EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6297 ISSN 1307-5543 – ejpam.com Published by New York Business Global Strong β-I-Submaximality and β-I-Paracompactness in Ideal Topological Spaces Chawalit Boonpok1 , Palin Raktaow2, Areeyuth Sama-Ae3,∗ 1 Mathematics and Applied Mathematics Research Unit, Department of Mathematics, Fac- ulty of Science, Mahasarakham University, Maha Sarakham, 44150, Thailand 2 Applied Mathematics and Innovation of Mathematics Teaching, Faculty of Science and Technology, Prince of Songkla University, Pattani Campus, Pattani, 94000, Thailand 3 Department of Mathematics and Computer Science, Faculty of Science and Technology, Prince of Songkla University, Pattani Campus, Pattani, 94000, Thailand Abstract. This work introduces and examines the concepts of strong β-I-submaximality and strong β-I-paracompactness in ideal topological spaces, presenting them as natural extensions of the classical notions of submaximality and paracompactness. The study emphasizes the analysis of submaximal spaces through the lens of strong β-I-open sets. Additionally, it offers several char- acterizations of strong β-I-paracompact spaces and investigates the preservation of this property under mappings. 2020 Mathematics Subject Classifications: 54A05, 54B05, 54C08 Key Words and Phrases: Ideal topological space, strong β-I-open, strong β-I-submaximal, strong β-I-paracompactness 1. Introduction and Preliminaries General topology has demonstrated its efficacy in both theoretical and practical do- mains. Topology’s significance has emerged in various domains, including computational topology, geometric design, computer-aided design, and engineering. Khalimsky et al. [1] and Kong and Kopperman [2] advanced digital topology and computer graphics by the application of connected topologies on finite ordered sets. Moore and Peters [3] examined computational topology in geometric and molecular design, whereas Rosen and Peters [4] employed topological approaches in engineering design research. The concept of submaximality in general topological spaces was first introduced by Hewitt [5], who defined a space as submaximal if every dense subset is open. This prop- erty plays a significant role in topology, often serving as a crucial condition in the study of ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6297 Email addresses: chawalit.b@msu.ac.th (C. Boonpok), palin.rt@gmail.com (P. Raktaow), 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, P. Raktaow, A. Sama-Ae / Eur. J. Pure Appl. Math, 18 (3) (2025), 6297 2 of 23 maximal topologies related to various topological invariants. The original idea of submax- imal spaces can be traced back to Bourbaki [6]. Hewitt’s foundational work laid the basis for further developments by Arhangel’skii and Collins [7], who provided necessary and sufficient conditions for submaximality and showed that all such spaces are left-separated. Their contributions also sparked interest in whether every submaximal space is σ-discrete. Additionally, they established that any connected Hausdorff space that does not admit a strictly finer connected topology is necessarily submaximal [7]. Dontchev [8] character- ized submaximal spaces using various topological notions studied the connections between submaximal and related spaces. Tokgoz and Yalvac [9] developed the concept of submax- imality and offer some new results. Paracompact spaces are considered one of the key classes of topological spaces, as they generalize both metrizable and compact spaces. These spaces were quickly recognized by topologists and analysts. A paracompact space is defined as a topological space in which every open cover has an open refinement that is locally