/compile/output.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 8, No. 4, 2015, 502-513 ISSN 1307-5543 – www.ejpam.com Weak Separation Axioms via e-I - Sets in Ideal Topological Spaces Wadei Faris Al-Omeri1, M.S. Md. Noorani1, A. AL-Omari2, T. Noiri3,∗ 1 School of Mathematical Sciences, Faculty of Science and Technology Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor DE, Malaysia 2 Department of Mathematics, Faculty of Science Al AL-Bayat University, P.O.Box 130095, Mafraq25113, Jordan 3 2949-1 Shiokita-cho, Hinagu, Yatsushiro-shi, Kumamoto-ken 869-5142, JAPAN. Abstract. In this paper, we use the notion of e-I -open sets to introduce and define some new weak separation axioms. Also we study some of their basic properties. Additionally, we investigate the relationship and implications of these axioms among themselves and with other known axioms. 2010 Mathematics Subject Classifications: 54A05 Key Words and Phrases: Ideal Topological Space, e-I -R0 Space, e-I -R1 Space, e-I -Open Set, e-I -R2 Space 1. Introduction The notion of R0 topological spaces is introduced by Shanin [15] in 1943. Later, Davis [4] rediscovered it and studied some properties of this weak separation axiom. Several topologists (e.g. [6, 10, 13]) further investigated properties of R0 topological spaces and many interesting results have been obtained in various contexts. In the same paper, Davis also introduced the notion of R1 topological spaces which are independent of both T0 and T1 but strictly weaker than T2. A subset A of a space (X ,τ) is said to be regular open (resp. regular closed) [16] if A = Int(Cl(A)) (resp. A = Cl(Int(A))). A is said to be δ-open [18] if for each x ∈ A, there exists a regular open set G such that x ∈ G ⊂ A. The complement of a δ-open set is said to be δ-closed. A point x ∈ X is called a δ-cluster point of A if Int(Cl(U))∩A 6= ; for each open set U containing x . The set of all δ-cluster points of A is called the δ-closure of A and is denoted by Clδ(A) [18]. The set δ-interior of A [18] is the union of all regular open sets of X contained in A and is denoted by Intδ(A). A is δ-open if Intδ(A) = A. The collection of all δ-open sets of (X ,τ) is denoted by δO(X ) and forms a topology τδ. ∗Corresponding author. Email addresses: wadeimoon1@hotmail.com (W. Al-omeri), msn@ukm.my (M. Noorani), omarimutah1@yahoo.com (A. AL-Omari), t.noiri@nifty.com (T. Noiri) http://www.ejpam.com 502 c© 2015 EJPAM All rights reserved. W. AL-Omeri, M. Noorani, A. AL-Omari, and T. Noiri / Eur. J. Pure Appl. Math, 8 (2015), 502-513 503 An ideal I on a topological space (X ,I ) is a nonempty collection of subsets of X which satisfies the following conditions: A ∈ I and B ⊂ A implies B ∈ I ; A ∈ I and B ∈ I implies A∪ B ∈ I . Applications to various fields were further investigated by Jankovic and Hamlett [11]; Dontchev [5]; Mukherjee et al. [12]; Arenas et al. [3]; Nasef and Mahmoud [14], etc. Given a topological space (X ,I ) 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 [11, 17] 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}. Furthermore Cl∗(A) = A∪ A∗(I ,τ) defines a Kuratowski clo- sure operator for the topology τ∗. When there is no chance for confusion, we will simply write A∗ for A∗(I ,τ). X ∗ is often a proper subset of X . 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 denote the closure and interior of A in (X ,τ), respectively. A subset A of a topological space (X ,τ) is said to be e-open [9] if A⊂ Int(δCl(A))∪ Cl(δInt(A)). The notion of e-open sets has been study extensively in recent years by many topologists. In this paper, we use the notion of e-I -open sets to introduce and define some new weak separation axioms. Also we study some of their basic properties. Additionally, we investigate the relationship and implications of these axioms among themselves and with other known axioms. 