Ratio Mathematica Volume 40, 2021, pp. 139-149 gpα- Kuratowski closure operators in topological spaces P. G. Patil* Bhadramma Pattanashetti† Abstract In this paper, we introduce and study topological properties of gpα- limit points, gpα-derived sets, gpα-interior and gpα-closure using the concept of gpα-open set. Further, the relationships between these concepts are investigated. Also, Kuratowski axioms are discussed. Keywords: gpα-open set, gpα-closed set, gpα-limit point, gpα-derived set, gpα-closure, gpα-interior points. 2020 AMS subject classifications: 54A05, 54C08. 1 *Department of Mathematics (Karnatak University, Dharwad, Karnataka, India.); pg- patil@kud.ac.in. †Department of Mathematics (Karnatak University, Dharwad, Karnataka, India.); gee- tagma@gmail.com. 1Received on January 12th, 2021. Accepted on May 12th, 2021. Published on June 30th, 2021. doi: 10.23755/rm.v40i1.578. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published under the CC-BY licence agreement. 139 P.G. Patil and Bhadramma Pattanashetti 1 Introduction Topology is an indispensable object of study in Mathematics with open sets as well as closed sets being the most fundamental concepts in topological spaces. General topology plays an important role in many branches of mathemat- ics as well as many fields of applied sciences. Introduction to the concept of pre-open sets and pre-closed sets were made by Mashhour et al. (7) and the idea of α-open sets was introduced by Njastad (6).The concepts and characterizations of semi open and semi pre open sets are studied in (4) and (1) repectively. The concept, generalized closed sets of Levine (5) opened the flood gates of research in generalizations of closed sets in general topology. Many researchers (3), (8), (10), (11) worked on weaker forms of closed sets. Recently, Benchalli et al.(2) and Patil et al. (9) introduced and studied the concept of ωα-open sets and gpα-open sets in topological spaces. The present authors continued the study of gpα-closed sets and their prop- erties. We study the gpα-closure, gpα-interior, gpα-neighbourhood, gpα-limit points and gpα-derived sets by using the concept of gpα-open sets and their topo- logical properties. We provide the relationship between gpα-derived set (resp. gpα-limit points, gpα-interior) and pre-derived set (resp. pre-limit points and pre- interior). Also, we studied Kuratowski closure axioms with respect to gpα-open sets. 2 Preliminaries Throught the paper, let X and Y (resp. (X,τ) and (Y,σ)) always denotes non-empty topological spaces on which no separation axioms are assumed unless explicitly mentioned. The following definitions are useful in the sequel: Definition 2.1. A subset A of X is said to be a (a) ωα-closed (2) if α-cl(A) ⊆ U whenever A ⊆ U and U is ω-open in X. (b) gpα-closed (9) if p-cl(A) ⊆ U whenever A ⊆ U and U is α-open in X. 3 gpα-Interior and gpα-closure in topological spaces This section deals with gpα-interior and gpα-closure and some of their properties. 