() @ Appl. Gen. Topol. 18, no. 2 (2017), 289-299doi:10.4995/agt.2017.6797 c© AGT, UPV, 2017 Generalized open sets in grill N-topology M. Lellis Thivagar a, I. L.Reilly b, M. Arockia Dasan a and V. Ramesh a a School of Mathematics, Madurai Kamaraj University, Madurai-625 021, Tamil Nadu, In- dia. (mlthivagar@yahoo.co.in, dassfredy@gmail.com, kabilanchelian@gmail.com) b Department of Mathematics, The University of Auckland, New Zealand. (i.reilly@auckland.ac.nz) Communicated by H.-P. A. Künzi Abstract The aim of this paper is to give a systematic development of grill N- topological spaces and discuss a few properties of local function. We build a topology for the corresponding grill by using the local function. Furthermore, we investigate the properties of weak forms of open sets in the grill N-topological spaces and discuss the relationships between them. 2010 MSC: 54A05; 54A99; 54C10. Keywords: Grill N-topological space; GNτ-α open sets; GNτ-semi open sets; GNτ-pre open sets; GNτ-β open sets. 1. Introduction The grill concept proved to be an important and useful tool like nets and filters, for studying some topological concepts such as proximity spaces, closure spaces, the theory of compactifications and other similar extension problems. The idea of grill on a topological space was first introduced by Choquet [4]. Later Chattopadhyay and Thorn [3] proved that grills are always unions of ultra filters. Further Roy and Mukherjee [13] defined and studied the typical topology associated with grill on a given topological space. Recently, Hatir and Jafari [6] and Al-Omari and Noiri [1] investigated new classes of general- ized open sets and the relevant generalizations of continuity in grill topological spaces. Many more researchers [5, 7, 9, 10, 11, 12] defined and established the Received 03 November 2016 – Accepted 09 March 2017 http://dx.doi.org/10.4995/agt.2017.6797 M. Lellis Thivagar, I. L. Reilly, M. Arockia Dasan and V. Ramesh properties of generalized open sets in classical topology. We note that Corson and Michael [5] used the term locally dense for pre open sets. Lellis Thivagar et al. [8] introduced the concept of N-topological space that is a set equipped with τ1, τ2, ..., τN, and also established its open sets. In this paper, we extend the notion of grill topological spaces into the grill N-topological spaces and we obtain a kind of topology by an operator which satisfies Kuratowski’s closure axioms for the corresponding grill. We also investigate the properties of some generalized open sets in grill N-topological spaces. 2. Preliminaries In this section we recall some known results of N-topological spaces and grill topological spaces which are used in the following sections. 