@ Appl. Gen. Topol. 15, no. 1 (2014), 33-42doi:10.4995/agt.2014.2126 c© AGT, UPV, 2014 On topological groups via a-local functions Wadei Al-Omeri a, Mohd. Salmi Md. Noorani a and Ahmad. Al-Omari b a Department of Mathematics, Faculty of Science and Technology Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor DE, Malaysia (wadeimoon1@hotmail.com,msn@ukm.my) b Department of Mathematics, Faculty of Science Al AL-Bayat University, P.O.Box 130095, Mafraq 25113, Jordan (omarimutah1@yahoo.com) Abstract An ideal on a set X is a nonempty collection of subsets of X which sat- isfies the following conditions (1)A ∈ I and B ⊂ A implies B ∈ I; (2) A ∈ I and B ∈ I implies A ∪ B ∈ I. Given a topological space (X, τ) an ideal I on X and A ⊂ X, ℜa(A) is defined as ∪{U ∈ τ a : U −A ∈ I}, where the family of all a-open sets of X forms a topology [5, 6], denoted by τa. A topology, denoted τa ∗ , finer than τa is generated by the basis β(I, τ) = {V − I : V ∈ τa(x), I ∈ I}, and a topology, denoted 〈ℜa(τ)〉 coarser than τa is generated by the basis ℜa(τ) = {ℜa(U) : U ∈ τ a}. In this paper A bijection f : (X, τ, I) → (X, σ, J ) is called a A∗- homeomorphism if f : (X, τa ∗ ) → (Y, σa ∗ ) is a homeomorphism, ℜa- homeomorphism if f : (X, ℜa(τ)) → (Y, ℜa(σ)) is a homeomorphism. Properties preserved by A∗-homeomorphism are studied as well as nec- essary and sufficient conditions for a ℜa-homeomorphism to be a A∗- homeomorphism. 2010 MSC: 54A05; 54C10. Keywords: ℜa-homeomorphism; topological groups; a-local function; ideal spaces; ℜa-operator; A∗-homeomorphism. 1. Introduction and Preliminaries Ideals in topological spaces have been considered since 1930. The sub- ject of ideals in topological spaces has been studied by Kuratowski [11] and Received 22 March 2013 – Accepted 28 January 2014 http://dx.doi.org/10.4995/agt.2014.2126 W. Al-Omeri, Mohd. Salmi Md. Noorani and A. Al-omari Vaidyanathaswamy [18]. Jankovic and Hamlett [10] investigated further prop- erties of ideal space. In this paper, we investigate a-local functions and its properties in ideals in topological space [1]. Also, the relationships among local functions such as local function [19, 10] and semi-local function [7] are investigated. A subset of a space (X, τ) is said to be regular open (resp. regular closed) [12] if A = Int(Cl(A)) (resp. A = Cl(Int(A))). A is called δ-open [20] if for each x ∈ A, there exists a regular open set G such that x ∈ G ⊂ A. The complement of δ-open set is called δ-closed. A point x ∈ X is called a δ-cluster point of A if int(Cl(U)) ∩ A 6= φ for each open set V containing x. The set of all δ-cluster points of A is called the δ-closure of A and is denoted by Clδ(A) [20]. The δ-interior of A is the union of all regular open sets of X contained in A and its denoted by Intδ(A) [20]. A is δ-open if Intδ(A) = A. δ-open sets forms a topology