈ U −A /∈ G. Thus to each x ∈ ΥG(A)−A, there exist U ∈ Nτ(x) such that U ∩(ΥG(A)−A) /∈ G. As Nτ is suitable for G, by theorem 4.2 we have ΥG(A) − A /∈ G. Conversely, let A ⊆ X, suppose that to each x ∈ A there corresponds some U ∈ Nτ(x) such that U ∩ A /∈ G. We need to show by virtue of theorem 4.2 that A /∈ G. Now, by theorem 3.8 (i) we have, ΥG(X − A) = {x ∈ X : there exists U ∈ Nτ(x) such that U − (X − A) /∈ G} = {x ∈ X : there exists U ∈ Nτ(x) such that U ∩ A /∈ G}. Thus A ⊆ ΥG(X − A) and hence A = ΥG(X − A) ∩ A = ΥG(X −A) − (X −A) /∈ G. Corolary 4.5. If the N-topology Nτ of a space X is suitable for a grill G on X, then ΥG is an idempotent operator i.e., ΥG(ΥG(A)) = ΥG(A) for any A ⊆ X. Proof. Since ΥG is Nτ-open in X, hence ΥG(A) ∈ Nτ for any A ⊆ X and so ΥG(A) ∈ NτG. Hence by theorem 3.5(i) , ΥG(A) ⊆ ΥG(ΥG(A)) for any 392 ΥG-operator in grill N-topology A ⊆ X. Also Nτ is suitable for G, so ΥG(A) ⊆ ΥG(A ∪ B) for some B /∈ G. Thus ΥG(ΥG(A)) ⊆ ΥG(A∪B) = ΥG(A). Theorem 4.6. In (X,Nτ,G), Nτ is suitable for G . Let A ⊆ X and V be a non- null open set such that V ⊆ ΦG(A) ∩ ΥG(A) . Then V −A /∈G and V ∩A ∈ G. Proof. V ⊆ ΦG(A)∩ΥG(A) =⇒ V ⊆ ΥG(A) =⇒ V −A ⊆ ΥG(A)−A /∈ G by theorem 4.3 we get V −A /∈ G. Again V ⊆ ΦG(A) and V 6= ∅ =⇒ V ∩A ∈ G. Theorem 4.7. In (X,Nτ,G), the following assertions are similar: (i) Nτ −{∅}⊆ G (ii) ΥG(∅) = ∅ (iii) If A is Nτ-closed then ΥG(A) −A = ∅ (iv) If A ⊆ X then Nτ-int(Nτ-cl(A)) = ΥG(Nτ-int(Nτ-cl(A))) (v) If A is Nτ-regular open in X then A = ΥG(A) (vi) If V ∈ Nτ then ΥG(V ) ⊆ Nτ-int(Nτ-cl(V )) ⊆ ΦG(V ) Proof. (i) =⇒ (ii) : ΥG(∅) = ∪{V ∈ Nτ : V −∅ /∈ G} by theorem 3.8 (ii) ΥG(∅) = ∪{V ∈ Nτ : V /∈ G} = ∅. (ii) =⇒ (iii) : Let x ∈ ΥG(A) −A then there exists V ∈ Nτ(x) such that x ∈ V −A /∈ G. Since A is Nτ-closed, we obtain x ∈ V −A ∈{U ∈ Nτ : U /∈ G}, a contradiction to ΥG(∅) = ∅. (iii) =⇒ (iv) : Since Nτ-int(Nτ-cl(A)) is Nτ-open, by theorem 3.5 (i) we get Nτ-int(Nτ-cl(A)) ⊆ ΥG(Nτ-int(Nτ-cl(A))). Again using (iii), we get ΥG(Nτ-cl(A)) ⊆ Nτ-cl(A). By remark 3.2, ΥG(Nτ-cl(A)) = Nτ-int(ΥG(Nτ- cl(A)) ⊆ Nτ-int(Nτ-cl(A)).Since ΥG(Nτ-int(Nτ-cl(A))) ⊆ ΥG(Nτ-cl(A)) ⊆ Nτ-int(Nτ-cl(A)). Thus Nτ-int(Nτ-cl(A)) = ΥG(Nτ-int(Nτ-cl(A))). (iv) =⇒ (v): It is trivial. (v) =⇒ (vi): Let V ∈ Nτ. By assumption, ∅ = ΥG(∅) = ∪{V ∈ Nτ : U /∈ G}. By theorem 3.8(ii) we obtain Nτ − {∅} ⊆ G. Then by theorem 2.6 we get V ⊆ ΦG(V ) and hence by Proposition 2.11, we have ΦG(V ) = Nτ-cl(V ). Now, V ⊆ Nτ-int(Nτ-cl(V )) ⊆ Nτ-cl(V ) = ΦG(V ). Since ΥG(V ) ⊆ ΥG(Nτ- int(Nτ-cl(V ))) = Nτ-int(Nτ-cl(V )) ⊆ ΦG(V ). (vi) =⇒ (i): If V ∈ Nτ − G, by theorem 3.5 (i), V ⊆ ΥG(V ) ⊆ ΦG(V ) = ∅. That is V = ∅ by theorem 2.4(i). 