finite. This concept was first intro- duced by Dieudonné [10] in 1944. A Hausdorff space is paracompact if and only if it admits partitions of unity that are subordinate to any open cover. Additionally, all paracompact Hausdorff spaces are normal, as shown in [11]. Various generalized forms of paracompact- ness, such as S-paracompactness [12], P3-paracompactness [13], and β-paracompactness [14], have been explored in the literature. In 2006, Al-Zoubi [12] used semi-open sets to define S-paracompact spaces, a generalization of paracompact spaces, and studied their relationships. Li and Song [15] constructed a Hausdorff S-paracompact space that is not paracompact and further investigated the characterizations of S-paracompact spaces. The concept of ideal topological spaces was first introduced by Kuratowski [16] and Vaidyanathaswamy [17]. The integration of ideals into topological structures has led to the generalization of certain classical topological ideas. The topology τ of a space can be augmented to a topology τ⋆ through an ideal I, resulting in the creation of an ideal topological space. This framework has been utilized in areas such as ideal resolvability [18], paracompactness concerning ideals [19], and continuity decomposition [20]. Addition- ally, novel topological constructs founded on ideals were presented in [21]. Jankovic and Hamlett [21] examined and elucidated the basic characteristics of these spaces, proposing the notion of I-open sets and performing comprehensive analyses of topologies employing ideals. Abd. El-Monsef et al. [22] conducted a comprehensive analysis of I-open sets. The notion of Ig-closed sets was established by Dontchev et al. in 1999 [18]. Further- more, Abd El-Monsef et al. [23] introduced the concept of the s-local function, which was subsequently examined by Khan and Noiri [24]. Ekici and Noiri [25] introduced the notion of I-submaximal ideal topological spaces and studied several characterizations and further properties of I-submaximal. Recently, Boonpok [26] presented the notion of semi-I-submaximal ideal topological spaces and examined their characterizations. Inspired by these advancements, this work aims to offer the concept of strong β-I-submaximal ideal topological spaces, which serves as a natural extension of the previously examined semi-I-submaximal spaces. The notion of paracompactness with respect to an ideal was first introduced by Za- hid [27] and later examined by Hamlett et al. [19]. Sathiyasundari and Renukadevi [28] C. Boonpok, P. Raktaow, A. Sama-Ae / Eur. J. Pure Appl. Math, 18 (3) (2025), 6297 3 of 23 investigated I-paracompactness and its properties, extending certain results from para- compact spaces to I-paracompact spaces. The concept of S-paracompactness in ideal topological spaces was studied by Sanabria et al. [29], who introduced I-S-paracompact spaces, a new type of space that includes both S-paracompact and I-paracompact spaces. In 2013, Demir and Ozbakir [14] proposed a modified version of expandable and paracom- pact spaces, termed β-expandable and β-paracompact spaces, respectively. They demon- strated that every β-paracompact space is essentially a β-expandable space. Yildirim et al. [30] introduced the concept of β-paracompactness within ideal topological spaces and compared it with existing forms of paracompactness. Recently, Boonpok et al. [31] pro- posed the concept of δ-βI-paracompactness in the context of ideal topological spaces as a weaker variant of β-paracompactness. Additionally, Boonpok and Sama-Ae [32] provided characterizations of δ1-βI-paracompactness with respect to an ideal. This paper constructs strong β-I-paracompact spaces using strong β-I-open sets, and the spaces under study are