2. Preliminaries A subset A of an ideal topological space (X ,τ,I ) is said to be e-I -open [2] if A⊂ Cl(δInt I(A))∪ Int(δClI(A)). The complement of an e-I -open set is called an e-I -closed set [2]. The intersection of all e-I -closed sets containing A is called the e-I -closure of A and is denoted by Cl∗e (A). The e- I -interior of A is defined by the union of all e-I -open sets contained in A and is denoted by Int∗e (A). The family of all e-I -open (resp. e-I -closed) sets of (X ,τ,I ) containing a point x ∈ X is denoted by EIO(X , x) (resp. EI C(X , x)). A subset U of X is called an e-I - neighborhood of a point x ∈ X if there exists an e-I -open set V of (X ,τ,I ) such that x ∈ V ⊂ U . A function f : (X ,τ,I )→ (Y,σ) is said to be e-I -continuous if f −1(V ) ∈ EIO(X ) for every open set V of Y . Definition 1. A topological space (X ,τ) is said to be: (i) R0 [4] if every open set contains the closure of each of its singletons. (ii) R1 [4] if for x, y in X with Cl({x}) 6= Cl({y}), there exist disjoint open sets U and V such that Cl({x}) ⊂ U and Cl({y} ⊂ V . Definition 2. A topological space (X ,τ) is said to be: (i) e-T1 [7, 8] if for each pair of distinct points x and y in X , there exist e-open sets U and V containing x and y, respectively, such that y /∈ U and x /∈ V . W. AL-Omeri, M. Noorani, A. AL-Omari, and T. Noiri / Eur. J. Pure Appl. Math, 8 (2015), 502-513 504 (ii) e-T2 [7, 8] if for each pair of distinct points x and y in X , there exist disjoint e-open sets U and V such that x ∈ U and y ∈ V . Definition 3. An ideal topological space (X ,τ,I ) is said to be: (i) e-I -T1 [1] if for each pair of distinct points x and y in X , there exist e-I -open sets U and V of X , such that x ∈ U and y /∈ U, y ∈ V and x /∈ V . (ii) e-I -T2 [1] if for each pair of distinct points x and y in X , there exist disjoint e-I -open sets U and V in X such that x ∈ U and y ∈ V . 3. On e-I -R0 Spaces Definition 4. Let (X ,τ,I ) be an ideal topological space and A ⊂ X . Then the e-I -kernel of A, denoted by IeKer(A), is defined to be the set IeKer(A) = ∩{G ∈ EIO(X )|A⊂ G}. Lemma 1. Let (X ,τ,I ) be an ideal topological space and x , y ∈ X . Then, y ∈ IeKer({x}) if and only if x ∈ Cl∗e ({y}). Proof. Suppose that y /∈ IeKer({x}). Then there exists U ∈ EIO(X , x) such that y /∈ U . Therefore, we have x /∈ Cl∗e ({y}). The proof of the converse case can be done similarly. Lemma 2. Let (X ,τ,I ) be an ideal topological space and S a subset of X . Then, IeKer(S) = {x ∈ X |Cl∗e ({x})∩ S 6= ;}. Proof. Let x ∈ IeKer(S). Suppose that Cl∗e ({x})∩S = ;. Hence x /∈ X\Cl∗e ({x})which is an e-I -open set containing S. Since x /∈ IeKer(S), this is a contradiction. Hence Cl∗e ({x})∩S 6= ;. Conversely, suppose that Cl∗e ({x}) ∩ S 6= ;. Next, let x ∈ X such that Cl∗e ({x}) ∩ S 6= ; and suppose that x /∈ IeKer(S). Then, there exists an e-I -open set U containing S and x /∈ U . Let y ∈ Cl∗e ({x})∩ S. Hence, U is an e-I -neighborhood of y which does not contains x . By this contradiction x ∈ IeKer(S) and hence the claim. Definition 5. An ideal topological space (X ,τ,I ) is called an e-I -R0 space if every e-I -open set contains the e-I -closure of each of its singletons. Definition 6. An ideal topological space (X ,τ,I ) is said to be e-I -T0 if for each pair of distinct points x and y in X , there exists an e-I -open set U such that x ∈ U and y /∈ U, or there exists an e-I -open set V such