EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 1, Article Number 5732 ISSN 1307-5543 – ejpam.com Published by New York Business Global Characterizations of δ1-βI-Paracompactness Concerning an Ideal Chawalit Boonpok1, Areeyuth Sama-ae2,∗ 1 Mathematics and Applied Mathematics Research Unit, Department of Mathematics, Faculty of Science, Mahasarakham University, Maha Sarakham, 44150, Thailand 2 Department of Mathematics and Computer Science, Faculty of Science and Technology, Prince of Songkla University, Pattani Campus, Pattani, 94000, Thailand Abstract. Al-Jarrah presented and examined the idea of β1-paracompactness in topological spaces, whereas Qahis extended the original idea of β1-paracompact spaces by further developing and in- vestigating the idea of β1-paracompact spaces with respect to an ideal. This work analyzes the properties, subsets, and subspaces of δ1-βI-paracompact spaces, which are wider in scope than the β1-paracompact spaces delineated by Qahis. Furthermore, we investigate the invariants of δ1-βI-paracompact spaces via the view of functions. 2020 Mathematics Subject Classifications: 54A05, 54B05, 54C08 Key Words and Phrases: Ideal topological space, δ1-βI-paracompact, δ-βI-open 1. Introduction Paracompact spaces, established considerably later than the two earlier classes, are seen to be one of the most important classes of topological spaces, concurrently generaliz- ing both metrizable and compact spaces. Paracompact spaces were promptly recognized by topologists and analysts. A paracompact space in mathematics is a topological space where every open cover has an open refinement that is locally finite. The notion of spaces was initially developed by Dieudonné [10] in 1944. In 1969, Singal and Arya [24] introduced a novel notion of paracompactness termed almost paracompactness, which is a weaker form of standard paracompactness that defines its fundamental topological properties. A Haus- dorff space is paracompact if and only if it allows partitions of unity that are subservient to any open cover. Every paracompact Hausdorff space is normal, as referenced in [12]. Var- ious forms of generalized paracompactness in literature, including S-paracompactness [4], P3-paracompactness [9], and β-paracompactness [5], are evaluated. In 2006, Al-Zoubi [4] employed semi-open sets to characterize S-paracompact spaces, a generalization of para- compact spaces, and analyzed the relationships between these spaces. Li and Song [18] ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5732 Email addresses: chawalit.b@msu.ac.th (C. Boonpok), areeyuth.s@psu.ac.th (A. Sama-ae) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) C. Boonpok, A. Sama-ae / Eur. J. Pure Appl. Math, 18 (1) (2025), 5732 2 of 13 developed a Hausdorff S-paracompact space that is not paracompact and looked at other characterizations of S-paracompact spaces. In 2013, Demir and Ozbakir [9] proposed a diminished variant of expandable and paracompact spaces, termed β-expandable spaces and β-paracompact spaces, respectively. Every β-paracompact space is a β-expandable space, as demonstrated in a provided argument. Yildirim et al. [25] came up with the idea of β-paracompactness in an ideal topological space and compared it to other types of paracompactness that are already known. In 2024, Alrababah et al. [6] studied the notion, attributes, and theorems associated with Dparacompact spaces. In 1930, Kuratowski proposed the concept of an ideal topological space [17]. Jankovic and Hamlett [15] have performed an investigation and offered a thorough explanation of the basic characteristics related to ideal topological spaces. In addition to elucidating the concept of I-open sets, they conducted exhaustive research on topologies that make use of ideals. A thorough investigation of the idea of I-open sets was carried out by Abd. El-Monsef et al. [1]. The concept of Ig-closed sets was originally presented by Dontchev et al. [11] in the year 1999. There was an early proposal made by Abd. El-Monsef et al. [2] about the concept of the s-local function. Khan and Noiri [16] subsequently performed an analysis on this concept. The concept of paracompactness in respect to an ideal was originally presented by Zahid [27], and it was subsequently investigated by Hamlett et