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Proceedings of IMM of NAS of Azerbaijan 27 Aysegul CAKSU GULER and Gulhan ASLIM b − I-OPEN SETS AND DECOMPOSITION OF CONTINUITY VIA IDEALIZATION Abstract We introduce the notion of b − I-open sets and strong BI -sets to obtain a decomposition of continuity via idealization. Additionally, we investigate properties of b − I-open sets and strong BI -sets. Introduction and preliminaries In 1990, Jankovic and Hamlett [7] introduced the notion of I-open sets in ideal topological spaces. Abd El-Monsef et al [2] further investigated I-open sets and Icontinuous functions. In 1999, Dontchev [4] has introduced the notion of pre-I-open sets which is weaker than that of I-open sets. At last Hatir at all [5] have introduced the notions of BI -sets, CI -sets, α − I-sets, semi-I-sets and β − I-sets. By using this sets, they provided decompositions of continuity. In this paper, we introduced the notions b − I-open and strong BI -sets to obtain decompositions of continuity. Throughout this paper for a subset A of a space (X, τ ), the closure of A and interior of A are denoted Cl(A) and Int(A) respectively. An ideal topological space is a topological space (X, τ ) with an ideal I on X, and is denoted (X, τ , I). The following collections form important ideals on a topological space (X, τ ): the ideal of all finite sets F, the ideal of all closed and discrete sets CD, the ideal of all nowhere dense sets N . A∗ (I) = {x ∈ X | U ∩ A ∈ / If or each neighborhood U of X} is called the local function of A with respect to I and τ [7]. When there is no chance for confusion A∗ (I) is denoted by A∗ . Note that often X ∗ is a proper subset of X. The hypothesis X = X ∗ was used by Hayashi in [5], while the hypothesis τ ∩ I = ∅ was used by Samuels in [9]. In fact, those two conditions are equivalent [7, Theorem 6.1] and so the ideal topological spaces satisfying this hypothesis are called as HayashiSamuels spaces. For every ideal topological space (X, τ , I), there exits a topology τ ∗ (I), finer than τ , generated by the base β(I, τ ) = {U \ I | U ∈ τ and I ∈ I}. In general β(I, τ ) is not always a topology [7]. Observe additionally that Cl∗ (A) = A∗ ∪ A defines a Kuratowski closure operator for τ ∗ (I). Now we recall some definitions and results, which are used in this paper. Definition 1. A subset A of a topological space X is called (a) α-open set [8] if A ⊂ Int(Cl(Int(A))), (b) t-set [9] if Int(A) = Int(Cl(A)), (c) b-open set [1] if A ⊂ Int(Cl(A)) ∪ Cl(Int(A)), (d) strong B-set [3] if A = U ∩ V where U is an open set and V is a t-set and Int(Cl(V )) = Cl(Int(V )). Definition 2. A subset A of an ideal topological space (X, τ , I) is said to be (a) I-open [7] A ⊂ Int(A∗ ), (b) α − I-open [5] if A ⊂ Int(Cl∗ (Int(A))), 28 Proceedings of IMM of NAS of Azerbaijan [A.C.Guler and G.Aslim] (c) semi-I-open [5] if A ⊂ Cl∗ (Int(A)), (d) pre-I-open [4] if A ⊂ Int(Cl∗ (A)), (e) t − I-set [5] if Int(Cl∗ (A)) = Int(A), (f ) BI -set [5] if A = U ∩ V , where U ∈ τ and V is a t − I-set. b − I-open set Definition 3. A subset A of an ideal topological space (X, τ , I) is said to be b − I-open if A ⊂ Int(Cl∗ (A)) ∪ Cl∗ (Int(A)). Proposition 1. Let A be a b − I-open set such