that y ∈ V and x /∈ V . Theorem 1. Let (X ,τ,I ) be an ideal topological space. Then X is e-I -T1 if and only if it is e-I -T0 and e-I -R0. Proof. Let X be an e-I -T1 space. By the definition of an e-I -T1 space, it is an e-I -T0 and e-I -R0 space. Conversely, let X be an e-I -T0 and e-I -R0 space. Let x , y be any two distinct points of X . Since X is e-I -T0, then there exists an e-I -open set U such that x ∈ U and y /∈ U or W. AL-Omeri, M. Noorani, A. AL-Omari, and T. Noiri / Eur. J. Pure Appl. Math, 8 (2015), 502-513 505 there exists an e-I -open set V such that y ∈ V and x /∈ V . Let x ∈ U and y /∈ U . Since X is e-I -R0, then Cl∗e (x) ⊂ U . We have y /∈ U and then y /∈ Cl∗e (x). We obtain y ∈ X\Cl∗e (x). Take S = X\Cl∗e (x). Thus, U and S are e-I -open sets containing x and y , respectively, such that y /∈ U and x /∈ S. Hence, X is e-I -T1. Remark 1. Since an ideal topological space (X ,τ,I ) is e-I -T1 if and only if the singletons are e-I -closed, it is clear that every e-I -T1 space e-I -R0. But the converse is not true in general. Example 1. Let X = {a, b, c}with a topologyτ= {;, X , {a}, {b, c}} andI = {O , {c}, {b}, {b, c}}. Since e-I -open={φ, X , {a}, {b, c}}. It is clear that every e-I -open set contains the e-I -closure of each of its singletons so the ideal topological space is e-I -R0, but none of e-I -T0 and e-I -T1. Remark 2. The following example and Example 1 show that the notions e-I -T0-ness and e-I -R0-ness are independent. Example 2. Let X = {a, b, c} with a topology τ = {;, X , {a}} and I = {O , {a}}. Now, we determine e-I -open={φ, X , {a}}. Then (X ,τ,I ) is e-I -T0 but it is not e-I -R0. Lemma 3. Let (X ,τ,I ) be an ideal topological space. Then for any points x and y in X , the following statements are equivalent: (i) IeKer({x}) 6= IeKer({y}). (ii) C l∗e ({x}) 6= Cl∗e ({y}). Proof. (i)⇒ (ii): Let IeKer({x}) 6= IeKer({y}), then there exists a point k in X such that k ∈ IeKer({x}) and k /∈ IeKer({y}). By Lemma 1, x ∈ Cl∗e ({x}) and y /∈ Cl∗e ({x}). Therefore, Cl∗e ({x}) ⊂ Cl∗e (Cl∗e ({k})) = Cl∗e ({k}) and hence y /∈ Cl∗e ({x}). Hence Cl∗e ({x}) 6= Cl∗e ({y}). By using IeKer({x}) 6= IeKer({y}), we obtain Cl∗e ({x}) 6= Cl∗e ({y}). (ii)⇒ (i): Let Cl∗e ({x}) 6= Cl∗e ({y}), then there exists a point k in X such that k ∈ Cl∗e ({x}) and k /∈ Cl∗e ({y}) and then there exists an e-I -open set containing k and therefore x but not y , namely, y /∈ IeKer({x}) and thus IeKer({x}) 6= IeKer({y}). Proposition 1. For an ideal topological space (X ,τ,I ), the following properties are equivalent: (i) (X ,τ,I ) is an e-I -R0 space, (ii) For any K ∈ EI C(X ), x /∈ K implies K ⊂ U and x /∈ U for some U ∈ EIO(X ), (iii) For any K ∈ EI C(X ), x /∈ K implies K ∩ Cl∗e ({x}) = ;, (iv) For any distinct points x and y of X , either C l∗e ({x}) = Cl∗e ({y}) or Cl∗e ({x})∩ Cl∗e ({y}) = ;. Proof. (i) ⇒ (ii): Let K ∈ EI C(X ) and x /∈ K . Then by (i), Cl∗e ({x}) ⊂ X\K . Set U = X\Cl∗e ({x}), then U ∈ EIO(X ), K ⊂ U and x /∈ U . (ii) ⇒ (iii): Let K ∈ EI C(X ) and x /∈ K . There exists U ∈ EIO(X ) such that K ⊂ U and W. AL-Omeri, M. Noorani, A. AL-Omari, and T. Noiri / Eur. J. Pure Appl. Math, 8 (2015), 502-513 506 x /∈ U . Since U ∈ EIO(X ), U ∩ Cl∗e ({x}) = ; and K ∩ Cl∗e ({x}) = ;. (iii) ⇒ (iv): Suppose that Cl∗e ({x}) 6= Cl∗e ({y}) for distinct points x , y ∈ X . There exists k ∈ Cl∗e ({x}) such that k /∈ Cl∗e ({y}) (or k ∈ Cl∗e ({y}) such that k /∈ Cl∗e ({x}). There exists V ∈ EIO(X ) such that y /∈ V and k ∈ V ; hence x ∈ V . Therefore, we have x /∈ Cl∗e ({y}). By (iii), we obtain Cl∗e ({x})∩ Cl∗e ({y}) = ;. The proof for otherwise is similar. (iv) ⇒ (i): Let V ∈ EIO(X , x). For