examined in detail with respect to β-paracompactness, as described in reference [30]. Multiple characterizations of strong β-I-paracompactness are presented to enhance the theory of ideal topology and give a wider framework for future research. Throughout this paper, unless otherwise specified, the symbols (X, τ), or simply X, refer to a general topological space without assuming any separation axioms. For a subset A of a topological space (X, τ), the closure and interior of A are denoted by cl(A) and Int(A), respectively. An ideal I on a set X is a nonempty family of subsets of X such that if A ∈ I and B ⊆ A, then B ∈ I, and if A ∈ I and B ∈ I, then A ∪ B ∈ I. A topological space (X, τ) together with an ideal I is called an ideal topological space and denoted by (X, τ, I). The set of all subsets of X is denoted as P (X). A set operator (·)∗ : P (X) → P (X), known as a local function [16], is defined with respect to a topology τ and an ideal I. For any subset A ⊆ X, it is given by A∗(I, τ) = {x ∈ X : U ∩A /∈ I for every U ∈ τ(x)} , where τ(x) = {U ∈ τ : x ∈ U} represents the set of open neighborhoods of x in the topology τ . This operator induces a finer topology on X, called the ∗-topology, denoted by τ∗(I). The ∗-topology is generated by the Kuratowski closure operator, which is defined as cl∗(A) = A∪A∗ [21]. For any ideal topological space, a finer topology τ∗(I), or simply τ∗, always exists. It is generated by the subbasis β(I, τ) = {U − I : U ∈ τ, I ∈ I}. However, β(I, τ) does not necessarily form a topology in general [21]. It is clear that A∗ ⊆ B∗ and cl∗(A) ⊆ cl∗(B) if A ⊆ B. Definition 1. [33] A subset A of an ideal topological space (X, τ, I) is called strong β-I- open if A ⊆ cl∗(Int(cl∗(A))). The complement of a strong β-I-open set is called a strong β-I-closed set. According to Definition 1, we have the following lemma. Lemma 1. In an ideal topological space, the following properties are satisfied: C. Boonpok, P. Raktaow, A. Sama-Ae / Eur. J. Pure Appl. Math, 18 (3) (2025), 6297 4 of 23 (1) The arbitrary union of strong β-I-open sets is itself a strong β-I-open set; and (2) The arbitrary intersection of strong β-I-closed sets remains a strong β-I-closed set. Definition 2. [33] The strong β-I-closure of a subset A of an ideal topological space (X, τ, I), denoted by sβ clI(A), is defined as the intersection of all strong β-I-closed sets containing A, while the strong β-I-interior of A, denoted by sβ IntI(A), is defined as the union of all strong β-I-open sets contained in A. Utilizing Definition 2, we have the following lemma. Lemma 2. Let A and B be subsets of an ideal topological space (X, τ, I). The following statements hold: (1) A ⊆ sβ clI(A); (2) sβ IntI(A) ⊆ A; (3) If A ⊆ B, then sβ IntI(A) ⊆ sβ IntI(B) and sβ clI(A) ⊆ sβ clI(B); (4) X − sβ clI(A) = sβ IntI(X −A); and (5) X − sβ IntI(A) = sβ clI(X −A). Proof. Statements (1), (2), and (3) follow directly from the definitions of sβ clI(A) and sβ IntI(A). To establish statement (4), observe that X − sβ clI(A) = X − ∩{F | A ⊆ F, F is strong β-I-closed} = ∪{X − F | X − F ⊆ X −A, X − F is strong β-I-open} = ∪{G | G ⊆ X −A, G is strong β-I-open} = sβ IntI(X −A), which confirms statement (4). Similarly, for statement (5), note that X − sβ IntI(A) = X − ∪{G | G ⊆ A, G is strong β-I-open} = ∩{X −G | X −A ⊆ X −G, X −G is strong β-I-closed} = ∩{F | X −A ⊆ F, F is strong β-I-closed} = sβ clI(X −A), thereby verifying statement (5). The following lemma outlines fundamental characteristics and conditions related to strong β-I-open and strong β-I-closed sets within the context of ideal topological spaces. Lemma 3. Let A be a subset of an ideal topological space (X, τ, I). The following prop- erties hold: (1) If A is