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South Asian Journal of Mathematics 2014 , Vol. 4 ( 6 ) : 252 ∼ 264 www.sajm-online.com ISSN 2251-1512 RESEARCH ARTICLE g ⋆-closed sets in ideal topological spaces O. Ravi ① , M. Paranjothi② ∗, S. Murugesan③, M. Meharin④ ① Department of Mathematics, P. M. Thevar College, Usilampatti, Madu-rai District, Tamil Nadu, India ② Department of Mathematics, Sowdambiga College of Engineering, Aruppukottai, Virudhunagar District, Tamil Nadu, India ③ Department of Mathematics, Sri S. Ramasamy Naidu Memorial Col-lege, Sattur-626 203, Tamil Nadu, India ④ School of Youth Empowerment, Madurai Kamaraj University, Madu- rai, Tamil Nadu, India E-mail: [email protected] Received: June-28-2014; Accepted: Oct-3-2014 *Corresponding author Abstract The notion of g ⋆ -closed sets is introduced in ideal topological spaces. Characterizations and properties of Ig⋆ -closed sets and Ig⋆ -open sets are given. A characterization of normal spaces is given in terms of Ig⋆ -open sets. Also, it is established that an Ig⋆ -closed subset of an I-compact space is I-compact. Key Words MSC 2010 1 g ⋆ -closed set, Ig⋆ -closed set and I-compact space 54A05, 54D15 Introduction and Preliminaries An ideal I on a topological space (X, τ ) is a nonempty collection of subsets of X which satisfies (i) A∈I and B⊆A⇒B∈I and (ii) A∈I and B∈I⇒A∪B∈I. Given a topological space (X, τ ) with an ideal I on X and if ℘(X) is the set of all subsets of X, a set operator (.)∗ : ℘(X)→℘(X), called a local function [16] of A with respect to τ and I is defined as follows: for A⊆X, A∗ (I,τ )={x∈X | U∩A∈I / for every U∈τ (x)} where τ (x)={U∈τ | x∈U}. We will make use of the basic facts about the local functions [[13], Theorem 2.3] without mentioning it explicitly. A Kuratowski closure operator cl∗ (.) for a topology τ ∗ (I,τ ), called the ⋆-topology, finer than τ is defined by cl∗ (A)=A∪A∗ (I,τ ) [31]. When there is no chance for confusion, we will simply write A∗ for A∗ (I,τ ) and τ ∗ for τ ∗ (I,τ ). If I is an ideal on X, then (X, τ , I) is called an ideal topological space. N is the ideal of all nowhere dense subsets in (X, τ ). A subset A of an ideal topological space (X, τ , I) is ⋆-closed [13] (resp. ⋆-dense in itself [11]) if A∗ ⊆A (resp. A⊆A∗ ). A subset A of an ideal topological space (X, τ , I) is Ig -closed [2] if A∗ ⊆U whenever A⊆U and U is open. By a space, we always mean a topological space (X, τ ) with no separation properties assumed. If A⊆X, cl(A) and int(A) will, respectively, denote the closure and interior of A in (X, τ ) and int∗ (A) will denote the interior of A in (X, τ ∗ ). A subset A of a space (X, τ ) is an α-open [24] (resp. semi-open [17], preopen [19], regular open [29]) set if A⊆int(cl(int(A))) (resp. A⊆cl(int(A)), A⊆int(cl(A)), A = int(cl(A))). Citation: O. Ravi, M. Paranjothi, S. Murugesan, M. Meharin, g⋆ -closed sets in ideal topological spaces, South Asian J Math, 2014, 4(6), 252-264. South Asian J. Math. Vol. 4 No. 6 The family of all α-open sets in (X, τ ), denoted by τ α , is a topology on X finer than τ . The closure of A in (X, τ α ) is denoted by clα (A). Definition 1.1. A subset A of a space (X, τ ) is said to be 1. g-closed [18] if cl(A)⊆U whenever A⊆U and U is open, 2. g-open [18] if its complement is g-closed, 3. g ⋆ -closed [33] or strongly g-closed [30] if cl(A)⊆U whenever A⊆U and U is g-open, 4. ĝ-closed [32] or ω-closed [28] or s⋆ g-closed [15, 20, 26] if cl(A)⊆U whenever