al. [13]. An investi- gation was studied by Sathiyasundari and Renukadevi [23] to investigate the idea of I-paracompactness and to assess its properties. The concept of I-paracompact spaces has been expanded to include some conclusions that were obtained from the concept of paracompact spaces. Within the context of ideal topological spaces, Sanabria et al. [22] conducted an investigation into the concept of S-paracompactness. The focus of their research was on the development and investigation of a new category of spaces, which they referred to as I-S-paracompact spaces. These spaces were constructed inside the framework of an ideal topological space. Spaces that are both S-paracompact and I- paracompact are included in this class. In 2016, Al-Jarrah [3] introduced the concept of β1-paracompactness, employing the definition of β-open as follows: A topological space (X, τ) is β1-paracompact if every β-open cover of X has a locally finite open refinement. In 2019, Qahis [21] developed a novel class of β1-paracompact spaces related to an ideal, analyzing their characterizations and exploring the corresponding invariants. This study presents a new classification of β1-paracompact spaces that is wider than that suggested by Qahis. 2. Preliminaries In this article, the notation (X, τ) denotes a topological space without any assumptions on separation axioms. For a subset A of a topological space (X, τ), CI(A) represents the closure of A in (X, τ), whereas Int(A) signifies the interior of A in (X, τ). An ideal I on a topological space (X, τ) is a nonempty collection of subsets of X that fulfills the following criteria: (i) A ∈ I and B ⊂ A implies B ∈ I, and C. Boonpok, A. Sama-ae / Eur. J. Pure Appl. Math, 18 (1) (2025), 5732 3 of 13 (ii) A ∈ I and B ∈ I implies A ∪B ∈ I. An ideal topological space (X, τ, I) is characterized as a topological space (X, τ) paired with an ideal I on the set X. The collection of all subsets of X is represented as P (X). A set operator (.)∗ : P (X) → P (X), characterized as a local function [17], is defined with respect to τ and I: for A ⊂ X, A∗(I, τ) = {x ∈ X : U ∩ A ̸∈ I for all U ∈ τ(x)}, where τ(x) = {U ∈ τ : x ∈ U}. We simply write A∗ instead of A∗(I, τ). A topology τ∗(I), or τ∗ for brevity that is finer than τ may be created for any ideal topological space, defined by β(I, τ) = {U − I : U ∈ τ and I ∈ I}. Nevertheless, β(I, τ) does not uniformly define a topology. Furthermore, Cl∗(A) = A ∪ A∗ provides a Kuratowski closure operator for τ∗. We simply write for τ∗ for τ∗(I, τ). If β(I, τ) = τ∗, then we say I is τ -simple [15]. If (X, τ, I) satisfies this condition, then I is said to be compatible [15] or I is said to be τ -local. Given an ideal topological space (X, τ, I), we say I is I-codense if I ∩ τ = {∅}. Definition 1. [14] Let A be a subset of an ideal topological space (X, τ, I). A point x ∈ X is called a δI-cluster point of A if Int(Cl∗(U)) ∩ A ̸= ∅ for every neighborhood U of x. The δClI(A) represents the δI-closure of A, which is the set of all δI-cluster points of A. A subset A of X is called a δI-closed [26] if δClI(A) = A, and the complement of a δI-closed set is called a δI-open set. The union of all δI-open sets included in A is the δI-interior of A, which will be represented by δIntI(A). Lemma 1. [14] Let A and B be subsets of an ideal topological space (X, τ, I). The fol- lowing statements are true: (i) If A ⊂ B then δClI(A) ⊂ δClI(B). (ii) If A is an open set, then δClI(A) = A. (iii) If A is a closed set, then δIntI(A) = A. Definition 2. [14] A subset A of an ideal topological space (X, τ, I) is called δ-βI-open if A ⊂ Cl(Int(δClI(A))) and is called δ-βI-closed if Int(Cl(δIntI(A))) ⊂ A. Definition 3. [14] Let (X, τ, I) be an ideal topological space. The union of all δ-βI- open sets contained in A is called the δ-βI-interior of A denoted by δ-βIntI(A). The intersection of all δ-βI-closed sets containing A is called the δ-βI-closure of A denoted by δ-βClI(A). The following lemma is easily derived from the definition 3. Lemma 2. Let A be a subset of an ideal topological space (X, τ, I). The following state- ments are true: (i) δ-βClI(A) ⊂ Cl(A). (ii) If A is open, then A is δ-βI-open. (iii) If A is closed, then A is δ-βI-closed and δ-βClI(A) = A. C. Boonpok, A. Sama-ae / Eur. J. Pure Appl. Math, 