open, then A is strong β-I-open; C. Boonpok, P. Raktaow, A. Sama-Ae / Eur. J. Pure Appl. Math, 18 (3) (2025), 6297 5 of 23 (2) If A is closed, then A is strong β-I-closed; (3) sβ IntI(A) is strong β-I-open, and Int(A) ⊆ sβ IntI(A); (4) sβ clI(A) is strong β-I-closed, and sβ clI(A) ⊆ cl(A); (5) A is strong β-I-open if and only if A = sβ IntI(A); (6) A is strong β-I-closed if and only if A = sβ clI(A); and (7) x ∈ sβ clI(A) if and only if U ∩A ̸= ∅ for every strong β-I-open set U containing x. Proof. (1): Let A be an open set. Then A = Int(A), and since A ⊆ cl∗(A), we have A = Int(A) ⊆ Int(cl∗(A)). It follows that A ⊆ cl∗(A) ⊆ cl∗(Int(cl∗(A))), which implies that A is strong β-I-open. (2): The result follows directly from (1). (3): Let Uα be any strong β-I-open set with Uα ⊆ A. By the definition of a strong β-I-open set, we have that Uα ⊆ sβ IntI(A). Since Uα is strong β-I-open, we have Uα ⊆ cl∗(Int(cl∗(Uα))). Then, sβ IntI(A) = ∪αUα ⊆ ∪α cl ∗(Int(cl∗(Uα))) ⊆ cl∗(Int(cl∗(sβ IntI(A)))), and therefore sβ IntI(A) is a strong β-I-open set. From the definitions of Int(A) and sβ IntI(A), we deduce that Int(A) ⊆ sβ IntI(A). (4): By part (4) of Lemma 2, it follows that sβ clI(A) is strong β-I-closed. Moreover, from the definitions of sβ clI(A) and cl(A), we have that sβ clI(A) ⊆ cl(A). (5): Since sβ IntI(A) is the union of all strong β-I-open sets contained in A, it follows that sβ IntI(A) ⊆ A. Therefore, A is strong β-I-open if and only if A = sβ IntI(A). (6): As sβ clI(A) is the intersection of all strong β-I-closed sets containing A, it implies that A ⊆ sβ clI(A). Hence, A is strong β-I-closed if and only if A = sβ clI(A). (7): Assume that x ∈ sβ clI(A). Suppose there exists a strong β-I-open set U con- taining x such that U ∩ A = ∅. Then it follows that A ⊆ X − U . Since X − U is strong β-I-closed, this would imply x /∈ sβ clI(A), which contradicts the assumption. Conversely, suppose that for every strong β-I-open set U containing x, we have U∩A ̸= ∅. Assume, for the sake of contradiction, that x /∈ sβ clI(A). Then there exists a strong β-I-closed set F such that A ⊆ F and x /∈ F . Consequently, x ∈ X − F , which is a strong β-I-open set disjoint from A, leading to a contradiction. Therefore, it follows that x ∈ sβ clI(A). 2. Strong β-I-Submaximality In this section, we explore a collection of equivalent conditions that provide a thor- ough characterization of when an ideal topological space (X, τ, I) can be regarded as strong β-I-submaximal, thereby establishing a foundational understanding of the under- lying properties that define this class of spaces. C. Boonpok, P. Raktaow, A. Sama-Ae / Eur. J. Pure Appl. Math, 18 (3) (2025), 6297 6 of 23 Definition 3. Let (X, τ, I) be an ideal topological space. A subset A ⊆ X is said to be: (1) Strong β-I-dense if sβ clI(A) = X; and (2) Strong β-I-codense if X −A is strong β-I-dense. Definition 4. An ideal topological space (X, τ, I) is called strong β-I-submaximal if each strong β-I-dense subset of X is strong β-I-open. Let X = {a, b, c}, τ = {∅, {a}, {a, b}, X}, and I = {∅, {c}}. It is clear that the only strong β-I-dense sets are {a}, {a, b}, and {a, b, c}, all of which are also strong β-I-open. Hence, the space (X, τ, I) is strong β-I-submaximal. Definition 5. Let (X, τ, I) be an ideal topological space. A subset A ⊆ X is said to be: (1) Locally strong β-I-closed if A is the intersection of a strong β-I-open set and a strong β-I-closed set; and (2) Co-locally strong β-I-closed if A is the union of a strong β-I-open set and a strong β-I-closed set. Theorem 1 presents five equivalent conditions that characterize when a subset A of an ideal topological space is locally strong β-I-closed. Theorem 1. For a subset A of an ideal topological