EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 12, No. 3, 2019, 960-977 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Operation on Fine Topology P. L. Powar1, Baravan A. Asaad2,3,∗, K. Rajak4, R. Kushwaha1 1 Department of Mathematics, Rani Durgawati University, Jabalpur, (M. P.), India 2 Department of Computer Science, College of Science, Cihan University-Duhok, Iraq 3 Department of Mathematics, Faculty of Science, University of Zakho, Iraq 4 Department of Mathematics, St. Aloysius College (Autonomous), Jabalpur, (M. P.), India Abstract. This paper introduces the concept of an operation γ on τf . Using this operation, we define the concept of fγ-open sets, and study some of their related notions. Also, we introduce the concept of fγg.closed sets and then study some of its properties. Moreover, we introduce and investigate some types of fγ-separation axioms and fγβ-continuous functions by utilizing the operation γ on τf . Finally, some basic properties of functions with fβ-closed graphs have been obtained. 2010 Mathematics Subject Classifications: 54A05, 54A10, 54C05, 54C10, 54D10 Key Words and Phrases: Fine-open sets, fγ-open sets, fγg.closed sets, fγ-separation axioms, fγβ-continuous functions, fβ-closed graphs 1. Introduction Kasahara [11] introduced the notion of an α operation approaches on a class τ of sets and studied the concept of α-continuous functions with α-closed graphs and α-compact spaces. After this, Jankovic [10] introduced the concept of α-closure of a set in X via α-operation and investigated further characterizations of function with α-closed graph. Later, Ogata [12] defined and studied the concept of γ-open sets, and applied it to inves- tigate operation-functions and operation-separation axioms. Asaad et al. [7] introduced the notion of γ-extremally disconnected spaces. Asaad et al. [5] studied further character- izations of γ-extremally disconnected spaces and investigated some relations of functions of γ-extremally disconnected spaces. Asaad [4] defined a γ operation on generalized open sets in X and studied its applications. In 2017-2018, Ahmad and Asaad ([1], [6]) intro- duced an operation γ on semi generalized open subsets of X and discussed some types of separation axioms, functions and closed spaces with respect to γ. Recently, Asaad and Ameen [8] introduced an operation on gα-open sets and studied some of its properties. On ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v12i3.3449 Email addresses: pvjrdvv@gamil.com (P. L. Powar), baravan.asaad@uoz.edu.krd (B. A. Asaad), kusumrajakrdvv@gmail.com (K. Rajak), kushwaharam786@gmail.com (R. Kushwaha) http://www.ejpam.com 960 c© 2019 EJPAM All rights reserved. B. A. Asaad et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 960-977 961 the other hand, Powar and Rajak [13] defined the concept of fine-open sets. They studied fine-irresolute homeomorphism and fine-quotient function. The aim of this paper is to introduce the concept of an operation γ on τf and to define the notion of fγ-open sets of (X, τ, τf ) by using the operation γ on τf . Also, some notions of fγ-open sets with their relationships are studied. In Section 4, we introduce the concept of fγg.closed sets and then investigate some of its properties. In Section 5, some types of fγ-separation axioms by utilizing the operation γ on τf are introduced and investigated. In the last two sections, some basic properties of fγβ-continuous functions with fβ-closed graphs have been obtained. 2. Preliminaries Throughout this paper, the space (X, τ) (or simply X) always mean topological space on which no separation axioms are assumed unless explicitly stated. Definition 2.1. [13] Let (X, τ) be a topological space, we define τ(Aα) = τα(say)= {Gα(6= X) : Gα ∩Aα 6= φ, for Aα ∈ τ and Aα 6= φ,X. for some α ∈ J , where J is the index set}. Now, define τf = {φ,X}∪α {τα}. The above collection τf of subsets of X is called the fine collection of subsets of X and (X, τ, τf ) is said to be the fine space X generated by the topology τ on X. Definition 2.2. [13] A subset U of a fine space X is said to be fine-open in X, if U belongs to the collection τf . It is clear that every open set of X is fine-open in X. The complement of a fine-open set of X is called the fine-closed in X. Remark 2.3. [13] Let (X, τ, τf ) be a fine space. Then the following are holds. (i) The arbitrary union of any fine-open sets in X is fine-open of X. (ii) The intersection of two fine-open sets need not be fine-open. Definition 2.4. [13] Let A be the subset of a fine space X, the fine interior of A is defined as the union of all fine-open sets contained in A. That means, the fine interior of A is the largest fine-open set contained in A and it is denoted by fint(A). Definition 2.5. [13] Let A be the subset of a fine space X, the fine closure of A is defined as the intersection of all fine-closed sets containing the set A. That means, the fine closure of A is the smallest fine-closed set containing A and it is denoted by fcl(A). Definition 2.6. [12] An operation γ on the topology τ on X is a mapping γ : τ → P (X) such