/compile/output.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 12, No. 2, 2019, 270-278 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Some characterizations of β-paracompactness in ideal topological space E. D. Yıldırım1, O. B. Özbakır2,∗and A.C. . Güler2 1 Department of Mathematics, Faculty of Science and Letters, Yaşar University, İzmir, Turkey 2 Department of Mathematics, Faculty of Science, Ege University, İzmir, Turkey Abstract. In this paper, we introduce β-paracompactness with respect to an ideal (I-β-paracompactness) as a weak form of β-paracompactness and I-paracompactness. We give some relations between this concept and some other types of paracompactness, and also we study some of its fundamental properties. 2010 Mathematics Subject Classifications: 54D20, 54A05, 54C10, 54G05 Key Words and Phrases: β-paracompact, ideal, I-β-paracompact, σ-β-locally finite 1. Introduction Paracompactness is one of the important concepts of general topology. In litera- ture, different kinds of generalized paracompactness such as S-paracompactness [5], P3- paracompactness [6] and β-paracompactness [11] are studied. The concept of I-paracompactness as generalization of paracompactness was given by Zahid [24]. Furthermore, this concept was studied by Hamlet et al. [13] and Sathiyasun- dari and Renukadevi [22]. Recently, S-paracompactness with respect to an ideal which is weaker form of I-paracompactness was studied by J. Sanabria et al. [21]. Here, we introduce I-β-paracompactness and we compare this concept with the other types of paracompactness. Then, we give counterexamples showing that the opposite di- rections of Proposition 1 and 2 do not hold. Furthermore, adding some conditions, we find that the reverse directions may happen to be true. Besides, we investigate some of its essential properties. Finally, we examine I-β-paracompactness under some functions. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v12i2.3394 Email addresses: esra.dalan@yasar.edu.tr (E. D. Yıldırım), oya.ozbakir@ege.edu.tr (O. B. Özbakır), aysegul.caksu.guler@ege.edu.tr (A.C. . Güler) http://www.ejpam.com 270 c© 2019 EJPAM All rights reserved. E. D. Yıldırım, O. B. Özbakır, A. C. . Güler / Eur. J. Pure Appl. Math, 12 (2) (2019), 270-278 271 2. Preliminaries Throughout this work, (X, τ) denotes a topological space on which no separation axioms are assumed unless clearly indicated. If A is a subset of (X, τ), then the closure of A and the interior of A will be denoted by cl(A) and int(A), respectively. Also, the class of all subsets of X will be denoted by P(X). A subset A of (X, τ) is said to be semi-open [16] if there exists U ∈ τ such that U ⊆ A ⊆ cl(U). This is equivalent to say that A ⊆ cl(int(A)). Also, A is said to be β-open [1] (preopen [18]) if A ⊆ cl(int(cl(A)))(A ⊆ int(cl(A))). The concept of β-open sets is equal to that of semi-preopen sets in [7]. The family of all semi- open (resp. β-open and preopen) sets of (X, τ) is denoted by SO(X, τ) (resp. βO(X, τ) and PO(X, τ)). The complement of a semi-open (resp. β-open and preopen) set is said to be semi-closed [10] (resp. β-closed [1, 7] and preclosed [18]). The semi-closure [10] (resp. β-closure [3, 7] and preclosure[18]) of A, denoted by scl(A) (resp. βcl(A) and pcl(A)), is the intersection of all semi-closed (resp. β-closed