EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 1, Article Number 5570 ISSN 1307-5543 – ejpam.com Published by New York Business Global Characterizations of Generalized Paracompactness in Ideal Topological Spaces Chawalit Boonpok1, Areeyuth Sama-Ae2,∗, Palin Raktaow3 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 3 Applied Mathematics and Innovation of Mathematics Teaching, Faculty of Science and Technology, Prince of Songkla University, Pattani Campus, Pattani, 94000, Thailand Abstract. The concept of δ-βI-paracompactness in ideal topological spaces is introduced as a weaker form of β-paracompactness, which was looked at in [17]. This study examines several characterizations of δ-βI-paracompact spaces and its subsets. Furthermore, we investigate the invariants of δ-βI-paracompactness through functions. 2020 Mathematics Subject Classifications: 54A05, 54B05, 54C08 Key Words and Phrases: Ideal topological space, δ-βI-paracompactness, locally finite collection 1. Introduction Established considerably later than the two earlier classes, paracompact spaces are considered one of the most important classes of topological spaces, concurrently generaliz- ing both metrizable and compact spaces. Topologists and analysts quickly acknowledged paracompact areas. The concept of a paracompact space in mathematics refers to a topological space in which each open cover possesses an open refinement that is locally finite. This concept of spaces was first developed by Dieudonné [4] in 1944. A Haus- dorff space is considered paracompact if and only if it allows partitions of unity that are subordinate to any open cover. All paracompact Hausdorff spaces are normal; see [6]. In literature, different kinds of generalized paracompactness, such as S-paracompactness [1], P3-paracompactness [3], and β-paracompactness [2], are studied. In 2006, Al-Zoubi [1] used semi-open sets to define S-paracompact spaces, which are a generalization of paracompact spaces, and studied the relationship between the spaces. Li and Song [14] ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5570 Email addresses: chawalit.b@msu.ac.th (C. Boonpok), areeyuth.s@psu.ac.th (A. Sama-Ae), palin.rt@gmail.com (P. Raktaow) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) C. Boonpok, A. Sama-Ae, P. Raktaow / Eur. J. Pure Appl. Math, 18 (1) (2025), 5570 2 of 12 constructed a Hausdorff S-paracompact space that is not a paracompact space and studied more characterizations of S-paracompact spaces. An ideal topological space was proposed by Kuratowski in 1930 [13]. Moreover, Jankovic and Hamlett [11] have examined and described the significant properties of ideal topological spaces. They established the concept of I-open sets and undertook comprehen- sive investigations into topologies utilizing ideals. Abd El-Monsef et al. [7] performed an advanced examination into the notions of I-open sets. Ig-closed sets were first introduced by Dontchev et al. in 1999 [5]. Abd El-Monsef et al. [8] first introduced the concept of the s-local function, which was later examined by Khan and Noiri [12]. The concept of paracompactness with respect to an ideal was initially introduced by Zahid [19] and later investigated by Hamlett et al. [9]. In addition, Sathiyasundari and Renukadevi [16] explored the concept of I-paracompact and examined its characteris- tics. They extended certain