space (X, τ, I), the following properties are equivalent: (1) A is locally strong β-I-closed; (2) A = U ∩ sβ clI(A) for some strong β-I-open set U ; (3) sβ clI(A)−A is strong β-I-closed; (4) A ∪ (X − sβ clI(A)) is strong β-I-open; and (5) A ⊆ sβ IntI(A ∪ (X − sβ clI(A))). Proof. (1) ⇒ (2): Suppose A is locally strong β-I-closed. Then there exists a strong β-I-open set U and a strong β-I-closed set F such that A = U ∩F . Given that F is strong β-I-closed, it follows that sβ clI(A) ⊆ sβ clI(F ) = F , so A ⊆ U ∩ sβ clI(A) ⊆ U ∩F = A. Consequently, A = U ∩ sβ clI(A). (2) ⇒ (3): Suppose A = U ∩ sβ clI(A) for some strong β-I-open set U . Since sβ clI(A)−A = sβ clI(A) ∩ (X −A) = (X − (U ∩ sβ clI(A))) ∩ sβ clI(A) = (X − U) ∩ sβ clI(A), sβ clI(A)−A is strong β-I-closed. C. Boonpok, P. Raktaow, A. Sama-Ae / Eur. J. Pure Appl. Math, 18 (3) (2025), 6297 7 of 23 (3) ⇒ (4): Assume that the set sβ clI(A) − A is strong β-I-closed. Given that X − (sβ clI(A) − A) = (X − sβ clI(A)) ∪ A, it follows that A ∪ (X − sβ clI(A)) is strong β-I-open. (4) ⇒ (5): The evidence is clear. (5) ⇒ (1): Suppose that A ⊆ sβ IntI(A ∪ (X − sβ clI(A))). Since X − sβ clI(A) is strong β-I-open, we have X − sβ clI(A) = sβ IntI(X − sβ clI(A)). It follows that X − sβ clI(A) = sβ IntI(X − sβ clI(A)) ⊆ sβ IntI(A ∪ (X − sβ clI(A))). Therefore, A ∪ (X − sβ clI(A)) ⊆ sβ IntI(A ∪ (X − sβ clI(A))), which shows that A ∪ (X − sβ clI(A)) is strong β-I-open. Since A = (A ∪ (X − sβ clI(A))) ∩ sβ clI(A), it follows that A is locally strong β-I-closed. The subsequent theorem presents five mutually equivalent conditions, each involving strong β-I-closed sets, strong β-I-dense sets, and B-sI- sets, which together provide a comprehensive characterization of when an ideal topological space qualifies as strong β-I- submaximal. To proceed, we begin by introducing the concept of a B-sI set within the context of an ideal topological space. Definition 6. A subset A of an ideal topological space (X, τ, I) is called a B-sI set if it can be written as A = U ∩ V , where U is a strong β-I-open set, and V is a set that satisfies the condition sβ IntI(V ) = sβ IntI(sβ clI(V )). Theorem 2. For an ideal topological space (X, τ, I), the following properties are equiva- lent: (1) (X, τ, I) is strong β-I-submaximal; (2) For every subset A of X, sβ clI(A)−A is strong β-I-closed; (3) Every subset of X is locally strong β-I-closed; (4) Every subset of X is a B-sI set; and (5) Every strong β-I-dense subset of X is a B-sI set. Proof. (1) ⇒ (2): Let (X, τ, I) be strong β-I-submaximal and let A ⊆ X. As X = sβ clI(A) ∪ (X − sβ clI(A)) ⊆ sβ clI(A) ∪ (X − sβ IntI(sβ clI(A)) C. Boonpok, P. Raktaow, A. Sama-Ae / Eur. J. Pure Appl. Math, 18 (3) (2025), 6297 8 of 23 = sβ clI(A) ∪ sβ clI(X − sβ clI(A)) ⊆ sβ clI(A ∪ (X − sβ clI(A))) = sβ clI(X − (sβ clI(A)−A)), we have sβ clI(X − (sβ clI(A) − A)) = X and hence X − (sβ clI(A) − A) is strong β-I- dense. According to the hypothesis, X−(sβ clI(A)−A) is strong β-I-open. Consequently, sβ clI(A)−A is strong β-I-closed. (2) ⇒ (3): By Theorem 1, a subset A is locally strong β-I-closed if and only if sβ clI(A)−A is strong β-I-closed. (3) ⇒ (4): If every subset of X is locally strong β-I-closed, then for any A ⊆ X, we can write A = U ∩ V where U is strong β-I-open and V is strong β-I-closed. Since V = sβ clI(V ), we get sβ IntI(V ) = sβ IntI(sβ clI(V )). Therefore A is a B-sI set. (4) ⇒ (5): It is obvious. (5) ⇒ (1): Let A be a strong β-I-dense subset, and assume that every strong β-I- dense subset of X is a B-sI set. Then A can be expressed as A = U ∩ V , where U is a strong β-I-open set and V is a set satisfying sβ IntI(V ) = sβ IntI(sβ clI(V )). Given that A ⊆ V , it implies that X = sβ clI(A) ⊆ sβ clI(V ), and