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 ([8]). Let X be a non empty set, τ1, τ2, ... , τN be N-arbitrary topologies defined on X and let the collection Nτ be defined by Nτ = {S ⊆ X : S = ( ⋃ N i=1 Ai) ∪ ( ⋂ N i=1 Bi), Ai, Bi ∈ τi}, satisfying the following axioms: (i) X, ∅ ∈ Nτ (ii) ⋃ ∞ i=1 Si ∈ Nτ for all {Si} ∞ i=1 ∈ Nτ (iii) ⋂ n i=1 Si ∈ Nτ for all {Si} n i=1 ∈ Nτ. Then the pair (X, Nτ) is called a N-topological space on X and the elements of the collection Nτ are known as Nτ-open sets on X. A subset A of X is said to be Nτ-closed on X if the complement of A is Nτ-open on X. The set of all Nτ-open sets on X and the set of all Nτ-closed sets on X are respectively denoted by NτO(X) and NτC(X). Definition 2.2 ([8]). Let (X, Nτ) be a N-topological space and S be a subset of X. Then (i) the Nτ-interior of S, denoted by Nτ-int(S), and is defined by Nτ-int(S) = ∪{G : G ⊆ S and G is Nτ-open}. (ii) the Nτ-closure of S, denoted by Nτ-cl(S), and is defined by Nτ-cl(S) = ∩{F : S ⊆ F and F is Nτ-closed}. Theorem 2.3 ([8]). Let (X, Nτ) be a N-topological space on X and A ⊆ X. Then x ∈ Nτ-cl(A) if and only if O ∩ A 6= ∅, for every Nτ-open set O containing x. Definition 2.4 ([4]). A non empty collection G of non empty subsets of a topological space (X, τ) is called a grill on X if (i) A ∈ G and A ⊂ B ⇒ B ∈ G and (ii) A, B ⊂ X and A ∪ B ∈ G ⇒ A ∈ G or B ∈ G. A topological space (X, τ) together with a grill G on X is called a grill topolog- ical space and is denoted by (X, τ, G). For any point x of a topological space (X, τ), τ(x) means the collection of all open sets containing x. c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 290 Generalized open sets in grill N-topology Definition 2.5 ([13]). Let (X, τ, G) be a grill topological space and for every A ⊆ X, the operator ΦG(A, τ) = {x ∈ X : A ∩ U ∈ G, ∀U ∈ τ(x)} is called the local function associated with the grill G and the topology τ. Definition 2.6 ([13]). Corresponding to a grill G on a topological space (X, τ), then the operator τG-cl : P(X) → P(X) defined by τG-cl(A) = A ∪ Φ(A) ∀A ⊆ X, satisfies Kuratowski’s closure axioms and also there exists a unique topology τG = {U ⊆ X : τG-cl(U c) = Uc} which is finer than τ. 3. Closure Operator in Grill N-Topological Spaces In this section we introduce grill N-topological spaces and investigate the properties of the local function ΦG(A, Nτ). Further we derive a topology by the closure operator τG-cl and we discuss some of its properties. Definition 3.1. A non empty collection G of non empty subsets of a N- topological space (X, Nτ) is called a grill on X if (i) A ∈ G