4_842_khan.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 4, No. 3, 2011, 237-243 ISSN 1307-5543 – www.ejpam.com On Is⋆g-Continuous Functions in Ideal Topological Spaces M. Khan1,∗,̧ T. Noiri 2 1 Department of Mathematics, COMSATS Institute of Information Technology, Park Road, Islam- abad, PAKISTAN 2 2949-1 Shiokita-cho, Hinagu, Yatsushiro-shi, Kumamoto-ken, 869-5142 JAPAN Abstract. By using Is⋆g -closed sets due to Khan and Hamza [5], we introduce the notion of Is⋆ g - continuous functions in ideal topological spaces. We obtain several properties of Is⋆ g -continuity and the relationship between this function and other related functions. 2000 Mathematics Subject Classifications: 45A05, 45A10 Key Words and Phrases: local-function, Is⋆g -closed set, Is⋆g -continuous, strong Is⋆g -continuous, weakly Is⋆g -continuous, T1/2-space. 1. Introduction Khan and Hamza [5] introduced and investigated the notion of Is⋆g -closed sets in ideal topological spaces as a generalization of Ig -closed sets due to Dontchev et al. [2]. In this pa- per, by using Is⋆g-closed sets we introduce Is⋆g -continuous functions, strongly Is⋆g -continuous functions and weakly Is⋆g -continuous functions. It turns out that weak Is⋆ g-continuity is weaker than weak I-continuity defined by Ackgoz et al. [1] . We obtain several properties of Is⋆g -continuity and the relationship between this function and other related functions. 2. Preliminaries Let (X ,τ) be a topological space with no separation properties assumed. For a subset A of a topological space (X ,τ) , cl(A) and Int(A) denote the closure and interior of A in (X ,τ), respectively. An ideal I on a set X is a non-empty collection of subsets of X which satisfies the following properties: (1) A∈ I and B ⊂ A implies B ∈ I , (2) A∈ I and B ∈ I implies A∪ B ∈ I . ∗Corresponding author. Email addresses: profmoiz001�yahoo. om (M. Khan), t.noiri�nifty. om (T. Noiri) http://www.ejpam.com 237 c© 2011 EJPAM All rights reserved. M. Khan,̧ T. Noiri / Eur. J. Pure Appl. Math, 4 (2011), 237-243 238 An ideal topological space is a topological space (X ,τ) with an ideal I on X and is denoted by (X ,τ, I). For a subset A ⊂ X , A⋆(I ,τ) = {x ∈ X : A∩ U /∈ I for every U ∈ τ(x)}, where, τ(x) = {U ∈ τ : x ∈ U}, is called the local function of A with respect to I and τ [4, 6]. We simply write A⋆ or A⋆X instead of A⋆(I ,τ) and B⋆A for B⋆(IA,τA) in case there is no chance for confusion. For every ideal topological space (X ,τ, I), there exists a topology τ⋆(I), finer than τ, generated by the base β(I ,τ) = {U − J : U ∈ τ and J ∈ I}. It is known in [4] that β(I ,τ) is not necessarily a topology. When there is no ambiguity, τ⋆(I) is denoted by τ⋆. Recall that A is said to be ()⋆-dense in itself (resp. τ⋆-closed, ()⋆-perfect) if A⊂ A⋆ (resp. A⋆ ⊂ A, A= A⋆). For a subset A of X , cl⋆(A) and Int⋆(A) will, respectively, denote the closure and interior of A in (X ,τ⋆). A subset A of X is said to be semi-open [7] if there exists an open set U in X such that U ⊂ A ⊂ cl(U). The complement of a semi-open set is