More on α-topological spaces

Article electronically published on April 05, 2017 Com mun.Fac.Sci.Univ.Ank.Series A1 Volum e 66, Numb er 2, Pages 323–331 (2017) DOI: 10.1501/Com mua1_ 0000000822 ISSN 1303–5991 http://com munications.science.ankara.edu.tr/index.php?series=A1 MORE ON -TOPOLOGICAL SPACES SHYAMAPADA MODAK AND MD. MONIRUL ISLAM Abstract. The aim of this paper is to introduce a new topology with the help of a-open sets. For this job, we shall de…ne two new types of set and discuss its properties in detail and characterize Njastad’s -open sets and Levine’s semi-open sets through these new types of set. 1. Introduction The study of ideal in topological space was introduced and studied by Kuratowski [15] and Vaidyanathaswamy [22] but in this study Jankovic and Hamlett gave a new dimension through their paper “New topologies from old via ideals" [14]. Now a days the authors like Navaneethakrishnan et al. [19], Hamlett and Jankovic [12], Arenas et al. [4], Nasef and Mahmoud [18], Mukherjee et al. [17] Dontchev et al. [6] and many others have enriched this study. The authors Al-Omari et al. [1, 2] in their papers “a-local function and its properties in ideal topological spaces" and “The <a operator in ideal topological spaces", have studied Ekici’s [7, 8, 9] a-open sets in terms of ideals. They have obtained a new topology with the help of two operators viz. <a and ()a , and have shown that this topology is …ner than Ekici’s a - topology. In this paper, we have further considered the space which is the joint venture of a-topology and an ideal as like Al-omari et al. have considered in [2, 1]. Through this paper we will solve the question “how much …ner is Noiri’s et al.’s topology than Ekici’s topology?" For solution of this question we have considered Njastad’s -open sets [20] from literature. 2. Preliminaries In this section we have discussed some preliminary concepts of literature and introduce some prime results for discussing the paper. Let A be a subset of a topological space (X; ), then ‘Int(A)’and ‘Cl(A)’will denote ‘interior of A’and ‘closure of A’respectively. Received by the editors: November 07, 2016, Accepted: February 02, 2017. 2010 Mathematics Subject Classi…cation. 540A, 54C10. Key words and phrases. set, a-open set, a-local function, <a operator. c 2 0 1 7 A n ka ra U n ive rsity C o m m u n ic a tio n s d e la Fa c u lté d e s S c ie n c e s d e l’U n ive rsité d ’A n ka ra . S é rie s A 1 . M a th e m a tic s a n d S ta tistic s. 323 324 SHYAM APADA M ODAK AND M D. M O NIRUL ISLAM We de…ne following as a mathematical tool for this research article: De…nition 1. Let A be a subset of a topological space (X; ). A is said to be regular open [21] (resp. semi-open [16, 11], semi-pre open [3], -open [20]) if A = Int(Cl(A)) (resp. A Cl(Int(A)); A Cl(Int(Cl(A))); A Int(Cl(Int(A)))). De…nition 2. [23] A subset A of a topological space (X; ) is said to be -open if, for each x 2 A, there exists a regular open set G such that x 2 G A. The complement of -open set is called -closed. Let (X; ) be a topological space, then the point x 2 X is called -cluster point of A if Int(Cl(V )) \ A 6= ;, for each open set V containing x. The -closure of A is denoted as Cl (A) [23] and it is a set of all -cluster point of A. In this regards, Int (A) [23] is the -interior of A and it is the union of all regular open sets of (X; ) contained in A. If Int (A) = A for a topological space (X; ), then A is -open and conversely [23]. It is remarkable