each y /∈ V , x 6= y and x /∈ Cl∗e ({y}). This shows that Cl∗e ({x}) 6= Cl∗e ({y}). By (iv), Cl∗e ({x}) ∩ Cl∗e ({y}) = ; for each y ∈ X\V and hence Cl∗e ({x}) ∩ (∪y∈X\V Cl∗e ({y})) = ;. On the other hand, since V ∈ EIO(X ) and y ∈ X\V , we have Cl∗e ({y}) ⊂ X\V and hence X\V = ∪y∈X\V Cl∗e ({y}). Therefore, we obtain (X\V )∩ Cl∗e ({x}) = ; and Cl∗e ({x}) ⊂ V . This shows that (X ,τ,I ) is an e-I -R0 space. Theorem 2. An ideal topological space (X ,τ,I ) is e-I -R0 space if and only if for any x and y in X , C l∗e ({x}) 6= Cl∗e ({y}) implies C l∗e ({x})∩ Cl∗e ({y}) = ;. Proof. Let (X ,τ,I ) is e-I -R0. By Proposition 1, we obtain the assertion. Conversely, let V ∈ EIO(X ; x). We will show that Cl∗e ({x}) ⊂ V . Let y ∈ X\V . Then x 6= y and x /∈ Cl∗e ({y}). This shows that Cl∗e ({x}) 6= Cl∗e ({y}). By assumption, Cl∗e ({x})∩Cl∗e ({y}) = ;. Hence y /∈ Cl∗e ({x}) and therefore Cl∗e ({x}) ⊂ V . Theorem 3. Let (X ,τ,I ) be an ideal topological space. Then the following properties are equiv- alent: (i) (X ,τ,I ) is an e-I -R0 space, (ii) x ∈ Cl∗e ({y}) if and only if y ∈ Cl∗e ({x}) for any points x and y in X . Proof. (i)⇒ (ii): Assume that (X ,τ,I ) is e-I -R0. Let x ∈ Cl∗e ({y}) and A ∈ EIO(X , y). Now by hypothesis, x ∈ Cl∗e ({y}) ⊂ A and x ∈ A. Therefore, every e-I -open set containing y contains x . Hence y ∈ Cl∗e ({x}). (ii) ⇒ (i): Let U ∈ EIO(X , x). If y /∈ U , then x /∈ Cl∗e ({y}) and hence y /∈ Cl∗e ({x}). This implies that Cl∗e ({x}) ⊂ U . Hence (X ,τ,I ) is e-I -R0 Theorem 4. For an ideal topological space (X ,τ,I ), the following properties are equivalent: (i) (X ,τ,I ) is an e-I -R0 space; (ii) For any nonempty set S of X and any G ∈ EIO(X ) such that S ∩ G 6= ;, there exists K ∈ EI C(X ) such that S ∩ K 6= ; and K ⊂ G; (iii) For any G ∈ EIO(X ), G = ∪{K ∈ EI C(X )|K ⊂ G}; (iv) For any K ∈ EI C(X ), K = ∩{G ∈ EIO(X )|K ⊂ G}; (v) For any x ∈ X , Cl∗e ({x}) ⊂ IeKer({x}). W. AL-Omeri, M. Noorani, A. AL-Omari, and T. Noiri / Eur. J. Pure Appl. Math, 8 (2015), 502-513 507 Proof. (i)⇒ (ii):Let S be a nonempty set of X and G ∈ EIO(X ) such that S∩G 6= ;. There exists x ∈ S ∩ G. Since x ∈ G ∈ EIO(X ), it follows that Cl∗e ({x}) ⊂ G. Take K = Cl∗e ({x}), then K ∈ EI C(X ), K ⊂ G and S ∩ K 6= ;. (ii)⇒ (iii): Let G ∈ EIO(X ). We have G ⊃ ∪{K ∈ EI C(X )|K ⊂ G}. Let x be any point of G. By (ii) there exists K ∈ EI C(X ) such that x ∈ K and K ⊂ G. Thus, we have x ∈ K ⊂ ∪{K ∈ EI C(X )|K ⊂ G} and hence G = ∪{K ∈ EI C(X )|K ⊂ G}. (iii)⇒ (iv): This is obvious. (iv)⇒ (v): Let x be any point of X and y /∈ IeKer({x}). There exists V ∈ EIO(X ) such that x ∈ V and y /∈ V ; hence Cl∗e ({y})∩ V = ;. By (iv), [∩{G ∈ EIO(X )|Cl∗e ({y}) ⊂ G}]∩ V = ; and there exists G ∈ EIO(X ) such that x /∈ G and Cl∗e ({y}) ⊂ G. Hence, Cl∗e ({x}) ∩ G = ; and y /∈ Cl∗e ({x}). Thus, Cl∗e ({x}) ⊂ IeKer({x}). (v) ⇒ (i): Let G ∈ EIO(X ) and x ∈ G. Let y ∈ IeKer({x}). We have x ∈ Cl∗e ({y}) and y ∈ G. It follows that IeKer({x}) ⊂ G. Thus, we obtain x ∈ Cl∗e ({x}) ⊂ IeKer({x}) ⊂ G. This shows that (X ,τ,I ) is an e-I -R0 space. Theorem 5. An ideal topological space (X ,τ,I ) is e-I -R0 if and only if for any pair of points x and y in X , IeKer({x}) 6= IeKer({y}) implies IeKer({x})∩IeKer({y}) = ;. Proof. Suppose that (X ,τ,I ) is an e-I -R0 space. Thus by Lemma 3, for any points x and y in X if IeKer({x}) 6= IeKer({y}), then Cl∗e ({x}) 6= Cl∗e ({y}). Now we prove that IeKer({x})∩IeKer({y}) = ;. Assume that z ∈ IeKer({x})∩IeKer({y}). By z ∈ IeKer({x}) and Lemma 1, it follows that x ∈ Cl∗e ({z}). Since x ∈ Cl∗e ({x}), by Theorem 2, Cl∗e ({x}) = Cl∗e ({z}). Similarly, we have Cl∗e ({x}) = Cl∗e ({z}) = Cl∗e ({y}). This is