results derived from paracompact spaces to the notion of I-paracompact spaces. The notion of S-paracompactness in ideal topological spaces was studied by Sanabria et al. [15]. Their work involved the introduction and examination of a new kind of space, namely I-S-paracompact spaces, which are defined on an ideal topological space. This class includes spaces that are S-paracompact and I-paracompact. In 2013, Demir and Ozbakir [3] introduced a diminished variant of expandable and paracompact spaces, termed β-expandable spaces and β-paracompact spaces, respectively. The proof was presented indicating that any β-paracompact space is essentially a β- expandable space. Yildirim et al. [17] introduced the notion of β-paracompactness in- side an ideal topological space and performed a comparison study with existing types of paracompactness. In this paper, the δ-βI-paracompact spaces are constructed using δ-βI- open sets. The spaces under examination are an extension of the β-paracompact spaces as delineated in reference [17]. 2. Preliminaries Throughout this paper, spaces (X, τ) and (Y, σ) (or simply X and Y ), always mean topological spaces on which no separation axiom is assumed. For a subset A of a topo- logical space (X, τ), CI(A) and Int(A) will denote the closure and interior of A in (X, τ), respectively. An ideal I on a topological space (X, τ) is a nonempty collection of subsets of X which satisfies: (i) A ∈ I and B ⊂ A implies B ∈ I, (ii) A ∈ I and B ∈ I implies A ∪B ∈ I. An ideal topological space (X, τ, I) is a topological space (X, τ) with an ideal I on X. The set of all subsets of X is denoted as P (X). A set operator (.)∗ : P (X) → P (X), which is a local function [13], is defined with respect to τ and I: for A ⊂ X, A∗(I, τ) = {x ∈ X : U ∩A ̸∈ I for every U ∈ τ(x)}, where τ(x) = {U ∈ τ : x ∈ U}. We simply write A∗ instead of A∗(I, τ). X∗ is often a proper subset of X and X = X∗ if τ ∩ I = {∅}. A topology τ∗(I), or more simply τ∗, finer than τ , exists for any ideal topological space and is generated by β(I, τ) = {U − I : U ∈ τ and I ∈ I}. However, in general, β(I, τ) is not always a topology. Additionally, Cl∗(A) = A ∪ A∗ defines a Kuratowski closure operator C. Boonpok, A. Sama-Ae, P. Raktaow / Eur. J. Pure Appl. Math, 18 (1) (2025), 5570 3 of 12 for τ∗(I). Lemma 1. [11] Let A and B be subsets of an ideal topological space (X, τ, I). Then the following statements are true: (i) If A ⊂ B, then A∗ ⊂ B∗; (ii) G ∩A∗ ⊂ (G ∩A)∗ for all G ∈ τ ; (iii) A∗ = Cl(A∗) ⊂ Cl(A). Definition 1. [10] 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 each neighborhood U of x. The set of all δI-cluster points of A is called the δI-closure of A and will be denoted by δClI(A). A is said to be δI-closed [18] if δClI(A) = A. The complement of a δI-closed set is called a δI-open set. δI-interior of A, will be denoted by δIntI(A), is the union of all δI-open sets contained in A. Lemma 2. [10] Let A and B be subsets of an ideal topological space (X, τ, I). Then the following 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. [10] A subset A of an ideal topological space (X, τ, I) is called δ-βI-open if A ⊂ Cl(Int(δClI(A))) and it is called δ-βI-closed if Int(Cl(δIntI(A))) ⊂ A. Definition 3. [10] 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). Lemma 3. Let A be a subset of an ideal topological space (X, τ, I). Then, (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. (iv) x ∈ δ-βClI(A) if and only if A ∩ V ̸= ∅ for every δ-βI-open set V containing x. Proof. (i) Applying the closure and δ-βI-closure definitions in X, we obtain (i). (ii) If A is open, then A = Int(A). Consequently, A = δClI(A) as stated in Lemma 2. Therefore, A = Int(A) = Int(δClI(A)) ⊂ Cl(Int(δClI(A))). Thus, A is characterized as a δ-βI-open set. C. Boonpok, A. Sama-Ae, P. Raktaow / Eur. J. Pure Appl. Math, 18 (1) (2025), 5570 4 of 12 (iii) In order to demonstrate (iii), we assume that A is closed. This implies that A = Cl(A), and by Lemma 2, δIntI(A) = A. Therefore, Cl(δIntI(A)) = Cl(A) = A, which implies that Int(Cl(δIntI(A))) = Int(A) ⊂ A. Consequently, A is δ-βI-closed. It implies that δ-βClI(A) ⊂ Cl(A) = A. We deduce that δ-βClI(A) = A. (iv) Let x ∈ δ-βClI(A). The point x is then included in all δ-βI-closed sets that contain A. Assume that V ∩ A = ∅ for a δ-βI-open set V that contains x. It implies that X − V is a δ-βI-closed set that contains A but x ̸∈ X − V . Consequently, we have a contraction. As a result, A ∩ V ̸= ∅ for any δ-βI-open set V that contains x. According to Theorem 1 in [10], if A ∩ V ̸= ∅ for every δ-βI-open sets V containing x, then x ∈ δ-βClI(A). 3. δ-βI-paracompactness and Characterizations This part talks about the idea of δ-βI-paracompactness, which is a weaker form of I-β-paracompactness that was studied by Yildirim et al. [17]. We will then look at how to describe it. Let U = {Uα : α ∈ Λ1} and V = {Vµ : µ ∈ Λ2} be two collections of subsets of a topological space X. The collection U is called a refinement of the collection V if for every α ∈ Λ1 there exists µ ∈ Λ2 such that Uα ⊂ Vµ. A collection V of subsets of a topological space (X, τ) is said to be β-locally finite [3] if for each x ∈ X, there exists a β-open set U containing x and U intersects at most finitely many members of V. Yildirim et al. [17] introduced the concept of paracompactness in an ideal topological space as follows: an ideal topological space (X, τ, I) is said to be I-β-paracompact if every open cover U of X has a β-locally finite β-open refinement V such that X −∪{V : V ∈ V} ∈ I. Definition 4. A collection V of subsets of an ideal topological space (X, τ, I) is said to be δ-βI-locally finite if for each x ∈ X, there exists a δ-βI-open set U containing x and U intersects at most finitely many members of V. Lemma 4. Let V be a collection of subsets of an ideal topological space (X, τ, I). If V is β-locally finite, then it is δ-βI-locally finite. Proof. Let V be β-locally finite. We will verify that V is δ-βI-locally finite. Let x ∈ X. Since V is β-locally finite, there exists a β-open set Gx containing x, which intersects at most finitely many elements of V. Given that Gx is β-open, it follows that it is δ-βI-open [10]. Consequently, V is δ-βI-locally finite. Definition 5. An ideal topological space (X, τ, I) is said to be δ-βI-paracompact if every open cover U of X has a δ-βI-locally finite δ-βI-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. A subset A of an ideal topological space (X, τ, I) is said to be δ-βI-paracompact if for any open cover U of A has a δ-βI-locally finite δ-βI-open refinement V such that A− ∪{V : V ∈ V} ∈ I. The two theorems that follow arise from the fact that every open set is β-open, every β-open is δ-βI-open and ∅ is in any