that U ⊆ γ(U) for each U ∈ τ , where P (X) is the power set of X and γ(U) denotes the value of γ at U . A nonempty subset A of a topological space (X, τ) with an operation γ on τ is said to be γ-open if for each x ∈ A, there exists an open set U containing x such that γ(U) ⊆ A. The complement of a γ-open subset of a space X as γ-closed. The family of all γ-open subsets of a space (X, τ) is denoted by τγ . B. A. Asaad et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 960-977 962 Definition 2.7. [10] A point x ∈ X is in the γ-closure of a set A ⊆ X if γ(U) ∩ A 6= φ for each open set U containing x. The set of all γ-closure points of A is called γ-closure of A and is denoted by Clγ(A). Definition 2.8. [12] A subset A of (X, τ) with an operation γ on τ is said to be γ-g-closed if Clγ(A) ⊆ U whenever A ⊆ U and U is γ-open in (X, τ). Definition 2.9. [12] A topological space (X, τ) with an operation γ on τ is said to be (i) γ-T0 if for any two distinct points x, y in X, there exists an open set U such that x ∈ U and y /∈ γ(U) or y ∈ U and x /∈ γ(U). (ii) γ-T1 if for any two distinct points x, y in X, there exist two open sets U and V containing x and y respectively such that y /∈ γ(U) and x /∈ γ(V ). (iii) γ-T2 if for any two distinct points x, y in X, there exist two open sets U and V containing x and y respectively such that γ(U) ∩ γ(V ) = φ. (iv) γ-T 1 2 if every γ-g-closed set in X is γ-closed. 3. fγ-Open Sets An operation γ on τf is a mapping γ : τf → P (X) such that U ⊆ γ(U) for every U ∈ τf , where P (X) is the power set of X and γ(U) is the value of γ at U . From this, we can easily to find γ(X) = X for any operation γ : τf → P (X). The operators defined by γ(U) = U , γ(U) = X, γ(U) = fcl(U) and γ(U) = fint(fcl(U)) are all examples of the operation γ. Definition 3.1. Let (X, τ, τf ) be a fine space and γ : τf → P (X) be an operation on τf . A nonempty set A of X is said to be fγ-open if for each x ∈ A, there exists a fine-open set U such that x ∈ U and γ(U) ⊆ A. The complement of a fγ-open set of X is fγ-closed. Suppose that the empty set φ is also fγ-open set for any operation γ : τf → P (X). The family of all fγ-open subsets of a fine space (X, τ, τf ) is denoted by τfγ . Theorem 3.2. The union of any collection of fγ-open sets in a fine space (X, τ, τf ) is a fγ-open set in (X, τ, τf ). Proof. Let x ∈ ⋃ α∈∆{Aα}, where {Aα}α∈∆ be a class of fγ-open sets in X. Then x ∈ Aα for some α ∈ ∆. Since Aα is fγ-open set in X, then there exists a fine-open set V such that x ∈ V ⊆ γ(V ) ⊆ Aα ⊆ ⋃ α∈∆{Aα}. Therefore, ⋃ α∈∆{Aα} is a fγ-open set in X. Example 3.3. The intersection of any two fγ-open sets in (X, τ, τf ) is generally not a fγ-open sets. To see this, let X = {a, b, c} and τ = P (X) = τf . Let γ : τf → P (X) be an B. A. Asaad et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 960-977 963 operation on τf defined as follows: For every A ∈ τf γ(A) = { A if A 6= {c} {b, c} if A = {c} Thus, τfγ = P (X)\{c}. Then {a, c} ∈ τfγ and {b, c} ∈ τfγ, but {a, c} ∩ {b, c} = {c} /∈ τfγ. Therefore, τfγ does not form a topology on X. It is clear from Definition 3.1 that every fγ-open set is fine-open in (X, τ, τf ) (That is, τfγ ⊆ τf ). But the converse need not be true as shown by the following example. Example 3.4. In Example 3.3, the set {c} is fine-open, but it is not fγ-open. Definition 3.5. A fine space (X, τ, τf ) with an operation γ on τf is said to be fγ-regular if for each x ∈ X and for each fine-open set U containing x, there exists a fine-open set W such that x ∈W and γ(W ) ⊆ U . Theorem 3.6. Let (X, τ, τf ) be a fine space and γ : τf → P (X) be an operation on τf . Then the following conditions are equivalent: (i) τf = τfγ. (ii) (X, τ, τf ) is a fγ-regular space. (iii) For every x ∈ X and for every fine-open set U of (X, τ, τf ) containing x, there exists a fγ-open set W of (X, τ, τf ) containing x such that W ⊆ U . Proof. (1) ⇒ (2) Let x ∈ X and U be a fine-open set in X such that x ∈ U . It follows from assumption that U is a fγ-open set. This implies that there exists a fine-open set W such that x ∈W and γ(W ) ⊆ U . Therefore, the fine space (X, τ, τf ) is fγ-regular. (2) ⇒ (3) Let x ∈ X and U be a fine-open set in (X, τ, τf ) containing x. Then by (2), there is a fine-open set W such that x ∈ W ⊆ γ(W ) ⊆ U . Again, by using (2) for the set W , it is shown that W is fγ-open. Hence W is a fγ-open set containing x such that W ⊆ U . (3) ⇒ (1) By applying the part (3) and Theorem 3.2, it follows that every fine-open set of X is fγ-open in X. That is, τf ⊆ τfγ . But in general, we have τfγ ⊆ τf . Therefore, τf = τfγ . Definition 3.7. Let (X, τ, τf ) be any fine space. An operation γ on τf is said to be (i) fine-open if for each x ∈ X and for every fine-open set U containing x, there exists a fγ-open set W containing x such that W ⊆ γ(U). (ii) fine-regular if for each x ∈ X and for every pair of fine-open sets U1 and U2 such that both containing x, there exists a fine-open set W containing x such that γ(W ) ⊆ γ(U1) ∩ γ(U2). B. A. Asaad et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 960-977 964 Lemma 3.8. Let a