τδ. A subset A of a space (X, τ) is said to be a-open (resp. a-closed) [5] if A ⊂ Int(Cl(Intδ(A))) (resp. Cl(Int(Clδ(A))) ⊂ A, or A ⊂ Int(Cl(Intδ(A))) (resp. Cl(Int(Clδ(A))) ⊂ A. The family of a-open sets of X forms a topology, denoted by τa [6]. The intersection of all a-closed sets contained A is called the a-closure of A and is denoted by aCl(A). The a-interior of A, denoted by aInt(A), is defined by the union of all a-open sets contained in A [5]. An ideal I on a topological space (X, I) is a nonempty collection of subsets of X which satisfies the following conditions: (1) A ∈ I and B ⊂ A implies B ∈ I; (2) A ∈ I and B ∈ I implies A ∪ B ∈ I. Applications to various fields were further investigated by Jankovic and Hamlett [10] Dontchev et al. [4]; Mukherjee et al. [13]; Arenas et al. [3]; Navaneethakrishnan et al. [14]; Nasef and Mahmoud [15], 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, 10] 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}. A Kuratowski closure operator Cl∗(.) for a topology τ∗(τ, I), called the ∗-topology, which is finer than τ is defined by Cl∗(A) = A ∪ A∗(τ, I), when there is no chance of confusion. A∗(I) is denoted by A∗ and τ∗ for τ∗(I, τ). X∗ is often a proper subset of X. The hypothesis X = X∗ [7] is equivalent to the hypothesis τ ∩ I = φ. If I is an ideal on X, then (X, τ, I) is called an ideal space. N is the ideal of all nowhere dense subsets in (X, τ). A subset A of an ideal space (X, τ, I) is ⋆-closed [4] (resp. ⋆-dense in itself [7]) if A∗ ⊆ A (resp A ⊆ A∗). A subset A of an ideal space (X, τ, I) is Ig−closed [20] if A ∗ ⊆ U whenever A ⊆ U and U is open. For every ideal topological space there exists a topology τ∗(I) finer than τ generated by β(I, τ) = {U − A | U ∈ τ and A ∈ I}, but in general β(I, τ) is not always topology [10]. Let (X, I, τ) ba an ideal topological space. We say that the topology τ is compatible with the I, denoted τ ∼ I, if the following holds for c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 34 On topological groups via a-local functions every A ⊂ X, if for every x ∈ A there exists a U ∈ τ such that U ∩ A ∈ I, then A ∈ I. Given a space (X, τ, I), (Y, σ, J ), and a function f : (X, τ, I) → (Y, τ, J ), we call f a ∗-homomorphism with respect to τ, I, σ, and J if f : (X, τ∗) → (Y, σ∗) is a homomorphism, where a homomorphism is a continuous injective function between two topological spaces, that is invertible with continuous inverse. We first prove some preliminary lemmas which lead to a theorem extending the theorem in [17] and apply the theorem to topological groups. Quite recently, in [2], the present authors defined and investigated the notions ℜa : ℘(X) → τ as follows, ℜa(A) = {x ∈ X : there exists Ux ∈ τ a containing x such that Ux −A ∈ I}, for every A ∈ ℘(X). In [16], Newcomb defined A = B[mod I] if (A−B)∪(B −A) ∈ I and observe that = [mod I] is an equivalence relation. In this paper a bijection f : (X, τ, I) → (X, σ, J ) is called a A∗-homeomorphism if f : (X, τa ∗ ) → (Y, σa ∗ ) is a homeomorphism, ℜa-homeomorphism if f : (X, ℜa(τ)) → (Y, ℜa(σ)) is a homeomorphism. Properties preserved by A∗- homeomorphism are studied as well as necessary and sufficient conditions for a ℜa-homeomorphism to be a A∗-homeomorphism. 