and preclosed) sets containing A. Note that, βcl(A) is β-closed [3, 7]. Lemma 1. [3, 7] For a subset A of a topological space (X, τ), the following conditions hold: (i) x ∈ βcl(A) if and only if A ∩ U 6= ∅ for every U ∈ βO(X, τ) containing x, (ii) A is β-closed if and only if A = βcl(A). Theorem 1. [19] Let (X, τ) be a space, A ⊆ Y ⊆ X and Y be β-open in (X, τ). Then A is β-open in (X, τ) if and only if A is β-open in the subspace (Y, τY ). A function f : (X, τ) → (Y, σ) is said to be pre β-closed [17] (pre β-open [17]) if for every β-closed (β-open)set A of (X, τ), f(A) is β-closed (β-open) in (Y, σ) and f : (X, τ) → (Y, σ) is said to be β-irresolute [17] if for every β-open set B of (Y, σ), f−1(B) is β-open in (X, τ). If f : (X, τ) → (Y, σ) is continuous and open, then f is β-irresolute and pre β-open. Lemma 2. [11] Let f : (X, τ) → (Y, σ) be a surjective function. Then f is pre β-closed if and only if for every y ∈ Y and every β-open set U in (X, τ) which contains f−1(y), there exists a V ∈ βO(Y, σ) such that y ∈ V and f−1(V ) ⊆ U . A space (X, τ) is called extremally disconnected[23](briefly, e. d.) if the closure of every open set in X is open and called submaximal [8] if each dense subset of X is open in X. Lemma 3. [20] (X, τ) is submaximal if and only if every pre-open set is open. Lemma 4. [9] (X, τ) is e.d. if and only if every β-open set is pre-open. A collection V of subsets of a space (X, τ) is said to be locally finite [23](resp. s-locally finite [4], β-locally finite [11] and p-locally finite[6]), if for each x ∈ X there exists Ux ∈ τ (resp. Ux ∈ SO(X, τ), Ux ∈ βO(X, τ) and Ux ∈ PO(X, τ)) containing x and Ux intersects at most finitely many members of V . Every locally finite collection of subsets of a space E. D. Yıldırım, O. B. Özbakır, A. C. . Güler / Eur. J. Pure Appl. Math, 12 (2) (2019), 270-278 272 (X, τ) is β-locally finite[11] and p-locally finite[6]. Also, a collection A of subsets of a space (X, τ) is said to be σ-locally finite if A = ∞⋃ n=1 An where each An is locally finite family [13]. Theorem 2. [11] Let (X, τ) be an e.d. submaximal space. Then every β-locally finite collection of subsets of X is locally finite. A space (X, τ) is said to be β-compact [2] if every cover of X by β-open sets has a finite subcover. Also a space (X, τ) is said to be paracompact [23] (resp. S-paracompact [5], β-paracompact [11] and P3-paracompact [6]), if every open cover of X has a locally finite open (resp. locally finite semi-open, β-locally finite β-open and p-locally finite preopen) refinement which covers to X. An ideal is defined as a nonempty collection I of subsets of X satisfying the following two conditions: (1) If A ∈ I and B ⊆ A, then B ∈ I, (2) If A ∈ I and B ∈ I, then A ∪B ∈ I. Given a topological space (X, τ) with an ideal I on X and if P(X) is the set of all subsets of X, a set operator (.)∗ : P(X) → P(X), called a local function [15] of A with respect to τ and I is defined as follows: for A ⊆ X, A∗(I, τ) = {x ∈ X : V ∩ A /∈ I for every V ∈ τ(x)} where τ(x) = {V ∈ τ : x ∈ V }. 