ideal. C. Boonpok, A. Sama-Ae, P. Raktaow / Eur. J. Pure Appl. Math, 18 (1) (2025), 5570 5 of 12 Theorem 1. If a topological space (X, τ) is paracompact, then (X, τ, I) is δ-βI-paracompact. Proof. It is evident, as ∅ ∈ I. Theorem 2. If (X, τ, I) is I-β-paracompact then it is δ-βI-paracompact. Proof. Every β-locally finite collection of subsets of X is δ-βI-locally finite, as demon- strated by Lemma 4. Furthermore, each β-open set is δ-βI-open [10]. We can con- tinue with the proof by following to the definitions of I-β-paracompactness and δ-βI- paracompactness. Consider the set X consisting of all positive integers. Let τ = {∅}∪{X}∪{{1, 2, ..., n} : n ∈ X} be a topology on X. Define I = {H ⊂ X : 1 ̸∈ H} as an ideal on the set X. An open cover of X is defined as W = {{1, 2, 3, ..., n} : n ∈ X}. However, there is no locally finite open refinement V that covers X. Thus, (X, τ) is not a paracompact space. But, X is a δ-βI-paracompact space, as for each open cover U of X has a δ-βI-locally finite δ-βI-open refinement V = {{1}} such that X − {1} = {2, 3, ...} ∈ I. Theorem 3. Let (X, τ, I) be an ideal topological space and let G be a δ-βI-open subset of X. Then G ∩ δ-βClI(A) = ∅ if and only if G ∩A = ∅, for all A ⊂ X. Proof. It follows from (iv) of Lemma 3 and the fact that A ⊂ δ-βClI(A). Theorem 4. Let V = {Vλ : λ ∈ Λ} be a collection of subsets of a topological space (X, τ). The following statements are true. (i) If V is δ-βI-locally finite and Hλ ⊂ Vλ for all λ ∈ Λ, then H = {Hλ : λ ∈ Λ} is δ-βI-locally finite. (ii) V is δ-βI-locally finite if and only if {δ-βClI(Vλ) : λ ∈ Λ} is δ-βI-locally finite. Proof. (i) Let x ∈ X. Since V is δ-βI-locally finite, there exists a δ-βI-open set U containing x, which intersects at most finitely many elements of V. As Hλ ⊂ Vλ for all λ ∈ Λ, U intersects with at most finitely many of the elements in H = {Hλ : λ ∈ Λ}. Hence, H = {Hλ : λ ∈ Λ} is δ-βI-locally finite. (ii) Let V be δ-βI-locally finite and let x ∈ X. Consequently, there exists a δ-βI-open set G that contains x and satisfies G ∩ Vλ = ∅ for every λ ̸= λ1, λ2, . . . , λn. According to Theorem 3, it follows that G ∩ δ-βClI(Vλ) = ∅ for every λ ̸= λ1, λ2, . . . , λn. Thus, {δ-βClI(Vλ) : λ ∈ Λ} is δ-βI-locally finite. If {δ-βClI(Vλ) : λ ∈ Λ} is δ-βI-locally finite, then V is δ-βI-locally finite, according to (i). Therefore, (ii) has been demonstrated. Theorem 5. If (X, τ, I) is δ-βI-paracompact and J is an ideal on X with I ⊂ J , then (X, τ,J ) is δ-βJ -paracompact. Proof. Let (X, τ, I) be δ-βI-paracompact and I ⊂ J , and let U = {Uα : α ∈ Λ} be an open cover ofX. Since (X, τ, I) is δ-βI-paracompact, U has a δ-βI-locally finite refinement V of δ-βI-open sets such that X −∪{V : V ∈ V} ∈ I. As I ⊂ J , X −∪{V : V ∈ V} ∈ J . Consequently, (X, τ,J ) is δ-βJ -paracompact. C. Boonpok, A. Sama-Ae, P. Raktaow / Eur. J. Pure Appl. Math, 18 (1) (2025), 5570 6 of 12 Lemma 5. If an open cover U = {Uλ : λ ∈ Λ} of an ideal topological space (X, τ, I) has a δ-βI-locally finite of δ-βI-open refinement V such that X −∪{V : V ∈ V} ∈ I, then there exists a precise δ-βI-locally finite δ-βI-open refinement H = {Hλ : λ ∈ Λ} of U such that X − ∪{Hλ : λ ∈ Λ} ∈ I. Proof. The proof is comparable to that of Lemma 1.3 in [15]. Definition 6. An ideal topological space (X, τ, I) is δ-βI-regular if for any closed subset F of X and x ̸∈ F , there exist disjoint δ-βI-open sets U and V such