and A ⊂ B ⇒ B ∈ G and (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 is denoted by (X, Nτ, G). Particularly, if N = 1, then (X, 1τ = τ, G) is called the grill topological space, if N = 2, then (X, 2τ, G) is called the grill bitopological space, if N = 3, then (X, 3τ, G) is called the grill tritopological space defined on X and so on. Remark 3.2. (i) The grill G = P(X) − {∅} is the 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 3.3. 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 3.4. 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) = ∅. Proof. It trivially follows from the Definition 3.3. � c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 291 M. Lellis Thivagar, I. L. Reilly, M. Arockia Dasan and V. Ramesh Theorem 3.5. Let (X, Nτ, G) be a grill N-topological 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). Proof. We prove the part (ii) only and part(i) is trivial. (ii). If x /∈ Nτ-cl(A), then there exists U ∈ Nτ(x) such that U ∩ A = ∅ /∈ G implies x /∈ ΦG(A). Thus ΦG(A) ⊆ Nτ-cl(A). Now we shall show that Nτ- cl(ΦG(A)) ⊆ ΦG(A). Suppose that x ∈ Nτ-cl(ΦG(A)), then there exists a U ∈ Nτ(x) such that U ∩ ΦG(A) 6= ∅. Let y ∈ U ∩ ΦG(A). Then U ∩ A ∈ G and so x ∈ ΦG(A). Thus Nτ-cl(ΦG(A)) = ΦG(A). Hence ΦG(ΦG(A)) ⊆ Nτ- cl(ΦG(A)) = ΦG(A) ⊆ Nτ-cl(A). � Remark 3.6. Equality does not always hold in (i) of Theorem 3.5. Let N = 2 and X = {a, b, c, d}, and consider τ1O(X) = {∅, X, {a}}, τ2O(X) = {∅, X, {a, b}}. Then 2τO(X) = {∅, X, {a}, {a, b}} is a bitopology and consider the grill G = {{a, b}, {a, b, c}, {a, b, d}, X}. Thus (X, 2τ, G) is a grill bitopological space on X. If A = {a} and B = {b, c}, then ΦG(A)∪ΦG(B) = ∅ ⊂ {b, c, d} = ΦG(A ∪ B). Definition 3.7. 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) ∀A ⊆ X, satisfies Kuratowski’s closure axioms and also there exists a unique topology NτG = {U ⊆ X : NτG-cl(U c) = Uc} which is finer than Nτ. Example 3.8. Let N = 3 and X = {a, b, c} and consider τ1O(X) = {∅, X, {a}}, τ2O(X) = {∅, X, {b}} and τ3O(X) = {∅, X, {a, b}}. Then 3τO(X) = {∅, X, {a}, {b}, {a, b}} is a tritopology and consider the grill G = {{a}, {a, b}, {a, c}, X}. Thus (X, 3τ, G) is a grill tritopological space on X and 3τG = {U ⊆ X : 3τG- cl(Uc) = Uc} = {∅, {a}, {b}, {a, b}, {a, c}, X} which is finer than 3τO(X). Theorem 3.9. (i) If G1 and G2 are two grills on a N-topological space (X, Nτ) with G1 ⊆ G2, then NτG2 ⊆ NτG1. (ii) If G is a grill on a N-topological space (X, Nτ) and B /∈ G, then B is NτG-closed set in (X, NτG). (iii) For any subset A of a N-topological space (X, Nτ) and any grill G on X, ΦG(A) is NτG-closed set in (X, NτG). (iv) If A is a NτG-closed, then ΦG(A) ⊆ A. Proof. (i) U ∈ NτG2 ⇒ NτG2-cl(U c) = Uc ⇒ ΦG2(U c) ⊆ Uc ⇒ ΦG1(U c) ⊆ Uc ⇒ NτG1-cl(U c) = Uc ⇒ U ∈ NτG1. (ii) If B /∈ G, then ΦG(B) = ∅ and NτG-cl(B) = B. (iii) We have, NτG-cl(ΦG(A)) = ΦG(A) ∪ ΦG(ΦG(A)) = ΦG(A) ⇒ ΦG(A) is NτG-closed. (iv) Assume that x /∈ A = NτG-cl(A) ⇒ x /∈ ΦG(A). Thus ΦG(A) ⊆ A. ✷ c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 292 Generalized open sets in grill N-topology Theorem 3.10. Let (X, Nτ, G) be a grill N-topological space. Then the col- lection β(G, Nτ) = {V − A : V ∈ Nτ and A /∈ G} is an open basis for NτG. Proof. Let (X, Nτ, G) be a grill N-topological space and U ∈ NτG and x ∈ U ⇒ X − U is NτG-closed ⇒ ΦG(X − U) ⊆ X − U ⇒ U ⊆ X − ΦG(X − U). Therefore, x ∈ U which implies that x /∈ ΦG(X − U). Then there exists a V ∈ Nτ(x) such that V ∩ (X − U) /∈ G. Let us take A = V ∩ (X − U) /∈ G and we have x ∈ V − A ⊆ U where V is Nτ-open set and A /∈ G. Thus U is the union of sets in β(G, Nτ). Clearly, β(G, Nτ) is closed under finite intersections, that is if V1 − A, V2 − B ∈ β(G, Nτ), then V1, V2 ∈ Nτ and A, B /∈ G and also V1 ∩V2 ∈ Nτ and A∪B /∈ G. Now, (V1 −A)∩(V2 −B) = (V1 ∩V2)−(A∪B) ∈ β(G, Nτ), and hence β(G, Nτ) = {V − A : V ∈ Nτ and A /∈ G} is an open base for NτG. � Theorem 3.11. 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. Proof. Let V ∈ Nτ. Then V = V − ∅ ∈ β(G, Nτ). Hence Nτ ⊆ β(G, Nτ). Now, let A ∈ β(G, Nτ), then there exists V ∈ Nτ and H /∈ G such that A = V − H. Then, NτG-cl(A c) = NτG-cl((V − H) c) = (V − H)c ∪ ΦG((V − H)c) = (V c ∪H)∪(ΦG(V c)∪ΦG(H)). But, H /∈ G, then, by Theorem 3.4(iii), ΦG(V c) ∪ ΦG(H) = ΦG(V c). Since V c is Nτ-closed and by Theorem 3.9(iv), ΦG(V c) ⊆ V c. Thus, NτG-cl(A c) ⊆ Ac and hence A ∈ NτG. In particular, if G = P(X) − {∅}, then NτG = Nτ. Now V ∈ β(G, Nτ) ⇒ V = U − A with U ∈ Nτ and A /∈ G, we have A = ∅, so that V = U ∈ Nτ and so Nτ = β(G, Nτ) = NτG. � Corollary 3.12. Let (X, Nτ, G) be a grill N-topological space. If U ∈ Nτ, then U ∩ ΦG(A) = U ∩ ΦG(U ∩ A), for any A ⊆ X. Proof. Clearly, U ∩ ΦG(A) ⊇ U ∩ ΦG(U ∩ A). On the other hand, let x ∈ U ∩ ΦG(A) and V ∈ Nτ(x). Then U ∩V ∈ Nτ(x) and x ∈ ΦG(A) ⇒ (U ∩V )∩A ∈ G, that is, (U ∩ A) ∩ V ∈ G ⇒ x ∈ ΦG(U ∩ A) ⇒ x ∈ U ∩ ΦG(U ∩ A). Thus U ∩ ΦG(A) = U ∩ ΦG(U ∩ A). � Corollary 3.13. Let (X, Nτ, G) be a grill N-topological space. If Nτ −{∅} ⊆ G, then U ⊆ ΦG(U) for all U ∈ Nτ. Proof. If U = ∅, then ΦG(U) = ∅ = U and if Nτ−{∅} ⊆ G, then ΦG(X) = X. By Corollary 3.12, we have for any U ∈ Nτ −{∅}, U ∩ΦG(X) = U ∩ΦG(U ∩X) and implies U = U ∩ ΦG(U). Thus, ΦG(U) ⊇ U. � Corollary 3.14. Let A be a subset of a grill N-topological space (X, Nτ, G). If U ∈ Nτ, then U ∩ NτG-cl(A) ⊆ NτG-cl(U ∩ A). Proof. Since U ∈ Nτ and by