140 gpα- Kuratowski closure operators in topological spaces Definition 3.1. (9) Let A ⊂ X, then gpα-interior of A is denoted by gpα-int(A), and is defined as gpα-int(A) = ∪{G : A ⊆ G : G is gpα-open in X }. Definition 3.2. (9) Let A ⊂ X, then gpα-closure of A is denoted by gpα-cl(A), and is defined as gpα-cl(A) = ∩{G : A ⊆ G : G is gpα-closed in X }. Theorem 3.1. If A is gpα-closed then gpα-cl(A) = A. Proof: Let A be gpα-closed in X. Since A ⊆ A, and A is gpα-closed in X. Then A ∈ {G : A ⊆ G and G is gpα-closed in X }, that is A = ∩{G : A ⊆ G and G is gpα-closed }. Hence gpα-cl(A) ⊆ A. But A ⊆ gpα-cl(A) is always true. Therefore gpα-cl(A) = A. In general the converse of Theorem 3.1 is not true. Example 3.1. Let X = {a,b,c,d,e} and τ = {X,φ,{a},{c,d},{a,c,d}}. Let A = {a,b}. Then we can observe that gpα-cl(A) = {a,b}. therefore gpα-cl(A) = A. But A is not gpα-closed in X. Remark 3.1. (9) Let A ⊆ X and A is gpα-closed in X. Then gpα-cl(A) is the smallest gpα-closed set containing A. However, the converse of the Remark 3.1 is not true in general. Example 3.2. Let X = {a,b,c} and τ = {X,φ,{a},{a,c}}. Consider A = {a,c}, then gpα-cl(A) = X , which is the smallest gpα-closed set containing A. But A is not gpα-closed in X. Remark 3.2. (9) For subsets A,B of X, then gpα-cl(A∩B)⊆gpα-cl(A) ∩ gpα- cl(B). Remark 3.3. (9) For subsets A,B of X, then gpα-cl(A∪B) = gpα-cl(A) ∪gpα- cl(B). Remark 3.4. For any A,B ⊆ X then we have the following properties: (i) gpα-int(X) = X and gpα-int(φ) = φ. (9) (ii) gpα-int(A) ⊆ A.(9) (iii) If B is any gpα-open set contained in A then B ⊆ gpα-int(A).(9) (iv) if A ⊆ B, gpα-int(A) ⊆ gpα-int(B). (v) gpα-int(gpα-int(A)) = gpα-int(A). Remark 3.5. For any A,B ⊆ X, gpα-int(A∪B) = gpα-int(A) ∪ gpα-int(B). 141 P.G. Patil and Bhadramma Pattanashetti 4 gpα-Neighbourhood points in topological spaces This section deals with the properties of gpα-neighbourhood points in topological spaces. Definition 4.1. (9) A subset N of X is said to be gpα-neighbourhood of a point x ∈ X, if there exists an gpα-open set G containing x such that x ∈ G ⊆ N. Theorem 4.1. Every neighbourhood N of X is a gpα-neighbourhood of X. Proof: Follows from the definition 4.1 and every open set is gpα-open (9). From the following example converse of the Theorem 4.1is not true. Example 4.1. Let X = {a,b,c,d,e} and τ = {X,φ,{a,b},{a,b,c}}. Let a ∈ X. Consider A = {a,d}. Since A is gpα-neighbourhood of the point a, but A is not a neighbourhood of the point a. Remark 4.1. If N ⊆ X is gpα-open then, N is gpα-neighbourhood of each of its points. Converse of the Theorem 4.1 need not true in general, as seen from the follow- ing example. Example 4.2. Let X = {a,b,c,d} and τ = {X,φ,{a},{c,d},{a,c,d}}. Here the gpα-open sets are: X, φ, {a}, {b}, {c}, {c,d}, {a,c,d}. Then the set A = {a,b} is gpα-neighbourhod of the points a and b, but the set A = {a,b} is not gpα-open in X. Theorem 4.2. Let A be gpα-closed set in X and x ∈ Ac. Then there exists gpα- neighbourhood N of x such that N ∩A = φ. Proof: Let A be gpα-closed in X and x ∈ Ac. Then Ac = X \ A which is gpα- open in X. Then from Remark 4.1, Ac is gpα-neighbourhood of each of its points. Hence, for every point x ∈ Ac, there exists gpα-neighbourhood N of X such that N ⊆ Ac. Hence N ∩A = φ. Theorem 4.3. Let x ∈ X and gpαN(x) be the collection of all gpα-neighbourhoods of X. Then the following results holds: (i) gpαN(x) 6= φ, ∀ x ∈ X. (ii) N ∈ gpαN(x) implies x ∈ N. (iii) Let N ∈ gpαN(x) and N ⊆ M, then M ∈ gpαN(x). (iv) N ∈ gpαN(x) and M ∈ gpαN(x) then N ∩M ∈ gpαN(x). Proof: (i) We have X is always gpα-open in X. Hence X is in gpαN(x) of every point x ∈ X. Therefore gpα-N(X) 6= φ for every point x ∈ X. (ii) From definition of N ∈ gpαN(x), it follows that x ∈ N. 142 gpα- Kuratowski closure operators in topological spaces (iii) Let N ∈ gpαN(x) and N ⊆ M. Then there exists gpα-open set G such that x ∈ G ⊆ N. Since N ⊆ M, then x ∈ G ⊆ M. So by definition of 4.1, M is a gpα-neighbourhood point of x. Hence M ∈ gpαN(x). (iv) Let N ∈ gpαN(x) and M ∈ gpαN(x). Then, there exist gpα-open sets U and V such that x ∈ U ⊆ N and x ∈ V ⊆ M. That is x ∈ U ∩ V ⊆ N ∩ M. Therefore, for every point x ∈ X, there exists gpα-open set U ∩ V such that x ∈ U ∩V ⊆ N ∩M. So, N ∩M is a gpα-neighbourhood of a point x. Hence, intersection of two gpα-neighbourhood of a point is again a gpα-neighbourhood of point. Corollary 4.1. For any subset A of X, every α-interior point of A is gpα-interior point of A. Proof: It follows from the fact that every α-open set is gpα-open in X (9). Theorem 4.4. For any subset A of X, every pre-interior point of A is gpα-interior point of A. Proof: For any pre-interior point x of A. Then there exists pre-open set G con- taining x such that G ⊆ A. Since every pre-open set is gpα-open (9), then G is gpα-open in X. Hence x is a gpα-interior point of A. 5 gpα-Kuratowski closure operators in topological spaces Theorem 5.1. If P-C(X,τ) is closed under finite union, then gpα-C(X,τ) is closed under finite union, where P-C(X,τ) and gpα-C(X,τ) are the families of pre-closed sets and gpα-closed sets in (X,τ) respectively. Proof: Let A and B are gpα-closed sets in X and A∪B ⊆ G, where G is ωα-open in X. Then A ⊆ G and B ⊆ G. Since A and B are gpα-closed, then pcl(A) ⊆ G and pcl(B) ⊆ G. Then pcl(A)∪pcl(B) = pcl(A∪B) ⊆ G from (7). Thus, from hypothesis, pcl(A∪B) ⊆ G. Hence A∪B is gpα-closed in X. Definition 5.1. Let τ∗gpα be the topology on X generated by gpα-closure in the usual manner, τ∗gpα = {G ⊂ X : gpα-cl(X \G) = X \G}. Definition 5.2. Let τ∗g∗p be the topology on X generated by g ∗p-closure in the usual manner, that is τ∗g∗p = {G ⊂ X : g∗p-cl(X \G) = X \G} Theorem 5.2. Let A ⊆ X. Then the following statements holds: (i) τ ⊆ τ∗gpα (ii) τ ⊆ τp ⊆ τ∗gpα (τp is family of pre-open sets.