that x ∈ U and F − V ∈ I. Theorem 6. Let (X, τ, I) be an ideal topological space. Assume that the subsequent as- sertions are true: (i) X is δ-βI-paracompact; (ii) X is Hausdorff; (iii) The arbitrary union of δ-βI-closed; sets remains δ-βI-closed. Then (X, τ, I) is δ-βI-regular. Proof. Let F be a closed subset of X, and let x ̸∈ F . Utilizing (ii), there exist disjoint open sets Vx and Ox that include x and y, respectively, for any point y ∈ F . It suggests that y ̸∈ Cl(Vx). Now we have that U = {Ox : x ∈ F} ∪ {X − F} is an open cover of X. By (i), there exists a δ-βI-locally finite δ-βI-open refinement H = {Hx : x ∈ F} ∪ {W} such that Hx ⊂ Ox for each x, W ⊂ X − F , and X − (∪{Hx : x ∈ F} ∪ {W}) ∈ I. Assume that V = ∪{Hx : x ∈ F} and U = X − ∪{δ-βClI(Hx) : x ∈ F}. Using (iii), U and V , therefore, are disjoint δ-βI-open sets in which x ∈ U and F − V ⊂ X − F ∈ I. Consequently, (X, τ, I) is δ-βI-regular. Theorem 7. Let (X, τ, I) be an ideal topological space. Suppose that the following state- ments hold: (i) X is δ-βI-paracompact; (ii) X is Hausdorff; (iii) δ-βClI(∪{Vλ : λ ∈ Λ}) = ∪{δ-βClI(Vλ) : λ ∈ Λ}, for any δ-βI-locally finite collec- tion V = {Vλ : λ ∈ Λ} of X. Then (X, τ, I) is δ-βI-regular. Proof. Let F be a closed set and x ̸∈ F . Using (ii), for any y ∈ F , there exists an open set Gy containing y such that x ̸∈ Cl(Gy). Then, G = {Gy : y ∈ F} ∪ {X − F} is an open cover of X. By (i) and Lemma 5, G has a precise δ-βI-locally finite δ-βI-open refinement W = {Wy : y ∈ F} ∪ {G} such that Wy ⊂ Gy for each y ∈ F , G ⊂ X − F C. Boonpok, A. Sama-Ae, P. Raktaow / Eur. J. Pure Appl. Math, 18 (1) (2025), 5570 7 of 12 and X − (∪{Wy : y ∈ F} ∪ {G}) ∈ I. As F − ∪{Wy : y ∈ F} = F − (∪{Wy : y ∈ F} ∪ {G}) ⊂ X − (∪{Wy : y ∈ F} ∪ {G}), we get that F −∪{Wy : y ∈ F} ∈ I. It follows that V = ∪{Wy : y ∈ F} is a δ-βI-open set in X and F − V ∈ I. Given that x ̸∈ Cl(Gy), it follows that x ̸∈ Cl(Wy), and consequently, x ̸∈ δ-βClI(Wy). By (iii), the fact that W is δ-βI-locally finite suggests that δ-βClI(∪{Wy : y ∈ F}) = ∪{δ-βClI(Wy) : y ∈ F}. We now obtain U ∩V = ∅ such that x ∈ U for a δ-βI-open set U = X−δ-βClI(V ). Therefore, (X, τ, I) is δ-βI-regular. Theorem 8. If an ideal topological space (X, τ, I) is δ-βI-paracompact and regular, then every open cover of X has a δ-βI-locally finite I-cover refinement of closed sets. Proof. Let U be an open cover of X. By regularity of X, for each x ∈ X and Ux ∈ U containing x, there exists an open set Gx of x such that Cl(Gx) ⊂ Ux. Thus U1 = {Gx : x ∈ X} is an open cover of X. Since X is δ-βI-paracompact, U1 has a δ-βI-locally finite refinement V1 = {Vλ : λ ∈ Λ} of δ-βI-open sets such that X − ∪{Vλ : λ ∈ Λ} ∈ I. As Vλ ⊂ δ-βClI(Vλ) and I is an ideal, X − ∪{δ-βClI(Vλ) : λ ∈ Λ} ∈ I. By Theorem 4, V = {δ-βClI(Vλ) : Vλ ∈ V1} is δ-βI-locally finite. Because V1 refines U1, for every λ ∈ Λ, there is some Gx ∈ U1 such that Vλ ⊂ Gx. Then δ-βClI(Vλ) ⊂ Cl(Vλ) ⊂ Cl(Gx), and hence δ-βClI(Vλ) ⊂ Ux. Therefore V = {δ-βClI(Vλ) : Vλ ∈ V1} refines U , and hence V is a δ-βI-locally finite I-cover refinement of closed sets. Theorem 9. Let (X, τ, I) be an ideal topological space. The following statements are equivalent: (i) For every closed subset F of X and every x ̸∈ F , there exist disjoint δ-βI-open sets U and V such that x ∈ U and F − V ∈ I. (ii) For every open subset G of X and every x ∈ G, there exists a δ-βI-open