Corollary 3.12, we obtain U ∩ NτG-cl(A) = (U ∩ A) ∪ (U ∩ ΦG(A)) ⊆ (U ∩ A) ∪ ΦG(U ∩ A) = NτG-cl(U ∩ A). � c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 293 M. Lellis Thivagar, I. L. Reilly, M. Arockia Dasan and V. Ramesh 4. Generalized Open Sets in Grill N-Topological Spaces In this section we introduce some weak forms of open sets in grill N-topological spaces and also we discuss the relationships between them. Definition 4.1. Let (X, Nτ, G) be a grill N-topological space and A ⊆ X. Then A is said to be (i) GNτ-open if A ⊆ Nτ-int(ΦG(A)). (ii) GNτ-α open if A ⊆ Nτ-int(NτG-cl(Nτ-int(A))). (iii) GNτ-semi open if A ⊆ NτG-cl(Nτ-int(A)). (iv) GNτ-pre open if A ⊆ Nτ-int(NτG-cl(A)). (v) GNτ-β open if A ⊆ Nτ-cl(Nτ-int(NτG-cl(A))). The set of all GNτ-open (resp. GNτ-α open, GNτ-semi open, GNτ-pre open, GNτ-β open) sets in a grill N-topological space (X, Nτ, G) is denoted by GNτO(X) (resp. GNταO(X), GNτSO(X), GNτPO(X), GNτβO(X)). The complements of GNτ-open ( resp. GNτ-α open, GNτ-semi open, GNτ- pre open, GNτ-β open) sets in a grill N-topological space (X, Nτ, G) are called their respective closed sets and the set of all GNτ-closed (resp. GNτ-α closed, GNτ-semi closed, GNτ-pre closed, GNτ-β closed) sets in a grill N-topological space (X, Nτ, G) is denoted by GNτC(X) (resp. GNταC(X), GNτSC(X), GNτPC(X), GNτβC(X)). For N = 1, then we take G1τO(X) (resp. GαO(X), GSO(X), GPO(X), GβO(X)). For N = 2, then we take G2τO(X) (resp. G2ταO(X), G2τSO(X), G2τPO(X), G2τβO(X)) and so on. We observe that part (iii) of the next theorem is analogous to the 1985 topological space result of Reilly and Vamanamurthy [12]. Theorem 4.2. Let A be a subset of a grill N-topological space (X, Nτ, G). (i) If A is Nτ-open, then A is GNτ-α open. (ii) If A is GNτ-open, then A is GNτ-pre open. (iii) A is GNτ-α open if and only if it is GNτ-semi open and GNτ-pre open. (iv) If A is GNτ-semi open, then A is GNτ-β open. (v) If A is GNτ-pre open, then A is GNτ-β open. Proof. Here we prove part (iii) only, and note that the remaining parts have similar proofs. (iii). Since A is GNτ-α-open, then A ⊆ Nτ-int(NτG-cl(Nτ-int(A))) ⊆ Nτ- int(NτG-cl(A)) and A ⊆ Nτ-int(NτG-cl(Nτ-int(A))) ⊆ NτG-cl(Nτ-int(A)). On the other hand, since A is GNτ-semi open and GNτ-pre open, then A ⊆ Nτ-int(NτG-cl(A)) ⊆ Nτ-int(NτG-cl(NτG-cl(Nτ-int(A)))) ⊆ Nτ-int(NτG- cl(Nτ-int(A))). � The following examples show that the converse of the above theorem need not be true, that GNτ-open sets and Nτ-open sets are independent, and that GNτ-semi open sets and GNτ-pre open sets are independent. c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 294 Generalized open sets in grill N-topology Example 4.3. Let N = 5 and X = {a, b, c} and consider τ1O(X) = {∅, X, {a}}, τ2O(X) = {∅, X, {b}}, τ3O(X) = {∅, X, {a, b}}, τ4O(X) = {∅, X, {a}, {a, b}} and τ5O(X) = {∅, X, {b}, {a, b}}. Then 5τO(X) = {∅, X, {a}, {b}, {a, b}} is a 5τ-topology and consider the grill G = {{a}, {a, b}, {a, c}, X}. Thus (X, 5τ, G) is a grill 5-topological space. Here the set {b} is a 5τ-open but not a G5τ-open set and the set {b, c} is G5τ-β-open but not G5τ-semi open, not G5τ-pre open and not G5τ-α open. Also the set {a, c} is G5τ-semi open but not G5τ-pre open and not G5τ-α open and the set {a, b} is G5τ-pre open but not G5τ-open. Example 4.4. Let N = 3 and X = {a, b, c, d} and consider τ1O(X) = {∅, X}, τ2O(X) = {∅, X, {a}} and τ3O(X) = {∅, X, {a, b}}. Then 3τO(X) = {∅, X, {a}, {a, b}} is a 3τ-topology and consider the grill G = {{a}, {a, b}, {a, c}, {a, d}, {a, b, c}, {a, b, d}, {a, c, d}, X}. Thus (X, 3τ, G) is a grill tritopological space. Here the set {a, b, c} is G3τ-open and G3τ-α open but not a 3τ-open set. Also if N = 2 and X = {a, b, c}, and consider τ1O(X) = {∅, X, {a}} and τ2O(X) = {∅, X}. Then 2τO(X) = {∅, X, {a}} is a 2τ-topology and consider the grill G = {{a, b}, X}. Thus (X, 2τ, G) is a grill bitopological space. Here the set {a, b} is G2τ-pre open but not G2τ-semi open and not G2τ-α open. Theorem 4.5. In a grill N-topological space (X, Nτ, G), the following are true: (i) If Nτ − {∅} ⊆ G, then every Nτ-open set is GNτ-open. (ii) If A ⊆ X is GNτ-open and NτG-closed, then A is Nτ-open. (iii) If A ⊆ X is GNτ-closed, then ΦG(Nτ-int(A)) ⊆ Nτ-cl(Nτ-int(A)) ⊆ A. Proof. (i) Let A be a Nτ-open set and by Corollary 3.13, A = Nτ-int(A) ⊆ Nτ- int(ΦG(A)). (ii) Since A is NτG-closed, A = NτG-cl(A) = A ∪ ΦG(A) ⇒ ΦG(A) ⊆ A and since A is GNτ-open, A ⊆ Nτ-int(ΦG(A)) ⊆ Nτ-int(A). Thus A = Nτ-int(A). (iii) Assume A is GNτ-closed, X − A is GNτ-open and X − A ⊆ Nτ- int(ΦG(X − A)) ⊆ ΦG(X − A). Then by Theorem 3.5, ΦG(X − A) = Nτ-cl(X − A) = X − Nτ-int(A) and now X − A ⊆ Nτ-int(ΦG(X − A)) ⊆ Nτ-int(X − Nτ-int(A)) ⊆ X − Nτ-cl(Nτ-int(A)). Thus Nτ- cl(Nτ-int(A)) ⊆ A and also ΦG(Nτ-int(A)) ⊆ NτG-cl(Nτ-int(A)) ⊆ Nτ-cl(Nτ-int(A)) ⊆ A. � Theorem 4.6. Let (X, Nτ, G) be a grill N-topological space and Ω be an index set. (i) If {Ai}i∈Ω ∈ GNτO(X), then ⋃ i∈Ω Ai ∈ GNτO(X). (ii) If {Ai}i∈Ω ∈ GNταO(X), then ⋃ i∈Ω Ai ∈ GNταO(X). (iii) If {Ai}i∈Ω ∈ GNτSO(X), then ⋃ i∈Ω Ai ∈ GNτSO(X). c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 295 M. Lellis Thivagar, I. L. Reilly, M. Arockia Dasan and V. Ramesh (iv) If {Ai}i∈Ω ∈ GNτPO(X), then ⋃ i∈Ω Ai ∈ GNτPO(X). (v) If {Ai}i∈Ω ∈ GNτβO(X), then ⋃ i∈Ω Ai ∈ GNτβO(X). Proof. We prove part (v) only, and note that the remaining parts have similar