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Asian Journal of Current Engineering and Maths 2: 3 May – June (2013) 205 - 207. Contents lists available at www.innovativejournal.in ASIAN JOURNAL OF CURRENT ENGINEERING AND MATHS Journal homepage: http://www.innovativejournal.in/index.php/ajcem ON Α-I ALEXANDROFF SPACES I. Arockiarani, A. A. Nithya Department of Mathematics with Computer Applications, Nirmala College for Women, Coimbatore, India ARTICLE INFO ABSTRACT Corresponding Author A. A. Nithya Department of Mathematics with Computer Applications, Nirmala College for Women, Coimbatore, India Key Words: P -space, P 0 -space, The objective of this paper is to explore the idea of ideal in α-IAlexandroff spaces. Further we have proved some equivalent conditions and arrived a condition in which τ and τα coincide. Alexandroff-space, α-I-open, αI- Alexandroff-space, semiAlexandroff-space. base B(I,τ )= {U \G : U ∈ τ and G ∈ I}. Additionally, cl*(A) =A∪A* defines ∗ 2000 Math. Subject Classification — Primary: 54G05, 54G10, Secondary: 54H05, 54G99. INTRODUCTION Intensive development and research have been conducted in the field of Alexandroff spaces in topology and also in digital topology. This research area which is currently gaining much interest, which paved the way for digital concept. A subcollection I is said to be ideal if it satisfies 1) If A ∈ I and B ⊂ A then B ∈ I. 2) If A,B ∈ I then A ∪ B ∈ I. Then the triple (X,τ ,I) is said to be ideal topological space. In an ideal topological space for a subset A of X, A*(I)={x ∈ X : U ∩ A ∉ I for each neighbourhood U of x } is called the local function of A with respect to I and τ . For every ideal topological space (X,τ ,I), there exists a topology τ * (I), finer than τ , generated by the a Kuratowski closure operator for τ *(I). A topological space (X, τ ) is said to be locally finite if its each element x of X is contained in a finite open set and a finite closed set. It is clear that all finite topological spaces are locally finite and all locally finite topological spaces are Alexandroff. In this paper, we consider a weaker form of Alexandroff spaces called α-I- Alexandroff. Definition 1.1. Let A be a subset of a space (X,τ ), then A is said to be [a] pre-open[12] if A ⊂ int(cl(A)), [b] semi-open[11] if A ⊂cl(int(A)), [c] β-open[1] if A ⊂ cl(int(cl(A))). [d] α-open if A ⊆ int(cl(int(A))). We denote the family of all pre-open sets, semi-open sets, β-open sets and α-open sets as PO(X,τ ),SO(X,τ ) βO(X,τ ) and αO(X, τ ). Definition 1.2 A subset S of a space (X, τ ,I) is said to be 1. I-open if S ⊆ int (S*), 2. pre-I-open if S⊆ int(cl*(S)) 3. semi-I-open if S⊆ cl* (int (S)). ©2013, AJCEM, All Right Reserved. 4. α-I-open if S ⊆ int (cl*(int(S))). Complements of α-I-open sets are called α-I-closed. The union of all α-I-open sets contained in A is said be α-I-int(A). The intersection of all α-I-closed sets containing A is said to be α-I-cl(A). A topological space (X, τ ) is a P ‘ -space if and only if arbitrary countable intersections of open sets are semiopen. A topological space (X, τ ) is called open hereditary irresolvable [4] if each open subset of X is irresolvable and quasi-maximal [4] if every dense set with non-empty interior has dense interior. A subset A ⊆ X is said to be simply- open [18] if A = U ∪ N , where U is open and N is nowhere dense (a set S is nowhere dense if int(cl(A))= φ. In [4], Chattopadhyay and Roy called A a δ-set if intcl(A) ⊆ cl(intA). It is easily observed that a set A is simply-open if and only if it is a δ-set. A topological space (X, τ ) is called door space if each A ⊆ X is either open or closed. α-I-Alexandroff Definition 2.1 A topological space (X, τ, I) is called α-IAlexandorff if arbitrary intersection of open sets is α-Iopen. Remark 2.2 An α-I Alexandorff space need not necessarily satisfy the con- dition ”Arbitrary intersections of α-I-open sets are α-I-open“. Lemma 2.3 If a space (X, τ, I) contains a point p such that {p} is open and dense, then (X, τ ,I) is α-IAlexandorff. Proof. Let {Oi : i ∈ I} be a collection of open sets and O be its intersection. Suppose that O is not α-I-open, i.e. O ⊄ int(cl ∗ (int(O))). Then there exists x ∈ O and an open set V containing x with V ∩ int O = ∅. Since {p} is open and dense, then 205 Nithiya et. al/On α-I Alexandroff spaces we have p ∈ V and p ∈ Oi for each i ∈ I, hence also p ∈ int O, w h i c h i s a contradiction. Thus O is α-I-open. Theorem 2.4 Let (X, τ ,I) be an ideal topological space then every Alexandorff space is α-I-Alexandorff space. Proof. Let Oi be the arbitary collection of open sets, since the space is Alexandorff therefore its intersection is open set. We know that every open set is α-I-open. Thus intersection of arbitary open sets is α-I-open. Hence the proof. Remark 2.5 Every α-I-Alexandorff is not always Alexandorff