A⊆U and U is semi-open. The g-closure of a subset A of X, denoted by gcl(A) or cl*(A) [4], is defined to be the intersection of all g-closed sets containing A. Definition 1.2. An ideal I is said to be 1. codense [3] or τ -boundary [23] if τ ∩ I={∅}, 2. completely codense [3] if PO(X)∩I={∅}, where PO(X) is the family of all preopen sets in (X, τ ). Lemma 1.1. Every completely codense ideal is codense but not conversely [3]. The following Lemmas, Result, Definitions, Remarks and Theorem will be useful in the sequel. Lemma 1.2. Let (X, τ , I) be an ideal topological space and A⊆X. If A⊆A∗ , then A∗ =cl(A∗ )=cl(A) =cl∗ (A) [[27], Theorem 5]. Lemma 1.3. Let (X, τ , I) be an ideal topological space. Then I is codense if and only if G⊆G∗ for every semi-open set G in X [[27], Theorem 3]. Lemma 1.4. Let (X, τ , I) be an ideal topological space. If I is completely codense, then τ ∗ ⊆τ α [[27], Theorem 6]. Result 1. For a subset of a topological space, the following properties hold: 1. Every closed set is g ⋆ -closed but not conversely [33]. 2. Every g ⋆ -closed set is g-closed but not conversely [33]. 3. Every closed set is ĝ-closed but not conversely [32]. 4. Every ĝ-closed set is g-closed but not conversely [32]. Lemma 1.5. If (X, τ , I) is a TI ideal topological space and A is an Ig -closed set, then A is a ⋆-closed set [[21], Corollary 2.2]. Lemma 1.6. Every g-closed set is Ig -closed but not conversely [[2], Theorem 2.1]. Lemma 1.7. [18] The intersection of a g-closed set and a closed set is g-closed. Lemma 1.8. [13] Let (X, τ , I) be an ideal topological space and A, B subsets of X. Then the following properties hold: 253 O. Ravi, et al : g ⋆ -closed sets in ideal topological spaces 1. A⊆B⇒A∗ ⊆B∗ , 2. A∗ =cl(A∗ )⊆cl(A), 3. (A∗ )∗ ⊆A∗ , 4. (A∪B)∗ =A∗ ∪B∗ , 5. (A∩B)∗ ⊆A∗ ∩B∗ . Definition 1.3. A subset G of an ideal topological space (X, τ , I) is said to be 1. Irg -closed [22] if G* ⊆ H whenever G ⊆ H and H is regular open in (X, τ , I). 2. pre∗I -open [5] if G ⊆ int*(cl(G)). 3. pre∗I -closed [5] if X\G is pre∗I -open. 4. I-R closed [1] if G = cl*(int(G)). Remark 1. [6] In any ideal topological space, every I-R closed set is ⋆-closed but not conversely. Definition 1.4. [6] Let (X, τ , I) be an ideal topological space. A subset G of X is said to be a weakly Irg -closed set if (int(G))* ⊆ H whenever G ⊆ H and H is a regular open set in X. Remark 2. [6] Let (X, τ , I) be an ideal topological space. The following diagram holds for a subset G ⊆ X: Ig -closed −→ Irg -closed −→ weakly Irg -closed ↑ ↑ ⋆-closed pre∗I -closed ↑ I-R-closed These implications are not reversible. Definition 1.5. [7, 8] A subset K of an ideal topological space (X, τ , I) is said to be 1. semi∗ -I-open if K ⊆ cl(int∗ (K)), 2. semi∗ -I-closed if its complement is semi∗ -I-open. Definition 1.6. [7] The semi∗ -I-closure of a subset K of an ideal topological space (X, τ , I), denoted by s∗I cl(K), is defined by the intersection of all semi∗ -I-closed sets of X containing K. Theorem 1.9. [7] For a subset K of an ideal topological space (X, τ , I), s∗I cl(K) = K ∪ int(cl∗ (K)). Definition 1.7. [9] Let (X, τ , I) be an ideal topological space and K ⊆ X. K is called 1. generalized semi∗ -I-closed (gs∗I -closed) in (X, τ , I) if s∗I cl(K) ⊆ O whenever K ⊆ O and O is an open set in (X, τ , I). 2. generalized semi∗ -I-open (gs∗I -open) in (X, τ , I) if X\K is a gs∗I -closed set in (X, τ , I). 