proofs. (v). Assume {Ai}i∈Ω ∈ GNτβO(X), then for each i ∈ Ω, Ai ⊆ Nτ-cl(Nτ- int(NτG-cl(Ai))) ⇒ ⋃ i∈Ω Ai ⊆ ⋃ i∈Ω (Nτ-cl(Nτ- int(NτG-cl(Ai)))) = Nτ-cl( ⋃ i∈Ω (Nτ-int(NτG-cl(Ai)))) ⊆ Nτ-cl(Nτ-int( ⋃ i∈Ω (NτG-cl(Ai)))) ⊆ Nτ-cl(Nτ-int (NτG-cl( ⋃ i∈Ω Ai))). This shows that ⋃ i∈Ω Ai ∈ GNτβO(X). � Theorem 4.7. Let (X, Nτ, G) be a grill N-topological space and A, B ⊆ X, then the following statements are true: (i) If A ∈ GNτSO(X) and B ∈ GNταO(X), then A ∩ B ∈ GNτSO(X). (ii) If A ∈ GNτPO(X) and B ∈ GNταO(X), then A ∩ B ∈ GNτPO(X). Proof. Here we prove part (i) only, and note that part (ii) has a similar proof. (i) Since A ⊆ NτG-cl(Nτ-int(A)), B ⊆ Nτ-int(NτG-cl(Nτ-int(A))) and by Corollary 3.14, A ∩ B ⊆ NτG-cl(Nτ-int(A)) ∩ Nτ-int(NτG-cl(Nτ-int(B))) ⊆ NτG-cl(Nτ-int(A)∩Nτ-int(NτG-cl(Nτ-int(B)))) ⊆ NτG-cl(Nτ-int(A)∩NτG- cl(Nτ-int(B))) ⊆ NτG-cl(NτG-cl(Nτ-int(A ∩ B))). This shows that A ∩ B ∈ GNτSO(X). � Lemma 4.8. Let (X, Nτ, G) be a grill N-topological space and A, B ⊆ X, then the following statements are true: (i) If A ∈ GNτSO(X) and B ∈ NτO(X), then A ∩ B ∈ GNτSO(X). (ii) If A ∈ GNτPO(X) and B ∈ NτO(X), then A ∩ B ∈ GNτPO(X). Example 4.9. Let N = 5 and X = {a, b, c} and consider τ1O(X) = {∅, X, {a}}, τ2O(X) = {∅, X, {b}}, τ3O(X) = {∅, X, {a, b}}, τ4O(X) = {∅, X, {a}, {a, b}} and τ5O(X) = {∅, X, {b}, {a, b}}. Then 5τO(X) = {∅, X, {a}, {b}, {a, b}} is a 5τ-topology and consider the grill G = {{a}, {b}, {a, b}, {a, c}, {b, c}, X}. Thus (X, 5τ, G) is a grill 5-topological space. The sets {a, c} and {b, c} are G5τ-open ( resp. G5τ-pre open, G5τ-β open) sets but their intersection {c} is not a G5τ- open ( resp. G5τ-pre open, G5τ-β open) set. In the same 5τ-topology, consider the maximal grill G = P(X) − {∅}. Thus (X, 5τ, G) is a grill 5-topological space. The set {a, c} and {b, c} are G5τ-semi open sets but their intersection {c} is not a G5τ-semi-open set. Theorem 4.10. Let (X, Nτ, G) be a grill N-topological space and A, B ∈ GNταO(X), then A ∩ B ∈ GNταO(X). Proof. Since A, B ∈ GNταO(X), then by using Theorem 4.2 and Theorem 4.7 we get A ∩ B ∈ GNτSO(X), A ∩ B ∈ GNτPO(X), and therefore A ∩ B ∈ GNταO(X). � c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 296 Generalized open sets in grill N-topology Theorem 4.11. In a grill N-topological space (X, Nτ, G), the family GNταO(X) is a topology and NτO(X) ⊆ GNταO(X). Proof. Clearly, ∅, X ∈ GNταO(X). The desired result follows from the Theorem 4.2, Theorem 4.6 and Theorem 4.10. � Theorem 4.12. Let (X, Nτ, G) be a grill N-topological space. Then A ⊆ X is (i) GNτ-semi open if and only if NτG-cl(A) = NτG-cl(Nτ-int(A)). (ii) GNτ-pre open, then Nτ-cl(Nτ-int(NτG-cl(A))) = Nτ-cl(A). Proof. (i) Assume that