a contradic- tion. Therefore, we have IeKer({x})∩IeKer({y}) = ;. Conversely, let (X ,τ,I ) be an ideal topological space such that for any points x and y in X , IeKer({x}) 6= IeKer({y}) implies IeKer({x})∩IeKer({y}) = ;. If Cl∗e ({x}) 6= Cl∗e ({y}), then by Lemma 3, IeKer({x}) 6= IeKer({y}). Hence, IeKer({x})∩IeKer({y}) = ; which implies Cl∗e ({x}) ∩ Cl∗e ({y}) = ;. Because z ∈ Cl∗e ({x}) implies that x ∈ IeKer({z}) and therefore IeKer({x}) ∩ IeKer({z}) 6= ;. By hypothesis, we have IeKer({x}) = IeKer({z}). Then z ∈ Cl∗e ({x}) ∩ Cl∗e ({y}) implies that IeKer({x}) = IeKer({z}) = IeKer({y}). This is a contradiction. Therefore, Cl∗e ({x}) ∩ Cl∗e ({y}) = ; and by Theorem 2 (X ,τ,I ) is an e-I -R0 space. Theorem 6. For an ideal topological space (X ,τ,I ), the following properties are equivalent: (i) (X ,τ,I ) is an e-I -R0 space, (ii) If F is an e-I -closed subset of X , then F = IeKer(F), (iii) If F is an e-I -closed subset of X and x ∈ F, then IeKer({x}) ⊂ F, (iv) If x ∈ X , then IeKer({x}) ⊂ Cl∗e ({x}). Proof. (i)⇒ (ii): Let F be an e-I -closed subset of X and x /∈ F . Thus X\F ∈ EIO(X x). Since (X ,τ,I ) is e-I -R0, Cl∗e ({x}) ⊂ X\F . Since F ⊂ X \ Cl∗e ({x}), IeKer(F) ⊂ X − Cl∗e ({x}) W. AL-Omeri, M. Noorani, A. AL-Omari, and T. Noiri / Eur. J. Pure Appl. Math, 8 (2015), 502-513 508 and x /∈ IeKer(F). Therefore, IeKer(F) = F . (ii)⇒ (iii): In general, A ⊂ B implies IeKer(A) ⊂ IeKer(B). Therefore, it follows from (ii) that IeKer({x}) ⊂ IeKer(F) = F . (iii)⇒ (iv): Since x ∈ Cl∗e ({x}) and Cl∗e ({x}) is e-I -closed, by (iii) IeKer({x}) ⊂ Cl∗e ({x}). (iv)⇒ (i): We show the implication by using Theorem 3. Let x ∈ Cl∗e ({y}). Then by Lemma 1 y ∈ IeKer({x}). By (iv), we obtain y ∈ IeKer({x}) ⊂ Cl∗e ({x}). Therefore, x ∈ Cl∗e ({y}) implies y ∈ Cl∗e ({x}). The converse is obvious and (X ,τ,I ) is an e-I -R0 space. Corollary 1. For an ideal topological space (X ,τ,I ), the following properties are equivalent: (i) (X ,τ,I ) is an e-I -R0 space, (ii) C l∗e ({x}) = IeKer({x}) for all x ∈ X . Proof. (i)⇒ (ii): Suppose that (X ,τ,I ) is an e-I -R0 space. By Theorem 4, Cl∗e ({x}) ⊂ IeKer({x}) for each x ∈ X . By Theorem 6, IeKer({x}) ⊂ Cl∗e ({x}). This shows that Cl∗e ({x}) = IeKer({x}). (ii)⇒ (i): This is obvious by Theorem 6. Corollary 2. Let (X ,τ,I ) be e-I -R0 and x ∈ X . If C l∗e ({x}) ∩ IeKer({x}) = {x}, then IeKer({x}) = {x}. Proof. The proof follows from Theorem 6 (iv). Definition 7. A net {xλ}λ∈∧ is said to be e-I -convergent to a point x in X , if for any U ∈ EIO(X , x), there exists λ0 ∈ ∧ such that xλ ∈ U for any λ ∈ ∧ such that λ ≥ λo. Lemma 4. Let (X ,τ,I ) be an ideal topological space and let x and y be any two points in X such that every net in X e-I -converging to y e-I -converges to x. Then x ∈ Cl∗e ({y}). Proof. Suppose that xn = y for each n ∈ N . Then {xn}n∈N is a net in Cl∗e ({y}). Since {xn}n∈N e-I -converges to y , then {xn}n∈N e-I -converges to x and this implies that x ∈ Cl∗e ({y}). Theorem 7. For an ideal topological space (X ,τ,I ), the following properties are equivalent: (i) (X ,τ,I ) is an e-I -R0 space, (ii) If x , y ∈ X , then y ∈ Cl∗e ({x}) if and only if every net in X e-I -converging to y e-I - converges to x. Proof. (i) ⇒ (ii): Let x , y ∈ X such that y ∈ Cl∗e ({x}). Suppose that {xα}α∈N be a net in X such that {xα}α∈N e-I -converges to y . Since y ∈ Cl∗e ({x}), by Theorem 2 we have Cl∗e ({x}) = Cl∗e ({y}). Therefore x ∈ Cl∗e ({y}). This means that {xα}α∈N e-I -converges to x . Conversely, let x , y ∈ X such that every net in X e-I -converging to y e-I converges to x . Then x ∈ Cl∗e ({y}) by Lemma 4. By Theorem 2, we have Cl∗e ({x}) = Cl∗e ({y}). Therefore W. AL-Omeri, M. Noorani, A. AL-Omari, and T. Noiri / Eur. J. Pure Appl. Math, 8 (2015), 502-513 509 y ∈ Cl∗e ({x}). (ii)⇒ (i): Assume that x and y are any two points of X such that Cl∗e ({x}) ∩ Cl∗e ({y}) 6= ;. Let z ∈ Cl∗e ({x}) ∩ Cl∗e ({y}). So there exists a net {xα}α∈N in Cl∗e ({x}) such that {xα}α∈N e-I -converges to z. Since z ∈ Cl∗e ({y}), then {xα}α∈N e-I -converges to y . It follows that y ∈ Cl∗e ({x}). By the same token we obtain x ∈ Cl∗e ({y}). Therefore Cl∗e ({x}) = Cl∗e ({y}) and by Theorem 2 (X ,τ,I ) is an e-I -R0 space. 