18 (1) (2025), 5732 4 of 13 A collection V of subsets of a topological space X is said to be locally finite if each point x ∈ X has a neighborhood U that contains x and U intersects only finitely many of the sets in the collection V. The upcoming theorems will use the following lemmas. Lemma 3. [7] The union of a finite family of locally finite collection of sets in a topological space is a locally finite family of sets. Lemma 4. [8] If {Uα : α ∈ Λ} is a locally finite family of subsets in a topological space X and if Vα ⊂ Uα for all α ∈ Λ, then the family {Vα : α ∈ Λ} is a locally finite in X. Lemma 5. [13] Let (X, τ) and (Y, σ) be topological spaces. If f : (X, τ) → (Y, σ) is a continuous surjective function and {Uα : α ∈ Λ} is a locally finite in Y , then {f−1(Uα) : α ∈ Λ} is a locally finite in X. Let (X, τ) and (Y, σ) denote topological spaces. A function f : (X, τ) → (Y, σ) is called almost closed [20] if for any regular closed set F in X, the image f(F ) is closed in Y . A subset K of the space X is defined as N -closed relative to X [20] if every cover of K by regular open sets of X possesses a finite subcover. Lemma 6. [19] Let (X, τ) and (Y, σ) be topological spaces and f : (X, τ) → (Y, σ) be almost closed surjection with N -closed point inverse. If {Uα : α ∈ Λ} is a locally finite open cover of X, then {f(Uα) : α ∈ Λ} is a locally finite cover of Y . 3. δ1-βI-paracompactness of spaces and subsets This section discusses the concept of δ1-βI-paracompactness, a less strict form of β1- paracompactness examined by Qahis [21], followed by an exploration of its characteriza- tion. Al-Jarrah [3] defined β1-paracompactness as follows: A topological space (X, τ) is called β1-paracompact if every β-open cover of X has a locally finite open refine- ment. Qahis [21] expanded the notion of β1-paracompactness to β1-paracompactness concerning an ideal as follows: An ideal topological space (X, τ, I) is said to be β1I- paracompact if every β-open cover U of X has a locally finite open refinement V such that X − ∪{V : V ∈ V} ∈ I. Utilizing δ-βI-open sets, we establish a new type of paracom- pactness that is weaker than the one that Qahis developed. Definition 4. An ideal topological space (X, τ, I) is said to be δ1-βI-paracompact if every δ-βI-open cover U of X has a locally finite open refinement V (not necessarily a cover) such that X − ∪{V : V ∈ V} ∈ I. The collection V of subsets of X such that X − ∪{V : V ∈ V} ∈ I is called an I-cover of X. A subset A of an ideal topological space (X, τ, I) is said to be δ1-βI-paracompact relative to X if for every δ-βI-open cover U of A has a locally finite open refinement V such that A− ∪{V : V ∈ V} ∈ I. We have the following result based on the definition that was previously given. Theorem 1. If an ideal topological space (X, τ, I) is δ1-βI-paracompact, then it is β1I- paracompact. C. Boonpok, A. Sama-ae / Eur. J. Pure Appl. Math, 18 (1) (2025), 5732 5 of 13 Proof. The theorem is established as every β-open cover of X serves as a δ-βI-open cover. Corollary 1. If an ideal topological space (X, τ, I) is δ1-βI-paracompact, then it is para- compact. Proof. Since any β1I-paracompact space is always a paracompact space, the corollary is obtained. Consider the ideal topological space (X, τ, I), where X = {1, 2, 3}, τ = {∅, X, {1}} and I = {∅, {2}, {3}, {2, 3}}. Hence, the set of all δ-βI-open sets of X is {A : A ⊂ X}. Every δ-βI-open cover U of X possesses a locally finite open refinement {{1}}, such that X − {1} = {2, 3} ∈ I. Consequently, (X, τ, I) is δ1-βI-paracompact; but, (X, τ) is not β1-paracompact, as there exists a β-open cover {{1, 2}, {1, 3}} of (X, τ) that lacks a locally finite open refinement [21]. In the subsequent theorem, we discuss a space endowed with two topologies; hence, to avoid ambiguity, we must redefine the concept of local finiteness. A collection V of subsets of an ideal topological space (X, τ, I) is said to be τ -locally finite if for each x ∈ X, there exists an open set U ∈ τ such that x ∈ U and U intersects with at most finitely many elements of V. As stated in [14], the intersection of any two δ-βI-open sets is not necessarily a δ-βI-open set; therefore, this assumption must be made in the subsequent theorem. Theorem 2. Let (X, τ, I) be an ideal topological space. If