therefore, X = sβ clI(V ). Thus X = sβ IntI(X) = sβ IntI(sβ clI(V )) = sβ IntI(V ). This indicates that V = X, hence A = U ∩ V = U ∩ X = U . Therefore, A is classified as strong β-I-open. Accordingly, (X, τ, I) is defined as strong β-I-submaximal. Theorem 3 presents three equivalent conditions involving co-locally strong β-I-closed sets and strong β-I-closed sets that characterize when a topological space is strong β-I- submaximal. Theorem 3. For an ideal topological space (X, τ, I), the following properties are equiva- lent: (1) (X, τ, I) is strong β-I-submaximal; (2) Every subset of X is co-locally strong β-I-closed; (3) For every subset A of X such that sβ IntI(A) = ∅ is strong β-I-closed; Proof. (1) ⇒ (2): Assume that (X, τ, I) is a strong β-I-submaximal space. For any subset A ⊆ X, Theorem 2 guarantees the existence of a strong β-I-open set U and a strong β-I-closed set V such that X −A = U ∩ V . It follows that A = (X − U) ∪ (X − V ), where X − U is strong β-I-closed and X − V is strong β-I-open. Therefore, A can be expressed as the union of a strong β-I-closed set and a strong β-I-open set, which means A is co-locally strong β-I-closed. (2) ⇒ (3): Suppose that every subset of X is co-locally strong β-I-closed. Let A ⊆ X be such that sβ IntI(A) = ∅. Then, by assumption, A can be written as A = U ∪V , where U is strong β-I-open and V is strong β-I-closed. Since U ⊆ A, it follows that U = sβ IntI(U) ⊆ sβ IntI(A) = ∅, C. Boonpok, P. Raktaow, A. Sama-Ae / Eur. J. Pure Appl. Math, 18 (3) (2025), 6297 9 of 23 which implies U = ∅. Thus, A = V , and since V is strong β-I-closed, it follows that A is strong β-I-closed. (3) ⇒ (1): Let A ⊆ X be a strong-I-dense subset. We aim to show that A is strong β-I-open. Since X − sβ clI(A) = sβ IntI(X −A) = ∅, it follows that the complement X−A has empty strong β-I-interior. By assumption, such a subset must be strong β-I-closed. Hence, X − A is strong β-I-closed, and therefore A is strong β-I-open. Definition 7. A subset A of an ideal topological space (X, τ, I) is referred to as strong β-I-discrete if for every point x ∈ A, the singleton set {x} is strong β-I-closed. Theorem 4 establishes five equivalent conditions involving strong β-I-discreteness and strong β-I-closed sets, which collectively characterize when an ideal topological space (X, τ, I) can be regarded as strong β-I-submaximal. The proof of Theorem 4 relies on the following lemma. Lemma 4. Let A be a subset of an ideal topological space (X, τ, I). Then, sβ IntI(sβ clI(A)−A) = ∅. Proof. Let A be a subset of X. Since sβ IntI(X −A) = X − sβ clI(A), we have sβ IntI(sβ clI(A)−A) = sβ IntI(sβ clI(A) ∩ (X −A)) ⊆ sβ IntI(sβ clI(A)) ∩ sβ IntI(X −A) = sβ IntI(sβ clI(A)) ∩ (X − sβ clI(A)) ⊆ sβ clI(A) ∩ (X − sβ clI(A)) = ∅. Theorem 4. For an ideal topological space (X, τ, I), the following properties are equiva- lent: (1) (X, τ, I) is strong β-I-submaximal; (2) For every subset A of X with sβ IntI(A) = ∅ is strong β-I-closed and strong β-I- discrete; (3) For every subset A of X, the set sβ clI(A) − A is strong β-I-closed and strong β-I-discrete; (4) Every strong β-I-codense subset of X is strong β-I-closed and strong β-I-discrete; and (5) Every strong β-I-codense subset of X is strong β-I-closed. C. Boonpok, P. Raktaow, A. Sama-Ae / Eur. J. Pure Appl. Math, 18 (3) (2025), 6297 10 of 23 Proof. (1) ⇒ (2): Assume that (X, τ, I) is strong β-I-submaximal. Let A be a subset of X such that sβ IntI(A) = ∅. Then, sβ clI(X −A) = X − sβ IntI(A) = X, so X − A is strong β-I-dense. By assumption, X − A is strong β-I-open. Thus, A is strong β-I-closed. Next, we verify that A is strong β-I-discrete. Let x ∈ A. Since {x} ⊆ A, we have sβ IntI({x}) ⊆ sβ IntI(A) = ∅, which implies sβ IntI({x}) = ∅. Then, sβ clI(X − {x}) = X − sβ IntI({x}) = X, so X −{x} is strong β-I-open, which means {x} is strong β-I-closed. Since this holds for each x ∈ A, it follows that A is strong β-I-discrete. (2) ⇒ (3): Suppose every subset A of X with sβ IntI(A) = ∅ is strong β-I-closed and strong β-I-discrete. By Lemma 4, we have sβ IntI(sβ clI(A)−A) = ∅. By the hypothesis, this implies that the set sβ clI(A)−A is strong β-I-closed and strong β-I-discrete. (3) ⇒ (4): Suppose every subset A of X satisfies that sβ clI(A)−A is strong β-I-closed and strong β-I-discrete. Let A be a strong β-I-codense subset of X. Then X −A is strong β-I-dense, i.e., X = sβ clI(X −A). Hence, A = X − (X −A) = sβ clI(X −A)− (X −A). By the assumption, A is strong β-I-closed and strong β-I-discrete. (4) ⇒ (5): This is obvious. (5)⇒ (1): Suppose that every strong β-I-codense subset ofX is strong β-I-closed. Let A be a strong β-I-dense subset of X. Then its complement X −A is strong β-I-codense. By the hypothesis, X − A is strong β-I-closed. Thus, A is strong β-I-open. Therefore, the space (X, τ, I) is strong β-I-submaximal. The final theorem in this section outlines three equivalent conditions that determine when an ideal topological space (X, τ, I) is strong β-I-submaximal. Theorem 5. For an ideal topological space (X, τ, I), the following properties are equiva- lent: (1) (X, τ, I) is strong β-I-submaximal; C. Boonpok, P. Raktaow, A. Sama-Ae / Eur. J. Pure Appl. Math, 18 (3) (2025), 6297 11 of 23 (2) Every subset of X is locally strong β-I-closed; and (3) Every strong β-I-dense subset of X is locally strong β-I-closed. Proof. (1) ⇒ (2): Assume that (X, τ, I) is strong β-I-submaximal. Let A ⊆ X. As the same proof in Theorem 2, we have sβ clI(X − (sβ clI(A) − A)) = X and hence X− (sβ clI(A)−A) is strong β-I-dense. By the hypothesis, X− (sβ clI(A)−A) is strong β-I-open. Therefore, sβ clI(A)−A is strong β-I-closed. By Theorem 1, A is locally strong β-I-closed. (2) ⇒ (3): It is clear. (3) ⇒ (1): Assume that every strong β-I-dense subset of X is locally strong β-I- closed. Let A be a strong β-I-dense subset of X. This implies sβ clI(A) = X. By assumption, A is locally strong β-I-closed, meaning there exist a strong β-I-open set U and a strong β-I-closed set V such that A = U ∩ V . Since A ⊆ V , we have that X = sβ clI(A) ⊆ sβ clI(V ) = V , which gives V = X. Thus, A = U ∩V = U ∩X = U and so A is strong β-I-open. Consequently, (X, τ, I) is strong β-I-submaximal. 3. Stong β-I-Paracompactness and Characterizations This section explores the notion of sβ-I-paracompactness, which is a variant of the I-β-paracompactness concept introduced by Yildirim et al. [30]. We aim to present a formal description of this concept. Let A = {Uα : α ∈ Λ1} and B = {Vµ : µ ∈ Λ2} be two families of subsets in a topological space X. We say that A is a refinement of B if for every α ∈ Λ1, there exists µ ∈ Λ2 such that Uα ⊆ Vµ. A family A of subsets of a topological space (X, τ) is called β-locally finite [14] if, for each point x ∈ X, there exists a β-open neighborhood U of x that intersects only finitely many sets from A. Yildirim et al. [30] defined an ideal topological space (X, τ, I) to be I-β-paracompact if every open cover of X has a β-locally finite refinement V consisting of β-open sets, and the set X − ∪{V : V ∈ V} is an element of I. Definition 8. A collection A of subsets of an ideal topological space (X, τ, I) is said to be sβ-I-locally finite if for each x ∈ X, there exists a strong β-I-open set U containing x and U intersects at most finitely many members of A. Lemma 5. Let A be a collection of subsets of an ideal topological space (X, τ, I). If A is sβ-I-locally finite, then it is β-locally finite. Proof. Assume that A is sβ-I-locally