393 Antony George A and Davamani Christober M 5 Υ̂G -sets In this segment, we discuss about a new open set Υ̂G in grill N-topological space and investigate some of its properties. Definition 5.1. In (X,Nτ,G) a subset S of X is called a Υ̂G-set if S ⊆ Nτ- cl(ΥG(S)). The group of all Υ̂G-sets in (X,Nτ,G) is signify by Υ̂G(X,Nτ). Proposition 5.2. If {Sα : α ∈ ∆} is a group of nonempty Υ̂G-sets in (X,Nτ,G), then ∪α∈∆Sα ∈ Υ̂G(X,Nτ). Proof. For each α ∈ ∆, Sα ⊆ Nτ-cl(ΥG(Sα)) ⊆ Nτ-cl(ΥG(∪α∈∆Sα)) This implies that ∪α∈∆Sα ⊆ Nτ-cl(ΥG(∪α∈∆Sα)). Thus ∪α∈∆Sα ∈ Υ̂G(X,Nτ). Remark 5.3. The intersection of two Υ̂G-sets need not be a Υ̂G-set and it is shown in the following example. Example 5.4. Let X = {1, 2, 3, 4}. Let N = 3. Consider τ1 = {∅,X,{2, 3}},τ2 = {∅,X,{1, 2, 3}} and τ3 = {∅,X,{1},{1, 2, 3}}. Then 3τ = {∅,X,{1},{1, 2, 3}, {2, 3}} and the grill G = {{1},{2},{1, 3},{1, 2},{1, 4},{2, 3},{2, 4},{1, 2, 3}, {2, 3, 4},{1, 2, 4},{1, 3, 4},{2, 3, 4},X}. Let A = {1, 4} and B = {2, 3, 4} are Υ̂G-sets but A ∩ B is not a Υ̂G-set. For Let A = {1, 4}, ΦG(X − A) = {2, 3, 4} and Υ̂G(A) = {1}. Hence A ⊆ Nτ-cl(Υ̂G(A)) implies that A is Υ̂G- set. For B = {2, 3, 4}, ΦG(X − B) = {1, 4} and Υ̂G(B) = {2, 3}. Hence B ⊆ Nτ-cl(Υ̂G(B)) implies that B is Υ̂G-set. On the other hand, since A∩B = {4}, ΦG(X−(A∩B)) = X and Υ̂G(B) = ∅. Hence A∩B * Nτ-cl(Υ̂G(A∩B)) implies that A∩B is not a Υ̂G-set. Remark 5.5. The intersection of an Nτα-set and Υ̂G-set is a Υ̂G-set. Corolary 5.6. In (X,Nτ,G), if for any S ∈ Nτ then S ⊆ ΥG(S) . Theorem 5.7. Let A ∈ Υ̂G(X,Nτ) on (X,Nτ,G). If U ∈ Nτα then U ∩ A ∈ Υ̂G(X,Nτ). Proof. Assume that A is Nτ-open for every A ⊆ X, G ∩ Nτ-cl(A) ⊆ Nτ- cl(G∩A). Let U ∈ Nτα and A ∈ Υ̂G(X,Nτ). By corollary 5.6, we have U∩A ⊆ 394 ΥG-operator in grill N-topology Nτ-int(Nτ-cl(Nτ-int(U)))∩Nτ-cl(ΥG(A)) ⊆ Nτ-int(Nτ-cl(ΥG(U)))∩Nτ- cl(ΥG(A)) ⊆ Nτ-cl(Nτ-int(Nτ-cl(ΥG(U))) ∩ ΥG(A) = Nτ-cl(Nτ-int(Nτ- cl(ΥG(U) ∩ ΥG(A))) = Nτ-cl(ΥG(U) ∩ ΥG(A)) = Nτ-cl(ΥG(U ∩A)). Hence U ∩A ∈ Υ̂G(X,Nτ). Corolary 5.8. Let A ∈ Υ̂G(X,Nτ) on (X,Nτ,G). If U ∈ Nτ then U ∩ A ∈ Υ̂G(X,Nτ). Definition 5.9. In (X,Nτ,G), if for every relatively nonempty open set L∩K,L ∈ Nτ and (L∩K) ∩E ∈ G then the set E is relatively G-dense in a set K. Next we prove a necessary and sufficient condition for A /∈ Υ̂G(X,Nτ). Theorem 5.10. A set A /∈ Υ̂G(X,Nτ) if and only if there exits x ∈ A such that there is a neighbourhood Vx ∈ Nτ(x) for which X −A is relatively NτG-dense in Vx. Proof. Let A /∈ Υ̂G(X,Nτ). We are to show that there exists x ∈ A and a neighbourhood Vx ∈ Nτ(x) satisfying that X − A is relatively G-dense in Vx. Since A * Nτ-cl(ΥG(A)) , there exits x ∈ X such that x ∈ A but x /∈ Nτ- cl(ΥG(A)). Hence there exists a neighbourhood Vx ∈ Nτ(x) such