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) [14]. A basis β(I, τ) for τ∗(I, τ) can be described as follows: β(I, τ) = {V − J : V ∈ τ and J ∈ I}[14]. We will simply write A∗ for A∗(I, τ), τ∗ or τ∗(I) for τ∗(I, τ) and β for β(I, τ). If I is an ideal on X, then (X, τ, I) is called an ideal topological space. A space (X, τ, I) is said to be I-paracompact [24] (I-S-paracompact [21]) if every open cover U of X has a locally finite open (semi-open) refinement V , not necessarily a cover, such that X − ⋃ {V : V ∈ V} ∈ I. A collection V of subsets of X such that X − ⋃ {V : V ∈ V} ∈ I is called an I-cover [24] of X. A space (X, τ, I) is said to be I-regular[12] if for each closed set F and a point p /∈ F , there exist disjoint open sets U and V such that p ∈ U and F − V ∈ I. 3. I-β-paracompactness Definition 1. A space (X, τ, I) is said to be I-β-paracompact or β-paracompact with respect to I if every open cover U of X has a β-locally finite β-open refinement V (not necessarily a cover) such that X − ⋃ {V : V ∈ V} ∈ I. A subset A of a space (X, τ, I) is called an I-β-paracompact set in (X, τ, I) if every open cover U of A has a β-locally finite (with respect to τ) β-open refinement V such that A− ⋃ {V : V ∈ V} ∈ I. Proposition 1. If (X, τ) is β-paracompact, then (X, τ, I) is I-β-paracompact. Proof. It is obvious since ∅ ∈ I. E. D. Yıldırım, O. B. Özbakır, A. C. . Güler / Eur. J. Pure Appl. Math, 12 (2) (2019), 270-278 273 Obviously, every compact space is I-β-paracompact since every compact space is β- paracompact [11]. The following example shows that the converse of Proposition 1 may not be true, in general. Example 1. Let X = N be the set of natural numbers with the topology τ = {G ⊆ N : 5 ∈ G} ∪ {∅} and the ideal I = {U ⊆ N : 5 /∈ U}. Observe that (X, τ, I) is I-β-paracompact space but (X, τ) is not β-paracompact since the collection {{5, x} : x ∈ N} is an open cover of X which admits no β-locally finite β-open refinement in X. Remark 1. (1) If I = {∅}, then (X, τ, I) is I-β-paracompact if and only if (X, τ) is β-paracompact. (2) If I = {∅} and (X, τ, I) is an e.d. space, then (X, τ, I) is I-β-paracompact if and only if (X, τ) is P3-paracompact. Proposition 2. If (X, τ, I) is I-S-paracompact then it is I-β-paracompact. Proof. Since every locally finite collection of subsets of X is β-locally finite and every semi-open set is β-open, it is clear. Clearly, every S-paracompact space is I-β-paracompact since every S-paracompact space is I-S-paracompact[21]. Also, every I-paracompact space is I-β-paracompact since every I-paracompact space is I-S-paracompact[21]. The following example shows that the converse of Proposition 2 may not be true, in general. Example 2. Let X = [0, 2]∪[3, 10] with the topology τ = {U ⊆ X : [0, 2] ⊆ U}∪{∅} and the ideal I = {A : A ⊆ [0, 2]}. Then (X, τ, I) is I-β-paracompact since every open cover of X has β-locally finite β-open refinement V = {{x} : x ∈ [0, 2]}∪{{y, z} : y ∈ [0, 2], z ∈ [3, 10]} such that X − ⋃ {V : V ∈ V} ∈ I. But it is not I-S-paracompact since τ = SO(X). Theorem 3. If (X, τ, I) is an e.d. submaximal I-β-paracompact space, then it is I-S- paracompact. Proof. It is obvious from Lemma 3, Lemma 4 and Theorem 2. Theorem 4. If (X, τ, I) is I-β-paracompact and J is an ideal on X with I ⊆ J , then (X, τ, J) is J-β-paracompact. Proof. Let (X, τ, I) be I-β-paracompact and I ⊆ J . And let U = {Uλ : λ ∈ Λ} be an open cover of X. Since (X, τ, I) is I-β-paracompact, U has a β-locally finite β-open refinement V such that X − ⋃ {V : V ∈ V} ∈ I. Since I ⊆ J , X − ⋃ {V : V ∈ V} ∈ J . Thus, (X, τ, J) is J-β-paracompact. Lemma 5. [11]Let V = {Vλ : λ ∈ Λ} be a collection of subsets of a space (X, τ). V is β-locally finite if and only if {βcl(Vλ) : λ ∈ Λ} is β-locally finite. E. D. Yıldırım, O. B. Özbakır, A. C. . Güler / Eur. J. Pure Appl. Math, 12 (2) (2019), 270-278 274 Lemma 6. If a cover U = {Uλ : λ ∈ Λ} of a space (X, τ, I) has a β- locally finite β-open refinement V such that X − ⋃ {V : V ∈ V} ∈ I then there exists a β-locally finite precise β- open refinement H = {Hλ : λ ∈ Λ} of U such that X − ⋃ {Hλ : Hλ ∈ H} ∈ I. Proof. The proof is similar to that of Lemma 1.3 in [21]. Definition 2. A collection A of subsets of a space (X, τ) is said to be σ-β-locally finite if A = ∞⋃ n=1 An where each collection An is a β- locally finite family. Lemma 7. Every β-locally finite collection of subsets of a space (X, τ) is σ-β-locally finite. Proof. It is obvious. Theorem 5. Let (X, τ) be a regular space. If (X, τ, I) is I-β-paracompact, then every open cover of X has a β-closed β-locally finite I-cover refinement. Proof. Let U be an open cover of X. By regularity of X , for each x ∈ X and Ux ∈ U containing x, there exists an open set Gx of x such that cl(Gx) ⊆ Ux. Then U1 = {Gx : x ∈ X} is an open cover of X. Since X is I-β-paracompact, U1 has β- locally finite β-open refinement V1 = {Vλ : λ ∈ Λ} such that X − ⋃ {Vλ : λ ∈ Λ} ∈ I. Then X − ⋃ {βcl(Vλ) : λ ∈ Λ} ∈ I. By Lemma 5, V = {βcl(Vλ) : Vλ ∈ V1} is β- locally finite. Since V1 refines U1, for every λ ∈ Λ, there is some Gx ∈ U1 such that Vλ⊆ Gx. Then βcl(Vλ)⊆ cl(Vλ)⊆ cl(Gx) implies βcl(Vλ)⊂ Ux. Hence V refines U . So, V = {βcl(Vλ) : Vλ ∈ V1} is β-closed β-locally finite I-cover refinement. Remark 2. If (X, τ, I) is considered to be e.d. submaximal regular space, then the Theo- rem 5 becomes the Theorem 2.20 in [22]. Theorem 6. If (X, τ, I) is I-β-paracompact, then every open cover of X has a β-open σ-β-locally finite I-cover refinement. Proof. It is obvious by Lemma 7. Theorem 7. Let (X, τ, I) be a regular space and βO(X, τ) be closed under finite intersec- tion. Then, (X, τ, I) is I-β-paracompact if and only if every open cover of X has a β-open σ-β-locally finite I-cover refinement. Proof. To show sufficiency, let U be an open cover of X. By hypothesis, there exists a σ-β-locally finite β-open refinement V of U such that X − ⋃ {V : V ∈ V} ∈ I. Also, V = ∞⋃ n=1 Vn where each collection Vn is a β- locally finite. For each n ∈ N, let Hn = ⋃ {V : V ∈ Vn} so that X − ⋃ {Hn : n ∈ N} ∈ I. For each n ∈ N, let Gn= Hn − n−1⋃ i=1 Hi. Then {Gn : n ∈ N} refines {Hn : n ∈ N}. Let x ∈ X, and let n be the smallest member of {n ∈ N : x ∈ Hn}. Then x ∈ Gn and X − ⋃ {Gn : n ∈ N} ∈ I. Also, Gnx is a β-open set containing x that intersects only finite family number of members of Gn so E. D. Yıldırım, O. B. Özbakır, A. C. . Güler / Eur. J. Pure Appl. Math, 12 (2) (2019), 270-278 275 that {Gn : n ∈ N} is β-locally finite. Let O= {V ∩ Gn : V ∈ Vn and n ∈ N}. Since {Gn : n ∈ N} is β-locally finite, O is β-locally finite. Also, since βO(X, τ) is closed under finite intersection and V is β-open refinement of U , O is β-open refinement of U . Then, X − ⋃ {V ∩ Gn : n ∈ N} ∈ I because X − ⋃ {Gn : n ∈ N} ∈ I. Thus, (X, τ, I) is I-β-paracompact. Remark 3. If (X, τ, I) is considered to be e.d. submaximal regular space, then Theorem 