2. a-Local Function and ℜa- operator Let (X, τ, I) an ideal topological space and A a subset of X. Then Aa ∗ (I, τ) = {x ∈ X : U ∩ A /∈ I, for every U ∈ τa(x)} is called a-local function of A [1] with respect to I and τ, where τa(x) = {U ∈ τa : x ∈ U}. We denote simply Aa ∗ for Aa ∗ (I, τ). Remark 2.1 ([1]). (1) The minimal ideal is considered {∅} in any topological space (X, τ) and the maximal ideal is considered P(X). It can be deduced that Aa ∗ ({∅}) = Cla(A) 6= Cl(A) and A a ∗ (P(X)) = ∅ for every A ⊂ X. (2) If A ∈ I, then Aa ∗ = ∅. (3) A * Aa ∗ and Aa ∗ * A in general. Theorem 2.2 ([1]). Let (X, τ, I) an ideal in topological space and A, B subsets of X. Then for a-local functions the following properties hold: (1) If A ⊂ B, then Aa ∗ ⊂ Ba ∗ , (2) For another ideal J ⊃ I on X, Aa ∗ (J) ⊂ Aa ∗ (I), (3) Aa ∗ ⊂ aCl(A), (4) Aa ∗ (I) = aCl(Aa ∗ ) ⊂ aCl(A) (i.e Aa ∗ is an a-closed subset of aCl(A)), (5) (Aa ∗ )a ∗ ⊂ Aa ∗ , (6) (A ∪ B)a ∗ = Aa ∗ ∪ Ba ∗ , (7) Aa ∗ − Ba ∗ = (A − B)a ∗ − Ba ∗ ⊂ (A − B)a ∗ , (8) If U ∈ τa, then U ∩ Aa ∗ = U ∩ (U ∩ A)a ∗ ⊂ (U ∩ A)a ∗ , (9) If U ∈ τa, then (A − U)a ∗ = Aa ∗ = (A ∪ U)a ∗ , (10) If A ⊆ Aa ∗ , then Aa ∗ (I) = aCl(Aa ∗ ) = aCl(A). c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 35 W. Al-Omeri, Mohd. Salmi Md. Noorani and A. Al-omari Theorem 2.3 ([1]). Let (X, τ, I) an ideal in topological space and A, B subsets of X.Then for a-local functions the following properties hold: (1) τa ∩ I = φ; (2) If I ∈ I, then aInt(I) = φ; (3) For every G ∈ τa, then G ⊆ Ga ∗ ; (4) X = Xa ∗ . Theorem 2.4 ([1]). Let (X, τ, I) be an ideal topological space and A subset of X. Then the following are equivalent: (1) I ∼a τ, (2) If a subset A of X has a cover a-open of sets whose intersection with A is in I, then A is in I, in other words Aa ∗ = φ, then A ∈ I, (3) For every A ⊂ X, if A ∩ Aa ∗ = φ, A ∈ I, (4) For every A ⊂ X, A − Aa ∗ ∈ I, (5) For every A ⊂ X, if A contains no nonempty subset B with B ⊂ Ba ∗ , then A ∈ I. Theorem 2.5 ([1]). Let (X, I, τ) be an ideal topological space. Then β(I, τ) is a basis for τa ∗ . β(I, τ) = {V − Ii : V ∈ τ a(x), Ii ∈ I} and β is not, in general, a topology. Theorem 2.6 ([2]). Let (X, τ, I) be an ideal topological space. Then the fol- lowing properties hold: (1) If A ⊂ X, then ℜa(A) is a-open. (2) If A ⊂ B, then ℜa(A) ⊆ ℜa(B). (3) If A, B ∈ ℘(X), then ℜa(A ∪ B) ⊂ ℜa(A) ∪ ℜa(B). (4) If A, B ∈ ℘(X), then ℜa(A ∩ B) = ℜa(A) ∩ ℜa(B). (5) If U ∈ τa ∗ , then