Convergence of the Associated Filters via Set-Operators

e-ISSN: 2564-7954 CUJSE 17(2): 101-107 (2020) Research Article Çankaya University Journal of Science and Engineering https://dergipark.org.tr/cankujse Convergence of the Associated Filters via Set-Operators Takashi Noiri 1 1 , Sk Selim 2 , Shyamapada Modak 2* 2949-1 Shiokita-cho, Hinagu, Yatsushiro-shi Kumomoto-ken, 869-5142 Japan 2 Department of Mathematics, University of Gour Banga, India Keywords Abstract Ideal Filter Associated filter Limit point of a filter Local function Let (X, τ) be a topological space. For a proper ideal I on (X, τ), the associated filter FI is defined and investigated in [1]. In this paper, we define several set-operators on an ideal topological space (X, τ, I) and investigate the relationship between the setoperators and limit points of the associated filter FI. 1. Introduction and Preliminaries Let (X, τ) be a topological space and I be an ideal on X, then for A ⊆ X, the local function is defined in [2] as A*(I, τ) ={x∈X:Ux∩A∈ /I for every Ux ∈ τ(x)}, where τ(x) is the collection of all open sets containing x. A*(I, τ) is simply denoted as A*(I) or A*. For the simplest ideals {∅} and P (X), we observe that A*({∅}) = cl(A) (cl(A) denotes the closure of A) and A*(P (X)) = ∅ for every A⊆X. Thus the study of the local function is interesting when the ideal I is a proper ideal (an ideal I does not contain whole set X) on the topological space. Otherwise, when an ideal contains the hole set X, then it contains all subsets of X and the value of local function on any set is always empty. The complementary set-operator ψ of the set-operator ()* is defined in [3] as ψ(A) = X \ (X \ A)∗ . It is notable that ()’ is not a closure operator and ψ is not an interior operator. However, the set operator C: P (X)→P (X) defined by C(A) = A∪A* makes a closure operator [2, 4] and it is denoted as ‘cl∗ ’, that is cl∗ (A) = A∪A*. This closure operator induces a topology on X and it is called the ∗-topology [5-12]. This topology is denoted as τ∗. Let X be a nonempty set and F ⊆ P (X). Then F is called a filter [13, 14] on X if it satisfies the following: 1. ∅ ∉ F 2. B ∈ F and B ⊆ A implies A ∈ F, 3. A, B ∈ F implies A ∩ B ∈ F. Let I be a proper ideal on a topological space (X, τ). Then F = {A  X : X \ A I } forms a filter on X. This filter is called the associated filter on X and denoted as FI. Definition 1. [1] An ideal Iu on a set X is called a universal ideal if for any A ⊆ X, either A ∈ Iu or X \A∈ Iu. * Corresponding Author: Received: April 11, 2020, Accepted: May 5, 2020 101 Noiri et al. CUJSE 17(2): 101-107 (2020) 2. Associated filters We define the operator ∧ on an ideal topological space (X, τ, I) in the following way: for a subset A of X, ∧(A) = ψ(A) \ A. Lemma 1. Let I be a proper ideal on a topological space (X, τ) and A ⊆ X. Then for x ∈ ψ(A) ∈ I, x is not a limit point of the associated filter FI. Proof. Given that x ∈ ψ(A) ∈ I. Thus for ψ(A) ∈ τ(x), X \ψ(A) ∈ FI. Hence x ∈ ψ(A) ∈ / FI. Therefore, x is not a limit point of FI. Corollary 1. Let I be a proper ideal on a topological space (X, τ) and A ⊆X. Then for x ∈∧(A) and ψ(A) ∈ I, x is not a limit point of the associated filter FI. Corollary 2. Let I be a proper ideal on a topological space (X, τ) and A ⊆X. Then for x ∈∧(A) and ψ(A) ∈ I, x is not a cluster point of the associated filter FI. For converse of the above corollary and lemma, we have followings: Theorem 1. Let Iu be a universal ideal on a topological space X. If x is not