254 South Asian J. Math. Vol. 4 No. 6 2 Ig⋆ -closed sets Definition 2.1. A subset A of an ideal topological space (X, τ , I) is said to be 1. Ig⋆ -closed if A∗ ⊆ U whenever A⊆U and U is g-open, 2. Ig⋆ -open if its complement is Ig⋆ -closed. Theorem 2.1. If (X, τ , I) is any ideal topological space, then every Ig⋆ -closed set is Ig -closed but not conversely. Proof. It follows from the fact that every open set is g-open. Example 2.2. Let X={a, b, c}, τ ={∅, X, {c}} and I={∅, {a}}. Then Ig⋆ -closed sets are ∅, X, {a}, {a, b} and Ig -closed sets are ∅, X, {a}, {b}, {a, b}, {a, c}, {b, c}. It is clear that {b} is Ig -closed but it is not Ig⋆ -closed. The following Theorem gives characterizations of Ig⋆ -closed sets. Theorem 2.3. If (X, τ , I) is any ideal topological space and A⊆X, then the following are equivalent. 1. A is Ig⋆ -closed, 2. cl∗ (A)⊆U whenever A⊆U and U is g-open in X, 3. For all x∈cl∗ (A), gcl({x})∩A6=∅. 4. cl∗ (A)−A contains no nonempty g-closed set, 5. A∗ −A contains no nonempty g-closed set. Proof. (1)⇒(2) If A is Ig⋆ -closed, then A∗ ⊆U whenever A⊆U and U is g-open in X and so cl∗ (A)=A∪A∗ ⊆U whenever A⊆U and U is g-open in X. This proves (2). (2)⇒(3) Suppose x∈cl∗ (A). If gcl({x})∩A=∅, then A⊆X−gcl({x}). By (2), cl∗ (A)⊆ X−gcl({x}), a contradiction, since x∈cl∗ (A). (3)⇒(4) Suppose F⊆cl∗ (A)−A, F is g-closed and x∈F. Since F⊆X−A and F is g-closed, then A⊆X−F and F is g-closed, gcl({x})∩A=∅, a contradiction, since x∈cl∗ (A) and by (3), gcl({x})∩A6=∅. Therefore cl∗ (A)−A contains no nonempty g-closed set. (4)⇒(5) Since cl∗ (A)−A=(A∪A∗ )−A=(A∪A∗ )∩Ac =(A∩Ac )∪(A∗ ∩Ac )=A∗ ∩Ac = A∗ −A. Therefore A∗ −A contains no nonempty g-closed set. (5)⇒(1) Let A⊆U where U is g-open set. Therefore X−U⊆X−A and so A∗ ∩(X−U) ⊆A∗ ∩(X−A)=A∗ −A. Therefore A∗ ∩(X−U)⊆A∗ −A. Since A∗ is always closed set, so A∗ ∩(X−U) is a g-closed set contained in A∗ −A. Therefore A∗ ∩(X−U)=∅ and hence A∗ ⊆U. Therefore A is Ig⋆ -closed. Theorem 2.4. Every ⋆-closed set is Ig⋆ -closed but not conversely. Proof. Let A be a ⋆-closed, then A∗ ⊆A. Let A⊆U where U is g-open. Hence A∗ ⊆U whenever A⊆U and U is g-open. Therefore A is Ig⋆ -closed. 255 O. Ravi, et al : g ⋆ -closed sets in ideal topological spaces Example 2.5. Let X={a, b, c, d}, τ ={∅, X, {d}, {a, c}, {a, c, d}} and I={∅, {a}, {d}, {a, d}}. Then Ig⋆ -closed sets are ∅, X, {a}, {b}, {d}, {a, b}, {a, d}, {b, c}, {b, d}, {a, b, c}, {a, b, d}, {b, c, d} and ⋆-closed sets are ∅, X, {a}, {b}, {d}, {a, b}, {a, d}, {b, d}, {a, b, c}, {a, b, d}. It is clear that {b, c} is Ig⋆ -closed set but it is not ⋆-closed. Theorem 2.6. Let (X, τ , I) be an ideal topological space. For every A∈I, A is Ig⋆ -closed. Proof. Let A⊆U where U is g-open set. Since A∗ =∅ for every A∈I, then cl∗ (A)=A∪A∗ =A⊆U. Therefore, by Theorem 2.3, A is Ig⋆ -closed. Theorem 2.7. If (X, τ , I) is an ideal topological space, then A∗ is always Ig⋆ -closed for every subset A of X. Proof. Let A∗ ⊆U where U is g-open. Since (A∗ )∗ ⊆A∗ [13], we have (A∗ )∗ ⊆U whenever A∗ ⊆U and U is g-open. Hence A∗ is Ig⋆ -closed. Theorem 2.8. Let (X, τ , I) be an ideal topological space. Then every Ig⋆ -closed, g-open set is ⋆-closed set. Proof. Since A is Ig⋆ -closed and g-open. Then A∗ ⊆A whenever A⊆A and A is g-open. Hence A is ⋆-closed. Corollary 2.9. If (X, τ , I) is a TI ideal topological space and A is an Ig⋆ -closed set, then A is ⋆-closed set. Proof. If (X, τ , I) is a TI ideal topological space, then every Ig -closed set is ⋆-closed. By Theorem 2.1, Ig⋆ -closed set is ⋆-closed. Corollary 2.10. Let (X, τ , I) be an ideal topological space and A be an Ig⋆ -closed set. Then the following are equivalent. 