mapping γ be fine-regular operation on τf . If A and B are fγ-open sets in a fine space (X, τ, τf ), then A ∩ B is also fγ-open set in (X, τ, τf ). Proof. Suppose x ∈ A ∩ B for any fγ-open sets A and B in (X, τ, τf ) both containing x. Then there exist fine-open sets U1 and U2 such that x ∈ U1 ⊆ A and x ∈ U2 ⊆ B. Since γ is a fine-regular operation on τf , then there exists a fine-open set W containing x such that γ(W ) ⊆ γ(U1) ∩ γ(U2) ⊆ A ∩B. Therefore, A ∩B is fγ-open set in (X, τ, τf ). Remark 3.9. By applying Lemma 3.8, it is easy to show that τfγ forms a topology on X for any fine-regular operation γ on τf . Definition 3.10. Let A be any subset of a fine space (X, τ, τf ) and γ be an operation on τf . The point x ∈ X is said to be fγ-closure of A if γ(U) ∩ A 6= φ for each U ∈ τf such that x ∈ U . We denote fclγ(A) by the fγ-closure of A which is the set of all fγ-closure points of A. Definition 3.11. Let A be any subset of a fine space (X, τ, τf ) and γ be an operation on τf . We define τfγ-cl(A) as the intersection of all fγ-closed sets of X containing A. i.e. τfγ-cl(A) = ⋂ {F : A ⊆ F , X\F ∈ τfγ}. Theorem 3.12. Let A be any subset of a fine space (X, τ, τf ) and γ be an operation on τf . Then x ∈ τfγ-cl(A) if and only if A ∩ U 6= φ for every fγ-open set U of X containing x. Proof. Let x ∈ τfγ-cl(A) and let A ∩ U = φ for some fγ-open set U of X containing x. Then A ⊆ X\U and X\U is fγ-closed set in X. So τfγ-cl(A) ⊆ X\U . Thus, x ∈ X\U . This is a contradiction. Hence A ∩ U 6= φ for every fγ-open set U of X containing x. Conversely, suppose that x /∈ τfγ-cl(A). So there exists a fγ-closed set F such that A ⊆ F and x /∈ F . Then X\F is a fγ-open set such that x ∈ X\F and A ∩ (X\F ) = φ. Contradiction of hypothesis. Therefore, x ∈ τfγ-cl(A). Lemma 3.13. The following statements are true for any subsets A and B of a fine space (X, τ, τf ) with an operation γ on τf . (i) fclγ(A) is fine-closed set in X and τfγ-cl(A) is fγ-closed set in X. (ii) A ⊆ fclγ(A) ⊆ τfγ-cl(A). (iii) τfγ-cl(φ) = fclγ(φ) = φ and τfγ-cl(X) = fclγ(X) = X. (iv) (a) A is fγ-closed if and only if τfγ-cl(A) = A and, (b) A is fγ-closed if and only if fclγ(A) = A. (v) If A ⊆ B, then τfγ-cl(A) ⊆ τfγ-cl(B) and fclγ(A) ⊆ fclγ(B). (vi) (a) τfγ-cl(A ∩ B) ⊆ τfγ-cl(A) ∩ τfγ-cl(B) and, B. A. Asaad et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 960-977 965 (b) fclγ(A ∩ B) ⊆ fclγ(A) ∩ fclγ(B). (vii) (a) τfγ-cl(A) ∪ τfγ-cl(B) ⊆ τfγ-cl(A ∪ B) and, (b) fclγ(A) ∪ fclγ(B) ⊆ fclγ(A ∪ B). (viii) τfγ-cl(τfγ-cl(A)) = τfγ-cl(A). Proof. Straightforward. Theorem 3.14. For any subsets A, B of a fine space (X, τ, τf ). If γ is a fine-regular operation on τf , then (i) τfγ-cl(A) ∪ τfγ-cl(B) = τfγ-cl(A ∪ B). (ii) fclγ(A) ∪ fclγ(B) = fclγ(A ∪ B). Proof. (1) It is enough to prove that τfγ-cl(A ∪ B) ⊆ τfγ-cl(A) ∪ τfγ-cl(B) since the other part follows directly from Lemma 3.13 (7). Let x /∈ τfγ-cl(A) ∪ τfγ-cl(B). Then by using Theorem 3.12, there exist two fγ-open sets U and V containing x such that A ∩ U = φ and B ∩ V = φ. Since γ is a fine-regular operation on τf , then by Lemma 3.8, U ∩ V is fγ-open in X such that (U ∩ V ) ∩ (A ∪ B) = φ. Therefore, we have x /∈ τfγ-cl(A ∪ B) and hence τfγ-cl(A ∪ B) ⊆ τfγ-cl(A) ∪ τfγ-cl(B). (2) Let x /∈ fclγ(A) ∪ fclγ(B). Then there exist fine-open sets U1 and U2 such that x ∈ U1, x ∈ U2, A ∩ γ(U1) = φ and A ∩ γ(U2) = φ. Since γ is a fine-regular operation on τf , then there exists a fine-open set W containing x such that γ(W ) ⊆ γ(U1) ∩ γ(U2). Thus, we have (A ∪ B) ∩ γ(W ) ⊆ (A ∪ B) ∩ (γ(U1) ∩ γ(U2)). This implies that (A ∪ B) ∩ γ(W ) = φ since (A ∪ B) ∩ (γ(U1) ∩ γ(U2)) = φ. This means that x /∈ fclγ(A ∪ B) and hence fclγ(A ∪ B) ⊆ fclγ(A) ∪ fclγ(B). Using Lemma 3.13 (7), we have the equality. Theorem 3.15. Let A be any subset of a fine space (X, τ, τf ). If γ is a fine-open operation on τf , then fclγ(A) = τfγ-cl(A), fclγ(fclγ(A)) = fclγ(A) and fclγ(A) is fγ-closed set in X. Proof. By Lemma 3.13 (2), we have fclγ(A) ⊆ τfγ-cl(A). Now, we need to show that τfγ-cl(A) ⊆ fclγ(A). Let x /∈ fclγ(A), then there exists a fine-open set U containing x such that A ∩ γ(U) = φ. Since γ is a fine-open on τf , then there exists a fγ-open set W containing x such that W ⊆ γ(U). So A ∩ W = φ and hence by Theorem 3.12, x /∈ τfγ- cl(A). Therefore, τfγ-cl(A) ⊆ fclγ(A). Hence fclγ(A) = τfγ-cl(A). Moreover, using the above result and by Lemma 3.13 (8), we get fclγ(fclγ(A)) = fclγ(A) and by Lemma 3.13 (4b), we obtain fclγ(A) is fγ-closed set in X. B. A. Asaad et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 960-977 966 Theorem 3.16. Let A be any subset of a fine space (X, τ, τf ) and γ be an operation on τf . Then the following statements are equivalent: (i) A is fγ-open set. (ii) fclγ(X\A) = X\A. (iii) τfγ-cl(X\A) = X\A. (iv) X\A is fγ-closed set. Proof. Clear. Lemma 3.17. Let (X, τ, τf ) be a fine space and γ be a fine-regular operation on τf . Then τfγ-cl(A) ∩ U ⊆ τfγ-cl(A ∩ U) holds for every fγ-open set U and every subset A of X. Proof. Suppose that x ∈ τfγ-cl(A) ∩ U for every fγ-open set U , then x ∈ τfγ-cl(A) and x ∈ U . Let V be any fγ-open set of X containing x. Since γ is fine-regular on τf . So by Lemma 3.8, U ∩ V is fγ-open set containing x. Since x ∈ τfγ-cl(A), then by Theorem 3.12, we have A ∩ (U ∩ V ) 6= φ. This means that (A ∩ U) ∩ V 6= φ. Therefore, again by Theorem 3.12, we obtain that x ∈ τfγ-cl(A ∩ U). Thus, τfγ-cl(A) ∩ U ⊆ τfγ-cl(A ∩ U). 