set U such that x ∈ U and δ-βClI(U)−G ∈ I. Proof. (i) ⇒ (ii) LetG be open and x ∈ G. Then, X−G remains closed, and x does not belong to X −G. By (i), there exist disjoint δ-βI-open sets U and V such that x ∈ U and (X −G)− V ∈ I. As U and V are disjoint, by Theorem 3, we have δ-βClI(U) ⊂ X − V . That implies δ-βClI(U) ∩ (X − G) ⊂ (X − G) − V . Hence δ-βClI(U) ∩ (X − G) = δ-βClI(U)−G ∈ I. (ii) ⇒ (i) Let F be closed and x ̸∈ F . It implies that X − F is open and x ∈ X − F . By (ii), there exists a δ-βI-open set U such that x ∈ U and δ-βClI(U) − (X − F ) ∈ I. Hence, V = X − δ-βClI(U) is a δ-βI-open set, and U ∩ V = ∅. That is F − V = F − (X − δ-βClI(U)) = δ-βClI(U)− (X − F ) ∈ I. 4. δ-βI-paracompactness of subsets In the preceding section, we presented the notion of δ-βI-paracompactness for subsets of an ideal topological space. Prior to delineating the characterizations, we note that the C. Boonpok, A. Sama-Ae, P. Raktaow / Eur. J. Pure Appl. Math, 18 (1) (2025), 5570 8 of 12 union of a finite family of δ-βI-locally finite collections of sets inside an ideal topological space remains δ-βI-locally finite. Theorem 10. Let A and B be subsets of an ideal topological space (X, τ, I). If A and B are δ-βI-paracompact in X, then A ∪B is also δ-βI-paracompact. Proof. Let U = {Uλ : λ ∈ Λ} be an open cover of A ∪ B. Then U is an open cover of A and B. Hence there are δ-βI-locally finite δ-βI-open families V = {Vα : α ∈ Λ1} of A and W = {Wµ : µ ∈ Λ2} of B which refine U such that A − ∪{Vα : α ∈ Λ1} ∈ I and B − ∪{Wµ : µ ∈ Λ2} ∈ I. It implies that A − ∪{Vα : α ∈ Λ1} = I1 and B − ∪{Wµ : µ ∈ Λ2} = I2, where I1, I2 ∈ I. Therefore, A∪B = ∪{Vα : α ∈ Λ1}∪{Wµ : µ ∈ Λ2}∪ (I1∪I2). It follows that A ∪ B − ∪{Vα ∪ Wµ : α ∈ Λ1, µ ∈ Λ2} ∈ I. We see that the collection H = {Vα ∪Wµ : α ∈ Λ1, µ ∈ Λ2} of δ-βI-open sets is a δ-βI-locally finite and H refines U . Consequently, A ∪B is δ-βI-paracompact. Theorem 11. Let (X, τ, I) be an ideal topological space. If A is δ-βI-paracompact and B is closed in X, then A ∩B is δ-βI-paracompact. Proof. Let U = {Uλ : λ ∈ Λ} be an open cover of A ∩ B. As X − B is open in X, U1 = {Uλ : λ ∈ Λ} ∪ {X − B} is an open cover of A. By assumption and Lemma 5, U1 has a δ-βI-locally finite precise δ-βI-open refinement V = {Vλ : λ ∈ Λ} ∪ {V } such that Vλ ⊂ Uλ for every λ ∈ Λ, and V ⊂ X −B such that A−∪({Vλ : λ ∈ Λ}∪ {V }) ∈ I. Since A∩B−∪{Vλ : λ ∈ Λ} = A∩B−∪({Vλ : λ ∈ Λ}∪{V }) ⊂ A−∪({Vλ : λ ∈ Λ}∪{V }), we have that A∩B−∪{Vλ : λ ∈ Λ} ∈ I. It is obvious that the collection V1 = {Vλ : λ ∈ Λ} of δ-βI-open sets is a δ-βI-locally finite and refines U . A∩B is therefore δ-βI-paracompact. Corollary 1. If A is a closed subset of ideal topological space (X, τ, I) which is δ-βI- paracompact, then A is δ-βI-paracompact. Corollary 2. If A is δ-βI-paracompact in X and B is an open contained A, then A−B is δ-βI-paracompact. Lemma 6. [9] Let I be an ideal on a topological space X. If Y is a subset of X, then IY = {I ∩ Y : I ∈ I} is an ideal on Y . Theorem 12. Let A and B be subsets of an ideal topological space (X, τ, I) such that A ⊂ B. If A is δ-βIB-paracompact in B and B is δ-βI-open in X, then A is δ-βI- paracompact in X. Proof. Let U = {Uλ : λ ∈ Λ} be an open cover of A in X. Then UA = {Uλ∩B : λ ∈ Λ} is an open cover of A in B. As A is δ-βIB -paracompact in B, UA has a δ-βIB -locally finite precise δ-βIB -open refinement VA = {Vλ∩B : λ ∈ Λ} such