1. A is a ⋆-closed set, 2. cl∗ (A)−A is a g-closed set, 3. A∗ −A is a g-closed set. Proof. (1)⇒(2) If A is ⋆-closed, then A∗ ⊆A and so cl∗ (A)−A=(A∪A∗ )−A=∅. Hence cl∗ (A)−A is g-closed set. (2)⇒(3) Since cl∗ (A)−A=A∗ −A and so A∗ −A is g-closed set. (3)⇒(1) If A∗ −A is a g-closed set, since A is Ig⋆ -closed set, by Theorem 2.3, A∗ −A=∅ and so A is ⋆-closed. Theorem 2.11. Let (X, τ , I) be an ideal topological space. Then every g ⋆ -closed set is an Ig⋆ -closed set but not conversely. Proof. Let A be a g ⋆ -closed set. Then cl(A)⊆U whenever A⊆U and U is g-open. We have cl∗ (A)⊆cl(A)⊆U whenever A⊆U and U is g-open. Hence A is Ig⋆ -closed. Example 2.12. In Example 2.1, g ⋆ -closed sets are ∅, X, {a, b}. It is clear that {a} is Ig⋆ -closed set but it is not g ⋆ -closed. 256 South Asian J. Math. Vol. 4 No. 6 Theorem 2.13. If (X, τ , I) is an ideal topological space and A is a ⋆-dense in itself, Ig⋆ -closed subset of X, then A is g ⋆ -closed. Proof. Suppose A is a ⋆-dense in itself, Ig⋆ -closed subset of X. Let A⊆U where U is g-open. Then by Theorem 2.3 (2), cl∗ (A)⊆U whenever A⊆U and U is g-open. Since A is ⋆-dense in itself, by Lemma 1.2, cl(A)=cl∗ (A). Therefore cl(A)⊆U whenever A⊆U and U is g-open. Hence A is g ⋆ -closed. Corollary 2.14. If (X, τ , I) is any ideal topological space where I={∅}, then A is Ig⋆ -closed if and only if A is g ⋆ -closed. Proof. From the fact that for I={∅}, A∗ =cl(A)⊇A. Therefore A is ⋆-dense in itself. Since A is Ig⋆ -closed, by Theorem 2.13, A is g ⋆ -closed. Conversely, by Theorem 2.11, every g ⋆ -closed set is Ig⋆ -closed set. Corollary 2.15. If (X, τ , I) is any ideal topological space where I is codense and A is a semi-open, Ig⋆ -closed subset of X, then A is g ⋆ -closed. Proof. By Lemma 1.3, A is ⋆-dense in itself. By Theorem 2.13, A is g ⋆ -closed. Example 2.16. In Example 2.1, g-closed sets are ∅, X, {a}, {b}, {a, b}, {a, c}, {b, c}. It is clear that {b} is g-closed set but it is not Ig⋆ -closed. Example 2.17. In Example 2.2, g-closed sets are ∅, X, {b}, {a, b}, {b, c}, {b, d}, {a, b, c}, {a, b, d}, {b, c, d}. It is clear that {a} is Ig⋆ -closed set but it is not g-closed. Remark 3. 1. By Example 2.16 and Example 2.17, g-closed sets and Ig⋆ -closed sets are independent. 2. By Example 2.6 and Example 2.7, g ⋆ -closed sets and ĝ-closed sets are independent. Example 2.18. Let X={a, b, c} and τ ={∅, X, {a}, {a, b}}. Then g ⋆ -closed sets are ∅, X, {c}, {a, c}, {b, c} and ĝ-closed sets are ∅, X, {c}, {b, c}. It is clear that {a, c} is g ⋆ -closed but it is not ĝ-closed. Example 2.19. Let X={a, b, c} and τ ={∅, X, {a}, {b, c}}. Then g ⋆ -closed sets are ∅, X, {a}, {b, c} and ĝ-closed sets are ∅, X, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}. It is clear that {b} is ĝ-closed but it is not g ⋆ -closed. Remark 4. We have the following implications for the subsets stated above. ĝ-closed 3 6 closed ? ∗-closed s ? - g ⋆ -closed - g-closed ? - Ig⋆ -closed ? - Ig -closed Theorem 2.20. Let (X, τ , I) be an ideal topological space and A⊆X. Then A is Ig⋆ -closed if and only if A=F−N where F is ⋆-closed and N contains no nonempty g-closed set. 257 O. Ravi, et al : g ⋆ -closed sets in ideal topological spaces Proof. If A is Ig⋆ -closed, then by Theorem 2.3 (5), N=A∗ −A contains no nonempty g-closed set. If F=cl∗ (A), then F is ⋆-closed such that