U ⊆ ℜa(U). (6) If A ⊂ X, then ℜa(A) ⊆ ℜa(ℜa(A)). (7) If A ⊂ X, then ℜa(A) = ℜa(ℜa(A)) if and only if (X − A)a ∗ = ((X − A)a ∗ )a ∗ . (8) If A ∈ I, then ℜa(A) = X − X a ∗ . (9) If A ⊂ X, then A ∩ ℜa(A) = Int a ∗ (A), where Inta ∗ is the interior of τa ∗ . (10) If A ⊂ X, I ∈ I, then ℜa(A − I) = ℜa(A). (11) If A ⊂ X, I ∈ I, then ℜa(A ∪ I) = ℜa(A). (12) If (A − B) ∪ (B − A) ∈ I, then ℜa(A) = ℜa(B). Theorem 2.7 ([1]). Let (X, τ, I) be an ideal topological space and A subset of X. If τ is a-compatible with I. Then the following are equivalent: (1) For every A ⊂ X, if A ∩ Aa ∗ = φ implies Aa ∗ = φ, (2) For every A ⊂ X, (A − Aa ∗ )a ∗ = φ, (3) For every A ⊂ X,(A ∩ Aa ∗ )a ∗ = Aa ∗ . Theorem 2.8 ([2]). Let (X, τ, I) be an ideal topological space with τ ∼a I. Then ℜa(A) = ∪{ℜa(U) : U ∈ τ a, ℜa(U) − A ∈ I}. c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 36 On topological groups via a-local functions Proposition 2.9 ([2]). Let (X, τ, I) be an ideal topological space with τa ∩I = φ.Then the following are equivalent: (1) A ∈ U(X, τ, I), (2) ℜa(A) ∩ aInt(A a ∗ ) 6= φ, (3) ℜa(A) ∩ A a ∗ 6= φ, (4) ℜa(A) 6= φ, (5) Inta ∗ (A) 6= φ, (6) There exists N ∈ τa − {∅} such that N − A ∈ I and N ∩ A /∈ I. Proposition 2.10 ([2]). Let (X, τ, I) be an ideal topological space. Then τ ∼a I, A ⊆ X. If N is a nonempty a-open subset of Aa ∗ ∩ ℜa(A), then N − A ∈ I and N ∩ A /∈ I. Theorem 2.11 ([2]). Let (X, τ, I) be an ideal topological space. Then τ ∼a I if and only if ℜa(A) − A ∈ I for every A ⊆ X. 3. A∗-homeomorphism Given an ideal topological space (X, τ, I) a topology τa finer than 〈ℜa(τ)〉 which 〈ℜa(τ)〉 is generated by the basis ℜa(τ) = {ℜa(U) : U ∈ τ a}. Definition 3.1 ([5]). A function f : (X, τ) → (X, σ) is called (1) a-continuous if the inverse image of a-open set is a-open. (2) a-open if the image of a-open set is a-open. Definition 3.2. Let (X, τ, I) and (X, σ, J ) be an ideal topological space. A bijection f : (X, τ, I) → (X, σ, J ) is called (1) A∗-homeomorphism if f : (X, τa ∗ ) → (Y, σa ∗ ) is a homeomorphism. (2) ℜa-homeomorphism if f : (X, ℜa(τ)) → (Y, ℜa(σ)) is a homeomor- phism. Theorem 3.3. Let (X, τ, I) and (X, σ, J ) be an ideal topological space with f : (X, ℜa(τ)) → (X, σ, J ) an a-open bijective, τ ∼ a I and f(I) ⊆ J . Then f(ℜa(A)) ⊆ ℜa(f(A)) for every A ⊆ X. Proof. Let A ⊆ X and let y ∈ f(ℜa(A)). Then f −1(y) ∈ ℜa(A) and there exists U ∈ τa such that f−1(y) ∈ ℜa(U) and ℜa(U) − A ∈ I by Theorem 2.8. Now f(ℜa(U)) ∈ σ a(y) and f(ℜa(U)) − f(A) = f[ℜa(U) − A] ∈ f(I) ⊆ J . Thus y ∈ ℜa(f(A), and the proof is complete. � Theorem 3.4. Let (X, τ, I) and (X, σ, J ) be an ideal topological space with f : (X, τ) → (X, ℜa(σ)) is a-continuous injection, σ ∼ a J and f−1(J ) ⊆ I. Then ℜa(f(A)) ⊆ f(ℜa(A)) for every A ⊆ X. Proof. Let y ∈ ℜa(f(A)) where A ⊆ X. Then by Theorem 2.8, there exists U ∈ σa such that y ∈ ℜa(U) and ℜa(U) − f(A) ∈ J . Now we have f −1(ℜa(U)) ∈ τa(f−1(y)) with f−1[ℜa(U)−f(A)] ∈ I then f −1[ℜa(U)]−A ∈ I and f −1(y) ∈ ℜa(A) and hence y ∈ f(ℜa(A)), and the proof is complete. � c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 37 W. Al-Omeri, Mohd. Salmi Md. Noorani and A. Al-omari Theorem 3.5. Let (X, τ, I) and (X, σ, J ) be a bijective with f(I) = J . Then the following properties are equivalent: (1) f is A∗-homeomorphism; (2) f(Aa ∗ ) = [f(A)]a ∗ for every A ⊆ X; (3) f(ℜa(A)) = ℜa(f(A)) for every A ⊆ X; Proof. (1) ⇒ (2) Let A ⊆ X. Assume y /∈ f(Aa ∗ ). This implies that f−1(y) /∈ Aa ∗ , and hence there exists U ∈ τa(f−1(y)) such that U ∩ A ∈ I. Conse- quently f(U) ∈ σa ∗ (y) and f(U) ∩ f(A) ∈ J , then y /∈ [f(A)]a ∗ (J , σa ∗ ) = [f(A)]a ∗ (J , σ). Thus [f(A)]a ∗ ⊆ f(Aa ∗ ). Now assume y /∈ [f(A)]a ∗ . This implies there exists a V ∈ σa ∗ (y) such that V ∩ f(A) ∈ J , then f−1(V ) ∈ τa ∗ (f−1(y)) and f−1(V ) ∩ A ∈ I. Thus f−1(y) /∈ Aa ∗ (I, τa ∗ ) = Aa ∗ (I, τa) and y /∈ f(Aa ∗ ). Hence f(Aa ∗ ) ⊆ [f(A)]a ∗ and f(Aa ∗ ) = [f(A)]a ∗ . (2) ⇒ (3) Let A ⊆ X. Then f(ℜa(A)) = f[X −(X −A) a ∗ ] = Y −f(X −A)a ∗ = Y − [Y − f(A)]a ∗ = ℜa(f(A)). (3) ⇒ (1) Let U ∈ τa ∗ . Then U ⊆ ℜa(U) by Theorem 2.6 and f(U) ⊆ f(ℜa(U)) = ℜa(f(U)). This shows that f(U) ∈ σ a ∗ and hence f : (X, τa ∗ ) → (Y, σa ∗ ) is τa ∗ -open. Similarly, f−1 : (Y, σa ∗ ) → (X, τa ∗ ) is σa ∗ -open and, f is A∗-homeomorphism. � Theorem 3.6. Let (X, τ, I) be an ideal topological space, then 〈ℜa(τ a ∗ )〉 = 〈ℜa(τ a)〉. Proof. Note that for every U ∈ τa and for every I ∈ I, we have ℜa(U − I) = ℜa(U). Consequently, ℜa(β) = ℜa(τ a) and 〈ℜa(β)〉 = 〈ℜa(τ a)〉, where β is a basis for τa. It follows directly from Theorem 11 of [9] that 〈ℜa(β)〉 = 〈ℜa(τ a ∗ )〉, hence the theorem is proved. � Theorem 3.7. Let f : (X, τ, I) → (Y, σ, J ) be a bijection with f(I) = J . Then the following are hold: (1) If f is a A∗-homeomorphism, then f is a ℜa-homeomorphism. (2) If τ ∼a I and σ ∼a J and f is a ℜa-homeomorphism, then f is a A∗-homeomorphism. Proof. (1) Assume f : (X, τa ∗ ) → (Y, σa ∗ ) is a A∗-homeomorphism, and let ℜa(U) be a basic open set in 〈ℜa(τ a)〉 with U ∈ τa. Then f(ℜa(U)) = ℜa(f(U)) by Theorem 3.5. Then f(ℜa(U)) ∈ ℜa(σ a ∗ ), but 〈ℜa(τ a ∗ )〉 = 〈ℜa(τ a)〉 by Theorem 3.6. Thus f : (X, ℜa(τ)) → (Y, ℜa(σ)) is a-open. Simi- larly, f−1 : (Y, ℜa(σ)) → (X, ℜa(τ)) is a-open and f is ℜa-homeomorphism. (2) Assume f is a ℜa-homeomorphism, then f(ℜa(A)) = ℜa(f(A)) for every A ⊆ X by Theorems 3.4 and 3.3. Thus f is a A∗-homeomorphism by Theorem 3.5. � 4. Results related to topological groups Given a topological group (X, τ, .) and an ideal I on X, denoted (X, τ, I, .) and x ∈ X, we denote by xI = {xI : I ∈ I}. We say that I is left translation c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 38 On topological groups via a-local functions invariant