that Vλ ⊂ Uλ for all λ ∈ Λ, and A− ∪{Vλ ∩B : λ ∈ Λ} ∈ IB. Since Vλ is δ-βI-open in X, the collection V = {Vλ : λ ∈ Λ} of δ-βI-open sets in X is δ-βI-locally finite that refines U , and A − ∪{Vλ : λ ∈ Λ} ⊂ A− ∪{Vλ ∩B : λ ∈ Λ} ∈ IB ⊂ I. Subsequently, A is δ-βI-paracompact in X. C. Boonpok, A. Sama-Ae, P. Raktaow / Eur. J. Pure Appl. Math, 18 (1) (2025), 5570 9 of 12 5. Preserving paracompactness This section will illustrate the preservation of δ-βJ -paracompactness under specific situations. We start by delineating the subsequent definitions. Definition 7. 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-open if f(G) is a δ-βJ -open set in Y for every δ-βI-open set G in X. (ii) f is called δ-βI-closed if f(F ) is a δ-βJ -closed set in Y for every δ-βI-closed set F in X. (iii) 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 . Definition 8. An ideal topological space (X, τ, I) is said to be δ-βI-compact if every cover V of δ-βI-open subsets of X has V1, V2, ..., Vn ∈ V such that X ⊂ V1 ∪ V2 ∪ · · · ∪ Vn 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 . Before establishing Theorem 13, we will first present the following lemma. Lemma 7. 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 a δ-βI-open set U in X containing {f−1(y)}, there exists a δ-βJ -open set V containing y such that f−1(V ) ⊂ U . Proof. Let y ∈ Y and U be a δ-βI-open set in X such that {f−1(y)} ⊂ U . We have that V = Y − f(X − U) is a δ-βJ -closed set such that y ∈ V and f−1(V ) ⊂ U . Subsequently, the necessity is verified. We next demonstrate sufficiency. Let F be a δ-βI- closed subset in X and y ∈ Y − f(F ). Thus, {f−1(y)} ⊂ X −F . By hypothesis, there is a δ-βJ -open set Vy such that f−1(Vy) ⊂ X − F , and hence, y ∈ Vy ⊂ Y − f(F ). Therefore Y − f(F ) = ∪{Vy : y ∈ Y } is a δ-βJ -open set in Y . Consequently, f(F ) is a δ-βJ -closed set. Theorem 13. Let (X, τ, I) and (Y, τ ′,J ) be ideal topological spaces, and f : X → Y be continuous, δ-βI-open, δ-βI-closed, surjective with {f−1(y)} being δ-βI-compact for every y ∈ Y and f(I) ⊂ J . If X is δ-βI-paracompact, then Y is δ-βJ -paracompact. Proof. Let U = {Uλ : λ ∈ Λ} be an open cover of Y . As f is continuous, H = {f−1(Uλ) : λ ∈ Λ} is an open cover of X. Given that X is δ-βI-paracompact, H has a precise δ-βI-locally finite refinement V = {Vλ : λ ∈ Λ} of δ-βI-open subsets such that X − ∪{Vλ : λ ∈ Λ} ∈ I. Since f is δ-βI-open, f(V) = {f(Vλ) : λ ∈ Λ} is a precise δ-βJ -open refinement of U , and Y − ∪{f(Vλ) : λ ∈ Λ} ∈ J . Next, we shall verify that f(V) is δ-βJ -locally finite. Let y ∈ Y . As V is δ-βI-locally finite, for x ∈ {f−1(y)}, there C. Boonpok, A. Sama-Ae, P. Raktaow / Eur. J. Pure Appl. Math, 18 (1) (2025), 5570 10 of 12 exists a δ-βI-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} is a δ-βI-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. As f is δ-βI-closed, using Lemma 7, 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 implies that Wy intersects at most finitely many members of f(V). Hence, f(V) is a δ-βJ -locally finite in Y . Therefore, (Y, τ ′,J ) is δ-βJ -paracompact. Theorem 14. Let (X, τ, I) be an ideal topological space and (Y, τ ′) be a topological space. Let f : X → Y be δ-βI-irresolute, continuous, δ-βI-open, surjective, and f(V) be a δ-βf(I)- locally finite in Y for every δ-βI-locally finite V in X. If (X, τ, I) is δ-βI-paracompact, then (Y, τ ′, f(I)) is