4. On e-I -R1 Spaces Definition 8. An ideal topological space (X ,τ,I ) is said to be e-I -R1 if for x, y in X with Cl∗e ({x}) 6= Cl∗e ({y}), there exist disjoint e-I -open sets U and V such that Cl∗e ({x}) is a subset of U and Cl∗e ({y}) is a subset of V . Proposition 2. If (X ,τ,I ) is e-I -R1, then it is e-I -R0. Proof. Let U ∈ EIO(X , x). If y /∈ U , since x /∈ Cl∗e ({y}), we have Cl∗e ({x}) 6= Cl∗e ({y}). So, there exists an e-I -open set Vy such that Cl∗e ({y}) ⊂ Vy and x /∈ Vy , which implies y /∈ Cl∗e ({x}). Thus Cl∗e ({x}) ⊂ U . Therefore (X ,τ,I ) is e-I -R0. Theorem 8. An ideal topological space (X ,τ,I ) is e-I -R1 if and only if for x , y ∈ X , IeKer({x}) 6= IeKer({y}), there exist disjoint e-I -open sets U and V such that Cl∗e ({x}) ⊂ U and Cl∗e ({y}) ⊂ V . Proof. It follows from Lemma 3. Remark 3. In the following diagram we denote by arrows the implications between the separation axioms which we have introduced and discussed in this paper and examples show that no other implications hold between them: R1 // �� R0 �� e-I -R1 �� // e-I -R0 �� e-R1 // e-R0 Example 3. Let X = {a, b, c}, τ= {φ, {a}, {b}, {a, b}, {b, c}, X } and I = {φ, {b}}. EIO = {φ, {a}, {b}, {a, b}, {b, c}, X }. Then (X ,τ,I ) is e-I -R0 but not R0 and e-I -R1. Example 4. Let X = {a, b, c} with a topology τ = {φ, X , {a}, {b}, {a, b}} and I = {φ, {a}}. Since EIO = {φ, X , {a}, {b}, {a, b}, {a, c}, {b, c}}. Then (X ,τ,I ) is e-I -R0 but not R0. Example 5. Let X = {a, b, c} with a topology τ = {φ, X , {a, b}} and I = {φ, {c}}. Since EIO = {φ, X , {a}, {b}, {c}, {a, b}, {a, c}, {b, c}}. Then (X ,τ,I ) is e-I -R1 but not R1. W. AL-Omeri, M. Noorani, A. AL-Omari, and T. Noiri / Eur. J. Pure Appl. Math, 8 (2015), 502-513 510 Example 6. Let X = {a, b, c} with a topology τ= {φ, X , {a}, {a, b}} and I = {φ, {a}, {b}, {a, b}}. Since EIO = {φ, X , {a}{a, b}, {a, c}} and e-open sets is {φ, X , {a}, {b}, {c}, {a, b}, {a, c}, {b, c}}. Then (X ,τ,I ) is e-R0 but not e-I -Ro. Theorem 9. The following properties are equivalent: (i) (X ,τ,I ) is e-I -R1, (ii) for each x , y ∈ X one of the following holds: • If U is e-I -open, then x ∈ U if and only if y ∈ U, • there exist disjoint e-I -open sets U and V such that x ∈ U and y ∈ V . (iii) If x , y ∈ X such that Cl∗e ({x}) 6= Cl∗e ({y}), then there exist e-I -closed sets F1 and F2 such that x ∈ F1, y /∈ F1, y ∈ F2, x /∈ F2, and X = F1 ∪ F2. Proof. (i) ⇒ (ii): Let x , y ∈ X . Then Cl∗e ({x}) = Cl∗e ({y}) or Cl∗e ({x}) 6= Cl∗e ({y}). If Cl∗e ({x}) = Cl∗e ({y}) and U is e-I -open, then x ∈ U implies y ∈ Cl∗e ({x}) ⊂ U and y ∈ U implies x ∈ Cl∗e ({y}) ⊂ U . Thus consider the case that Cl∗e ({x}) 6= Cl∗e ({y}). Then there exist disjoint e-I -open sets U and V such that x ∈ Cl∗e ({x}) ⊂ U and y ∈ Cl∗e ({y}) ⊂ V . (ii)⇒ (iii): Let x , y ∈ X such that Cl∗e ({x}) 6= Cl∗e ({y}). Then x /∈ Cl∗e ({y}) or y /∈ Cl∗e ({x}), say x /∈ Cl∗e ({y}). Then there exists an e-I -open set A such that x ∈ A and y /∈ A, which implies there exist disjoint e-I -open sets U and V such that x ∈ U and y ∈ V . Then F1 = X\V and F2 = X\U are e-I -closed sets such that