I is codense and τ -simple, (X, τ∗, I) is δ1-βI-paracompact, any δ-βI-open set in (X, τ, I) is a δ-βI-open set in (X, τ∗, I), and the intersection of two δ-βI-open sets in (X, τ, I) remains a δ-βI-open set in (X, τ, I), then every δ-βI-open cover of (X, τ, I) has a locally finite δ-βI-open I-cover refinement. Proof. Let U = {Uα : α ∈ Λ1} be a δ-βI-open cover of (X, τ, I). Then, U is a δ-βI-open cover of (X, τ∗, I). Since (X, τ∗, I) is δ1-βI-paracompact, U has τ∗-locally finite refinement V = {Gλ : λ ∈ Λ2} of open sets in (X, τ∗, I) such that X − ∪{Gλ : λ ∈ Λ2} ∈ I, where Gλ = Vλ − Iλ, Vλ ∈ τ and Iλ ∈ I for all λ ∈ Λ2. Because V is τ∗-locally finite, for every x ∈ X, there exists H ∈ τ∗ containing x such that H ∩ Gλ = ∅ for λ ̸= λ1, λ2, ..., λn. Given that I is τ -simple, H = U − I, for some U ∈ τ and I ∈ I. Hence, (U − I)∩Gλ = ∅ for λ ̸= λ1, λ2, ..., λn, which implies that (U ∩ Vλ) − (I ∪ Iλ) = ∅ for λ ̸= λ1, λ2, ..., λn. As I is codense, U ∩ Vλ = ∅ for λ ̸= λ1, λ2, ..., λn, and therefore U ∩ (Vλ ∩ Uα) = ∅ for λ ̸= λ1, λ2, ..., λn and for all α ∈ Λ1. Consequently, W = {Vλ ∩ Uα : α ∈ Λ1, λ ∈ Λ2} is τ -locally finite, and W is a δ-βI-open refinement of U , following assumption. Next, we will show that W refines U . Because V refines U , for every Gλ ∈ V, there exists Uα ∈ U such that Gλ ⊂ Uα. Therefore Gλ = Uα∩Gλ = Uα∩(Vλ−Iλ) = (Vλ∩Uα)−Iλ ⊂ Vλ∩Uα ⊂ Uα. It implies that X−∪{Vλ∩Uα : α ∈ Λ1, λ ∈ Λ2} ⊂ X−∪{Gλ : λ ∈ Λ2} ∈ I, which implies that X − ∪{Vλ ∩ Uα : α ∈ Λ1, λ ∈ Λ2} ∈ I. The following lemma is required for the proof of Proposition 1. Lemma 7. Let (X, τ, I) be an ideal topological space and A ⊂ X. The following statements are true: C. Boonpok, A. Sama-ae / Eur. J. Pure Appl. Math, 18 (1) (2025), 5732 6 of 13 (i) x ∈ δ-βClI(A) if and only if G ∩A ̸= ∅ for every δ-βI-open set G containing x. (ii) G ∩ δ-βClI(A) = ∅ if and only if G ∩A = ∅, for every δ-βI-open set G. Proof. (i) It is derived from Theorem 1 in [14]. (ii) This is a consequence of (i). Proposition 1. Let (X, τ, I) be a δ1-βI-paracompact space. If for any δ-βI-open set U that contains x, there exists a δ-βI-open set V such that x ∈ V ⊂ δ-βClI(V ) ⊂ U , then every δ-βI-open cover of X has a δ-βI-locally finite δ-βI-closed I-cover refinement. Proof. Let U be a δ-βI-open cover of X. For each x ∈ X, let Ux be a δ-βI-open set in U containing x. By assumption, there exists a δ-βI-open set Vx such that x ∈ Vx ⊂ δ-βClI(Vx) ⊂ Ux. Thus V = {Vx : x ∈ X} is a δ-βI-open cover refinement of U . As (X, τ, I) is δ1-βI-paracompact, there exists a locally finite open refinement H = {Hα : α ∈ Λ} which refines V and X − ∪{Hα : α ∈ Λ} ∈ I. By (ii) of Lemma 7, H1 = {δ-βClI(Hα) : α ∈ Λ} is δ-βI-locally finite. As Hα ⊂ δ-βClI(Hα) for all α ∈ Λ, X − ∪{δ-βClI(Hα) : α ∈ Λ} ⊂ X − ∪{Hα : α ∈ Λ}, and it therefore implies that X − ∪{δ-βClI(Hα) : α ∈ Λ} ∈ I. Hence, H1 is an I-cover. Next, we shall verify that H1 refines U . Let δ-βClI(Hα) ∈ H1. Since H refines V, there exists Vx ∈ V such that Hα ⊂ Vx, it implies that δ-βClI(Hα) ⊂ δ-βClI(Vx) ⊂ Ux. Consequenty, H1 refines U . Therefore, the proposition has been established. Theorem 3. If an ideal topological space (X, τ, I) is δ1-βI-paracompact, then (X, τ∗, I) is δ1-βI-paracompact. Proof. Let U = {Uα : α ∈ Λ1} be a δ-βI-open cover of (X, τ∗, I), where Uα = Vα − Iα, Vα ∈ τ and Iα ∈ I for all α ∈ Λ1. Then, V = {Vα : α ∈ Λ1} is a δ-βI-open cover of (X, τ, I) and therefore there exists a τ -locally finite open refinement W = {Wλ : λ ∈ Λ2} such that X − ∪{Wλ : λ ∈ Λ2} ∈ I, as (X, τ, I) is δ1-βI-paracompact. Now, we have {Wλ ∩ Iα′ : λ ∈ Λ2}, for some α′ ∈ Λ1, is a set of subset of I and hence, by assumption, ∪λ∈Λ2(Wλ ∩ Iα′) ∈ I. Hence, X − ∪λ∈Λ2(Wλ − Iα′) ⊂ (X − ∪λ∈Λ2Wλ) ∪ (∪λ∈Λ2Wλ ∩ Iα′) ∈ I, which implies that X − ∪λ∈Λ2(Wλ − Iα′) ∈ I. As W is τ -locally finite, W ′ = {Wλ − Iα′ : λ ∈ Λ2} is τ -locally finite. Because τ∗ is finer than τ , W ′ is a τ∗-locally finite τ∗-open which refines U . Consequently, (X, τ∗, I) is δ1-βI-paracompact. A topological space (X, τ) is a T2 space if any two distinct points x and y in X, there exist disjoint open neighborhoods U and V such that x ∈ U and y ∈ V . The following theorem establishes a characteristic of a δ1-βI-paracompact subset within a T2 ideal topological space X. Theorem 4. If an ideal topological space (X, τ, I) is a T2 space and A is δ1-βI-paracompact relative to X, then A is closed in (X, τ∗, I). C. Boonpok, A. Sama-ae / Eur. J. Pure Appl. Math, 18 (1) (2025), 5732 7 of 13 Proof. We shall verify that A∗ ⊂ A. Suppose x ̸∈ A. As (X, τ, I) is T2, for each y ∈ A, there exists an open set Uy such that y ∈ Uy and x ̸∈ Cl(Uy). Thus, the family U = {Uy : y ∈ A} is an open cover of A, and hence it is a δ-βI-open cover of A. As A is δ1-βI-paracompact, U has a τ -locally finite open refinement V = {Vα : α ∈ Λ} of U such that A−∪{Vα : α ∈ Λ} ∈ I. Now x ̸∈ Cl(Vα) for all α implies that x ̸∈ ∪{Cl(Vα) : α ∈ Λ}. Because of ∪{Cl(Vα) : α ∈ Λ} = Cl(∪{Vα : α ∈ Λ}), x is not in Cl(∪{Vα : α ∈ Λ}). Let G1 = X − Cl(∪{Vα : α ∈ Λ}) and G2 = A − Cl(∪{Vα : α ∈ Λ}). We know that G1 ∈ τ , G2 ∈ I, (G1 −G2) ∩A = ∅, and x ∈ G1 −G2 ∈ τ∗, it follows that x ̸∈ A∗. Consequently, A∗ is closed in the topological space (X, τ∗). The theorem has been shown. Theorem 5. If an ideal topological space (X, τ, I) is δ1-βI-paracompact and A ⊂ X is δ-βI-closed in X, then A is δ1-βI-paracompact. Proof. Let U = {Uα : α ∈ Λ} be a δ-βI-open cover of A. As X − A is a δ-βI-open subset of X, H = {Uα : α ∈ Λ} ∪ {X − A} is a δ-βI-open cover of X. By assumption, H has a locally finite open refinement V = {Vλ : λ ∈ Λ1} ∪ {V } such that for each λ ∈ Λ1, Vλ ⊂ Uα for some α ∈ Λ, V ⊂ X − A and X − (∪{Vλ : λ ∈ Λ1} ∪ {V }) ∈ I. As A − ∪{Vλ : λ ∈ Λ1} = A ∩ X − ∪{Vλ : λ ∈ Λ1} = A ∩ X − ∪({Vλ : λ ∈ Λ1} ∪ {V }) ⊂ X − ∪({Vλ : λ ∈ Λ1} ∪ {V }), we have A − ∪{Vλ : λ ∈ Λ1} ∈ I. For any λ ∈ Λ1, there exists α ∈ Λ such that Vλ ⊂ Uα, showing that {Vλ : λ ∈ Λ1} represents a locally finite open refinement of U . This demonstrates that A is δ1-βI-paracompact. Theorem 6. Let A and B be subsets of an ideal topological space (X, τ, I). If A is δ1-βI- paracompact and B is δ-βI-closed in X, then A ∩B is δ1-βI-paracompact. Proof. Let U = {Uα : α ∈ Λ} be a δ-βI-open cover of A∩B. Since X−B is a δ-βI-open subset in X, U1 = {Uα : α ∈ Λ} ∪ {X − B} is a δ-βI-open cover of A. Since A is δ1-βI- paracompact, U1 has a locally finite open refinement V = {Vλ : λ ∈ Λ1} ∪ {V } such that for each λ ∈ Λ1, Vλ ⊂ Uα for some α ∈ Λ, V ⊂ X−B and A−(∪{Vλ : λ ∈ Λ1}∪{V }) ∈ I. As A∩B−∪{Vλ : λ ∈ Λ1} = A∩B−∪({Vλ : λ ∈ Λ1}∪{V }) ⊂ A−∪({Vλ : λ ∈ Λ1}∪{V }), we have A ∩ B − ∪{Vλ : λ ∈ Λ1} ∈ I. For any λ ∈ Λ1, there exists a α ∈ Λ such that Vλ ⊂ Uα, indicating that the collection {Vλ : λ ∈ Λ1} constitutes a locally finite open refinement of U . This suggests that A ∩B is δ1-βI-paracompact. Theorem 7. Let A and B be subsets of an ideal topological space (X, τ, I). If A and B are δ1-βI-paracompact in X, then A ∪B is δ1-βI-paracompact. Proof. Let U = {Uα : α ∈ Λ} be a δ-βI-open cover of A ∪ B. It follows that U is an δ-βI-open cover of A and B. By hypothesis, there are locally finite open families V = {Vλ : λ ∈ Λ1} of A and W = {Wµ : µ ∈ Λ2} of B which refine U such that A−∪{Vλ : λ ∈ Λ1} ∈ I and B − ∪{Wµ : µ ∈ Λ2} ∈ I. It suggests that A− ∪{Vλ : λ ∈ Λ1} = I1 and B − ∪{Wµ : µ ∈ Λ2} = I2, for some I1, I2 ∈ I. Hence, A ∪B ⊂ (∪{Vλ : λ ∈ Λ1} ∪ I1) ∪ (∪{Wµ : µ ∈ Λ2} ∪ I2) C. Boonpok, A. Sama-ae / Eur. J. Pure Appl. Math, 18 (1) (2025), 5732 8 of 13 = ∪{Vλ ∪Wµ : λ ∈ Λ1, µ ∈ Λ2} ∪ (I1 ∪ I2), which implies that A ∪ B ⊂ ∪{Vλ ∪Wµ : λ ∈ Λ1, µ ∈ Λ2} ∈ I. As V and W are locally finite, for any point x ∈ X there exist δ-βI-open sets G1 and G2 such that both G1 and G2 intersect at most finitely members of U and V, respectively. Thus, at most finitely many members of {Vλ ∪Wµ : λ ∈ Λ1, µ ∈ Λ2}, does G1 ∩G2 intersect. Therefore, A ∪ B is δ1-βI-paracompact. 4. Preserving δ1-βI-paracompactness In this section, we will illustrate how δ1-βI-paracompactness is maintained under spe- cific conditions. We start by introducing the subsequent definition. Definition 5. Let (X, τ, I), and (Y, τ ′,J ) be ideal topological spaces, and f : X → Y be a function. (i) f is called δ-βI-irresolute if f−1(V ) is a δ-βI-open set in X for every δ-βJ -open set V in Y . (ii) f is called δ-βI-open if f(U) is a δ-βJ -open set in Y for every δ-βI-open set U in X. (iii) f is called δ-βI-closed if f(U) is a δ-βJ -closed set in Y for every δ-βI-closed set U in X. Note that f−1(J ) is an ideal on X if f : X → Y is a function, (X, τ) is a topological space, and (Y, τ ′) is a topological space with an