that Vx ∩ ΥG(A) = ∅. This implies that Vx ∩ (X − ΦG(X − A)) = ∅ and hence Vx ⊆ ΦG(X − A). Let U be any non empty open set in Vx. Since Vx ⊆ ΦG(X − A), therefore U ∩ (X −A) ∈ G which implies that (X −A) is relatively G-dense in Vx. Converse is obvious. Definition 5.11. A space (X,Nτ,G) is said to be anti-co-dense grill if Nτ-{∅}⊆ G Theorem 5.12. Let G be a anti-co dense grill on a space (X,Nτ,G). Then SO(X,NτG) = Υ̂G(X,Nτ). Definition 5.13. A set A ⊆ X in (X,Nτ,G), A is called a ΥA-set if A ⊆ Nτ- int(Nτ-cl(ΥG(A))). The collection ΥA-sets in (X,Nτ,G) is denoted by NτA. From Definitions of 5.1 and 5.13 it follows that NτA ⊆ Υ̂G(X,Nτ). The collec- tion NτA forms a topology finer than Nτ. Theorem 5.14. Let G be a anti-co dense grill on (X,Nτ,G). Then the collection NτA = {A ⊆ X : A ⊆ Nτ-int(Nτ-cl(ΥG(A)))} forms a N-topology on X. 395 Antony George A and Davamani Christober M Proof. (i) It is observed that ∅ ⊆ Nτ-int(Nτ-cl(ΥG(∅)) and X ⊆ Nτ-int(Nτ- cl(ΥG(X), and thus ∅ and X ∈ Nτα. (ii) Let {Aα : α ∈ ∆}⊆ NτA, then ΥG(Aα) ⊆ ΥG(∪Aα) for every α ∈ ∆. Thus Aα ⊆ Nτ-int(Nτ-cl(ΥG(Aα))) ⊆ Nτ-int(Nτ-cl(ΥG((∪Aα))) for every α ∈ ∆, which implies that ∪Aα ⊆ Nτ-int(Nτ-cl(ΥG(∪Aα))). Therefore, ∪Aα ∈ NτA. (iii) Let A,B ∈ NτA. Since ΥG(A) is open in (X,Nτ), we obtain A ∩ B ⊆ Nτ-int(Nτ-cl(ΥG(A))) ∩ Nτ-int(Nτ-cl(ΥG(A))) = Nτ-int(Nτ-cl(ΥG(A) ∩ ΥG(B))). Therefore A∩B ⊆ Nτ-cl(Nτ-int(ΥG(A∩B))). Proposition 5.15. Let (X,Nτ,G) be a grill N-topological space. Then ΥG(A) 6= ∅ if and only if A contains a nonempty NτG-interior. Corolary 5.16. Let (X,Nτ,G) be a grill N-topological space. Then {x} ∈ Υ̂G(X,Nτ) if and only if {x}∈ NτA . Proof. Let {x} ∈ Υ̂G(X,Nτ), therefore by proposition 5.15, {x} is open in (X,Nτ,G). Since {x} ⊆ ΥG({x}) and ΥG({x}) is Nτ-open in (X,Nτ), there- fore {x}⊆ Nτ-int(Nτ-cl(ΥG{x}). 6 Conclusion In this paper, we introduced a new operator in grill N-topological space, using this operator some important properties and equivalent expressions are derived. We arrived a topology NτG using the introduced operator. In addition, suitability condition of a grill with the N-topological space X is formulated. We discuss the characterizing condition for a N-topology to be a suitable for a grill G on X. Also we introduce and study Υ̂G–sets and utilize the ΥG-operator to define a generalized open set and their properties. This concept can be extended to other applicable research areas of topology such as Nano topology, Fuzzy topology, Intuitionistic topology, Digital topology and so on. 396 ΥG-operator in grill N-topology References [1] G. Choquet, Sur les notions de filter et grill, Comptes Rendus Acad. Sci. 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