δ-βf(I)-paracompact. Proof. Let U = {Uλ : λ ∈ Λ} represent an open cover of Y . Consequently, we have that H = {f−1(Uλ) : λ ∈ Λ} creates an open cover of X. Since X is δ-βI-paracompact, H has a δ-βI-locally finite precise δ-βI-open refinement V = {Vλ : λ ∈ Λ} such that X − ∪{Vλ : λ ∈ Λ} ∈ I. As Y − ∪{f(Vλ) : λ ∈ Λ}) ⊂ f(X − ∪{Vλ : λ ∈ Λ}) and f(X − ∪{Vλ : λ ∈ Λ}) ∈ f(I), we have Y − ∪{f(Vλ) : λ ∈ Λ}) ∈ f(I). Given that f is surjective, f(I) is an ideal of Y . By assumption, we have that f(V) = {f(Vλ) : λ ∈ Λ} is a precise δ-βf(I)-locally finite of δ-βf(I)-open subsets in Y . Next, we will confirm that f(V) refines U . Let f(Vλ) ∈ f(V). Thus, Vλ ⊂ f−1(Uλ) for some Uλ ∈ H, as V refines H. This indicates that f(Vλ) ⊂ f(f−1(Uλ)) ⊂ Uλ. Subsequently, (Y, τ ′, f(I)) is δ-βf(I)- paracompact. Theorem 15. Let (X, τ) be a topological space and (Y, τ ′,J ) be an ideal topological space. Let f : X → Y be open, δ-βf−1(J )-irresolute, and bijective. If (Y, τ ′,J ) is δ-βJ - paracompact, then (X, τ, f−1(J )) is δ-βf−1(J )-paracompact. Proof. Let U = {Uλ : λ ∈ Λ} be an open cover of X. As f is open, f(U) = {f(Uλ) : λ ∈ Λ} is an open cover of Y . By hypothesis, f(U) has a δ-βJ -locally finite precise δ-βJ -open refinement H = {Vλ : λ ∈ Λ} such that Y −∪{Vλ : λ ∈ Λ} ∈ J . It implies that Y −∪{Vλ : λ ∈ Λ} = J for some J ∈ J , which follows that f−1(Y )− ∪{f−1(Vλ) : λ ∈ Λ} = f−1(J). Then, X − ∪{f−1(Vλ) : λ ∈ Λ} ∈ f−1(J ). As f is δ-βf−1(J )-irresolute, V = {f−1(Vλ) : λ ∈ Λ} is a δ-βf−1(J )-locally finite δ-βf−1(J )-open collection. The refinement of U by V will be asserted. Let f−1(Vλ) ∈ V. Hence, Vλ ∈ H, and there exists Uλ ∈ U such that Vλ ⊂ f(Uλ) as H refines f(U). Therefore f−1(Vλ) ⊂ f−1(f(Uλ)) = Uλ ∈ U . Accordingly, X is δ-βf−1(J )-paracompact. Theorem 16. Let (X, τ, I) and (Y, τ ′,J ) be ideal topological spaces, and let f : (X, τ, I) → (Y, τ ′,J ) be open, δ-βI-irresolute, bijective, and f(I) = J . If A ⊂ Y is δ-βJ -paracompact in Y , then f−1(A) ⊂ X is δ-βI-paracompact. Proof. Let U = {Uλ : λ ∈ Λ} be an open cover of f−1(A). Given that f is an open mapping, f(U) = {f(Uλ) : λ ∈ Λ} forms an open cover of A. By hypothesis, f(U) C. Boonpok, A. Sama-Ae, P. Raktaow / Eur. J. Pure Appl. Math, 18 (1) (2025), 5570 11 of 12 possesses a δ-βJ -locally finite precise δ-βJ -open refinement H = {Vλ : λ ∈ Λ} such that A − ∪{Vλ : λ ∈ Λ} ∈ J . Then, f−1(A) − ∪{f−1(Vλ) : λ ∈ Λ} ∈ f−1(J ) = I. As f is δ-βI-irresolute, V = {f−1(Vλ) : λ ∈ Λ} is a δ-βI-locally finite δ-βI-open collection. Let f−1(Vλ) ∈ V. There exists f(Uλ) ∈ f(U) such that Vλ ⊂ f(Uλ), since H refines f(U). It follows that f−1(Vλ) ⊂ f−1(f(Uλ)) = Uλ, thereby indicating that H refines U . Therefore f−1(A) is δ-βI-paracompact in X. 6. Conclusion This paper looks at different ways to describe the δ-βI-paracompactness of an ideal topological space as a weaker form of β-paracompactness compared to an ideal I (or I-β- paracompactness). We found that every I-β-paracompact space is a δ-βI-paracompact space, and every Hausdorff δ-βI-paracompact space under some conditions is δ-βI-regular. The union of two δ-βI-paracompact subsets is a δ-βI-paracompact subset, and the intersec- tion of a δ-βI-paracompact subset and a δ-βI-closed set is δ-βI-paracompact. In addition, we illustrate that δ-βI-paracompactness is preserved under certain conditions. 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