(12)) 143 P.G. Patil and Bhadramma Pattanashetti Proof: (i) Let A ∈ τ, then Ac is closed in X. We have Ac ⊆ gpα-cl(Ac) ⊆ cl(Ac). Since, Ac is closed, then cl(Ac) is also closed in X. Hence Ac ⊆ gpα-cl(Ac) ⊆ Ac. Therefore gpα-cl(Ac) ⊆ Ac. But Ac ⊆ gpα-cl(Ac) is always true. Thus gpα- cl(Ac) = Ac. Hence A ∈ τ∗gpα. (ii) Since, every pre-closed set is gpα-closed in X, proof follows. Theorem 5.3. For any subset A of a topological space X, τ ⊆ τ∗gpα ⊆ τ∗g∗p. Proof: Let us consider A ∈ τ, then Ac is closed in X. Then Ac ⊆ gpα-cl(Ac) ⊆ cl(Ac). Since, Ac is closed, then cl(Ac) = Ac. Therefore Ac ⊆ gpα-cl(Ac) ⊆ Ac. Hence, gpα-cl(Ac) ⊆ Ac. But (Ac) ⊆ gpα-cl(Ac) is always true. Hence (Ac) = gpα-cl(Ac). Thus A ∈ τ∗gpα and hence τ ⊆ τ∗gpα. Let A ∈ τ∗gpα. Then gpα-cl(Ac) = Ac. But Ac ⊆ g∗p-cl(Ac) ⊆ gpα-cl(Ac) = Ac from (9). Hence g∗p-cl(Ac) = Ac. Hence A ∈ τ∗g∗p. Thus A ∈ τ∗gpα implies that A ∈ τ∗g∗p. Hence τ ⊆ τ∗gpα ⊆ τ∗g∗p. Theorem 5.4. The following statements are equal for the space X: (i) Every gpα-closed set is pre-closed. (ii) τp = τ∗gpα. (iii) For each x ∈ X, {x} is ωα-open or pre-open. Proof: (i) → (ii) Let G ∈ τ∗gpα. Then from (9) and by the Theorem 3.1, we have gpα-cl(A) = A. Hence X \G = gpα-cl(X \G) = p-cl(X \G). Therefore X \G is pre-closed and so G is pre-open. Therefore τ∗gpα ⊆ τp and from (9), τp ⊆ τ∗gpα. Hence τp = τ∗gpα. (ii) → (iii) Let {x} ∈ X. By (9), we have X \{x} = gpα-cl(X \{x}) is true only when {x} is not ωα-closed. Hence {x}∈ ωα-C(X,τ) or x ∈ τp. (iii) → (i) Let A be gpα-closed in X and x ∈ p− cl(A). Then, we have x ∈ A. case I: If {x} is ωα-closed. Suppose x /∈ A, then p−cl(A)\A contains ωα-closed set {x}, which is contradiction. Hence x ∈ A. case II: If {x} is pre-open. Since x ∈ pcl(A), then {x}∩A = φ. Hence, we have pcl(A) = A and thus A is pre-closed. Therefore gpα-C(X,τ) ⊂ p-C(X,τ). Theorem 5.5. Every gpα-closed set closed if and only if τ = τ∗gpα. Proof: Suppose every gpα-closed set is closed. Let A be gpα-closed then, gpα- cl(A) = cl(A). Thus τ = τ∗gpα. Conversely, let A be gpα-closed then from (9), A = gpα-cl(A). Hence X \ A ∈ τ∗gpα. Hence, A is closed in X. Theorem 5.6. Every gpα-closed set is pre-closed if and only if τp = τ∗gpα. Proof: Suppose that every gpα-closed set is pre-closed. Let A be gpα-closed in X. Then from hypothesis, gpα-cl(A) = pcl(A). Thus τp = τ∗gpα. Conversely, let A be gpα-closed in X. Then A = gpα-cl(A). Thus X \ A ∈ τ∗gpα. Hence, A is pre-closed in X. 144 gpα- Kuratowski closure operators in topological spaces Remark 5.1. Let A be any subset of X. Then gpα-int(A) is the largest gpα-open set contained in A if A is gpα-open. Theorem 5.7. gpα-closure is a Kuratowski closure operator on X. Proof:Follows from the Definition 3.2 and (9) 6 Characterizations of gpα-closed sets in topologi- cal spaces Definition 