Characterizations of Hayashi-Samuel Spaces via Boundary Points

Communications in Advanced Mathematical Sciences Vol. II, No. 3, 219-226, 2019 Research Article e-ISSN:2651-4001 DOI: 10.33434/cams.546925 Characterizations of Hayashi-Samuel Spaces via Boundary Points Sk Selim1 , Shyamapada Modak2 *, Md. Monirul Islam3 Abstract Some new closure operators in topological spaces with ideals are a part of this paper. A comparative study of a new type of boundary point, which is defined with the help of the local function and the boundary points will be discussed through this paper. Characterizations of Hayashi-Samuel spaces are also an object of this paper. Keywords: Ideal topological space, Hayashi-Samuel space, Local function, ψ-operator. 2010 AMS: Primary 54A10, Secondary 54A05, 54A99 1 Department of Mathematics, University of Gour Banga, P.O. Mokdumpur, Malda, ORCID: 0000-0002-4226-2004 2 Department of Mathematics, University of Gour Banga, P.O. Mokdumpur, Malda, ORCID:0000-0002-0226-2392 3 Department of Mathematics, University of Gour Banga, P.O. Mokdumpur, Malda, ORCID: 0000-0003-4748-4690 *Corresponding author: Received: 29 March 2019, Accepted: 1 August 2019, Available online: 30 September 2019 1. Introduction and preliminaries A modification of closure operator in topological space is the local function in ideal topological space. This study was introduced by Kuratowski [1] and Vaidyanathswamy [2]. An ideal [1] I on a topological space (X, τ) 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 . A topological space (X, τ) with an ideal I on X is called an ideal topological space and is denoted by (X, τ, I ). For a subset A of an ideal topological space (X, τ, I ), the local function A∗ is defined as: A∗ = {x ∈ X : Ux ∩ A ∈ / I , Ux ∈ τ(x)} (where τ(x) is the collection of all open sets which contains x) and it was defined by imposing extra condition on the closure operator. As a result, the mathematicians like Samuel [3], Pavlović [4], Hayashi [5], Hashimoto [6], Janković and Hamlett [7, 8], Ekici [9, 10, 11], Hatir [12], Noiri [11, 12, 13] have reached to obtain a new topology known as ∗-topology and it is finer topology than the original topology. In an ideal topological space (X, τ, I ), the structures-“topology” and“ideal” played important roles simultaneously. The condition τ ∩ I = {0} / is a remarkable part in ideal topological space and such ideal topological space is called Hayashi-Samuel space [14]. Modak and his associates studied this ideal topological space and introduced different types of generalized open sets and operators with the help of ideals (see [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26]). The complement operator of the local function is known as ψ-operator [8, 27] and it is defined by: ψ(A) = X \ (X \ A)∗ , for a subset A of an ideal topological space (X, τ, I ). ψ-operator is an important part for the study of ideal topological space. In this paper, we introduce a new type of boundary points in ideal topological spaces by using ∗ - operator. We consider a comparative study of these boundary points with the boundary points in topological spaces. We also explore the characterizations of Hayashi-Samuel space which was established in [18, 19, 24]. We also obtain more closure operators in ideal topological Characterizations of Hayashi-Samuel Spaces via Boundary Points — 220/226 spaces through this paper. 2. ∗ boundary points Boundary operator [28] is a set valued set-function and we may consider it by the following way: Let (X, τ) be a topological space and A ⊆ X. The boundary operator Bd : ℘(X) → C(τ) is defined as Bd(A) = Cl(A) ∩ Cl(X \ A), where C(τ) denotes the collection of all closed sets and Cl(A) denotes the closure of A in (X, τ). Thus boundary point of a set A ⊆ X is a common point between closure of A and closure of (X \ A). We modify the boundary operator with the help of the local function and call it ∗-boundary operator. Definition 2.1. Let (X, τ, I ) be an ideal topological space. The operator Bd ∗ : ℘(X) → C(τ), defined by: Bd ∗ (A) = A∗ ∩ (X \ A)∗ , for A ∈ ℘(X), is called ∗-boundary operator on (X, τ, I ). The point x ∈ Bd ∗ (A) is called ∗-boundary point of A and it is the common point of A∗ and (X \ A)∗ . We start with the following example which shows that there is some common points in A∗ and (X \ A)∗ . Example 2.2. Let X = R, Ru be the usual topology on R and I = {0}. / Then Q∗ = Cl(Q) = R and (R \ Q)∗ = Cl(R \ Q) = R. ∗ ∗ This shows that there are common points between Q and (R \ Q) . We know that boundary points of a set depends on the topology. For this, if we consider the indiscrete topology on R, then Bd(Q) = R, where Q denotes the set of all rational numbers. But if we consider the discrete topology on R, then Bd(Q) = 0. / ∗-boundary point of a set depends on not only the topology but the ideal also. Followings examples show the role of ideal in ∗-boundary points: Let (X, τ, I ) be an ideal topological space and A ⊆ X. (i) If we take I = {0}, / then Bd ∗ (A) = Bd(A). (ii) If the ideal I = ℘(X), Bd ∗ (A) = 0. / Note that in discrete topological space, boundary points of any set is always empty. But in any ideal topological space, if the ideal is the collection of all subsets of the set then ∗-boundary points of any set is always empty. (iii) When the ideal I = I f , the ideal of finite subsets of X, then Bd ∗ (A) is the ω-accumulation points of A and X \ A. (iv) If one choose the ideal I = Ic , the ideal of countable subsets of X, then A∗ is precisely the set of condensation points of A and boundary points accordingly. (v) Let In be the collection of all nowhere dense subsets of (X, τ), then In is an ideal on X. If we take I = In , then A∗ = Cl(Int(Cl(A))) and Bd ∗ (A) = Cl(Int(Cl(A))) \ Int(Cl(Int(A))). (vi) Let (X, τ) be a topological space and Im be the collection of all meager sets (or sets of first category). Then it forms an ideal on X and A∗ is set the points of second category of A. Note that for a subset A ⊆ X in a topological space (X, τ) with an ideal I , x ∈ Bd ∗ (A) implies Ux ∈ / I for all Ux ∈ τ(x) but converse statement is not true in general. / X, {a}, {a, b}} and I = {0, / {a}}. Then ({b})∗ = {b, c} and all open sets containing Example 2.3. Let X = {a, b, c}, τ = {0, ∗ a do not belongs to I but a ∈ / Bd ({b}). One of the characterizations of ∗-boundary point is: Theorem 2.4. Let (X, τ, I ) be an ideal topological space and A ⊆ X. Then x ∈ Bd ∗ (A) if and only if x ∈ A∗ \ ψ(A). Similar characterization of boundary point is: Theorem 2.5. [28] Let (X, τ) be a topological space and A ⊆ X. Then x ∈ Bd(A) if and only if x ∈ Cl(A) \ Int(A), where Int(A) denotes the interior of A. Theorem 2.6. Let (X, τ, I ) be an ideal topological space and A ⊆ X. Then Bd ∗ (A) = 0/ if and only if A∗ ⊆ ψ(A). Similar characterization of boundary point is: Theorem 2.7. [28] Let (X, τ) be a topological space and A ⊆ X. Then Bd(A) = 0/ if and only if A (...truncated)