that Int(A) = ∅, then A is pre-I-open. For a subset of an ideal topological space the following hold: (a) Every open set is b − I-open. (b) Every semi-I-open set is b − I-open. (c) Every pre-I-open set is b − I-open. (d) Every b − I-open set is β − I-open. (e) Every b − I-open is b-open. Proof. (a) The proof is obvious. (b) The proof is obvious. (c) The proof is obvious. (d) Let A be a b − I-open set. Then we have A ⊂ Int(Cl∗ (A)) ∪ Cl∗ (Int(A)) ⊂ Cl(Int(Cl∗ (A)))∪((Int(A))∗ ∪Int(A)) ⊂ Cl(Int(Cl∗ (A)))∪(Cl(Int(A))∪Int(A)) ⊂ Cl(Int(Cl∗ (A))) ∪ Cl(Int(A)) ⊂ Cl(Int(Cl∗ (A)). This shows that A is an β − I-open set. (e) Let A be a b − I-open set. Then we have A ⊂ Int(Cl∗ (A)) ∪ Cl∗ (Int(A)) = Int(A∗ ∪ A) ∪ ((Int(A))∗ ∪ Int(A)) ⊂ Int(Cl(A) ∪ A) ∪ (Cl(Int(A)) ∪ Int(A)) = Int(Cl(A)) ∪ Cl(Int(A)). This shows that A is a b-open set. Remark 1. From above the following implication and none of these implications is reversible as shown by examples given below and well -known facts open - α-I-open - semi-I-open ? pre-I-open b − I-open - @ @ R @ β − I-open Example 1. Let X = {a, b, c, d}, τ = {X, ∅, {b}, {a, d}, {a, b, d}} and I = {∅, {b}}. Then A = {b, d} is b − I-open, but it is not semi-I-open. Because Int(Cl∗ (A)) ∪ Cl∗ (Int(A)) = Int({b, d}∗ ∪ {b, d}) ∪ Cl∗ (Int({b, d})) = Int(X) ∪ Proceedings of IMM of NAS of Azerbaijan 29 [b-I-open sets and decom.of continuity...] ({b}∗ ∪ {b}) = X ∪ {b} = X ⊃ A and hence A is b − I-open. Since Cl∗ (Int(A)) = Cl∗ (Int({b, d})) = ({b}∗ ∪ {b}) = {b} 6⊃ A. So A is not semi-I-open. Example 2. Let X = {a, b, c}, τ = {X, ∅, {a}, {b}, {a, b}} and I = {∅, {b}}. Then A = {a, c} is b−I-open, but it is not pre-I-open. For Int(Cl∗ (A))∪Cl∗ (Int(A)) = Int({a, c}∗ ∪ {a, c}) ∪ Cl∗ ({a}) = Int({a, c}) ∪ ({a}∗ ∪ {a}) = {a, c} ⊃ A and hence A is b − I-open. Since Int(Cl∗ (A)) = Int({a, c}∗ ∪ {a, c}) = Int({a, c}) = {a} 6⊃ A. Hence A is not pre-I-open. Example 3. Let X = {a, b, c}, τ = {X, ∅, {a}, {b}, {a, b}} and I = {∅, {b}}. Then A = {b, c} is β − I-open, but it is not b − I-open. Since Cl(Int(Cl∗ (A))) = Cl(Int({b, c}∗ ∪ {b, c})) = Cl(Int({b, c})) = Cl({b}) = {b, c} ⊃ A. Hence A is not β − I-open. For Int(Cl∗ (A)) ∪ Cl∗ (Int(A)) = Int({b, c}∗ ∪ {b, c}) ∪ Cl∗ ({b}) = Int({b, c}) ∪ ({b}∗ ∪ {b}) = {b} 6⊃ A and hence A is not b − I-open. Proposition 2. For an ideal topological space (X, τ , I) and A ⊆ X we have: (a) If I = ∅, then A is b − I-open if and only if A is b-open. (b) If I = P(X), then A is b − I-open if and only open A ∈ τ . (c) If I = N , then A is b − I-open if and only if A is b-open. Proof (a) Necessity is shown in Proposition 2.2. For sufficiency note that in case of the minimal ideal A∗ = Cl(A). (b) Necessity: If A is a b − I-open set, then A ⊂ Int(Cl∗ (A)) ∪ Cl∗ (Int(A)) = Int(A∗ ∪ A) ∪ ((Int(A)∗ ) ∪ Int(A)) = Int(∅ ∪ A) ∪ (∅ ∪ Int(A)) = Int(A) ∪ Int(A) = Int(A). Hence A is open. Sufficiency: It is shown in Proposition 2.2 (c) Necessity is given in Proposition 3.2. Sufficiency: Note that the local function of A with respect to N and τ can be given explicitly [10]. We have A∗ (N ) = Cl(Int(Cl(A))). Hence A is b − I-open if and only if A ⊂ Int(Cl(Int(Cl(A))) ∪ A)∪ Cl(Int(Cl(Int(A))) ∪ A. Suppose that A is b-open. Since always Int(Cl(A)) ∪ Cl(Int(A)) ⊆ A∪ Cl(Int(Cl(A))) ∪ Cl(Int(A)), then A ⊆ Int(A∪Cl(Int(Cl(A))))∪ Cl(Int(A)) = Int(A ∪ A∗ (N )) ∪ CI(Int(A)). Hence A is a b-I-open set. The intersection of even two b − I-open sets need not to be b − I-open as shown in the following examples. Example 4. Let X = {a, b, c, d}, τ = {X, ∅, {a}, {b, d}, {a, b, d}} and I = {∅, {c}, {d}, {c, d}}. Then A = {a, c} and B = {b, c} are b − I-open but A ∩ B = {c} is not b−I-open. Since Int(Cl∗ (A)) ∪ Cl∗ (Int(A)) = {a, c} ⊃ A and Int(Cl∗ (B)) ∪ Cl∗ (Int(B)) = {b, c, d} ⊃ B. But Int(Cl∗ (A ∩ B))∪ Cl∗ (Int(A ∩ B)) = ∅ 6⊃ A ∩ B. Lemma 1 [7, Theorem 2.3 (g)] Let (X, τ , I) be ideal topological space and let A ⊆ X. Then U ∈ τ ⇒ U ∩ A∗ = U ∩ (U ∩ A)∗ ⊆ (U ∩ A)∗ . Proposition 3. Let (X, τ , I) be ideal topological space and let A, U ⊆ X. If A is a b − I-open set and U ∈ τ . Then A ∩ U is a b − I-open set. Proof. By assumption A ⊆ Int(Cl∗ (A)) ∪ Cl∗ (Int(A)) and U ⊆ Int(U ). Thus applying Lemma 2.1, 30 Proceedings of IMM of NAS of Azerbaijan [A.C.Guler and G.Aslim] A ∩ U ⊆ (Int(Cl∗ (A)) ∪ Cl∗ (Int(A))) ∩ Int(U ) ⊆ (Int(Cl∗ (A)) ∩ Int(U )) ∪ (Cl∗ (Int(A)) ∩ Int(U )) = Int((A∗ ∩ U ) ∪ (A ∩ U )) ∪ ((Int(A))∗ ∩ Int(U )) ∪ (Int(A) ∩ Int(U )) ⊆ Int((A ∩ U )∗ ∪ (A ∩ U )) ∪ ((Int(A ∩ U ))∗ ∪ (Int(A ∩ U ))) = Int(Cl∗ (A ∩ U )) ∪ Cl∗ (Int(A ∩ U )). Thus A ∩ U is a b − I-open set. Strong BI -set Definition 4. Let (X, τ , I) be an ideal topological space and be A ⊆ X is called a strong BI -set if A = U ∩ V , where U ∈ τ and V is a t − I-set and Int(Cl∗ (V )) = Cl∗ (Int(V)). Proposition 4. Let (X, τ , I) be an ideal topological space and be A ⊆ X. The following hold: (a) If A is a strong BI -set, then a BI -set. (b) If A is a strong B-set, then A is a strong BI -set. Proof. The proofs are obvious by Proposition 2.3 of [5] Remark 2. The converses of proposition (a) and (b) need not to be true as the following examples show. Example 5. Let X = {a, b, c, d}, τ = {X, ∅, {a}, {b, c}, {a, b, c}} and I = {∅, {c}, {a, c}}. Then A = {a, c} is a BI -set, but it is not a strong BI -set. For Int(Cl∗ (A)) = Int({a, c}∗ ∪ {a, c}) = Int({a, c}) = {a} = Int(A) and hence A is a t − I-set. It is obvious that A is a BI -set. But Int(Cl∗ (A)) = Int({a, c}∗ ∪ {a, c}) = Int({a, c}) = {a} and Cl∗ (Int(A)) = Cl∗ (Int({a, c})) = {a}∗ ∪ {a} = {a, d} i.e Int(Cl∗ (A)) 6= Cl∗ (Int(A)). So A is not a strong BI -set. Example 6. Let X = {a, b, c, d}, τ = {X, ∅, {b}, {c, d}, {b, c, d}} and I = {∅, {b}, {c}, {b, c}}. Then A = {b, c} is a strong BI -set but not a strong B-set. Int(Cl∗ (A)) = Int({b, c}∗ ∪ {b, c}) = Int({b, c}) = {b} = Int(A). Then A is a t − I-set. Besides Int(Cl∗ (A)) = {b} = {b}∗ ∪ {b} = Cl∗ (Int(A)). So A is a strong BI -set. But, Int(Cl(A)) = Int({X}) = X 6= Int(A). Therefore A is not a strong B-set. Proposition 5. For a subset A ⊆ (X, τ , I) the following conditions are equivalent: (a) A is open. (b) A is b − I-open and a strong BI -set. Proof. (a) ⇒ (b) By Proposition 3.3, every open set is b − I-open. On the other hand every open set is strong BI -set, because X is t − I-set and Int(Cl∗ (X)) = Cl∗ (Int(X)). (b) ⇒ (a) Let A is b − I-open and strong BI -set. Then, A ⊂ Int(Cl∗ (A)) ∪ Cl∗ (Int(A)) = Int(Cl∗ (U ∩ V )) ∪ Cl∗ (Int(U ∩ V )), where U is open and V is t − Iset and Int(Cl∗ (V )) = Cl∗ (Int(V )). Hence A ⊂ (Int(Cl∗ (U )) ∩ Int(Cl∗ (V ))) ∪ (Cl∗ (Int(U )) ∩ Cl∗ (Int(V ))) A ⊂ U ∩ (Int(CI ∗ (V ))) ∪ CI ∗ (Int(V ))) A ⊂ U ∩ (Int(CI ∗ (V )) ∪ CI ∗ (Int(V ))) A ⊂ U ∩ Int(CI ∗ (V )) A ⊂ U ∩ Int(V ) = Int(A). Proceedings of IMM of NAS of Azerbaijan 31 [b-I-open sets and decom.of continuity...] So A is open. Remark 3. The notion of b−I-openness is different from that of strong BI -sets. Because (i) In Example 2.3, A = {b, c} is not b−I-open. But Int(Cl∗ (A)) = Cl∗ (Int(A)) = Int(A) = {b}. So A is a strong BI -set. (ii) In Example 2.1, A = {b, d} is b − I-open. But Int(Cl∗ (A)) = X 6= Int(A). So A is not a strong BI -set. Decomposition of Continuity Definition 5. A function f : (X, τ ) → (Y, σ) is said to be b-continuous [1] (resp. strong B−continuous [3]) if for each open set V of (Y, σ), f −1 (V ) is b−open (resp. strong B-set)in (X, τ ). Definition 6. A function f : (X, τ , I) → (Y, σ) is said to be b − I-continuous (resp. semi-I-continuous [4], pre-I-continuous [3] strong BI -continuous) if for each open set V of (Y, σ), f −1 (V ) is b-open (resp. semi-I-open, pre-I-open, strong BI -set) in (X, τ ). Proposition 6. If a function f : (X, τ , I) → (Y, σ) is semi-I-continuous (pre-Icontinuous), then f is b − I-continuous. Proof. This is an immediate consequences of Proposition 2.2. (b) and (c). Proposition 7. If a function f : (X, τ , I) → (Y, σ) is b − I-continuous, then f is b-continuous. Proof. This is an immediate consequences of Proposition 2.2 (e). Proposition 8. If a function f : (X, τ , I) → (Y, σ) is strong B-continuous, then f is strong BI -continuous. Proof. This is an immediate consequences of Proposition 3.1 (b). Theorem 1. For a function f : (X, τ , I) → (Y, σ) the following conditions are equivalent: (a) f is continuous, (b) f is b − I-continuous and strong BI -continuous. Proof. This is an immediate consequences of Proposition 3.2. References [1]. Dimitrije Andrijevi ć, On b-open Sets, MATHEMAT, 48 (1996), 59-64. [2] M. E. Abd El-Monsef, E. F. Lashien and A. A. Nasef, On I-open sets and I-continuous function, Kyungpook Math. J., 32 (1992), 21-30. [3] J. Dontchev, Strong B-sets and another decomposition of continuity, Acta Math. Hungar., 75 (1997), 259-265. [4] J. Dontchev, Idealization of Ganster-Reilly decomposition theorems, Math. GN/9901017, 5 Jan 1999 (Internet). [5] E. Hatir and T. Noiri, On decompositions of continuity via idealization, Acta Math. Hungar, 96 (4) (2002), 341-349. 32 Proceedings of IMM of NAS of Azerbaijan [A.C.Guler and G.Aslim] [6] E. Hayashi, Topologies defined by local properties, math. Ann., 156 (1964), 205-215. [7] D. Jankovic̆ and T. R. Hamlett, New topologies from old via ideals, Amer. Math. Monthly, 97 (1990), 295-310. [8] O. N j ȧstad, On some clasess of nearly open sets, Pacific J. Math., 15 (1965), 961-970. [9] P. Samuels, A topology formed from a given topology and ideal, J. London Math. Soc., 10 (1975), 409-416. [10] R. Vaidyanathaswamy, Proc. Indian Acad Sci., 20 (1945),51-61 Gulhan ASLIM and Aysegul CAKSU GULER Ege University, Department of Mathematics 35100 Izmir/TURKEY e-mail : [email protected] and. e-mail : [email protected]. Received November 04, 2004; Revised January 18, 2005. Translated by Mamedova V.A.