6.1. (9) A point x ∈ X is a gpα-limit point of a subset A of X, if and only if every gpα-neighbourhood of x contains a point of A distinct from x. That is, [N \{x}]∩A 6= φ for each gpα-neighbourhood N of x. Definition 6.2. (9) The set of all gpα-limit points of A is a gpα-derived set of A and is denoted by gpα-d(A). Example 6.1. Let X = {a,b,c} and τ = {X,φ,{a}}. Let A = {c}. Then the only limit point with respect to the set A = c is point b. Therefore d(A) = {b}. But gpα-limit point with respect to the set A is φ. Therefore gpα-d(A) = φ. Theorem 6.1. Let A be any subset of X, Then,A is gpα-closed if and only if gpα- d(A)⊆ A. Proof: Let A be gpα-closed in X, then Ac is gpα-open in X such that x ∈ Ac. Then for each point x ∈ X and from Definition 4.1 , there exist gpα-open set G such that x ∈ G ⊆ X \A. Then A∩ (X \A) = φ. Therefore, gpα-neighbourhood of G contains no points of A. Hence, x is not a gpα-limit point of A. Thus, no point of X \ A is a gpα-limit point of A, that is A contains all the gpα-limit points. Therefore A contains the gpα-derived points. Hence gpα-d(A) ⊆ A. Conversely, suppose gpα-d(A)⊆ A and let x ∈ Ac. So x /∈ A. Hence x /∈ gpα- d(A). Therefore x is not a limit point of A. Then, there exists gpα-open set G such that G∩ (A\{x}) = φ, that is G ⊆ X \A. Therefore for each x ∈ X \A, there exists gpα-open set G such that x ∈ G ⊆ X \A. Therefore X \A is gpα-open in X and hence A is gpα-closed. Theorem 6.2. Let τ1 and τ2 be any two topologies on a set X such that gpα- O(X,τ1) ⊆ gpα-O(X,τ2). Then for every subset A of X, every gpα-limit point of A with respect to τ2 is gpα-limit point of A with respect to τ1. Proof: Let x be a gpα-limit point of A with respect to τ2. Then by definition of gpα- limit point (G∩A)\{x} 6= φ, this is true for every G ∈ gpα-O(X,τ2) and x ∈ G. 145 P.G. Patil and Bhadramma Pattanashetti But by hypothesis, gpα-O(X,τ1) ⊆ gpα-O(X,τ2). Hence (G∩A)\{x} 6= φ for every G ∈ gpα-O(X,τ1) such that x ∈ G. Hence x is a gpα-limit point of A with respect to the topology τ1. Theorem 6.3. Let A and B be any two subsets of (X,τ). Then the following assertions are valid: (i) gpα-d(A) ⊆ dp(A), where dp is a pre-derived set (12). (ii) gpα-d(A∪gpα-d(A)) ⊆ A∪gpα-d(A). Proof: (i) It clearly observed from the fact that every pre-open set is gpα-open in X. Then y ∈ G and y ∈ gpα-d(A) \{x}. That is y ∈ G and y ∈ gpα-d(A). Hence G ∩ (A \ {y}) 6= φ. Let z ∈ G ∩ (A \ {y}), then x 6= z as x /∈ A. Thus G∩ (A\{x}) 6= φ. (ii) Let x ∈ gpα-d(A ∪ gpα-d(A)). If x ∈ A, then x ∈ gpα-d(A). Therefore x ∈ A∪gpα-d(A). On the contrary assume that x /∈ A. Then G∩(A∪gpα-d(A)) \{x}) 6= φ, is true for all G ∈ gpα-d(A) and x ∈ G. Therefore (G∩A)\{x} 6= φ or G∩ (gpα-d(A)) \{x}) 6= φ. Thus x ∈ gpα-d(A). If G ∩ (gpα-d(A)) \{x} 6= φ, then will get x ∈ gpα-d(gpα-d(A)). Since x /∈ A, then x ∈ gpα-d(gpα-d(A)) \A. Therefore gpα-d(A∪gpα-d(A)) ⊆ A∪gpα-d(A). Remark 6.1. We