4. fγg.Closed Sets Definition 4.1. A subset A of a fine space (X, τ, τf ) with an operation γ on τf is said to be fγ-generalized closed (briefly fγg.closed) if fclγ(A) ⊆ U whenever A ⊆ U and U is a fγ-open set in X. Lemma 4.2. Let (X, τ, τf ) be a fine space and γ be an operation on τf . A set A in (X, τ, τf ) is fγg.closed if and only if A ∩ τfγ-cl({x}) 6= φ for every x ∈ fclγ(A). Proof. Suppose A is fγg.closed set in X and suppose (if possible) that there exists an element x ∈ fclγ(A) such that A∩ τfγ-cl({x}) = φ. This follows that A ⊆ X\τfγ-cl({x}). Since τfγ-cl({x}) is fγ-closed implies X\τfγ-cl({x}) is fγ-open and A is fγg.closed set in X. Then, we have that fclγ(A) ⊆ X\τfγ-cl({x}). This means that x /∈ fclγ(A). This is a contradiction. Hence A ∩ τfγ-cl({x}) 6= φ. Conversely, let U ∈ τfγ such that A ⊆ U . To show that fclγ(A) ⊆ U . Let x ∈ fclγ(A). Then by hypothesis, A ∩ τfγ-cl({x}) 6= φ. So there exists an element y ∈ A ∩ τfγ-cl({x}). Thus y ∈ A ⊆ U and y ∈ τfγ-cl({x}). By Theorem 3.12, {x} ∩ U 6= φ. Hence x ∈ U and so fclγ(A) ⊆ U . Therefore, A is fγg.closed set in (X, τ, τf ). Theorem 4.3. Let A be a subset of fine space (X, τ, τf ) and γ be an operation on τf . If A is fγg.closed, then fclγ(A)\A does not contain any non-empty fγ-closed set. B. A. Asaad et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 960-977 967 Proof. Let F be a non-empty fγ-closed set in X such that F ⊆ fclγ(A)\A. Then F ⊆ X\A implies A ⊆ X\F . Since X\F is fγ-open set and A is fγg.closed set, then fclγ(A) ⊆ X\F . That is F ⊆ X\fclγ(A). Hence F ⊆ X\fclγ(A) ∩ fclγ(A)\A ⊆ X\fclγ(A) ∩ fclγ(A) = φ. This shows that F = φ. This is a contradiction. Therefore, F 6⊆ fclγ(A)\A. Theorem 4.4. If γ : τf → P (X) is a fine-open operation, then the converse of the Theo- rem 4.3 is true. Proof. Let U be a fγ-open set in (X, τ, τf ) such that A ⊆ U . Since γ : τf → P (X) is a fine-open operation, then by Theorem 3.15, fclγ(A) is fγ-closed set in X. Thus, using Theorem 3.2, we have fclγ(A) ∩X\U is a fγ-closed set in (X, τ, τf ). Since X\U ⊆ X\A, fclγ(A) ∩X\U ⊆ fclγ(A)\A. Using the assumption of the converse of the Theorem 4.3, fclγ(A) ⊆ U . Therefore, A is fγg.closed set in (X, τ, τf ). Corollary 4.5. Let A be a fγg.closed subset of fine space (X, τ, τf ) and let γ be an operation on τf . Then A is fγ-closed if and only if fclγ(A)\A is fγ-closed set. Proof. Let A be a fγ-closed set in (X, τ, τf ). Then by Lemma 3.13 (4b), fclγ(A) = A and hence fclγ(A)\A = φ which is fγ-closed set. Conversely, suppose fclγ(A)\A is fγ-closed and A is fγg.closed. Then by Theorem 4.3, fclγ(A)\A does not contain any non-empty fγ-closed set and since fclγ(A)\A is fγ-closed subset of itself, then fclγ(A)\A = φ implies fclγ(A) ∩ X\A = φ. Hence fclγ(A) = A. This follows from Lemma 3.13 (4b) that A is fγ-closed set in (X, τ, τf ). Theorem 4.6. Let (X, τ) be a fine space and γ be an operation on τf . If a subset A of X is fγg.closed and fγ-open, then A is fγ-closed. Proof. Since A is fγg.closed and fγ-open set in X, then fclγ(A) ⊆ A and hence by Lemma 3.13 (4b), A is fγ-closed. Theorem 4.7. In any fine space (X, τ, τf ) with an operation γ on τf . For an element x ∈ X, the set X\{x} is fγg.closed or fγ-open. Proof. Suppose that X\{x} is not fγ-open. Then X is the only fγ-open set containing X\{x}. This implies that fclγ(X\{x}) ⊆ X. Thus X\{x} is a fγg.closed set in X. Corollary 4.8. In any fine space (X, τ, τf ) with an operation γ on τf . For an element x ∈ X, either the set {x} is fγ-closed or the set X\{x} is fγg.closed. Proof. Suppose {x} is not fγ-closed, then X\{x} is not fγ-open. Hence by Theo- rem 4.7, X\{x} is fγg.closed set in X. Definition 4.9. Let A be any subset of a fine space (X, τ, τf ) and γ be an operation on τf . Then the τfγ-kernel of A is denoted by τfγ-ker(A) and is defined as follows: B. A. Asaad et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 960-977 968 τfγ-ker(A)= ∩{U : A ⊆ U and U ∈ τfγ} In other words, τfγ-ker(A) is the intersection of all fγ-open sets of (X, τ, τf ) containing A. Theorem 4.10. Let A ⊆ (X, τ, τf ) and γ be an operation on τf . Then A is fγg.closed if and only if fclγ(A) ⊆ τfγ-ker(A). Proof. Suppose that A is fγg.closed. Then fclγ(A) ⊆ U , whenever A ⊆ U and U is fγ-open. Let x ∈ fclγ(A). Then by Lemma 4.2, A ∩ τfγ-cl({x}) 6= φ. So there exists a point z in X such that z ∈ A ∩ τfγ-cl({x}) implies that z ∈ A ⊆ U and z ∈ τfγ- cl({x}). By Theorem 3.12, {x} ∩ U 6= φ. Hence we show that x ∈ τfγ-ker(A). Therefore, fclγ(A) ⊆ τfγ-ker(A). Conversely, let fclγ(A) ⊆ τfγ-ker(A). Let U be any fγ-open set containing A. Let x be a point in X such that x ∈ fclγ(A). Then x ∈ τfγ-ker(A). Namely, we have x ∈ U , because A ⊆ U and U ∈ τfγ}. That is fclγ(A) ⊆ τfγ-ker(A)⊆ U . Therefore, A is fγg.closed set in X. 5. On fγ-Separation Axioms Definition 5.1. A fine