if for every x ∈ X we have xI ⊆ I. Observe that if I is left translation invariant then xI = I for every x ∈ X. We define I to be right translation invariant if and only if Ix = I for every x ∈ X [8]. Given a topological group (X, τ, I), I is said to be τa-boundary [2], if τa ∩I = {φ}. Note that if I is left or right translation invariant, X /∈ I, and I ∼a I, then I is τa-boundary. Definition 4.1 ([2]). Let (X, τ, I) be an ideal topological space. A subset A of X is called a Baire set with respect to τa and I, denoted A ∈ Br(X, τ, I), if there exists a a-open set U such that A = U [mod I]. Let U(X, τ, I) be denoted {A ⊆ X : there exists B ∈ Br(X, τ, I) − I such that B ⊆ A}. Lemma 4.2. Let (X, τ) and (X, σ) be two topological spaces and F be a col- lection of a-open mappings from X to Y . Let U ∈ τa − {φ} and let A be a non empty subset of U. If f(U) ∈ F(A) = {f(A) : f ∈ F} for every f ∈ F, Then F(A) ∈ σa − {φ}. Proof. Let y ∈ F(A), then there exist f ∈ F such that y ∈ f(A). Now, A ⊆ U, then f(A) ⊆ f(U) and y ∈ f(U). Then f(U) is a-open in (Y, σ) (as f is a- open map). So there exists V ∈ σa(y) such that y ∈ V ⊆ f(U) ⊆ F(A). So F(A) ∈ σa − {φ}. � Theorem 4.3. Let (X, τ) and (X, σ) be two topological spaces and I be an ideal (X, τ) with τ ∼a I and τa ∩ I = {φ}. Moreover, let U ∈ τa − {φ}, A ⊆ X, U ⊆ Aa∗ ∩ ℜa(A) and F be a non-empty collection of a-open mappings from X to Y . Suppose y ∈ F(U) ⇒ U ∩ F−1(y) /∈ I, where F−1(y) = ∪{f−1(y) : f ∈ F}. Then F(U ∩ A) ∈ σa − {φ}. Proof. Since U is a non-empty a-open set contained in Aa∗∩ℜa(A) and τ ∼ a I, by Proposition 2.10 it follows that U −A ∈ I and U ∩A /∈ I. For any y ∈ F(U), U ∩ F−1(y) /∈ I (by hypothesis) and we have U ∩ F−1(y) = U ∩ F−1(y) ∩ (A ∪ Ac) = [U ∩ F−1(y) ∩ A] ∪ [U ∩ F−1(y) ∩ Ac] ⊆ [U ∪ F−1(y) ∩ A] ∪ (U − A) (where Ac = complement of A). Since U ∩ F−1(y) /∈ I and U − A ∈ I, we have U ∩ F−1(y) ∩ A /∈ I. Then for any y ∈ F(U), U ∩ F−1(y) ∩ A 6= {φ}. Now for a given f ∈ F, k ∈ f(U) ⇒ k ∈ F(U), then there exist x ∈ U ∩ A and x ∈ g−1(k) for some g ∈ F, where k = g(x) ⇒ k ∈ g(U ∩A), and k ∈ F(U ∩A). Hence f(U) ⊆ F(U ∩ A), for all f ∈ F. Then F(U ∩ A) ∈ σa − {φ} by Lemma 4.2. � Lemma 4.4. Let I be a left (right) translation invariant ideal on a topological group (X, τ, .) and x ∈ X. Then for any A ⊆ X the following hold: (1) xℜa(A) = ℜa(xA), and ℜa(A)x = ℜa(Ax), (2) xAa ∗ = (xA)a ∗ (resp.Aa ∗ x = (Ax)a ∗ ). c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 39 W. Al-Omeri, Mohd. Salmi Md. Noorani and A. Al-omari Proof. We assume that I is right translation invariant, the proof is similar for the case when I is left translation invariant would be . (1) We first note that for any two subsets A and B of X, (A− B)x = Ax− Bx. In fact, y ∈ (A − B)x, then y = tx, for some t ∈ A − B. Now t ∈ A then tx ∈ Ax. But tx ∈ Bx ⇒ tx = bx for some b ∈ B ⇒ t = b ∈ B