x ∈ F1, y /∈ F1, y ∈ F2, x /∈ F2, and X = F1 ∪ F2. (iii)⇒ (i): First, we show that (X ,τ,I ) is e-I -R0. Let U be e-I -open and let x ∈ U . Suppose that Cl∗e ({x}) 6⊂ U . Let y ∈ Cl∗e ({x}) ∩ (X\U). Then Cl∗e ({x}) 6= Cl∗e ({y}) and there exist F1, F2 ∈ EI C(X ) such that x ∈ F1, y ∈ F2, y /∈ F1, x /∈ F2, and X = F1 ∪ F2. Then y ∈ F2\F1 = X\F1, which is e-I -open, and x /∈ X\F1, which is a contradiction. Hence, (X ,τ,I ) is e-I -R0. To show X to be e-I -R1 assume that a, b ∈ X such that Cl∗e ({a}) 6= Cl∗e ({b}). Then there exist P1, P2 ∈ EI C(X ) such that a ∈ P1, b /∈ P1, a /∈ P2, b ∈ P2 and X = P1 ∪ P2. Thus a ∈ P1\P2 and b ∈ P2\P1, which are e-I -open. This implies Cl∗e ({a}) ⊂ P1\P2 = X − P2 ∈ EIO(X ) and Cl∗e ({b}) ⊂ P2\P1. Thus, (X ,τ,I ) is e-I -R1. Theorem 10. The following properties are equivalent: (i) (X ,τ,I ) is e-I -T2, (ii) (X ,τ,I ) is e-I -R1 and e-I -T1, (iii) (X ,τ,I ) is e-I -R1 and e-I -T0. Proof. (i) ⇒ (ii): Since (X ,τ,I ) is e-I -T1, then it is e-I -T1. If x , y ∈ X such that Cl∗e ({x}) 6= Cl∗e ({y}), then x 6= y and there exist disjoint e-I -open sets U and V such that x ∈ U and y ∈ V . Therefore, Cl∗e ({x}) = {x} ⊂ U and Cl∗e ({y}) = {y} ⊂ V . Hence (X ,τ,I ) is e-I -R1. (ii)⇒ (iii): Since (X ,τ,I ) is e-I -T1, then (X ,τ,I ) is e-I -T0. W. AL-Omeri, M. Noorani, A. AL-Omari, and T. Noiri / Eur. J. Pure Appl. Math, 8 (2015), 502-513 511 (iii) ⇒ (i): Since (X ,τ,I ) is e-I -R1, then (X ,τ,I ) is e-I -R0 and e-I -T0 and hence by Theorem 1 (X ,τ,I ) is e-I -T1. Let x , y ∈ X such that x 6= y . Since Cl∗e ({x}) = {x} 6= {y} = Cl∗e ({y}), then there exist disjoint e-I -open sets U and V such that x ∈ U and y ∈ V . Hence, (X ,τ,I ) is e-I -T2. In view of Definition 3, it follows that Theorem 11. An ideal topological space (X ,τ,I ) is e-I -T2 if and only if for x , y ∈ X such that x 6= y, there exist e-I -closed sets F1 and F2 such that x ∈ F1, y /∈ F1, y ∈ F2, x /∈ F2, and X = F1 ∪ F2. Remark 4. Let {xλ}λ∈λ be a net in (X ,τ,I ) and eI l im({xλ}λ∈λ) denote {x ∈ X : I − converges to x}. Theorem 12. The following properties are equivalent: (i) (X ,τ,I ) is e-I -R1, (ii) for x , y ∈ X Cl∗e ({x}) = Cl∗e ({y}), whenever there exists a net {xλ}λ∈A such that x , y ∈ eI l im({xλ}λ∈A), (iii) (X ,τ,I ) is e-I -R0, and for every e-I -convergent net {xλ}λ∈A in X , eI l im({xλ}λ∈A) = Cl∗e ({x}) for some x ∈ X . Proof. (i)⇒ (ii): Let x , y ∈ X such that there exists a net {xλ}λ∈A in X such that x , y ∈ eI l im({xλ}λ∈A). Then, by Theorem 9, (a) if U is e-I -open, then x ∈ U if and only if y ∈ U or (b) there exist disjoint e-I -open sets U and V such that x ∈ U and y ∈ V . Since x , y ∈ eI l im({xλ}λ∈A), then (a) is satisfied, and we obtain Cl∗e ({x}) = Cl∗e ({y}). (ii)⇒ (iii): Let U ∈ EIO(X , x). Let y /∈ U . For each n ∈ N let xn = x . Then {xn}n∈N e-I - converges to x and since Cl∗e ({x}) 6= Cl∗e ({y}), by (ii) {xn} does not e-I -converge to y and there exists A ∈ EIO(X ) such that y ∈ A and x /∈ A. Thus, y /∈ Cl∗e ({x}) and Cl∗e ({x}) ⊂ U . Hence (X ,τ,I ) is e-I -R0. Let {xλ}λ∈A be an e-I -convergent net in X . Let x ∈ X such that {xλ}λ∈A e-I -converges to x . If y ∈ Cl∗e ({x}), then {xλ}λ∈A e-I -converges to y , which implies Cl∗e ({x}) ⊂ eI l im({xλ}λ∈A). Let y ∈ eI l im({xλ}λ∈A), then x , y ∈ eI l im({xλ}λ∈A), which implies y ∈ Cl∗e ({y}) = Cl∗e ({x}). Hence eI l im({xλ}λ∈A) = Cl∗e ({x}). (iii)⇒ (i): Assume that (X ,τ,I ) is not e-I -R1. Then there exist x , y ∈ X such that Cl∗e ({x}) 6= Cl∗e ({y}) and every e-I -open set containing Cl∗e ({x}) intersects every e-I -open set containing Cl∗e ({y}). Since (X ,τ,I ) is e-I -R0, then every e-I -open set containing x