said to be semi-closed. A subset A is said to be semi-regular if A is semi-open and semi-closed. A subset A of X is said to be generalized closed [8] (briefly, g-closed) if cl(A) ⊂ U whenever A ⊂ U and U is open in X . The complement of a g-closed set is said to be g-open. A space X is called a T1/2-space [3] if every g-closed set in X is closed. Recall that if (X ,τ, I) is an ideal topological space and A is a subset of X , then (A,τA, IA) is an ideal topological space, where τA is the relative topology on A and IA = {A∩ J : J ∈ I}. 3. Is⋆ g-Closed Sets The notion of Is⋆ g-closed sets was defined by Khan and Hamza [5]. In this section we will obtain further properties of Is⋆g-closed sets in ideal topological spaces. Definition 1. A subset A of a space (X ,τ, I) is said to be Is⋆g -closed [5] if A⋆ ⊂ U whenever A ⊂ U and U is semi-open in X . The complement of an Is⋆g-closed set is said to be Is⋆g -open, equivalently if F ⊂ Int⋆(A) whenever F ⊂ A for every semi-closed set F in X . Lemma 1. Every open set is Is⋆g-open. Lemma 2 ([Lemma 2.7, 2]). Let (X ,τ, I) be an ideal topological space and B ⊂ A ⊂ X . Then B⋆(IA,τA) = B⋆(I ,τ)∩ A. Lemma 3. If U is open and A is Is⋆g-open, then U ∩ A is Is⋆g -open. Proof. We prove that X − (U ∩ A) is Is⋆g-closed. Let X − (U ∩ A) ⊂ G where G is semi- open in X . This implies (X − U) ∪ (X − A) ⊂ G. Since (X − A) ⊂ G and (X − A) is Is⋆g- closed in X , therefore (X − A)⋆ ⊂ G. Moreover X − U is closed and contained in G, therefore, (X−U)⋆ ⊂ cl(X−U)⊂ G. Hence (X−(U∩A))⋆ = ((X−U)∪(X−A))⋆ = (X−U)⋆∪(X−A)⋆ ⊂ G. This proves that U ∩ A is Is⋆g-open. Theorem 1. Let (X ,τ, I) be an ideal topological space and B ⊂ A⊂ X . If B is an Is⋆g -closed set relative to A, where A is open and Is⋆g -closed in X , then B is Is⋆g-closed in X . Proof. Let B ⊂ G, where G is semi-open in X . Then B ⊂ A∩ G and A∩ G is semi-open in X and hence in A. Therefore B⋆A ⊂ A∩ G. It follows from Lemma 2 that A∩ B⋆X ⊂ A∩ G or M. Khan,̧ T. Noiri / Eur. J. Pure Appl. Math, 4 (2011), 237-243 239 A⊂ G∪(X −B⋆X ). By Theorem 2.3 of [4], B⋆X is closed in X and G∪(X −B⋆X ) is semi-open in X . Since A is Is⋆g -closed in X , A⋆X ⊂ G∪(X−B⋆X ) and hence B⋆ = B⋆∩A⋆ ⊂ B⋆∩[G∪(X−B⋆X )]⊂ G. Therefore, we obtain B⋆X ⊂ G. This proves that B is Is⋆g -closed in X . Theorem 2. Let A be a semi-open set in a space (X ,τ, I) and B ⊂ A⊂ X . If B is Is⋆g-closed in X , then B is Is⋆g -closed relative to A. Proof. Let B ⊂ U where U is semi-open in A. Then there exists a semi-open set V in X such that U = A∩V . Thus B ⊂ A∩V . Now B ⊂ V implies that B⋆X ⊂ V . It follows that A∩B⋆X ⊂ A∩V . By Lemma 2, B⋆A ⊂ A∩ V = U . This proves that B is a Is⋆ g-closed relative to A. Corollary 1. Let B ⊂ A ⊂ X and A be open and Is⋆g-closed in (X ,τ, I). Then B is Is⋆g-closed relative to A if and only if B is Is⋆g -closed in X . Theorem 3. If B is a subset of a space (X ,τ, I) such that A⊂ B ⊂ A⋆ and A is Is⋆ g-closed in X , then B is also Is⋆ g-closed in X . Proof. Let G be a semi-open set in X containing B, then A ⊂ G. Since A is Is⋆g -closed, therefore A⋆ ⊂ G and hence B⋆ ⊂ (A⋆)⋆ ⊂ A⋆ ⊂ G. This implies that B is Is⋆g-closed in X . Theorem 4. Let B ⊂ A⊂ X and suppose that B is Is⋆ g-open in X and A is a semi-regular set in X . Then B is Is⋆g-open relative to A. Proof. We prove that A−B is Is⋆g-closed relative to A. Let U ∈ SO(A) such that (A−B)⊂ U . Now (A− B) ⊂ (X − B) ⊂ U ∪ (X − A), where U ∪ (X − A) ∈ SO(X ) because A∈ SR(X ). Since X−B is Is⋆g -closed in X , therefore (X−B)⋆X ⊂ U∪(X−A) or (X−B)⋆X∩A⊂ (U∪(X−A))∩A⊂ U . By Lemma 2, (A−B)⋆A = (A−B)⋆X ∩A⊂ (X −B)⋆X ∩A⊂ U and hence (A−B)⋆A ⊂ U . This proves that B is Is⋆g -open relative to A. Theorem 5. Let B ⊂ A⊂ X . B is Is⋆ g-open in A and A is open in X then B is Is⋆ g-open in X . Proof. Let F be a semi-closed subset of B in X . Since A is open, therefore F ∈ SC(A). Since B is Is⋆ g-open in A, therefore F ⊂ Int⋆A(B) = A∩ Int⋆X (B) ⊂ Int⋆X (B). This proves that B is Is⋆g -open in X . 4. Is⋆ g-Continuous Functions Definition 2. A function f : (X ,τ, I) → (Y,Ω, J) is said to be weakly I-continuous [1] if for each x ∈ X and each open set V in Y containing f (x), there exists an open set U containing x such that f (U)⊂ cl⋆(V ). Definition 3. A function f : (X ,τ, I)→ (Y,Ω) is said to be Is⋆g -continuous if for every U ∈ Ω, f −1(U) is Is⋆ g-open in (X ,τX , I). M. Khan,̧ T. Noiri / Eur. J. Pure Appl. Math, 4 (2011), 237-243 240 Remark 1. Every continuous function is Is⋆g -continuous and the converse need not be true as seen from Example 2 (below). Definition 4. A function f : (X ,τ)→ (Y,Ω, J) is said to be strongly Is⋆g-continuous if for every Is⋆g -open set U in Y , f −1(U) is open in X . Remark 2. Every strongly Is⋆g -continuous function is continuous but the converse is not true in general. Example 1. Let X = {a, b, c, d} with τ = {φ, X , {a}, {b}, {a, b}, {b, c}, {a, b, c}}. Let Y = {a, b, c, d} with Ω = {φ, Y, {a}, {b, c}, {a, b, c}} and J = {φ, {a}}. Let f : (X ,τ)→ (Y,Ω, J) be defined by f (a) = b, f (b) = a, f (c) = a and f (d) = d. Then f is continuous. Let U = {a, c} then U is Is⋆ g-open in Y but f −1(U) = {a, c} is not open in X . Hence f is not strongly Is⋆g- continuous. Definition 5. A function f : (X ,τ, I)→ (Y,Ω, J) is said to be weakly Is⋆g -continuous if for each x ∈ X and each open set V in Y containing f (x), there exists an Is⋆g-open set U containing x such that f (U)⊂ cl⋆(V ). Remark 3. (1) Every weakly I-continuous function is weakly Is⋆g-continuous but the converse is not true in general. (2) Every Is⋆g -continuous function is weakly Is⋆g-continuous. By the above definitions, for a function f : (X ,τ, I) → (Y,Ω, J) we obtain the following implications: strong Is⋆g -continuity ⇒ continuity ⇒ Is⋆g-continuity ⇓ ⇓ Weak I-continuity ⇒ weak Is⋆g-continuity Remark 4. Is⋆ g-continuity and weak I-continuity are independent of each other. Example 2. Let X = Y = {a, b, c, d} and τ = Ω = {φ, X , {a, b}} with I = {φ, {a}, {b}, {a, b}}. Define f : (X ,τ, I) → (Y,Ω, I) by f (a) = a, f (b) = c, f (c) = b and f (d) = b. Then f is Is⋆g -continuous but not weak I-continuous. Since for c ∈ X , f (c) = b and an open set V = {a, b} containing f (c), the only open set containing c is U = X and f (U)* cl⋆(V ) = {a, b}. Example 3. Let X = {a, b, c, d}, τ = {φ, X , {a, b}} and I = {φ, {a}, {b}, {a, b}}. Let Y = {1,2,3,4}, Ω = {φ, {1,2}, Y } and J = {φ, {3}, {4}, {3,4}}. Define f : (X ,τ, I)→ (Y,Ω, I) by f (a) = 1, f (b) = 3, f (c) = 2 and f (d) = 4. f is weak I-continuous but not Is⋆g-continuous. Since V = {1,2} is open in Y but f −1(V ) = {a, c} is not Is⋆ g-open in X . Theorem 6. Let f : (X ,τ, I)→ (Y,Ω) be a function. Then, the following statements are equiva- lent: (1) f is Is⋆ g-continuous. (2) The inverse image of each closed set in Y is Is⋆g -closed in X. M. Khan,̧ T. Noiri / Eur. J. Pure Appl. Math, 4 (2011), 237-243 241 (3) The inverse image of each open set in Y is Is⋆ g-open in X. Definition 6. An ideal topological space (X ,τ, I) is said to be T-dense if every subset of X is ⋆-dense in itself. Definition 7. Let N be a subset of a space (X ,τ, I) and x ∈ X . Then N is called an Is⋆ g-open neighborhood of x if there exists an Is⋆g -open set U containing x such that U ⊂ N. Theorem 7. Let (X ,τ, I) be T-dense. Then, for a function f : (X ,τ, I) → (Y,Ω) the following statements are equivalent: (1) f is Is⋆ g-continuous. (2) For each x ∈ X and each open set V in Y with f (x) ∈ V , there exists an Is⋆ g-open set U containing x such that f (U)⊂ V . (3) For each x ∈ X and each open set V in Y with f (x) ∈ V , f −1(V ) is an Is⋆ g-open neighbor- hood of x. Proof. (1) ⇒ (2) Let x ∈ X and let V be an open set in Y such that f (x) ∈ V . Since f is Is⋆ g-continuous, f −1(V ) is Is⋆g -open in X . By putting U = f −1(V ), we have x ∈ U and f (U)⊂ V . (2) ⇒ (3) Let V be an open set in Y and let f (x) ∈ V . Then by (2), there exists an Is⋆g -open set U containing x such that f (U) ⊂ V . So x ∈ U ⊂ f −1(V ). Hence f −1(V ) is an Is⋆g -open neighbourhood of x . (3) ⇒ (1) Let V be an open set in Y and let f (x) ∈ V . Then by (3), f −1(V ) is an Is⋆g- neighborhood of x . Thus for each x ∈ f −1(V ), there exists an Is⋆g -open set Ux containing x such that x ∈ Ux ⊂ f −1(V ). Hence f −1(V ) = ∪x∈ f −1(V )Ux and so by Theorem 2.12 [5], f −1(V ) is Is⋆g-open in X . Theorem 8. A function f : (X ,τ)→ (Y,Ω, J) is strongly Is⋆ g-continuous if and only if the inverse image of every Is⋆g -closed set in Y is closed in X . Theorem 9. (1) Let f : (X ,τ)→ (Y,Ω, J) be strongly Is⋆ g-continuous and h : (Y,Ω, J)→ (Z ,σ) be Is⋆g -continuous, then h ◦ f is continuous. (2) Let f : (X ,τ, I) → (Y,Ω) be Is⋆g -continues and g : (Y,Ω) → (Z ,σ) be continuous, then g ◦ f : (X ,τ, I)→ (Z ,σ) is Is⋆g -continuous. Theorem 10. Let f : (X ,τ, I)→ (Y,Ω) be Is⋆g -continuous and U ∈ RO(X ). Then the restriction f | U : (U ,τU , IU )→ (Y,Ω) is Is⋆g-continuous. Proof. Let V be any open set of (Y,τY ). Since f is Is⋆ g-continuous, f −1(V ) is Is⋆ g-open in X . By Theorem 2.14 of [5], f −1(V )∩ U is Is⋆g -open in X . Thus by Theorem 4 ( f | U)−1(V ) = f −1(V )∩ U is Is⋆ g-open in U because U is regular-open in X . This proves that f | U : (U ,τ | U , I | U)→ (Y,τY ) is Is⋆g-continuous. M. Khan,̧ T. Noiri / Eur. J. Pure Appl. Math, 4 (2011), 237-243 242 Theorem 11. Let f : (X ,τ, I) → (Y,Ω, J) be a function and {Uα : α ∈ ∇} be an open cover of a T-dense space X . If the restriction f | Uα is Is⋆g-continuous for each α ∈ ∇, then f is Is⋆g -continuous. Proof. Suppose F is an arbitrary open set in (Y,Ω, J). Then for each α ∈ ∇, we have ( f | Uα) −1(V ) = f −1(V )∩Uα. Because f | Uα is Is⋆g -continuous, therefore, f −1(V )∩Uα is Is⋆g- open in X for each α ∈ ∇. Since for each α ∈ ∇, Uα is open in X , by Theorem 5, f −1(V )∩Uα is Is⋆g -open in X . Now since X is T -dense, by [Theorem 2.12 5], ∪α∈∇ f −1(V )∩ Uα = f −1(V ) is Is⋆g-open in X . This implies f is Is⋆g -continuous. Theorem 12. If (X ,τ, I) is a T -dense space and f : (X ,τ, I) → (Y,Ω) is Is⋆g -continuous, then graph function g : X → X × Y , defined by g(x) = (x , f (x)) for each x ∈ X , is Is⋆ g-continuous. Proof. Let x ∈ X and W be any open set in X × Y containing g(x) = (x , f (x)). Then there exists a basic open set U × V such that g(x) ⊂ U × V ⊂ W . Since f is Is⋆g-continuous, there exits an Is⋆g -open set U1 in X containing x such that f (U1)⊂ V . By Lemma 3, U1 ∩U is Is⋆g -open in X and we have x ∈ U1 ∩U ⊂ U and g(U1 ∩U)⊂ U × V ⊂W . Since X is T -dense, therefore by Theorem 7, g is Is⋆g -continuous. Theorem 13. A function f : (X ,τ, I)→ (Y,Ω) is Is⋆g-continuous if the graph function g : X → X × Y is Is⋆g -continuous. Proof. Let V be an open set in Y containing f (x). Then X × V is an open set in X × Y and by the Is⋆ g-continuity of g, there exists an Is⋆g-open set U in X containing x such that g(U)⊂ X × V . Therefore, we obtain f (U)⊂ V . This shows that f is Is⋆g-continuous. Theorem 14. Let {Xα : α ∈ ∇} be any family of topological spaces. If f : (X ,τ, I)→ Πα∈∇Xα is an Is⋆g-continuous function, then Pα ◦ f : X → Xα is Is⋆g -continuous for each α ∈ ∇, where Pα is the projection of ΠXα onto Xα. Proof. We will consider a fixed α0 ∈ ∇. Let Gα0 be an open set of Xα0 . Then (Pα0 )−1(Gα0 ) is open in ΠXα. Since f is Is⋆g -continuous, f −1((Pα0 )−1(Gα0 )) = (Pα0 ◦ f )−1(Gα0 ) is Is⋆ g-open in X . Thus Pα ◦ f is Is⋆g -continuous. Corollary 2. For any bijective function f : (X ,τ)→ (Y,Ω, J), the following are equivalent: (1) f −1 : (Y,Ω, J)→ (X ,τ) is Is⋆g -continuous. (2) f (U) is Is⋆g-open in Y for every open set U in X . (3) f (U) is Is⋆g-closed in Y for every closed set U in X . Proof. It is trivial. Definition 8. An ideal topological space (X ,τ, I) is an RI-space [1], if for each x ∈ X and each open neighbourhood V of x, there exists an open neighbourhood U of x such that x ∈ U ⊂ cl⋆(U)⊂ V . REFERENCES 243 Theorem 15. Let (Y,Ω, J) be an RI-space and (X ,τ, I) be a T-dense space. Then f : (X ,τ, I)→ (Y,Ω, J) is weak Is⋆g-continuous if and only if f is Is⋆g -continuous. Proof. The sufficiency is clear. Necessity. Let x ∈ X and V be an open set of Y containing f (x). Since Y is an RI-space, there exists an open set W of Y such that f (x) ∈ W ⊂ cl⋆(W ) ⊂ V . 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