space (X, τ, τf ) with an operation γ on τf is said to be (i) fγ-T0 if for any two distinct points x, y in X, there exists a fine-open set U such that x ∈ U and y /∈ γ(U) or y ∈ U and x /∈ γ(U). (ii) fγ-T ∗0 if for each pair of distinct points x, y in X, there exists a fγ-open set U containing one of the points but not the other. Definition 5.2. A fine space (X, τ, τf ) with an operation γ on τf is said to be (i) fγ-T1 if for any two distinct points x, y in X, there exist two fine-open sets U and V such that x ∈ U , y /∈ γ(U), y ∈ V and x /∈ γ(V ). (ii) fγ-T ∗1 if for each pair of distinct points x, y in X, there exist two fγ-open sets U and V such that x ∈ U but y /∈ U and y ∈ V but x /∈ V . Definition 5.3. A fine space (X, τ, τf ) with an operation γ on τf is said to be (i) fγ-T2 if for any two distinct points x, y in X, there exist two fine-open sets U and V such that x ∈ U , y ∈ V and γ(U) ∩ γ(V ) = φ. (ii) fγ-T ∗2 if for each pair of distinct points x, y in X, there exist fγ-open sets U and V such that x ∈ U , y ∈ V and U ∩ V = φ. Definition 5.4. A fine space (X, τ, τf ) with an operation γ on τf is said to be fγ-T 1 2 if every fγg.closed set in X is fγ-closed set. Theorem 5.5. For any fine space (X, τ, τf ) with an operation γ on τf . Then (X, τ, τf ) is fγ-T 1 2 if and only if for each point x ∈ X, the set {x} is fγ-closed or fγ-open. Proof. Let X be a fγ-T 1 2 space and let {x} is not fγ-closed set in (X, τ, τf ). By Corollary 4.8, X\{x} is fγg.closed. Since (X, τ, τf ) is fγ-T 1 2 , then X\{x} is fγ-closed set which means that {x} is fγ-open set in X. B. A. Asaad et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 960-977 969 Conversely, let F be any fγg.closed set in the fine space (X, τ, τf ). We have to show that F is fγ-closed (that is fclγ(F ) = F (by Lemma 3.13 (4b))). It is sufficient to show that fclγ(F ) ⊆ F . Let x ∈ fclγ(F ). By hypothesis {x} is fγ-closed or fγ-open for each x ∈ X. So we have two cases: Case (1): If {x} is fγ-closed set. Suppose x /∈ F , then x ∈ fclγ(F )\F contains a non-empty fγ-closed set {x}. A contradiction since F is fγg.closed set and according to the Theorem 4.3. Hence x ∈ F . This follows that fclγ(F ) ⊆ F and hence fclγ(F ) = F . This means from by Lemma 3.13 (4b) that F is fγ-closed set in (X, τ, τf ). Thus (X, τ, τf ) is fγ-T 1 2 space. Case (2): If {x} is fγ-open set. Then by Theorem 3.12, F ∩{x} 6= φ which implies that x ∈ F . So fclγ(F ) ⊆ F . Thus by Lemma 3.13 (4b), F is fγ-closed. Therefore, (X, τ, τf ) is fγ-T 1 2 space. Theorem 5.6. For any fine space (X, τ, τf ) with an operation γ on τf , we have (i) Let γ be a fine-open operation on τf . Then a space X is a fγ-T0 space if and only if fclγ({x}) 6= fclγ({y}), for every pair x, y of X with x 6= y. (ii) A space X is fγ-T ∗0 if and only if τfγ-cl({x}) 6= τfγ-cl({y}), for every pair of distinct points x, y of X. Proof. (1) Let x, y be any two distinct points of a fγ-T0 space (X, τ, τf ). Then by definition, we assume that there exists a fγ-open set U such that x ∈ U and y /∈ γ(U). Since γ is a fine-open operation on τf , then there exists a fγ-open set W such that x ∈W and W ⊆ γ(U). Hence y ∈ X\γ(U) ⊆ X\W . Since X\W is a fγ-closed set in (X, τ, τf ). Then we obtain that fclγ({y}) ⊆ X\W and therefore fclγ({x}) 6= fclγ({y}). Conversely, suppose for any x, y ∈ X with x 6= y, we have fclγ({x}) 6= fclγ({y}). Now, we assume that there exists z ∈ X such that z ∈ fclγ({x}), but z /∈ fclγ({y}). If x ∈ fclγ({y}), then {x} ⊆ fclγ({y}), which implies that fclγ({x}) ⊆ fclγ({y}) (by Lemma 3.13 (5)). This implies that z ∈ fclγ({y}). This contradiction shows that x /∈ fclγ({y}). This means that by Definition 3.10, there exists a fine-open set U such that x ∈ U and γ(U)∩{y} = φ. Thus, we have that x ∈ U and y /∈ γ(U). It gives that the fine space (X, τ, τf ) is fγ-T0. (2) Let X be a fγ-T ∗0 space and x, y be any two distinct points of X. Then there exists a fγ-open set G containing x or y (say x, but not y). So X\G is a fγ-closed set, which does not contain x, but contains y. Since τfγ-cl({y}) is the smallest fγ-closed set containing y, τfγ-cl({y}) ⊆ X\G, and so x /∈ τfγ-cl({y}). Therefore, τfγ-cl({x}) 6= τfγ-cl({y}). Conversely, suppose for any x, y ∈ X with x 6= y, τfγ-cl({x}) 6= τfγ-cl({y}). Now, let z ∈ X such that z ∈ τfγ-cl({x}), but z /∈ τfγ-cl({y}). Now, we claim that x ∈ τfγ-cl({y}). For, if x ∈ τfγ-cl({y}), then {x} ⊆ τfγ-cl({y}), which implies that τfγ-cl({x}) ⊆ τfγ- cl({y}). This is a contradiction to the fact that z /∈ τfγ-cl({y}). Hence x belongs to the fγ-open set X\τfγ-cl({y}) to which y does not belong. It gives that X is fγ-T ∗0 space. Corollary 5.7. Suppose that γ is a fine-open operation on τf . A fine space (X, τ, τf ) is fγ-T0 if and only if (X, τ, τf ) is fγ-T ∗0 . B. A. Asaad et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 960-977 970 Proof. This follows from Theorem 5.6 and the fact that fclγ(A) = τfγ-cl(A) for anyA ⊆ X holds under the assumption that γ is a fine-open operation on τf (see Theorem 3.15). Theorem 5.8. For a fine space (X, τ, τf ) with an operation γ on τf . Then the following statements