that the collection of all -open sets in a topological space (X; ) forms a topology and it is denoted as [23]. De…nition 3. [8, 9, 10] A subset A of (X; ) is said to be a-open (resp. a-closed) if A Int(Cl(Int (A))) (resp. Cl(Int(Cl (A))) A). The family of a-open sets in (X; ) forms a topology on X. This collection is denoted as a [8], and a (x) is denoted as the collection of all a-open sets containing x. In this paper we also denote ‘aCl’ by the means of closure operator of Ekici’s a topology [7, 8]. Hereditary class and a-local function are also the mathematical tool for this paper: De…nition 4. [15] A collection I }(X) is said to be an ideal on X if B implies B 2 I and A; B 2 I implies A [ B 2 I. A2I Let I be an ideal on the topological space (X; ), then (X; ; I) is called an ideal topological space. According to Al-Omari et al. [2, 1], we give the following: The a-local function ()a : }(X) ! }(X) for a subset A of an ideal topological space (X; ; I) is de…ned as (A)a = fx 2 X : U \ A 2 = I; for every U 2 a (x)g, a and as like complement operator of () , <a : }(X) ! }(X) is de…ned as <a (A) = X n (X n A)a = fx 2 X : there exists Ux 2 a (x) such that Ux n A 2 Ig. Due to the operator ()a , we have a topology a [1] whose one of the basis is (I; ) = fV n I : V 2 a ; I 2 Ig [1]. In this respect, we will denote ‘Inta ’and ‘Cla ’as ‘interior’operator and ‘closure’operator of (X; a ) respectively. Following results help us for repairing the paper: Theorem 1. [1] Let (X; ; I) be an ideal topological space and U 2 U <a (U ). a . Then M ORE ON -TOPOLOGICAL SPACES 325 Corollary 2. Let A be a subset of an ideal topological space (X; ; I), then aInt(A) <a (A). Theorem 3. [1] Let A be a subset of an ideal topological space (X; ; I) with ;. Then <a (A) (A)a . Corollary 4. Let A be a subset of an ideal topological space (X; ; I) with Then <a (A) aCl(A). Lemma 5. Let (X; ; I) be an ideal topological space and O 2 if and only if (O)a = aCl(O). a . Then a \I = a \I = ;. a \I = ; Proof. Let a \ I = ; and ; 6= O 2 a . Now Oa aCl(O) always. For reverse inclusion, let x 2 aCl(O). Therefore all neighbourhoods Ux 2 a (x), Ux \ O 6= ; implies Ux \ O 2 = I, since a \ I = ;. Therefore x 2 (O)a . Hence (O)a = aCl(O). Conversely let O 2 a , (O)a = aCl(O). Then X a = X and this implies I \ a = ; [2]. Proposition 6. Let (X; ; I) be an ideal topological space with following hold: (1) For A X, <a (A) aInt(aCl(A)). (2) For a-closed subset A; <a (A) A. (3) For A X; aInt(aCl(A)) = <a (aInt(aCl(A))). (4) For any a -regular open subset A; A = <a (A). (5) For any O 2 a ; <a (O) aInt(aCl(O)) (O)a . a \ I = ;. Then Proof. (1) From Theorem 3, <a (A) (A)a . Then <a (A) aCl(A), and since <a (A) is open, <a (A) aInt(aCl(A)). (3) <a (aInt(aCl(A))) (aInt(aCl(A))a = aCl(aInt(aCl(A))) (from Lemma 5) aCl(A). Thus <a (aInt(aCl(A))) aInt(aCl(A)). Reverse inclusion: aInt(aCl(A)) <a (aInt(aCl(A))) (from Theorem 1). Thus aInt(aCl(A)) = <a (aInt(aCl(A))). 3. <a aCl sets De…nition 5. Let (X; ; I) be an ideal topological space and A be a <a aCl set if A aCl(<a (A)). The collection of all <a X, A is said to aCl sets in (X; ; I) is denoted by <a (X; Note 3.1. Let (X; ; I) be an ideal topological space. <a (X; a ). If A 2 a a ). , then A 2 Later, we shall given the example for the converse of this note. Theorem 7. Let fAi : i 2 g be aScollection of nonempty <a ideal topological space (X; ; I), then i2 Ai 2 <a (X; a ). aCl sets in an 326 SHYAM APADA M ODAK AND M D. M O NIRUL ISLAM (...truncated)