can see the following implification with respect to gpα-open sets. Example 6.2. Let X = {a,b,c,d,e} and τ = {X,φ,{a},{c,d},{a,c,d}}. In (X,τ) we have, pre-open sets are: X, φ, {a},{c},{d},{a,c},{b,c},{c,d},{d,e},{a,c,d}, {a,c,e},{a,d,e},{a,b,c,d},{a,b,c,e},{a,b,d,e},{a,c,d,e}. gpα-open sets are: X, φ, {a},{c},{d},{a,b},{b,d},{c,d},{a,c}, {b,c},{d,e},{a,b,d},{a,c,d},{b,c,d},{a,b,c},{a,c,e},{a,d,e}, {a,b,c,d},{a,b,c,e},{a,b,d,e},{a,c,d,e} . (i)Let A = {b,c,d}. Then pre-limit point of the set A is {b,e} and gpα-limit point of A is {e}. Hence dp(A) = {b,e} and gpα-d(A) = {e}. Hence, gpα-d(A) ⊆ dp(A). (ii)Let A = {a,c,d}, then gpα-d(A) = {b,e}. Consider A∪gpα-d(A) = {a,c,d}∪{b,e} = X. But gpα-d(X) = {b,e}. Therefore gpα-d(A∪gpα-d(A)) = gpα-d(X) = {b,e}. Now consider A∪gpα-d(A) = {a,c,d}∪{b,e} = X. Hence gpα-d(A ∪ gpα-d(A)) 6= A ∪ gpα-d(A), that is {b,e} 6= X, but gpα- d(A∪gpα-d(A) ⊂ A∪gpα-d(A). 146 gpα- Kuratowski closure operators in topological spaces Remark 6.2. if A ⊆ B, then gpα-d(A) ⊆ gpα-d(B). Example 6.3. Consider the Example 6.2, Let A = {b,c,d}. Then we have gpα-d(A) = {e}. Let B = {a,e}. Then gpα-limit point of the set B is φ, that is gpα-d(B) = φ. Thus we can observe that gpα-d(B) ⊂ gpα-d(A) but B * A. Remark 6.3. gpα-d(A∩B) ⊆ gpα-d(A) ∩ gpα-d(B). Example 6.4. From the Example 6.2, Let A = {b,c,d} and B = {b,c} are any two subsets of X. Then gpα-d(A) = {e} and gpα-d(B) = φ. Also gpα-d(A∩B) = φ. Therefore gpα-d(A∩B) ⊆ gpα-d(A) ∩ gpα-d(B) Theorem 6.4. Let A be any subset of X and x ∈ X. Then the following statements are equal: (i) For each x ∈ X, A∩G 6= φ where G is gpα-open in X. (ii) x ∈ gpα-cl(A). Proof: Let A be any subset of X. (i) → (ii): On the contrary assume that x /∈ gpα-cl(A). Then there exists gpα- closed set F such that A ⊆ F and x /∈ F . Then X\F is gpα-open in X containing a point x. Hence A∩ (X \F) ⊆ A∩ (X \A) = φ, which is contradiction to the assumption. Hence x ∈ gpα-cl(A). (ii) → (i): Follows from the Definition 3.2. Corollary 6.1. For any subset A of a space X, gpα-d(A) ⊆ gpα-cl(A). Theorem 6.5. Let A be any subset of X, then gpα-cl(A) = A∪gpα-d(A). Proof: Let x ∈ gpα-cl(A). On the contrary assume that x /∈ A. Let G be any gpα- open set containing a point x. Then (G\{x}) ∩A 6= φ. Therefore x is gpα-limit point of A and hence x is gpα-derived set of A, that is x ∈ gpα-d(A). Hence gpα-cl(A) ⊆ A∪gpα-d(A). From the Corollary 6.1, we have gpα-d(A) ⊆ gpα-cl(A) and A ⊆ gpα-cl(A) is always true. Hence A∪gpα-d(A) ⊆ gpα-cl(A). Therefore gpα-cl(A) = A∪gpα-d(A). Theorem 6.6. Let A be gpα-open set in X and B be any subset of X. Then A∩gpα- cl(B) ⊆ gpα-cl(A∩B). Proof: Let x ∈ A∩gpα-cl(B). Then x ∈ A and x ∈ gpα-cl(B). From the Theorem 6.5, we have gpα-cl(B) = B ∪gpα-d(B). If x ∈ B then x ∈ A∩B. Then A∩B ⊆ gpα-cl(A∩B). If x /∈ B then x ∈ gpα-d(B). From the definition of gpα-limit point, we have 147 P.G. Patil and Bhadramma Pattanashetti