F−N=(A∪A∗ )−(A∗ −A)=(A∪ A∗ )∩(A∗ ∩Ac )c =(A∪A∗ )∩((A∗ )c ∪A) =(A∪A∗ )∩(A∪(A∗ )c )= A∪(A∗ ∩(A∗ )c )=A. Conversely, suppose A=F−N where F is ⋆-closed and N contains no nonempty g-closed set. Let U be an g-open set such that A⊆U. Then F−N⊆U which implies that F∩(X−U)⊆N. Now A⊆F and F∗ ⊆F then A∗ ⊆F∗ and so A∗ ∩(X−U)⊆F∗ ∩(X−U)⊆F∩ (X−U)⊆N. By hypothesis, since A∗ ∩(X−U) is g-closed, A∗ ∩(X−U)=∅ and so A∗ ⊆U. Hence A is Ig⋆ -closed. Theorem 2.21. Let (X, τ , I) be an ideal topological space and A⊆X. If A⊆B⊆A∗ , then A∗ =B∗ and B is ⋆-dense in itself. Proof. Since A⊆B, then A∗ ⊆B∗ and since B⊆A∗ , then B∗ ⊆(A∗ )∗ ⊆A∗ . Therefore A∗ =B∗ and B⊆A∗ ⊆B∗ . Hence proved. Theorem 2.22. Let (X, τ , I) be an ideal topological space. If A and B are subsets of X such that A⊆B⊆cl∗ (A) and A is Ig⋆ -closed, then B is Ig⋆ -closed. Proof. Since A is Ig⋆ -closed, then by Theorem 2.3 (4), cl∗ (A)−A contains no nonempty g-closed set. Since cl∗ (B)−B⊆cl∗ (A)−A and so cl∗ (B)−B contains no nonempty g-closed set. Hence B is Ig⋆ -closed. Corollary 2.23. Let (X, τ , I) be an ideal topological space. If A and B are subsets of X such that A⊆B⊆A∗ and A is Ig⋆ -closed, then A and B are g ⋆ -closed sets. Proof. Let A and B be subsets of X such that A⊆B⊆A∗ which implies that A⊆B ⊆A∗ ⊆cl∗ (A) and A is Ig⋆ -closed. By Theorem 2.22, B is Ig⋆ -closed. Since A⊆B⊆A∗ , then A∗ =B∗ and so A and B are ⋆-dense in itself. By Theorem 2.13, A and B are g ⋆ -closed. The following Theorem gives a characterization of Ig⋆ -open sets. Theorem 2.24. Let (X, τ , I) be an ideal topological space and A⊆X. Then A is Ig⋆ -open if and only if F⊆int∗ (A) whenever F is g-closed and F⊆A. Proof. Suppose A is Ig⋆ -open. If F is g-closed and F⊆A, then X−A⊆X−F and so cl∗ (X−A)⊆X−F by Theorem 2.3 (2). Therefore F⊆X−cl∗ (X−A)=int∗ (A). Hence F⊆int∗ (A). Conversely, suppose the condition holds. Let U be a g-open set such that X−A⊆U. Then X−U⊆A and so X−U⊆int∗ (A). Therefore cl∗ (X−A)⊆U. By Theorem 2.3 (2), X−A is Ig⋆ -closed. Hence A is Ig⋆ -open. Corollary 2.25. Let (X, τ , I) be an ideal topological space and A⊆X. If A is Ig⋆ -open, then F⊆int∗ (A) whenever F is closed and F⊆A. The following Theorem gives a property of Ig⋆ -closed. Theorem 2.26. Let (X, τ , I) be an ideal topological space and A⊆X. If A is Ig⋆ -open and int∗ (A)⊆B⊆A, then B is Ig⋆ -open. Proof. Since A is Ig⋆ -open, then X−A is Ig⋆ -closed. By Theorem 2.3 (4), cl∗ (X−A)− (X−A) contains no nonempty g-closed set. Since int∗ (A)⊆int∗ (B) which implies that cl∗ (X−B)⊆cl∗ (X−A) and so cl∗ (X−B)−(X−B)⊆cl∗ (X−A)−(X−A). Hence B is Ig⋆ -open. The following Theorem gives a characterization of Ig⋆ -closed sets in terms of Ig⋆ -open sets. 258 South Asian J. Math. Vol. 4 No. 6 Theorem 2.27. Let (X, τ , I) be an ideal topological space and A⊆X. Then the following are equivalent. 1. A is Ig⋆ -closed, 2. A∪(X−A∗ ) is Ig⋆ -closed, 3. A∗ −A is Ig⋆ -open. Proof. (1)⇒(2) Suppose A is Ig⋆ -closed. If U is any g-open set such that A∪(X−A∗ )⊆U, then X−U⊆ X−(A∪(X−A∗ ))=X∩(A∪(A∗ )c )c =A∗ ∩Ac =A∗ −A. Since A is Ig⋆ -closed, by Theorem 2.3 (5), it follows that X−U=∅ and so X=U. Therefore A∪(X−A∗ )⊆U which implies that A∪(X−A∗ )⊆X and so (A∪(X−A∗ ))∗ ⊆X∗ ⊆X=U. Hence A∪(X−A∗ ) is Ig⋆ -closed. (2)⇒(1) Suppose A∪(X−A∗ ) is Ig⋆ -closed. If F is any g-closed set such that F⊆A∗ −A, then F⊆A∗ and F⊆X\A which implies that X−A∗ ⊆X−F and A⊆X−F. Therefore A∪(X−A∗ )⊆A∪(X−F)=X−F and X−F is g-open. Since (A∪(X−A∗ ))∗ ⊆X−F which implies that