contains Cl∗e ({x}) and every e-I -open set containing y contains Cl∗e ({y}), which implies that every e-I -open set containing x intersects every e-I -open set containing y . Let Dx = {U ⊂ X |U ∈ EIO(X , x)}. Let≥x be the binary relation on Dx defined by U1 ≥x U2 if and only if U1 ⊂ U2. Then, clearly (Dx ,≥x) is a directed set. Let Dy = {U ⊂ X |U ∈ EIO(X , y)} and let ≥y be the binary relation on Dy defined by U1 ≥y U2 if and only if U1 ⊂ U2. Then, (Dx ,≥y) is also a directed set. Let D = {(U1, U2)|U1 ∈ Dx and U2 ∈ Dy} and let≥ be the binary relation on D defined by (U1, U2)≥ (V1, V2) if and only if U1 ≥x V1 and U2 ≥y V2. Then, (D,≥) is a directed set. For each (U1, U2) ∈ D, let x(U1,U2) ∈ (U1, U2). REFERENCES 512 Then {x(U1,U2) }(U1,U2) ∈ D is a net in X that e-I -converges to both x and y . Thus, there exists z ∈ X such that eI l im({x(U1,U2) }(U1,U2)∈D) = Cl∗e ({z}), which implies x , y ∈ Cl∗e ({z}). Since {Cl∗e ({w}) : w ∈ X } is a decomposition of X , then Cl∗e ({x}) = Cl∗e ({z}) = Cl∗e ({y}), which is a contradiction. Hence (X ,τ,I ) is e-I -R1. ACKNOWLEDGEMENTS The authors would like to acknowledge the grant: UKM Grant DIP- 2014-034 and Ministry of Education, Malaysia grant FRGS/1/2014/ST06/UKM/01/1 for fi- nancial support. References [1] W. Al-Omeri, M. Noorani, and A. Al-Omari. New forms of contra-continuity in ideal topology spaces. Missouri Journal of Mathematical Sciences, 26(1):33–47, 2014. [2] W. Al-Omeri, M. Noorani, and A. Al-Omari. on e-I -open sets, e-I -continuoues functions and decomposition of continuity. Journal of Mathematics and Applications, (38):15–31, 2014. [3] F. G. Arenas, J. Dontchev, and M. L. Puertas. Idealization of some weak separation axioms. Acta Mathematica Hungarica, 89(1):47–53, 2000. [4] A. S. Davis. Indexed systems of neighborhoods for general topological spaces. American Mathematical Society, 68:886–893, 1961. [5] J. Dontchev. Strong B-sets and another decomposition of continuity. Acta Mathematica Hungarica, 75:259–265, 1997. [6] K. K. Dube. A note on R0 topological spaces. Matematicki vesnik, 11:203–208, 1974. [7] E. Ekici. Some generalizations of almost contra-super-continuity. Filomat, 21(2):31–44, 2007. [8] E. Ekici. New forms of contra-continuity. Carpathian Journal Mathmatics, 24(1):37–45, 2008. [9] E. Ekici. On e-open sets, DP∗-sets and DPE∗ and decompositions of continuity. Arabian Journal for Science and Engineering, 33(2A):269–282, 2008. [10] D. W. Hall, S. K. Murphy, and J. Rozycki. On spaces which are essentially T1. Journal of the Australian Mathematical Society, 12:451–455, 1971. [11] D. Jankovic and T. R. Hamlett. New topologies from old via ideals. American Mathemat- ical Monthly, 97(4):295–310, 1990. [12] M. N. Mukherjee, R. Bishwambhar, and R. Sen. On extension of topological spaces in terms of ideals. Topology and its Applications, 154(18):3167–3172, 2007. REFERENCES 513 [13] S. A. Naimpally. On R0-topological spaces. Annales Universitatis Scientiarum Budapesti- nensis, Sectio Mathematica, 10:53–54, 1967. [14] A. A. Nasef and R. A. Mahmoud. Some applications via fuzzy ideals. Chaos, Solitons and Fractals, 13(4):825–831, 2002. [15] N. A. Shanin. On separation in topological spaces. Doklady Akademii Nauk SSSR, 38:110– 113, 1943. [16] M. H. Stone. Application of the theory of boolean rings to general topology. Transactions of the American Mathematical Society, 41:375–481, 1937. [17] R. Vaidyanathaswamy. The localization theory in set-topology. Proceedings of the Indian Academy of Science, 20:51–61, 1945. [18] N. V. Veliĉko. H-closed topological spaces. Transactions of the American Mathematical Society, 78(2):103–118, 1968.