a contradiction. So y = tx ∈ Ax−Bx. Again, y ∈ Ax−Bx ⇒ y ∈ Ax and y /∈ Bx ⇒ y = ax for some a ∈ A and ax /∈ Bx ⇒ a /∈ B ⇒ y = ax, where a ∈ A−B ⇒ y ∈ (A−B)x. Now, y ∈ ℜa(Ax) ⇒ y ∈ Ux for some U ∈ τ a with U − A ∈ I. Then Ux = V ∈ τa and (U − A)x = Ux − Ax ∈ I where Ux ∈ τa. Then y ∈ V , where V ∈ τa and V − Ax ∈ I ⇒ y ∈ ∪{V ∈ τa : V − Ax ∈ I} = ℜa(Ax). Thus xℜa(A) ⊆ ℜa(Ax). Conversely, let y ∈ ℜa(Ax) = ∪{U ∈ τ a : U − Ax ∈ I} ⇒ y ∈ U ∈ τa, where U − Ax ∈ I. Put V = Ux−1. Then V ∈ τa. Now yx−1 ∈ V and V −A = Ux−1 −A = (U −Ax)x−1 ∈ I ⇒ yx−1 ∈ ℜa(A) ⇒ y ∈ ℜa(A)x. Thus ℜa(Ax) ⊆ ℜa(A)x and hence ℜa(A)x = ℜa(Ax) (2) In view of (1) ℜa(X −A)x = ℜa((X −A)x), then [X −A a ∗ ]x = X −(Ax)a ∗ and X − Aa ∗ x = X − (Ax)a ∗ thus Aa ∗ x = (Ax)a ∗ . � Lemma 4.5. Let I be an ideal space on a topological group (X, τ, .) such that I is left or right translation invariant and τ ∼a I. Then I ∩ τa = {φ}. Proof. Since X /∈ I and τ ∼a I, by Theorem 2.4 there exist x ∈ X such that for all U ∈ τa(x), (4.1) U = U ∩ X /∈ I Let V ∈ I ∩ τa. If V = {φ} we have nothing to show. Suppose V 6= {φ}. Without loosing of generality we may assume that i ∈ V (i denoted the identity of X). For y ∈ V then y −1 V ∈ τa and y−1V ∈ y−1I so that y−1V ∈ I where i ∈ y−1V . Thus xV ∈ τa and xV ∈ xI and hence xV ∈ I. Thus xV ∈ τa ∩ I, where xV is a neighborhood of x, which is contradicting (4.1) and hence I ∩ τa = {φ}. � Theorem 4.6. Let (X, τ, .) be a topological group and I be an ideal on X such that τ ∼a I. Let P ∈ U(X, τ, I) and S ∈ P(X) − I. Let U, V ∈ τa such that U ∩ Sa ∗ 6= {φ}, V ∩ aInt(P a ∗ ) ∩ ℜa(P) 6= {φ}. If A = U ∩ S ∩ S a ∗ and B = V ∩ aInt(P a ∗ ) ∩ P ∩ ℜa(P) then the following hold: (1) If I is left translation invariant, then BA−1 is a non-empty a-open set contained in PS−1. (2) If I is right translation invariant, then A−1B is a non-empty a-open set contained in S−1P. Proof. (1) Since X is a topological group, τ ∼a I and I is right translation invariant, we have by Lemma 4.5, I ∩ τa = {φ}. Now by Theorem 2.2 (U ∩S ∩Sa ∗ )a ∗ ⊆ (U ∩S)a ∗ and by Theorem 2.7 we get (U ∩S ∩(U ∩S)a ∗ )a ∗ = (U ∩ S)a ∗ . Hence (4.2) (U ∩ S ∩ Sa ∗ )a ∗ = (U ∩ S)a ∗ c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 40 On topological groups via a-local functions Thus by Theorem 2.2 we have U ∩ Sa ∗ = U ∩ (U ∩ S)a ∗ ⊆ (U ∩ S)a ∗ = (U ∩ S ∩ Sa ∗ )a ∗ by (*). Since U ∩ Sa ∗ 6= {φ}, we have A 6= {φ}. Again, Aa ∗ = (U ∩ S ∩ Sa ∗ )a ∗ ⊇ U ∩ Sa ∗ ⊇ U ∩ Sa ∗ ∩ S = A i.e. A ⊆ Aa ∗ . For each a ∈ A, define fa : X → X given by fa(x) = xa −1, and F = {fa : a ∈ A}. Since A 6= {φ}, F 6= {φ} and each fa is a homeomorphism. Let G = V ∩aInt((P) a ∗ )∩ ℜa(P). Now it is sufficient to show that G ∩ F −1(y) /∈ I for every y ∈ F(G). Because then by Theorem 4.3, F(G ∩ P) = F(B) = BA−1 is a non-empty a-open set in X contained in PS−1. Let y ∈ F(G). Then y = xa−1 for some a ∈ A