are true: (i) (X, τ, τf ) is fγ-T1. (ii) For every point x ∈ X, the set {x} is fγ-closed. (iii) (X, τ, τf ) is fγ-T ∗1 . Proof. (1) ⇒ (2) Let x be a point of an fγ-T1 space (X, τ, τf ). Then for any point y ∈ X such that x 6= y, there exists a fine-open set Vy such that y ∈ Vy but x /∈ γ(Vy). Thus, y ∈ γ(Vy) ⊆ X\{x}. This implies that X\{x} = ∪{γ(Vy) : y ∈ X\{x}}. It is shown that X\{x} is fγ-open set in (X, τ, τf ). Hence {x} is fγ-closed set in (X, τ, τf ). (2) ⇒ (3) Suppose every singleton set in X is fγ-closed. Let x, y ∈ X such that x 6= y. This implies that x ∈ X\{y}. By hypothesis, we get X\{y} is a fγ-open set contains x but not y. Similarly X\{x} is a fγ-open set contains y but not x. Therefore, X is fγ-T ∗1 space. (3) ⇒ (1) It is shown that if x ∈ U , where U ∈ τfγ , then there exist a fine-open set V such that x ∈ V ⊆ γ(V ) ⊆ U . Applying the part (3), we obtain (X, τ, τf ) is fγ-T1. Theorem 5.9. For any fine space (X, τ, τf ) and any operation γ on τf , the following properties hold. (i) Every fγ-T2 space is fγ-T1. (ii) Every fγ-T1 space is fγ-T 1 2 . (iii) Every fγ-T 1 2 space is fγ-T ∗0 . (iv) Every fγ-T ∗n space is fγ-T ∗n−1, where n ∈ {2, 1}. (v) Every fγ-T ∗n space is fγ-Tn, where n ∈ {2, 0}. Proof. Follows directly from their definitions. Remark 5.10. By Theorem 5.9 and Theorem 5.8, we obtain the following diagram of implications. Moreover, the following Examples 5.11, 5.12, 5.13 and 5.14 below show that the reverse implications are not true in general. fγ-T ∗2 �� // fγ-T ∗1 �� // fγ-T ∗0 �� fγ-T2 // fγ-T1 OO // fγ-T 1 2 ;;wwwwwwwww // fγ-T0 γ-T2 OO // γ-T1 OO // γ-T 1 2 OO // γ-T0 OO B. A. Asaad et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 960-977 971 Example 5.11. Let X = {a, b, c} and τ = {φ,X, {a}, {b}, {a, b}}. Then τf = τ ∪ {{a, c}, {b, c}}. (i) Define an operation γ on τf as follows: For every A ∈ τf γ(A) =  {a, b} if A = {b} {b, c} if A = {c} or {b, c} X otherwise Obviously, the space (X, τ, τf ) is fγ-T0, but it is not fγ-T ∗0 . Hence the fine space (X, τ, τf ) is not fγ-T 1 2 . (ii) Let γ : τf → P (X) be an operation on τf defined as follows: For every set A ∈ τf γ(B) = { B if B = {a} or {b} or {a, b} or {b, c} X otherwise Thus, τfγ = {φ,X, {a}, {b}, {a, b}, {b, c}}. Clearly, the space (X, τ, τf ) is fγ-T 1 2 , but it is not fγ-T1. Example 5.12. Let X = {a, b, c} and τ = {φ,X, {b}}. Then τf = {φ, X, {b}, {a, b}, {a, c}}. Let γ : τf → P (X) be an operation on τf defined as follows: For every set A ∈ τf γ(A) = { A if A = {b} or {a, b} fcl(A) otherwise Thus, τfγ = {φ,X, {b}, {a, b}}. Then the fine space (X, τ, τf ) is fγ-T ∗0 , but it is not fγ- T 1 2 . Since {b, c} is fγg.closed set in (X, τ, τf ), but {b, c} is not fγ-closed set in (X, τ, τf ). Therefore, (X, τ, τf ) is not a fγ-T ∗1 space. Example 5.13. Suppose X = {a, b, c} and τ = all subsets of X. Define an operation γ on τf as follows: For every A ∈ τf γ(A) = { A if A = {a, b} or {a, c} or {b, c} X otherwise Therefore, (X, τ, τf ) is fγ-T ∗1 space, and by Theorem 5.8, it is fγ-T1, but (X, τ, τf ) is not fγ-T2 and hence it is not fγ-T ∗2 . Example 5.14. Let X = {a, b, c} and τ = {φ,X, {a, b}}. Then τf = τ ∪ {{a}, {b}, {a, c}, {b, c}}. Define an operation γ : τf → P (X) by γ(A) = A for all A ∈ τf . Here, τfγ = τf and τγ = τ . Then the fine space (X, τ, τf ) is fγ-Ti, but it is not γ-Ti for i = 0, 1 2 , 1, 2. B. A. Asaad et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 960-977 972 6. fγβ-Continuous Functions Throughout Section 6 and Section 7, let (X, τ, τf ) and (Y, σ, σf ) be two fine spaces and let γ : τf → P (X) and β : σf → P (Y ) be operations on τf and σf respectively. In this sec- tion, we introduce a new class of functions called fγβ-continuous. Some characterizations and properties of this function are investigated. Definition 6.1. A function h : (X, τ, τf ) → (Y, σ, σf ) is said to be fγβ-continuous if for each x ∈ X and each fine-open set V containing h(x), there exists a fine-open set U containing x such that h(γ(U)) ⊆ β(V ). Theorem 6.2. Let h : (X, τ, τf )→ (Y, σ, σf ) be a fγβ-continuous function, then, (i) h(fclγ(A)) ⊆ fclβ(h(A)), for every A ⊆ (X, τ, τf ). (ii) h−1(F ) is fγ-closed set in (X, τ, τf ), for every fβ-closed set F of (Y, σ, σf ). Proof. (1) Let y ∈ h(fclγ(A)) and V be any fine-open set containing y. Then by hypothesis, there exists x ∈ X and fine-open set U containing x such that h(x) = y and h(γ(U)) ⊆ β(V ). Since x ∈ fclγ(A), we have γ(U) ∩ A 6= φ. Hence φ 6= h(γ(U) ∩ A) ⊆ h(γ(U)) ∩ h(A) ⊆ β(V ) ∩ h(A). This implies that y ∈ fclβ(h(A)). Therefore, h(fclγ(A)) ⊆ fclβ(h(A)). (2) Let F be any fβ-closed set of (Y, σ, σf ). By using (1), we have h(fclγ(h−1(F ))) ⊆ fclβ(F ) = F . Therefore, fclγ(h−1(F )) = h−1(F ). Hence h−1(F ) is fγ-closed set in (X, τ, τf ). Theorem 6.3. In Theorem 6.2, the properties of fγβ-continuity of f , (1) and (2) are equivalent to each other if either the fine space (Y, σ, σf ) is fβ-regular or the operation β is fine-open. Proof. It follows from the proof of Theorem 6.2 that we know the following implications: ”fγβ-continuity of h” ⇒ (1) ⇒ (2). Thus, when the fine space (Y, σ, σf ) is fβ-regular, we prove the implication: (2) ⇒ fγβ-continuity of h. Let x ∈ X and let V ∈ σf such that h(x) ∈ V . Since (Y, σ, σf ) is a fβ-regular space, then by Theorem 3.6, V ∈ σgβ. By using (2) of Theorem 6.2, h−1(V ) ∈ τfγ such that x ∈ h−1(V ). So there exists a fine-open set U such that x ∈ U and γ(U) ⊆ h−1(V ). This implies that h(γ(U)) ⊆ V ⊆ β(V ). Therefore, h is fγβ-continuous. Now, when β is a fine-open operation, we show the implication: (2) ⇒ fγβ-continuity of h. Let x ∈ X and let V ∈ σf such that h(x) ∈ V . Since β is a fine-open operation, then there exists W ∈ σgβ such that h(x) ∈ W and W ⊆ β(V ). By using (2) of Theorem 6.2, h−1(W ) ∈ τfγ such that x ∈ h−1(W ). So there exists a fine-open set U such that x ∈ U and γ(U) ⊆ h−1(W ) ⊆ h−1(β(V )). This implies that h(γ(U)) ⊆ β(V ). Hence h is fγβ-continuous. Definition 6.4. A function h : (X, τ, τf )→ (Y, σ, σf ) is said to be B. A. Asaad et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 960-977 973 (i) fγβ-closed if the image of each fγ-closed set of X is fβ-closed in Y . (ii) fβ-closed if the image of each fine-closed set of X is fβ-closed in Y . Theorem 6.5. Suppose that a function h : (X, τ, τf ) → (Y, σ, σf ) is both fγβ-continuous and fβ-closed, then: (i) For every fγg.closed set A of (X, τ, τf ), the image h(A) is fβg.closed in (Y, σ, σf ). (ii) For every fβg.closed set B of (Y, σ, σf ), the inverse set h−1(B) is fγg.closed in (X, τ, τf ). Proof. (1) Let G be any fβ-open set in (Y, σ, σf ) such that h(A) ⊆ G. Since h is fγβ-continuous function, then by using Theorem 6.2 (2), h−1(G) is fγ-open set in (X, τ, τf ). Since A is fγg.closed and A ⊆ h−1(G), we have fclγ(A) ⊆ h−1(G), and hence h(fclγ(A)) ⊆ G. Thus, by Lemma 3.13 (1), fclγ(A) is fine-closed set and since h is fβ- closed, then h(fclγ(A)) is fβ-closed set in Y . Therefore, fclβ(h(A)) ⊆ fclβ(h(fclγ(A))) = h(fclγ(A)) ⊆ G. This implies that h(A) is fβg.closed in (Y, σ, σf ). (2) Let H be any fγ-open set of a fine space (X, τ, τf ) such that h−1(B) ⊆ H. Let C = fclγ(h−1(B)) ∩ (X\H), then by Lemma 3.13 (1), C is fine-closed set in (X, τ, τf ). Since h is fβ-closed function. Then h(C) is fβ-closed in (Y, σ, σf ). Since h is fγβ-continuous function, then by using Theorem 6.2 (1), we have h(C) = h(fclγ(h−1(B))) ∩ h(X\H) ⊆ fclβ(B) ∩ h(X\H) ⊆ fclβ(B) ∩ (Y \B) = fclβ(B)\B. This implies from Theorem 4.3 that h(C) = φ, and hence C = φ. So fclγ(h−1(B)) ⊆ H. Therefore, h−1(B) is fγg.closed in (X, τ, τf ). Theorem 6.6. Let h : (X, τ, τf )→ (Y, σ, σf ) be an injective, fγβ-continuous and fβ-closed function. If (Y, σ, σf ) is fβ-T 1 2 , then (X, τ, τf ) is fγ-T 1 2 . Proof. Let G be any fγg.closed set of (X, τ, τf ). Since h is fγβ-continuous and fβ-closed function. Then by Theorem 6.5 (1), h(G) is fβg.closed in (Y, σ, σf ). Since (Y, σ, σf ) is fβ-T 1 2 , then h(G) is fβ-closed in Y . Again, since h is fγβ-continuous, then by Theorem 6.2 (2), h−1(h(G)) is fγ-closed in X. Hence G is fγ-closed in X since h is injective. Therefore, (X, τ, τf ) is a fγ-T 1 2 space. Theorem 6.7. Let a function h : (X, τ, τf )→ (Y, σ, σf ) be surjective, fγβ-continuous and fβ-closed. If (X, τ, τf ) is fγ-T 1 2 , then (Y, σ, σf ) is fβ-T 1 2 . Proof. Let H be a fβg.closed set of (Y, σ, σf ). Since h is fγβ-continuous and fβ-closed function. Then by Theorem 6.5 (2), h−1(H) is fγg.closed in (X, τ, τf ). Since (X, τ, τf ) is fγ-T 1 2 , then we have, h−1(H) is fγ-closed set in X. Again, since h is fβ-closed function, then h(h−1(H)) is fβ-closed in Y . Therefore, H is fβ-closed in Y since h is surjective. Hence (Y, σ, σf ) is fβ-T 1 2 space. B. A. Asaad et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 960-977 974 Theorem 6.8. If a function h : (X, τ, τf )→ (Y, σ, σf ) is injective fγβ-continuous and the fine space (Y, σ, σf ) is fβ-T2, then the fine space (X, τ, τf ) is fγ-T2. Proof. Let x1 and x2 be any distinct points of a fine space (X, τ, τf ). Since h is an injective function and (Y, σ, σf ) is fβ-T2. Then there exist two fine-open sets U1 and U2 in Y such that f(x1) ∈ U1, h(x2) ∈ U2 and β(U1) ∩ β(U2) = φ. Since h is fγβ-continuous, there exist fine-open sets V1 and V2 in X such that x1 ∈ V1, x2 ∈ V2, h(γ(V1)) ⊆ β(U1) and h(γ(V2)) ⊆ β(U2). Therefore β(U1) ∩ β(U2) = φ. Hence (X, τ, τf ) is fγ-T2. Theorem 6.9. If a function h : (X, τ, τf )→ (Y, σ, σf ) is injective fγβ-continuous and the fine space (Y, σ, σf ) is fβ-Ti, then the fine space (X, τ, τf ) is fγ-Ti for i ∈ {0, 1}. Proof. The proof is similar to Theorem 6.8. Definition 6.10. A function h : (X, τ, τf )→ (Y, σ, σf ) is said to be fγβ-homeomorphism if h is bijective, fγβ-continuous and h−1 is fβγ-continuous. Theorem 6.11. Assume that a function h : (X, τ, τf )→ (Y, σ, σf ) is fγβ-homeomorphism. If (X, τ, τf ) is fγ-T 1 2 , then (Y, σ, σf ) is fβ-T 1 2 . Proof. Let {y} be any singleton set of (Y, σ, σf ). Then there exists an element x of X such that y = h(x). So by hypothesis and Theorem 5.5, we have {x} is fγ-closed or fγ-open set in X. By using Theorem 6.2, {y} is fβ-closed or fβ-open set. Hence the fine space by Theorem 5.5, (Y, σ, σf ) is fβ-T 1 2 . 