A∗ ∪(X−A∗ )∗ ⊆X−F and so A∗ ⊆X−F which implies that F⊆X−A∗ . Since F⊆A∗ , it follows that F=∅. Hence A is Ig⋆ -closed. (2)⇔(3) Since X−(A∗ −A)=X∩(A∗ ∩Ac )c =X∩((A∗ )c ∪A)=(X∩ (A∗ )c )∪(X∩A)=A∪ (X−A∗ ), the equivalence is clear. Theorem 2.28. Let (X, τ , I) be an ideal topological space. Then every subset of X is Ig⋆ -closed if and only if every g-open set is ⋆-closed. Proof. Suppose every subset of X is Ig⋆ -closed. If U⊆X is g-open, then U is Ig⋆ -closed and so U∗ ⊆U. Hence U is ⋆-closed. Conversely, suppose that every g-open set is ⋆-closed. If U is g-open set such that A⊆U⊆X, then A∗ ⊆U∗ ⊆U and so A is Ig⋆ -closed. The following Theorem gives a characterization of normal spaces in terms of Ig⋆ -open sets. Theorem 2.29. Let (X, τ , I) be an ideal topological space where I is completely codense. Then the following are equivalent. 1. X is normal, 2. For any disjoint closed sets A and B, there exist disjoint Ig⋆ -open sets U and V such that A⊆U and B⊆V, 3. For any closed set A and open set V containing A, there exists an Ig⋆ -open set U such that A⊆U⊆cl∗ (U)⊆V. Proof. (1)⇒(2) The proof follows from the fact that every open set is Ig⋆ -open. (2)⇒(3) Suppose A is closed and V is an open set containing A. Since A and X−V are disjoint closed sets, there exist disjoint Ig⋆ -open sets U and W such that A⊆U and X−V⊆W. Since X−V is g-closed and W is Ig⋆ -open, X−V⊆int∗ (W) and so X−int∗ (W)⊆V. Again U∩W=∅ which implies that U∩int∗ (W)=∅ and so U⊆X−int∗ (W) which implies that cl∗ (U)⊆X−int∗ (W)⊆V. U is the required Ig⋆ -open sets with A⊆U⊆cl∗ (U)⊆V. (3)⇒(1) Let A and B be two disjoint closed subsets of X. By hypothesis, there exists an Ig⋆ -open set U such that A⊆U⊆cl∗ (U)⊆X−B. Since U is Ig⋆ -open, A⊆int∗ (U). Since I is completely codense, by Lemma 1.4, τ ∗ ⊆τ α and so int∗ (U) and X−cl∗ (U)∈τ α . Hence A⊆int∗ (U)⊆int(cl(int(int∗ (U))))=G and 259 O. Ravi, et al : g ⋆ -closed sets in ideal topological spaces B⊆X−cl∗ (U)⊆int(cl(int(X−cl∗ (U)))) =H. G and H are the required disjoint open sets containing A and B respectively, which proves (1). Definition 2.2. A subset A of an ideal topological space (X, τ , I) is said to be an g # α-closed set [25] if clα (A)⊆U whenever A⊆U and U is g-open. The complement of g # α-closed is said to be an g # α-open set. If I=N , it is not difficult to see that Ig⋆ -closed sets coincide with g # α-closed sets and so we have the following Corollary. Corollary 2.30. Let (X, τ , I) be an ideal topological space where I=N . Then the following are equivalent. 1. X is normal, 2. For any disjoint closed sets A and B, there exist disjoint g # α-open sets U and V such that A⊆U and B⊆V, 3. For any closed set A and open set V containing A, there exists an g # α-open set U such that A⊆U⊆clα (U)⊆V. Definition 2.3. A subset A of an ideal topological space is said to be I-compact [10] or compact modulo I [23] if for every open cover {Uα | α∈∆} of A, there exists a finite subset ∆0 of ∆ such that A−∪{Uα | α∈∆0 }∈I. The space (X, τ , I) is I-compact if X is I-compact as a subset. Theorem 2.31. Let (X, τ , I) be an ideal topological space. If A is an Ig -closed subset of X, then A is I-compact [[21], Theorem 2.17]. Corollary 2.32. Let (X, τ , I) be an ideal topological space. If A is an Ig⋆ -closed subset of X, then A is I-compact. Proof. The proof follows from the fact that every Ig⋆ -closed is Ig -closed. Remark 5. Let (X, τ , I) be an ideal topological space. By Remark 2, Definition 1.7, Definition 