and x ∈ G ⇒ F−1(y) = xa−1A. Thus x ∈ xa−1A ⊆ xa−1Aa ∗ (as A ⊆ Aa ∗ ) ⊆ (xa−1A)a ∗ (by Lemma 4.4) = (F−1(y))a ∗ ⇒ Nx ∩ F −1(y) /∈ I for some Nx ∈ τ a(x). Thus BA−1 is a nonempty a-open subset of PS−1. So in particular, as (2) is similar to (1). � Corollary 4.7. Let (X, τ, .) be a topological group and I be an ideal on X such that τ ∼a I. Let A ∈ U(X, τ, I) and B ∈ P(X) − I. (1) If I is right translation invariant, then [B ∩ Ba ∗ ]−1[A ∩ aInt(Aa ∗ ) ∩ ℜa(A)] is a non-empty a-open set contained in B −1A. (2) If I is left translation invariant, then [A∩aInt(Aa ∗ )∩ℜa(A)][B∩B a ∗ ]−1 is a non-empty a-open set contained in AB−1. Proof. We only show that Ba ∗ 6= {φ} and A∩aInt(Aa ∗ )∩ℜa(A) 6= {φ}, the rest follows from Theorem 4.6 by taking U = V = X. In fact, if Ba ∗ = {φ}, then B ∩ Ba ∗ = {φ} which gives in view of Theorem 2.4, B ∈ I, a contradiction. Again, A ∈ U(X, τ, I) ⇒ aInt(Aa ∗ ) ∩ ℜa(A) 6= {φ} (by Lemma 4.5 and Proposition 2.9) ⇒ aInt(Aa ∗ )∩ℜa(A) ∈ τ a −{φ}. Now, aInt(Aa ∗ )∩ℜa(A) = [A∩aInt(Aa ∗ )∩ℜa(A)]∪[A c ∩aInt(Aa ∗ )∩ℜa(A)] /∈ I (by Lemma 4.5). Then [Ac ∩ aInt(Aa ∗ ) ∩ ℜa(A)] ⊆ [A c ∩ ℜa(A)] = ℜa(A) − A ∈ I by Theorem 2.11. Thus A ∩ aInt(Aa ∗ ) ∩ ℜa(A) /∈ I and hence A ∩ aInt(A a ∗ ) ∩ ℜa(A) 6= {φ}. � Corollary 4.8. Let (X, τ, .) be a topological group and I be an ideal on X such that I ∩ τa = {φ} and A ∈ U(X, τ, I). (1) If I is left translation invariant, then e ∈ aInt(A−1A). (2) If I is right translation invariant, then e ∈ aInt(AA−1). (3) If I is left as well as right translation invariant, then e ∈ aInt(AA−1 ∩ A−1A). Proof. It suffices to prove (1) only. We have, A ∈ U(X, τ, I) then there exists B ∈ Br(X, τ, I) − I such that B ⊆ A. Now for any x ∈ X, ℜa(B)x ∩ ℜa(B) = ℜa(Bx) ∩ ℜa(B) = ℜa(Bx ∩ B) (by Lemma 4.4 and Theorem 2.6). Thus if ℜa(B)x ∩ ℜa(B) 6= {φ}, then Bx ∩ B 6= {φ}. Now, if x ∈ [ℜa(B)] −1[ℜa(B)] then x = y−1z for some y, z ∈ ℜa(B), then yx = z = t (say) ⇒ t ∈ ℜa(B)x and t ∈ ℜa(B) ⇒ ℜa(B)x ∩ ℜa(B) 6= {φ} ⇒ x ∈ {x ∈ X : ℜa(B)x ∩ ℜa(B) 6= {φ}} then [ℜa(B)] −1[ℜa(B)] ⊆ {x ∈ X : ℜa(B)x ∩ ℜa(B) 6= {φ}} ⊆ {x ∈ X : Bx ∩ B 6= {φ}} ⊆ B−1B ⊆ A−1A. Since ℜa(B) 6= {φ} by Proposition 2.9 as B ∈ U(X, τ, I) and ℜa(B) is a-open for any B ⊆ X, we have e ∈ [ℜa(B)] −1[ℜa(B) ⊆ aInt(A −1A). � c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 41 W. Al-Omeri, Mohd. Salmi Md. Noorani and A. Al-omari Acknowledgements. The authors would like to acknowledge the grant from ministry of high education Malaysia UKMTOPDOWN-ST-06-FRGS0001-2012 for financial support. References [1] W. Al-Omeri, M. Noorani and A. Al-Omari, a-local function and its properties in ideal topological space,Fasciculi Mathematici, to appear. [2] W. AL-Omeri, M. Noorani and A. AL-Omari, On ℜa- operator in ideal topological spaces, submitted. [3] F. G. Arenas, J. Dontchev and M. L. 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