7 becomes Theorem 2.22 in [22]. Theorem 8. For any ideal topological space (X, τ, I), the following are equivalent: (i) For every closed subset A of X and every x /∈ A, there exist disjoint β-open sets U and V such that x ∈ U and A− V ∈ I. (ii) For every open subset G of X and every x ∈ G, there exists a β-open set U such that x ∈ U and βcl(U)−G ∈ I. Proof. (i) ⇒ (ii) Let G ⊆ X be open and x ∈ G. Then X − G = A is closed and x /∈ A. From (i), there exist disjoint β-open sets U and V such that x ∈ U and A−V ∈ I. Since U and V are disjoint, we have βcl(U) ⊆ X − V . Thus, A∩ βcl(U) ⊆ A− V . Then, βcl(U) ∩ (X −G) ∈ I. Therefore, βcl(U)−G ∈ I. (ii) ⇒ (i) Let A ⊆ X be closed and x /∈ A. Then, X − A = G is open and x ∈ G. From (ii), there exists a β-open set U such that x ∈ U and βcl(U) − G ∈ I. Thus, X−βcl(U) = V ∈ βO(X) and U∩V = ∅. Furthermore, A−V = (X−G)−(X−βcl(U)) = βcl(U)−G ∈ I. The following example reveals that for a locally finite collection of subsets of V = {Vλ : λ ∈ Λ} of a space (X, τ), the equality cl( ⋃ {Vλ : λ ∈ Λ}) = ⋃ {cl(Vλ) : λ ∈ Λ} always holds whereas for β-locally finite collection of subsets U = {Uλ : λ ∈ Λ} of a space (X, τ), the equality βcl( ⋃ {Uλ : λ ∈ Λ}) = ⋃ {βcl(Uλ) : λ ∈ Λ} does not hold in general. Example 3. Consider the real number R with usual topology τ . Let V = {[0, 1), (1, 2]}. Then V is β-locally finite in (R, τ) since it is finite. But βcl([0, 1) ∪ (1, 2]) 6= βcl([0, 1)) ∪ βcl((1, 2]). Theorem 9. Suppose that for a β-locally finite collection of subsets V = {Vλ : λ ∈ Λ} of a space (X, τ, I), the equality βcl( ⋃ {Vλ : λ ∈ Λ}) = ⋃ {βcl(Vλ) : λ ∈ Λ} holds. If (X, τ, I) is Hausdorff I-β-paracompact, then for every closed subset A of X and every x /∈ A, there exist disjoint β-open sets U and V such that x ∈ U and A− V ∈ I. Proof. Let A ⊆ X closed and x /∈ A. Since X is Hausdorff space, there exists an open set Hy containing y for each y ∈ A such that x /∈ cl(Hy). Thus, H = {Hy : y ∈ A}∪{X−A} is an open cover of X. By hypothesis and Lemma 6, H has a β-locally finite precise β-open refinement W = {Wy : y ∈ A} ∪ {G} such that Wy ⊆ Hy for each y ∈ A, G ⊆ X−A and X−( ⋃ {Wy : y ∈ A}∪{G}) ∈ I. Since A−( ⋃ {Wy : y ∈ A}) = A−( ⋃ {Wy : y ∈ A} ∪ {G}) ⊆ X − ( ⋃ {Wy : y ∈ A} ∪ {G}), we have A − ( ⋃ {Wy : y ∈ A}) ∈ I. Let E. D. Yıldırım, O. B. Özbakır, A. C. . Güler / Eur. J. Pure Appl. Math, 12 (2) (2019), 270-278 276 we say V = ⋃ {Wy : y ∈ A}. Then, V is β-open set in X and A − V ∈ I. Since x /∈ cl(Hy), we have x /∈ cl(Wy). This implies that x /∈ βcl(Wy). Since W is β-locally finite, βcl(V ) = βcl( ⋃ {Wy : y ∈ A}) = ⋃ {βcl(Wy) : y ∈ A} by hypothesis. Thus, for a β-open set U = X − βcl(V ), we have U ∩ V = ∅ such that x ∈ U . From Theorem 8 and Theorem 9, we have the following Corollary. Corollary 1. If (X, τ, I) is an e.d. submaximal Hausdorff I-β-paracompact space, then (X, τ, I) is I-regular. Theorem 10. Let A and B be subsets in ideal topological space (X, τ, I). If A is I-β- paracompact set in X and B is closed in X, then A ∩B is I-β-paracompact set in X. Proof. Let U = {Uλ : λ ∈ Λ} be an open cover of A ∩ B. Since X − B is open in X, U ′ = {Uλ : λ ∈ Λ} ∪ {X − B} is open cover of A. By hypothesis and Lemma 6, U ′ has a β- locally finite precise β-open refinement {Vλ : λ ∈ Λ} ∪ {V } such that Vλ ⊆ Uλ for each λ ∈ Λ, V ⊆ X − B and A − ( ⋃ {Vλ : λ ∈ Λ} ∪ {V }) ∈ I. Since (A∩B)−( ⋃ {Vλ : λ ∈ Λ}) = (A∩B)−( ⋃ {Vλ : λ ∈ Λ}∪{V }) ⊆ A−( ⋃ {Vλ : λ ∈ Λ}∪{V }), we have (A ∩B)− ( ⋃ {Vλ : λ ∈ Λ}) ∈ I. Hence, A ∩B is I-β-paracompact set in X. Corollary 2. Let (X, τ, I) be an I-β-paracompact space and A ⊆ X. If A is closed in X, then A is an I-β-paracompact set in X. Lemma 8. [13] If I 6= ∅ is an ideal on X and Y is a subset of X, then IY = {Y ∩G|G ∈ I }= {G ∈ I|G ⊆ Y } is an ideal on Y. Theorem 11. Let A and B be subsets in ideal topological space (X, τ, I) such that B ⊆ A. If A is β-open in X and B is an IA-β-paracompact set in A then B is an I-β-paracompact set in X. Proof. Let U = {Uλ : λ ∈ Λ} be an open cover of B in X. Then, UB = {Uλ∩A : λ ∈ Λ} is an open cover of B in A. Since B is an IA-β-paracompact set in A, UB has a β-locally finite precise β-open refinement VB in A such that B − ⋃ {Vλ : Vλ ∈ VB} ∈ IA. Thus, VB is a β-locally finite precise β-open refinement in X by Theorem 1. Also, B − ⋃ {Vλ : Vλ ∈ VB} ∈ I. Hence, B is an I-β-paracompact set in X. Theorem 12. Let f : (X, τ, I) → (Y, σ, J) be a continuous, open and pre β-closed surjec- tion with f−1(y) β-compact for every y ∈ Y and f(I) ⊆ J . If (X, τ, I) is I-β-paracompact, then (Y, σ, J) is J-β-paracompact. Proof. Let U = {Uλ : λ ∈ Λ} be an open cover of Y . Then, {f−1(Uλ) : λ ∈ Λ} is an open cover of X. Since (X, τ, I) is I-β-paracompact, this open cover has a β-locally finite precise β-open refinement V = {Vλ : λ ∈ Λ} such that X − ⋃ {Vλ : Vλ ∈ V} ∈ I. Since f is pre β-open, f(V) = {f(Vλ) : λ ∈ λ} is a precise β-open refinement of U . Also, Y − ⋃ {f(Vλ) : λ ∈ Λ} ∈ J . Now, let we prove that f(V) is β-locally finite. Let y ∈ Y . Since V is β-locally finite, for x ∈ f−1(y), there exists a β-open set Gx containing x such that Gx intersects at most finitely members of V. Since f−1(y) is β-compact, {Gx : x ∈ REFERENCES 277 f−1(y)} has a finite subcollection Hy such that f−1(y) ⊆ ⋃ Hy and ⋃ Hy intersects at most finitely members of V. By Lemma 2, there exists a β-open set Wy containing y such that f−1(Wy) ⊆ ⋃ Hy. Then, f−1(Wy) intersects at most finitely members of V. This implies that Wy intersects at most finitely members of f(V). Hence, f(V) is β-locally finite in Y . So, (Y, σ, J) is J-β-paracompact. Theorem 13. Let f : (X, τ, I) → (Y, σ, J) be an open, β- irresolute bijective mapping and I = f−1(J). If A is J-β-paracompact in Y , then f−1(A) is I-β-paracompact in X. Proof. Let U = {Uλ : λ ∈ Λ} be an open cover of f−1(A). Since f is open, U1 = {f(Uλ) : λ ∈ Λ} is an open cover of A. By hypothesis, this open cover has a β-locally finite precise β-open refinement V1 = {Vλ : λ ∈ Λ} such that A− ⋃ {Vλ : λ ∈ Λ} ∈ J . Then, f−1(A) − ⋃ {f−1(Vλ) : λ ∈ Λ} ∈ f−1(J) = I. Since f is β-irresolute, V = {f−1(Vλ) : λ ∈ Λ} is β-locally finite β-open. Let f−1(Vλ) ∈ V. Since V1 refines U1, there exists f(Uλ) ∈ U1 such that Vλ ⊆ f(Uλ). Then f−1(Vλ) ⊆ f−1(f(Uλ)) = Uλ. Hence V refines U . Therefore f−1(A) is I-β-paracompact in X. Acknowledgements The author would like to thank the referees for their helpful suggestions. References [1] Abd El-Monsef, M. E., El-Deeb, S. N. and Mahmoud, R. A. β-open sets and β- continuous mapping, Bull. Fac. Sci. Assiut Univ. 12, 77-90, 1983. [2] Abd El-Monsef, M. E. and Kozae, A. M. 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