7. Functions with fβ-Closed Graphs For a function h : (X, τ, τf )→ (Y, σ, σf ), the subset {(x, h(x)) : x ∈ X} of the product space (X × Y, τ × σ) is called the graph of h and is denoted by G(h) [9]. In this section, we further investigate general operator approaches of closed graphs of functions. Let λ : (τ × σ)f → P (X × Y ) be an operation on (τ × σ)f . Definition 7.1. The graph G(h) of h : (X, τ, τf )→ (Y, σ, σf ) is called fβ-closed if for each (x, y) ∈ (X × Y )\G(h), there exist fine-open sets U ⊆ X and V ⊆ Y containing x and y, respectively, such that (U × β(V )) ∩ G(h) = φ. The proof of the following lemma follows directly from the above definition. Lemma 7.2. A function h : (X, τ, τf ) → (Y, σ, σf ) has fβ-closed graph if and only if for each (x, y) ∈ (X × Y )\G(h), there exist U ∈ τf containing x and V ∈ σf containing y such that h(U) ∩ β(V ) = φ. Definition 7.3. An operation λ : (τ × σ)f → P (X × Y ) is said to be fine-associated with γ and β if λ(U × V ) = γ(U)× β(V ) holds for each U ∈ τf and V ∈ σf . B. A. Asaad et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 960-977 975 Definition 7.4. The operation λ : (τ × σ)f → P (X × Y ) is said to be fine-regular with respect to γ and β if for each (x, y) ∈ X × Y and each fine-open set W containing (x, y), there exist fine-open sets U in X and V in Y such that x ∈ U , y ∈ V and γ(U)× β(V ) ⊆ λ(W ). Theorem 7.5. Let λ : (τ × τ)f → P (X ×X) be a fine-associated operation with γ and γ. If h : (X, τ, τf ) → (Y, σ, σf ) is a fγβ-continuous function and (Y, σ, σf ) is a fβ-T2 space, then the set A = {(x, y) ∈ X ×X : h(x) = h(y)} is a fλ-closed set of (X ×X, τ × τ). Proof. We want to prove that fclλ(A) ⊆ A. Let (x, y) ∈ (X ×X)\A. Since (Y, σ, σf ) is fβ-T2. Then there exist two fine-open sets U and V in (Y, σ, σf ) such that h(x) ∈ U , h(y) ∈ V and β(U) ∩ β(V ) = φ. Moreover, for U and V there exist fine-open sets R and S in (X, τ, τf ) such that x ∈ R, y ∈ S and h(γ(R)) ⊆ β(U) and h(γ(S)) ⊆ β(V ) since h is fγβ-continuous. Therefore we have (x, y) ∈ γ(R) × γ(S) = λ(R × S) ∩ A = φ because R× S ∈ (τ × τ)f . This shows that (x, y) /∈ fclλ(A). Corollary 7.6. Suppose λ : (τ × τ)f → P (X ×X) is fine-associated operation with γ and γ, and it is fine-regular with γ and γ. A fine space (X, τ, τf ) is fγ-T2 if and only if the diagonal set 4 = {(x, x) : x ∈ X} is fλ-closed of (X ×X, τ × τ). Theorem 7.7. Let λ : (τ × σ)f → P (X × Y ) be a fine-associated operation with γ and β. If h : (X, τ, τf )→ (Y, σ, σf ) is fγβ-continuous and (Y, σ, σf ) is fβ-T2, then the graph of h, G(h) = {(x, h(x)) ∈ X × Y } is a fλ-closed set of (X × Y, τ × σ). Proof. The proof is similar to Theorem 7.5. Definition 7.8. Let (X, τ, τf ) be a fine space and γ be an operation on τf . A subset S of X is said to be fγ-compact if for every fine-open cover {Ui, i ∈ N} of S, there exists a finite subfamily {U1, U2, ..., Un} such that S ⊆ γ(U1) ∪ γ(U2) ∪ ... ∪ γ(Un). Theorem 7.9. Suppose that γ is fine-regular and λ : (τ ×σ)f → P (X ×Y ) is fine-regular with respect to γ and β. Let h : (X, τ, τf ) → (Y, σ, σf ) be a function whose graph G(h) is fλ-closed in (X × Y, τ × σ). If a subset S is fβ-compact in (Y, σ, σf ), then h−1(S) is fγ-closed in (X, τ, τf ). Proof. Suppose that h−1(S) is not fγ-closed then there exist a point x such that x ∈ fclγ(h−1(S)) and x 6∈ h−1(S). Since (x, s) 6∈ G(h) and each s ∈ S and fclλ(G(h)) ⊆ G(h), there exists a fine-open set W of (X × Y, τ × σ) such that (x, s) ∈ W and β(W ) ∩ G(h) = φ. By fine-regularity of λ, for each s ∈ S we can take two fine-open sets U(s) and V (s) in (Y, σ, σf ) such that x ∈ U(s), s ∈ V (s) and γ(U(s)) × β(V (s)) ⊆ λ(W ). Then we have h(γ(U(s))) ∩ β(V (s)) = φ. Since {V (s) : s ∈ S} is fine-open cover of S, then by fγ-compactness there exists a finite number s1, s2, ..., sn ∈ S such that S ⊆ β(V (s1)) ∪ β(V (s2)) ∪ ... ∪ β(V (sn)). By the fine-regularity of γ, there exist a fine-open set U such that x ∈ U , γ(U) ⊆ γ(U(s1)) ∩ γ(U(s2)) ∩ ... ∩ γ(U(sn)). Therefore, we have γ(U)∩ h−1(S) ⊆ U(si)∩ h−1(β(V (si))) = φ. This shows that x 6∈ fclγ(h−1(S)). This is a contradiction. Therefore, h−1(S) is fγ-closed. REFERENCES 976 Theorem 7.10. Suppose that the following condition hold: (i) γ : τf → P (X) is fine-open (ii) β : σf → P (Y ) is fine-regular, and (iii) λ : (τ×σ)f → P (X×Y ) is associated with γ and β, and λ is fine-regular with respect to γ and β. Let h : (X, τ, τf )→ (Y, σ, σf ) be a function whose graph G(h) is fλ-closed in (X×Y, τ×σ). If every cover of A by fγ-open sets of (X, τ, τf ) has finite sub cover, then h(A) is fβ-closed in (Y, σ, σf ). Proof. Similar to Theorem 7.9. 8. Conclusion In the present paper, the concepts of an operation γ on τf are introduced. Also, the concept of fγ-open sets are defined, and some of their properties are studied via this operation. Moreover, the concept of fγg.closed sets are studied. Furthermore, some types of fγ-separation axioms and fγβ-continuous functions are investigated. In addition, some basic properties of functions with fβ-closed graphs are obtained. References [1] N. 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