2.1 and Theorem 2.1, the following diagram holds for a subset G ⊆ X: gs∗I -closed ↑ Ig⋆ -closed −→ Ig -closed −→ Irg -closed −→ weakly Irg -closed These implications are not reversible. Example 2.33. In Example 2.2, gs∗I -closed sets are ∅, X, {a}, {b}, {c}, {d}, {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {a, b, c}, {a, b, d}, {b, c, d}. It is clear that {c} is gs∗I -closed set but not Ig⋆ -closed. Definition 2.4. A subset A of an ideal topological space (X, τ , I) is said to be an s∗ CI -set if A = L ∩ M, where L ∈ τ and M is a semi∗ -I-closed set in X. Theorem 2.34. Let (X, τ , I) be an ideal topological space and V ⊆ X. Then V is an s∗ CI -set in X if and only if V = G ∩ s∗I cl(V) for an open set G in X. 260 South Asian J. Math. Vol. 4 No. 6 Proof. If V is an s∗ CI -set, then V = G ∩ M for an open set G and a semi∗ -I-closed set M. But then V ⊆ M and so V ⊆ s∗I cl(V) ⊆ M. It follows that V = V ∩ s∗I cl(V) = G ∩ M ∩ s∗I cl(V) = G ∩ s∗I cl(V). Conversely, it is enough to prove that s∗I cl(V) is a semi∗ -I-closed set. But s∗I cl(V) ⊆ M, for any semi∗ I-closed set M containing V. So, int(cl∗ (s∗I cl(V))) ⊆ int(cl∗ (M)) ⊆ M. It follows that int(cl∗ (s∗I cl(V))) ⊆ ∩ V ⊆ M, M is semi∗ −I−closed M = s∗I cl(V). Theorem 2.35. Let (X, τ , I) be an ideal topological space and A ⊆ X. The following properties are equivalent. 1. A is a semi∗ -I-closed set in X. 2. A is an s∗ CI -set and an gs∗I -closed set in X. Proof. (1) ⇒ (2): It follows from the fact that any semi∗ -I-closed set in X is an s∗ CI -set and an gs∗I -closed set in X. (2) ⇒ (1): Suppose that A is an s∗ CI -set and an gs∗I -closed set in X. Since A is an s∗ CI -set, then by Theorem 2.18, A = G ∩ s∗I cl(A) for an open set G in (X, τ , I). Since A ⊆ G and A is gs∗I -closed set in X, then s∗I cl(A) ⊆ G. It follows that s∗I cl(A) ⊆ G ∩ s∗I cl(A) = A. Thus, A = s∗I cl(A) and hence A is semi∗ -I-closed. 3 g-I-locally closed sets Definition 3.1. A subset A of an ideal topological space (X, τ , I) is called g-I-locally closed set (briefly, g-I-LC) if A=U∩V where U is g-open and V is ⋆-closed. Definition 3.2. [14] A subset A of an ideal topological space (X, τ , I) is called weakly I-locally closed set (briefly, weakly I-LC) if A=U∩V where U is open and V is ⋆-closed. Proposition 3.1. Let (X, τ , I) be an ideal topological space and A a subset of X. Then the following hold. 1. If A is g-open, then A is g-I-LC-set. 2. If A is ⋆-closed, then A is g-I-LC-set. 3. If A is weakly I-LC-set, then A is g-I-LC-set. The converses of the above Proposition 3.1 need not be true as shown in the following examples. Example 3.2. 1. In Example 2.1, g-I-LC-sets are ∅, X, {a}, {b}, {c}, {a, b}, {a, c}, {b, c} and ⋆-closed sets are ∅, X, {a}, {a, b}. It is clear that {b} is g-I-LC-set but it is not ⋆-closed. 2. In Example 2.1, g-open sets are ∅, X, {a}, {b}, {c}, {a, c}, {b, c}. It is clear that {a, b} is g-I-LC-set but it is not g-open. Example 3.3. In Example 2.1, weakly I-LC-sets are ∅, X, {a}, {c}, {a, b}. It is clear that {b} is g-I-LC-set but it is not weakly I-LC-set. Theorem 3.4. Let (X, τ , I) be an ideal topological space. If A is a g-I-LC-set and B is a ⋆-closed set, then A∩B is a g-I-LC-set. 261 O. Ravi, et al : g ⋆ -closed sets in ideal topological spaces Proof. Let B be ⋆-closed, then A∩B=(U∩V)∩B=U∩(V∩B), where V∩B is ⋆-closed. Hence A∩B is g-ILC-set. Theorem 3.5. A subset of an ideal topological space (X, τ , I) is ⋆-closed if and only if it is (i) weakly I-LC and I g -closed [12] (ii) g-I-LC and I g⋆ -closed. Proof. (ii) Necessity is trivial. We prove only sufficiency. Let A be g-I-LC-set and I g⋆ -closed set. Since A is g-I-LC, A=U∩V, where U is g-open and V is ⋆-closed. So, we have A=U∩V⊆U. Since A is I g⋆ -closed, A*⊆U. Also A=U∩V⊆V and V is ⋆-closed, then A*⊆V. Consequently, we have A*⊆U∩V=A and hence A is ⋆-closed. Remark 6. 