ideal J . Furthermore, given that f is surjective and X possesses an ideal I, f(I) is an ideal on Y . The proof of Theorem 8 requires the following lemma. Lemma 8. Let (X, τ, I) and (Y, τ ′,J ) be ideal topological spaces, and f : X → Y be surjective. Then f is δ-βI-closed if and only if for every y ∈ Y and for every δ-βI-open set U in X that contains {f−1(y)}, there exists a δ-βJ -open set V containing y in Y such that f−1(V ) ⊂ U . Proof. Let y ∈ Y and U be a δ-βI-open set inX such that {f−1(y)} ⊂ U . We have that V = Y −f(X−U) is a δ-βI-open set such that y ∈ V and f−1(V ) ⊂ U . Subsequently, the necessity is verified. Sufficiency will now be demonstrated. Let F be a δ-βI-closed subset of X and y ∈ Y − f(F ). Consequently, {f−1(y)} ⊂ X − F . According to the hypothesis, there exists a δ-βJ -open set Vy such that f−1(Vy) ⊂ X − F , thereby suggesting that y ∈ Vy ⊂ Y − f(F ). Therefore, Y − f(F ) = ∪{Vy : y ∈ Y } forms a δ-βI-open set in Y . Thus, f(F ) represents a δ-βI-closed set. We conclude that f is δ-βI-closed. Next, we provide the characteristics of a function that maps between two ideal topolog- ical spaces, where one space conveys identical properties to the other. Initially, we define the concept of δ-βI-compactness and offer a lemma utilized in the proof of the theorem 8. C. Boonpok, A. Sama-ae / Eur. J. Pure Appl. Math, 18 (1) (2025), 5732 9 of 13 Definition 6. An ideal topological space (X, τ, I) is said to be δ-βI-compact if every cover V of δ-βI-open sets of X has V1, V2, ..., Vn ∈ V such that X ⊂ V1 ∪ V2 ∪ · · · ∪ Vn. The next theorem describe characterizations of a function that maps from a δ1-βI- paracompact ideal topological space (X, τ, I) to an ideal topological space (Y, τ ′,J ), en- suring that Y has the same properties as X. Theorem 8. Let (X, τ, I) and (Y, τ ′,J ) be ideal topological spaces. Suppose that f : X → Y satisfies the following statements: (i) f is open; (ii) f is δ-βI-irresolute; (iii) f is δ-βI-closed; (iv) f is a surjective function with {f−1(y)} is δ-βI-compact for every y ∈ Y ; and (v) f(I) ⊂ J . If (X, τ, I) is δ1-βI-paracompact, then (Y, τ ′,J ) is δ1-βJ -paracompact. Proof. Let U = {Uλ : λ ∈ Λ} be a δ-βI-open cover of Y . Assuming that f is δ-βI- irresolute, it follows that H = {f−1(Uλ) : λ ∈ Λ} forms a δ-βI-open cover of X. Given that X is δ1-βI-paracompact, the collection H possesses a τ -locally finite refinement V = {Vα : α ∈ Λ1} such that X−∪{Vα : α ∈ Λ1} ∈ I. As f is open, f(V) = {f(Vα) : α ∈ Λ1} is an open refinement of U and Y −∪{f(Vα) : α ∈ Λ1} ∈ J . Next, we shall verify that f(V) is τ ′-locally finite. Let y ∈ Y . Since V is τ -locally finite, for x ∈ {f−1(y)}, there exists an open set Gx containing x such that Gx intersects at most finitely many members of V. Because {f−1(y)} is δ-βI-compact and {Gx : f(x) = y} forms an open cover of {f−1(y)}, there exists a finite subcollection Hy such that {f−1(y)} ⊂ ∪Hy, and ∪Hy intersects at most finitely many members of V. Assuming f is δ-βI-closed, by applying Lemma 8, there exists a δ-βJ -open set Wy containing y such that f−1(Wy) ⊂ ∪Hy. Hence, f −1(Wy) intersects at most finitely many members of V. This indicates that Wy intersects at most finitely many members of f(V). Consequently, since f(V) is a τ ′-locally finite in Y , it implies that (Y, τ ′,J ) is δ1-βJ -paracompact. The subsequent theorem and corollaries establish characterizations of a function that maps from a δ1-βI-paracompact ideal topological space (X, τ, I) to a topological space (Y, τ ′), guaranteeing that Y possesses the same characteristics as X. Theorem 9. Let (X, τ, I) be an ideal topological space and (Y, τ ′) be a topological space. Suppose that f : X → Y satisfies the following statements: (i) f is open; (ii) f is δ-βI-irresolute; (iii) f is almost closed; and C. Boonpok, A. Sama-ae / Eur. J. Pure Appl. Math, 18 (1) (2025), 5732 10 of 13 (iv) f is a surjective function with N -closed point inverse. If (X, τ, I) is δ1-βI-paracompact, then (Y, τ ′, f(I)) is δ1-βf(I)-paracompact. Proof. As f : X → Y is surjective, f(I) is an ideal on Y . Let U = {Uλ : λ ∈ Λ} be a δ-βf(I)-open cover of Y . As f is δ-βI-irresolute, H = {f−1(Uλ) : λ ∈ Λ} is a δ-βI-open cover of X. Since X is δ1-βI-paracompact, H has a locally finite open refinement H1 = {Hα : α ∈ Λ1} such that X −∪{Hα : α ∈ Λ1} ∈ I. Thus, f(X −∪{Hα : α ∈ Λ1}) ∈ f(I). We know that Y − ∪{f(Hα) : α ∈ Λ1} ⊂ f(X − ∪{Hα : α ∈ Λ1}), we therefore have that Y − ∪{f(Hα) : α ∈ Λ1} ∈ f(I). Since f is open, almost closed, surjective with N -closed point inverse, andH1 is locally finite, f(H1) = {f(Hα) : α ∈ Λ1} is locally finite by Lemma 6. Next, we shall verify that f(H1) refines U . Let f(Hα) ∈ f(H1). Then Hα ∈ H1. As H1 refines H, there exists f−1(Uλ) ∈ H such that Hα ⊂ f−1(Uλ) for some λ ∈ Λ. Therefore, f(Hα) ⊂ f(f−1Uλ)) ⊂ Uλ. This shows that (Y, τ ′, f(I)) is δ1-βf(I)-paracompact. As any compact set is a N -closed set and any closed map is an almost closed map, by Theorem 9, we have the following corollary. Corollary 2. Let a function f : (X, τ, I) → (Y, τ ′) be open, δ-βI-irresolute, closed, and surjective with compact point inverse. If (X, τ, I) is δ1-βI-paracompact, then (Y, τ ′, f(I)) is δ1-βf(I)-paracompact. By observing the proof of Theorem 9, we will obtain the following corollary. Corollary 3. Let a function f : (X, τ, I) → (Y, τ ′) be open, δ-βI-irresolute, almost closed, surjective, and f(V) is a τ ′-locally finite in Y for every τ -locally finite V in X. If (X, τ, I) is δ1-βI-paracompact, then (Y, τ ′, f(I)) is δ1-βf(I)-paracompact. The following theorem provides properties of a function that maps from a topological space X to a δ1-βI-paracompact ideal topological space Y guarantees that X exhibits identical characteristics to Y . Theorem 10. Let (X, τ) be a topological space and (Y, τ ′,J ) be an ideal topological space. Suppose that f : X → Y satisfies the following statements: (i) f is δ-βI-open; (ii) f is continuous; and (iii) f is bijective. If (Y, τ ′,J ) is δ1-βJ -paracompact, then (X, τ, f−1(J )) is δ1-βf−1(J )-paracompact. Proof. Let I = f−1(J ). Let U = {Uα : α ∈ Λ1} be a δ-βI-open cover of X. As f is δ-βI-open, f(U) = {f(Uα) : α ∈ Λ1} is a δ-βJ -open cover of Y . By hypothesis, f(U) has a locally finite open refinement H = {Hλ : λ ∈ Λ2} such that Y − ∪{Hλ : λ ∈ Λ2} ∈ J . It implies that Y − ∪{Hλ : λ ∈ Λ2} = J for some J ∈ J , which follows that Y = ∪{Hλ : λ ∈ Λ2} ∪ J . Hence, X = f−1(Y ) = f−1(∪{Hλ : λ ∈ Λ2} ∪ J) = C. Boonpok, A. Sama-ae / Eur. J. Pure Appl. Math, 18 (1) (2025), 5732 11 of 13 (∪{f−1(Hλ) : λ ∈ Λ2}) ∪ f−1(J). As f−1(J) ∈ I, X − ∪{f−1(Hλ) : λ ∈ Λ2} ∈ I. Consequently, V = {f−1(Hλ) : λ ∈ Λ2} is locally finite, as demonstrated by Lemma 5. Now we will verify that V refines U . Let f−1(Hλ) ∈ V. Hence Hλ ∈ H. Given that H refines f(U), there exists an element f(Uλ) ∈ f(U) such that Hλ ⊂ f(Uλ). This implies that f−1(Hλ) ⊂ f−1(f(Uλ)) = Uλ ∈ U . Therefore, X is δ1-βI-paracompact. The final result of this section presents attributes of a function whereby the inverse image of a δ1-βJ -paracompact subset is a δ1-βI-paracompact subset. Theorem 11. Let (X, τ, I) and (Y, τ ′,J ) be ideal topological spaces. Suppose that f : X → Y is δ-βI-open, continuous, and bijective with f(I) = J . If A ⊂ Y is a δ1-βJ - paracompact subset of Y , then f−1(A) is a δ1-βI-paracompact subset of X. Proof. Let U = {Uλ : λ ∈ Λ} be a δ-βI-open cover of f−1(A) in X. Since f is δ-βI-open, f(U) = {f(Uλ) : λ ∈ Λ} is a δ-βI-open cover of A in Y . By hypothesis, f(U) has a locally finite open refinement H = {Vα : α ∈ Λ1} of A such that A − ∪{Vα : α ∈ Λ1} ∈ J . Then, f−1(A) − ∪{f−1(Vα) : α ∈ Λ1} ∈ f−1(J ) = I. As f is continuous, V = {f−1(Vα) : α ∈ Λ1} is locally finite. Next, we will prove that V refines U . Let f−1(Vα) ∈ V. As H refines f(U), there exists f(Uλ) ∈ f(U) such that Vα ⊂ f(Uλ). It follows that f−1(Vα) ⊂ f−1(f(Uλ)) = Uλ. Therefore f−1(A) is a δ1-βI-paracompact subset of X. 5. Conclusion This paper examines the characteristics of δ1-βI-paracompact spaces, which are broader in scope than the β1-paracompact spaces defined by Qahis [21]. Additionally, we investi- gate the invariants of δ1-βI-paracompactness via functions. We established that if (X, τ, I) is δ1-βI-paracompact, then (X, τ∗, I) is δ1-βI-paracompact, and that every δ1-βI-closed subset of a δ1-βI-paracompact space is a δ1-βI-paracompact subset. The union of two δ1-βI-paracompact subsets is a δ1-βI-paracompact subset, and the intersection of a δ1-βI- paracompact subset with a δ1-βI-closed set is δ1-βI-paracompact subset. Furthermore, we demonstrate that δ1-βI-paracompactness is preserved