G ∩ B 6= φ for every gpα-open set G containing x. Therefore G ∩ (A ∩ B) = (G ∩ A) ∩ B 6= φ. Hence x ∈ gpα-d(A ∩ B) ⊆ gpα-cl(A ∩ B). Therefore A∩gpα-cl(B) ⊆ gpα-cl(A∩B). However the equality does not holds in general Example 6.5. Let X = {a,b,c,d,e} and τ = {X,φ,{a}}. Let A = {a,b} and B = {a,c} are two subsets of X. Then A∩gpα-cl(B) = {a,b} and gpα-cl(A∩B) = X This implies A∩gpα-cl(B) 6= gpα-cl(A∩B). But , A∩gpα-cl(B) ⊂ gpα-cl(A∩B). 7 Conclusions In this present work, we have analyzed the notion of generalized pre α-closed sets in topological spaces. We have established the results of gpα-closure, gpα- interior, gpα-neighbourhood and gpα-limit points. Moreover, we have character- ized these concepts with suitable examples. Finally, we apply gpα-open sets for Kuratowski closure axioms. There is a scope to study and extend these newly defined concepts. 8 Acknowledgment The first author is grateful to the University Grants Commission, New Delhi, India for financial support under UGC SAP DRS-III:F-510/3/DRS-III/2016(SAP- I) dated 29th Feb. 2016 to the Department of Mathematics, Karnatak University, Dharwad, India. The second author is grateful to Karnatak University Dharwad for the financial support to research work under URS scheme. Authors are very thankful to the anonymous reviewer/s for their suggestions in the improvement of this paper. References [1] D. Andrijevic,(1986), Semi-pre open sets, Mat. Vensik, 38(1), 24-32. [2] S. S. Benchalli, P. G. Patil and T. D. Rayanagoudar, (2009), ωα-closed sets in topological spaces, The Global Jl. of Appl. Maths and Mth. Sciences, (2), 53-63. 148 gpα- Kuratowski closure operators in topological spaces [3] S. Jafari, S. S. Benchalli, P.G. Patil and T. D. Rayanagoudar, (2012), Pre g∗- closed sets in topological spaces, Jl. of Adv.Studies in Topology, (3), 55-59. [4] N. Levine,(1963), Semi-open and semi-continuity in topological space, Amer. Math. Monthly 70, 36-41. [5] N. Levine, (1970), Generalized closed sets in topology, Rend. Circ. Math. Palermo, 19, (2), 89-96. [6] O. Njastad, (1965), On some classes of nearly open sets, Pacific Jl. of Math., 15, 961-970. [7] A.S. Mashhour, M.E.Abd El-Monsef and S.N.Deeb, (1982), On pre- continuous and weak pre continuous mappings, Proc. Math. Phys. Soc. Egypt, 53, 47-53. [8] P.G. Patil ,S. S. Benchalli, Pallavi S. Mirajakar, (2015), Generalized star ωα-closed sets in topological spaces, Jl. of New Results in Sci, (9), 37-45. [9] Praveen H. Patil and P.G. Patil (2018), Generalized pre α-closed sets in topo- logical spaces, Jl. of New Theory, (20), 48-56. [10] P. Sundaram and M. Sheik John, (2002),On ω-closed sets, Acta. Ciencia Indica, 4 389-392. [11] M.K.R.S. Veerakumar, (2002), g∗pre-closed sets, Acta. Cienia Indica, 28, 51-60 [12] Young Bae Jun, Seong W. Jeong, Hyeon J. Lee and Joon Woo Lee, (2008), Applications of pre-open sets, Appl. General Topology, 9(2), 213-228. 149