is shown in the following example. Example 2.6 Let X=R be the real line with the topology τ = {∅, X, {0}} ∪{G ⊆ X : 0 ∈ G and X \G is finite}, I={φ}. Clearly {0} is open and dense in (X, τ, I) and so (X, τ, I ) is α-I-Alexandorff by Lemma.2.3. On the other hand, the intersection of all open sets X \ {x}, where x is irrational, is not open. Thus (X, τ, I ) is not Alexandorff. Theorem 2.7 For an ideal topological space (X, τ .I) the following conditions are equivalent: (1) X is T1 and α-I-Alexandorff. (2) X is discrete. Proof. (1) ⇒ (2) For each x ∈ X and each y = x, there exists an open set Ux containing x such that y ∉ Ux (X is T1 ). Since X is αI-Alexandorff and since {x} = ∩y=x Ux , then {x} is αI-open and hence open, since a singleton is α-I-open if and only if it is open. Thus X is discrete. (2) ⇒ (1) Let X be discrete space, then every singleton set is open. Therefore for any two points x and y we can find two open sets such that x ∈ U and y 6∈ U similarly y ∈ V and x ∉ V . Hence it is T1 space. Also, since the underlying space is discrete arbitrary intersection of open sets is α-Iopen. Hence the proof. Lemma 2.8 [10]For a subset A of a space X the following conditions are equivalent: (1) A is a simply-open set. (2) A is the intersection of a α-I-open and a α-Iclosed set (ie) A∈LC(X, τ α ). Lemma 2.9 [4] For a topological space X the following conditions are equivalent: (1) Every subset of X is simply-open. (2) X is open hereditary irresolvable and quasi-maximal. Theorem 2.10 Let (X, τ ) be T 1 and α-I-Alexandorff. Then X is open hereditary irresolvable and quasimaximal. Proof. According to Lemma 2.9 and 2.8, we need to show that every subset of X is simply-open. Let A ⊆ X. Set A = A1 ∪ A2 (with A1 ∩ A2 = ∅), where each point of A1 is closed in X and each point of A2 is open in X (this is possible, since X is T 1 ). Since X is α-I-Alexandorff, then A1 is α-I-closed and moreover A2 is α-I-open. Using Lemma 2.8, each subset of X is the complement of a simply-open set, i.e. each subset of X is simply-open. Theorem 2.11 Every α-I-Alexandorff space is a P ‘ - space. Proof. Let us take a countable collection of open set in X, Since it is α-I Alexandorff every arbitrary intersection of open sets is α-I-open and thus it is semi open which implies X is P ‘ space. Hence the proof. Remark 2.12 However the converse need not be true is shown by the fol- lowing example. Example 2.13 Let X be the real line where the nontrivial open sets are all sets containing the zero point and having countable complements. Clearly this is a P space and hence a P 0 -space. The intersection of all open sets of the form X \ {x}, for x = 0, is not α-Iopen. This shows that X is not α-I-Alexandorff. Theorem 2.14 Let X be a door space then the following conditions are equivalent: (1) X is Alexandorff. (2) X is α-I Alexandorff. Proof. (1)⇒(2) It is straightforward. (2)⇒(1) Let us consider a collection of open sets whose intersection is α-I open. Since X is a door space all subsets of X will be either open set or closed set. In this case αIO(X) = τ . Therefore in this case every α-I open set is open set. Thus it is Alexandorff space. Theorem 2.15 Let f : (X, τ, I ) → (Y, σ, J ) be a surjective continuous α-I- open function. If X is Alexandorff, then Y is α-I-Alexandorff. Proof. To prove Y is α-I-Alexandorff, let {Oi |i ∈ I} be the arbitrary collection of open sets in Y. Since f is continuous, we have f−1 (Oi ) is open X. Since X is Alexandorff intersection of this collection is open set G, then using f is surjective and α-I open we get f(G) is α-I-open in Y. f(G)=f ∩ f −1 (Oi ). =(∩(f f −1 (Oi |i ∈ I)) =∩(Oi |i ∈ I) in Y. Thus Y is α-I- Alexandorff. Definition 2.16 Let (X, τ ,I) be a ideal topological space, then if cl∗ (int(A)) = int(A), then it is said to be ic*-set. Definition 2.17 A function f : (X, τ, I) → (Y, σ, J ) is said to be ic*- continuous if for every open set in Y, then its inverse image is an ic* set. Theorem 2.18 A function f : (X, τ, I) → (Y, σ, J ) is continuous if and only if it is α-I-continuous and ic*continuous. proof. If f is continuous then it is always α-I-continuous and ic*-continuous. Conversely, Since f is α-I-continuous and ic*-continuous, for every open set U in Y f −1 (U ) is α-I-open and ic*-set. Therefore f−1 (U ) ⊆ int(cl∗(int(f −1 (U ))) and cl ∗ (int(f −1 (U ))) = int(f −1 (U )). Thus we get f −1 (U ) ⊆ int(f −1 (U )) and always int(f −1 (U )) ⊆ f −1 (U ).Hence f −1 (U ) = int(f −1 (U )), (i.e)f −1 (U ) is open in X. Theorem 2.19 Let X be α-I-Alexandorff and let f : (X, τ, I) → (Y, σ, J ) be α-I-open function, continuous and onto. Then Y is α-IAlexandorff. Proof. To prove Y is α-I-Alexandorff, let {Oi |i ∈ I } be the arbitrary collection of open sets in Y. Since f is continuous, we have f−1 (Oi ) is open X. 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