1. The notions of weakly I-LC-set and I g -closed set are independent [12]. 2. The notions of g-I-LC-set and I g⋆ -closed set are independent. Example 3.6. In Example 2.1, it is clear that {b} is g-I-LC-set but not I g⋆ -closed. Example 3.7. In Example 2.2, g-I-LC-sets are ∅, X, {a}, {b}, {c}, {d}, {a, b}, {a, c}, {a, d}, {b, d}, {c, d}, {a, b, c}, {a, b, d}, {a, c, d}. It is clear that {b, c} is I g⋆ -closed set but not g-I-LC-set. Definition 3.3. Let A be a subset of a topological space (X, τ ). Then the g-kernel of the set A, denoted by g-ker(A), is the intersection of all g-open supersets of A. Definition 3.4. A subset A of a topological space (X, τ ) is called ∧g -set if A=g-ker(A). Definition 3.5. A subset A of an ideal topological space (X, τ , I) is called λg -I-closed if A=L∩F where L is a ∧g -set and F is ⋆-closed. Lemma 3.8. 1. Every ⋆-closed set is λg -I-closed but not conversely. 2. Every g-I-LC-set is λg -I-closed. Example 3.9. In Example 2.1, λg -I-closed sets are ∅, X, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}. It is clear that {b} is λg -I-closed set but not ⋆-closed. Lemma 3.10. For a subset A of an ideal topological space (X, τ , I), the following are equivalent. 1. A is λg -I-closed. 2. A=L∩cl*(A) where L is a ∧g -set. 3. A=g-ker(A)∩cl*(A). Lemma 3.11. A subset A⊆(X, τ , I) is I g⋆ -closed if and only if cl*(A)⊆g-ker(A). Proof. Suppose that A ⊆ X is an I g⋆ -closed set. Let x ∈ cl*(A). Suppose x ∈ / g-ker(A). Then there exists an g-open set U containing A such that x ∈ / U. Since A is an I g⋆ -closed set, A ⊆ U and U is g-open implies that cl*(A) ⊆ U and so x ∈ / cl*(A), a contradiction. Therefore cl*(A) ⊆ g-ker(A). Conversely, suppose cl*(A) ⊆ g-ker(A). If A ⊆ U and U is g-open, then cl*(A) ⊆ g-ker(A) ⊆ U. Therefore, A is I g⋆ -closed. Theorem 3.12. For a subset A of an ideal topological space (X, τ , I), the following are equivalent. 1. A is ⋆-closed. 262 South Asian J. Math. Vol. 4 No. 6 2. A is I g⋆ -closed and g-I-LC. 3. A is I g⋆ -closed and λg -I-closed. Proof. (1)⇒(2)⇒(3) Obvious. (3)⇒(1) Since A is I g⋆ -closed, so by Lemma 3.3, cl*(A)⊆g-ker(A). Since A is λg -I-closed, so by Lemma 3.2, A=g-ker(A)∩cl*(A)=cl*(A). Hence A is ⋆-closed. The following two examples show that the concepts of I g⋆ -closed set and λg -I-closed set are independent. Example 3.13. In Example 2.2, λg -I-closed sets are ∅, X, {a}, {b}, {c}, {d}, {a, b}, {a, c}, {a, d}, {b, d}, {c, d}, {a, b, c}, {a, b, d}, {a, c, d}. It is clear that {b, c, d} is I g⋆ -closed but not λg -I-closed. Example 3.14. In Example 2.2, it is clear that {c} is λg -I-closed but not I g⋆ -closed. 4 Decompositions of ⋆-continuity Definition 4.1. A function f : (X, τ , I)→(Y, σ) is said to be ⋆-continuous [12] (resp. I g -continuous [12], g-I-LC-continuous, λg -I-continuous, I g⋆ -continuous, weakly I-LC-continuous [14]) if f−1 (A) is ⋆-closed (resp. I g -closed, g-I-LC-set, λg -I-closed, I g⋆ -closed, weakly I-LC-set) in (X, τ , I) for every closed set A of (Y, σ). Theorem 4.1. 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