under specific conditions: If f : X → Y is open, δ-βI-irresolute, almost closed, surjective with N -closed point in- verse, and if (X, τ, I) is δ1-βI-paracompact, then (Y, τ ′, f(I)) is δ1-βf(I)-paracompact. If f : X → Y is δ-βI-open, continuous, bijective, and if (Y, τ ′,J ) is δ1-βJ -paracompact, then (X, τ, f−1(J )) is δ1-βf−1(J )-paracompact. Acknowledgements The authors would like to express their gratitude to the reviewers for their helpful comments and suggestions that assisted enhance this paper better. This research was supported by the National Science, Research, and Innovation Fund (NSRF) and Prince of Songkla University (Ref. No. SAT6701343S). C. Boonpok, A. Sama-ae / Eur. J. Pure Appl. Math, 18 (1) (2025), 5732 12 of 13 References [1] M. E. Abd.El-Monsef, E. F. Lashien, and A. A. Nasef. On I-open sets and I- continuous functions. Kyungpook Mathematical Journal, 32(2):21–30, 1992. [2] M. E. Abd.El-Monsef, E. F. Lashien, and A. A. Nasef. Some topological operators via ideals. Kyungpook Mathematical Journal, 32(2):273284, 1992. [3] H. H. Al-Jarrah. β1-paracompact spaces. Journal of Nonlinear Sciences and Appli- cations, 9:1728–1734, 2016. [4] K. Al-Zoubi. S-paracompact spaces. Acta Mathematica Hungarica, 110:203–212, 2006. [5] K. Al-Zoubi and S. Al-Ghour. On P3-paracompact spaces. International Journal of Mathematics and Mathematical Sciences, 2007:1–16, 2007. [6] R. Alrababah, A. Amourah, J. Salah, and R. Ahmad. Paracompactness in Topological Spaces. European Journal of Pure and Applied Mathematic, 17(4):2990–3003, 2024. [7] A. V. Arkhangelskii and V. I. Ponomarev. Fundamentals of General Topology: Prob- lems and Exercises. International Hindustan Publishing, India, 1984. [8] N. Bourbaki. General Topology. Hermann Addison Wesley, Massachusets, 1966. [9] I. Demir and O.B. Ozbakir. On β-paracompact spaces. Filomat, 27(6):971–976, 2013. [10] J. Dieudonné. Une généralisation des espaces compacts. Journal de Mathématiques Pures et Appliquées, 23(9):65–76, 1944. [11] J. Dontchev, M. Ganster, and T. Noiri. Unified operation approach of generalized closed sets via topological ideals. Mathematica Japonica, 49:395–402, 1999. [12] J. Dugundji. Topology. Allyn and Bacon, Boston, 1966. [13] T. R. Hamlett, D. Rose, and D. Jankovic. Paracompactness with respect to an ideal. International Journal of Mathematics and Mathematical Sciences, 20:433–442, 1997. [14] E. Hatir. On Decompositions of continuity and complete continuity in ideal Topo- logical Spaces. European Journal of Pure and Applied Mathematics, 6(3):352–362, 2013. [15] D. Janković and T. R. Hamlett. New topologies from old via ideals. The American Mathematical Monthly, 97(4):295–310, 1990. [16] M. Khan and T. Noiri. Semi-local functions in ideal topological spaces. Journal of Advanced Research in Pure Mathematics, 2(1):36–42, 2010. [17] K. Kuratowski. Topology I. Państwowe Wydawnictwo Naukowe, Warszawa, 1933. [18] P. Y. Li and Y. K. Song. Some remarks on S-paracompact spaces. Acta Mathematica Hungarica, 118:345–355, 2008. [19] T. Noiri. Completely continuous image of nearly paracompact space. Matematichki Vesnik, 29:59–64, 1977. [20] T. Noiri. A note on inverse-preservations of regular open sets. Publications de l’Institut Mathématique. Nouvelle Série, 50:99–102, 1984. [21] A. Qahis. β1-paracompact spaces with respect to an ideal. European Journal of Pure and Applied Mathematics, 12(1):135–145, 2019. [22] J. Sanabria, E. Rosas, C. Carpintero, M. Salas-Brown, and O. Garćıa. S- paracompactness in ideal topological spaces. Matematički Vesnik, 68(3):192–203, C. Boonpok, A. Sama-ae / Eur. J. Pure Appl. Math, 18 (1) (2025), 5732 13 of 13 2016. [23] N. Sathiyasundari and V. Renukadevi. Paracompactness with respect to an ideal. Filomat, 20(2):333–339, 2013. [24] M. K. Singal and S. P. Arya. On nearly paracompact spaces. Matematički Vesnik, 6(47):3–16, 1969. [25] E. D. Yildirim, O. B. Ozbakir, and A. C. Guler. β-paracompactness in ideal topo- logical space. European Journal of Pure and Applied Mathematics, 12(2):270–278, 2019. [26] S. Yuksel, A. Acikgoz, and T. Noiri. On δ-I-continuous functions. Acta Mathematica Hungarica, 29:39–51, 2005. [27] M. I. Zahid. Para H-closed Spaces, Locally Para H-closed Spaces and Their Minimal Topologies. PhD thesis, University of Pittsburgh, 1981.