A is GNτ-semi open, then A ⊆ NτG-cl(Nτ-int(A)) ⇒ NτG-cl(A) ⊆ NτG-cl(Nτ-int(A)) ⊆ NτG-cl(A). Thus NτG-cl(A) = NτG-cl(Nτ-int(A)). Converse is obvious, since A ⊆ NτG-cl(A). (ii) Assume that A is GNτ-pre open, then A ⊆ Nτ-int(NτG-cl(A)) ⇒ Nτ-cl(A) ⊆ Nτ-cl(Nτ-int(NτG-cl(A))) ⊆ Nτ-cl(NτG-cl(A)) = Nτ- cl(A∪ΦG(A)) = Nτ-cl(A)∪Nτ-cl(ΦG(A)) = Nτ-cl(A)∪ΦG(A) ⊆ Nτ- cl(A). Thus Nτ-cl(Nτ-int(NτG-cl(A))) = Nτ-cl(A). � Theorem 4.13. Let (X, Nτ, G) be a grill N-topological space and A ⊆ X. (i) Then A is GNτ-semi open if and only if there exists a U ∈ Nτ such that U ⊆ A ⊆ NτG-cl(U). (ii) If A is a GNτ-semi open and A ⊆ B ⊆ NτG-cl(A), then B is GNτ- semi open. Proof. (i) Since A is GNτ-semi open, then A ⊆ NτG-cl(Nτ-int(A)). Take U = Nτ-int(A). Then we have U ⊆ A ⊆ NτG-cl(U). On the other hand, assume U ⊆ A ⊆ NτG-cl(U) for some U ∈ Nτ. Since U ⊆ A, then U ⊆ Nτ-int(A) ⇒ NτG-cl(U) ⊆ NτG-cl(Nτ-int(A)). Thus A ⊆ NτG- cl(Nτ-int(A)). (ii) Since A is GNτ-semi open, then there exists a U ∈ Nτ such that U ⊆ A ⊆ NτG-cl(U). Then U ⊆ A ⊆ B ⊆ NτG-cl(A) ⊆ NτG-cl(NτG- cl(U)) = NτG-cl(U). By part(i), we have B is GNτ-semi open. � Theorem 4.14. Let (X, Nτ, G) be a grill N-topological space and A ⊆ X. (i) If A is GNτ-α closed, then Nτ-cl(Nτ-int(NτG-cl(A))) ⊆ A. (ii) If A is GNτ-semi closed, then Nτ-int(NτG-cl(A)) ⊆ A. (iii) If A is GNτ-pre closed, then NτG-cl(Nτ-int(A)) ⊆ A. (iv) If A is GNτ-β-closed, then Nτ-int(NτG-cl(Nτ-int(A))) ⊆ A. Proof. (i). Assume A is GNτ-α closed, then X − A is GNτ-α open and implies X−A ⊆ Nτ-int(NτG-cl(Nτ-int(X−A))) ⊆ Nτ-int(NτG-cl(X−Nτ-cl(A))) ⊆ Nτ-int(Nτ-cl(X−NτG-cl(A))) ⊆ Nτ-int(X−Nτ-int(NτG-cl(A))) ⊆ X−Nτ- cl(Nτ-int(NτG-cl(A))). Thus Nτ-cl(Nτ-int(NτG-cl(A))) ⊆ A. Similarly we can prove the remaining parts. � c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 297 M. Lellis Thivagar, I. L. Reilly, M. Arockia Dasan and V. Ramesh The proof of the next theorem is straightforward. Theorem 4.15. Let A be a subset of a grill N-topological space (X, Nτ, G). (i) If A is Nτ-closed, then A is GNτ-α closed. (ii) If A is GNτ-closed, then A is GNτ-pre closed. (iii) A is GNτ-α closed if and only if it is GNτ-semi closed and GNτ-pre closed. (iv) If A is GNτ-semi closed, then A is GNτ-β closed. (v) If A is GNτ-pre closed, then A is GNτ-β closed. Remark 4.16. From the above theorems, lemmas and examples we have the following diagram. We depict by arrow the implications between the classes of generalized open sets. (1) Nτ-open, (2) GNτ-α open, (3) GNτ-semi open, (4) GNτ-open, (5) GNτ-pre open, (6) GNτ-β open. Conclusion A set is merely an amorphous collection of elements, without coherence or form. 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