a limit point of the associated filter U, then x ∈ ψ(X \ A) for any A ∈ P (X). Proof. Let x ∈ X and x be not a limit point of U. Then there exists Ux ∈ τ(x) such that Ux ∈ / U and hence Ux ∩ A ∈ / U, for any A ∈ P (X). Therefore, X \ (Ux ∩ A) ∈ / Iu and hence A ∩ Ux ∈ Iu. Thus x ∈ / A* and hence x ∈ (X\A*) = ψ(X \ A). We define the operator ∆1 on an ideal topological space (X, τ, I) in the following way: for a subset A of X, ∆1 (A) = ∨(A) ∪ ∧ (A) = (ψ(A) \A*) ∪ (ψ(A)\A). Theorem 2. Let I be a proper ideal on a topological space (X, τ) and A ⊆ X. Then for x ∈ ∆1(A) and ψ(A) ∈ I, x is not a limit (resp. cluster) point of the associated filter FI. Proof. Case (i): Suppose x ∈ (ψ(A)\A*). Thus x ∈ X \ (X \A)∗ and x ∈ / A*.Thus x ∈ / (X \ A)∗∪A* and hence x ∈ / X ∗ . Thus there exists Ux ∈ τ(x) such that Ux ∩ X ∈ I, hence Ux ∈ I. Thus X \ Ux ∈FI. Therefore, x is not a limit point of FI. Case (ii): Suppose x ∈ (ψ(A) \ A). Then from Lemma 1, x is not a limit point of FI. We define the operator ∆2 on an ideal topological space (X, τ, I) in the following way: ̅ (A) = (ψ(A) \ A*)∪(A \ A*). ∆2 (A) = ∨(A)∪∧ For convergence of the associated filter to a point x ∈ ∆2(A), we get followings: Theorem 3. Let I be a proper ideal on a topological space (X, τ). Then for x ∈ (ψ(A) \ A*) but x ∈ / (A \ A*), x is not a limit point of the associated filter FI. Proof. Similar to Theorem 2 Case (i). Theorem 4. Let I be a proper ideal on a topological space (X, τ). Then for x ∈ / (ψ(A) \ A*) but x ∈ (A\A*), x is not a limit point of the associated filter FI. Proof. Given that x ∈ / (ψ(A)\A*) and x ∈ (A \ A*), then x ∈ / A*, x ∈ A and x ∈ / ψ(A) 102 Noiri et al. CUJSE 17(2): 101-107 (2020) (since x ∈ / A*, if x ∈ ψ(A) then x ∈ ψ(A) \ A*, contradiction). Therefore, x ∈ / A*, x ∈ A and x ∈ (X \ A)*. Then there exists Ux ∈ τ(x) such that Ux∩A∈ I, and for all Wx ∈ τ(x), Wx∩(X\A) ∈ / I and x ∈ A. Thus, for the particular Ux , Ux∩A∈ I and Ux \ A ∈ / I. Thus for the particularUx , Ux ∈ / I and hence Ux ∈ / FI. Therefore, x is not a limit point of FI. Moreover, if x ∈ (ψ(A) \ A*) and x ∈ (A \ A*), then x is not a limit of the associated filter We define the operator ∆3 on an ideal topological space (X, τ, I) in the following way: for any subset A of X, ̅ (A) ∪ ∧(A) = (A\A*) ∪ (ψ(A)\A). ∆3(A) = ∧ For convergence of the associated filter to a point of ∆3(A), we shall take the help of next section. 3. Complementary set-functions In this section we consider the complementary function to the earlier section and discuss the convergence of the associated filter in terms of the complementary function. We define the operator ∨ and ∇i on an ideal topological space (X, τ, I) in the following way: for any subset A of X, ∨(A) = X \ ∧(X \ A) and ∇i(A) = X \ ∆i(X \ A) for i = 1, 2, 3. Theorem 5. Let (X, τ, I) be an ideal topological space. Then for A∈ P (X), 1. ∨(A) = A*∪(X \ A). 2. ∨(X \ A) = (X \ A)∗∪A. 3. ∨(X \ A) = (X \ ψ(A)) ∪A 4. ∨(A) ∪A = X. Proof. 1. ∨(A) = X \ (ψ(X \ A) \ (X \ A)) = X \ ((X \ A*) \ (X \ A)) = X \ [(X \ A*)∩A] = A* ∪ (X \ A). Proof of 2,3 and 4 follows from 1. If x ∈∨(A), then x may be a limit point of the associated filter FI. Example 3.1. Let X = {a, b, c}, τ = {∅, {a}, {b}, {a, b}, X} and I = {∅, {a}}. Let A = {a, c}. Then c, b ∈ ∨(A) = A* ∪ (X \ A) = {c} ∪ (X \ {a, c}) = {b ,c}. But the associated filter FI = {{b, c}, X} does not converges to b. Moreover, FI ⟶c. Theorem 6. Let (X, τ, I) be an ideal topological space. Then for A ∈ P (X), 1. ∆1 (A) = ψ(A)\(A*∩A). 2. ∆1 (...truncated)