finite. We aim to show that A is β-locally finite. Let x ∈ X. Since A is sβ-I-locally finite, there exists a strong β-I-open set Gx containing x that intersects only finitely many members of A. Because every strong β-I-open set is also β-open, Gx is a β-open neighborhood of x with the same finiteness property. Thus, A is β-locally finite. C. Boonpok, P. Raktaow, A. Sama-Ae / Eur. J. Pure Appl. Math, 18 (3) (2025), 6297 12 of 23 Definition 9. An ideal topological space (X, τ, I) is said to be sβ-I-paracompact if every open cover of X has an sβ-I-locally finite refinement A consisting of strong β-I-open sets (not necessarily a cover) such that X − ∪{V : V ∈ A} ∈ I. The collection A of subsets of X such that X − ∪{V : V ∈ A} ∈ I is called an I-cover. A subset A of an ideal topological space (X, τ, I) is said to be sβ-I-paracompact if for any open cover of A has an sβ-I-locally finite refinement A consisting of strong β-I-open sets such that A− ∪{V : V ∈ A} ∈ I. The two theorems that follow arise from the fact that every open set is strong β-I-open, every strong β-I-open set is β-open and ∅ is in any ideal. Theorem 6. If a topological space (X, τ) is paracompact, then (X, τ, I) is sβ-I-paracompact. Proof. It is evident, as ∅ ∈ I. Theorem 7. If (X, τ, I) is sβ-I-paracompact then it is I-β-paracompact. Proof. Every sβ-I-locally finite collection of subsets of X is β-locally finite, as demon- strated by Lemma 5. Furthermore, each strong β-I-open set is a β-open set. We can continue with the proof by following to the definitions of β-I-paracompactness and sβ-I- paracompactness. Consider the set N of all positive integers. Define a topology τ on N by τ = {∅} ∪ {N} ∪ {{1, 2, . . . , n} : n ∈ N} . Let I = {A ⊆ N : 1 /∈ A}. It is easy to see that I is an ideal on N. Consider the open cover U = {{1, 2, 3, . . . , n} : n ∈ N} of N. It is evident that there does not exist a locally finite open refinement V of U that still covers N. Therefore, the space (N, τ) is not paracompact. Nevertheless, N is an sβ-I-paracompact space. For any open cover U of N, there exists refinement V = {{1}} of a strong β-I-open set {1} that V is sβ-I-locally finite. This holds because the complement N− {1} = {2, 3, . . . } ∈ I, satisfying the required condition. Theorem 8. Let (X, τ, I) be an ideal topological space and A ⊆ X. Then G∩sβ clI(A) = ∅ if and only if G ∩A = ∅, for all strong β-I-open subset G of X. Proof. It follows from part (7) of Lemma 3, together with the fact that A ⊆ sβ clI(A). Theorem 9. Let A = {Vλ : λ ∈ Λ} be a collection of subsets of an ideal topological space (X, τ, I). The following statements are true. (1) If A is sβ-I-locally finite and Hλ ⊆ Vλ for all λ ∈ Λ, then B = {Hλ : λ ∈ Λ} is sβ-I-locally finite. (2) A is sβ-I-locally finite if and only if {sβ clI(Vλ) : λ ∈ Λ} is sβ-I-locally finite. C. Boonpok, P. Raktaow, A. Sama-Ae / Eur. J. Pure Appl. Math, 18 (3) (2025), 6297 13 of 23 Proof. (1): Let x ∈ X. Since A is sβ-I-locally finite, there exists a strong β-I-open set U containing x, which intersects at most finitely many elements of A. As Hλ ⊆ Vλ for all λ ∈ Λ, it follows that U intersects at most finitely many of the sets in B = {Hλ : λ ∈ Λ}. Hence, B = {Hλ : λ ∈ Λ} is sβ-I-locally finite. (2): Let A be sβ-I-locally finite and let x ∈ X. Then, there exists a strong β-I-open set G containing x that satisfies G∩Vλ = ∅ for every λ ̸= λ1, λ2, . . . , λn. By Theorem 8, we obtain that G∩sβ clI(Vλ) = ∅ for every λ ̸= λ1, λ2, . . . , λn. Therefore, {sβ clI(Vλ) : λ ∈ Λ} is sβ-I-locally finite. The converse follows from (1). Theorem 10. If (X, τ, I) is sβ-I-paracompact and J is an ideal on X with I ⊆ J , then (X, τ,J ) is sβ-J -paracompact. Proof. Let (X, τ, I) be sβ-I-paracompact, and suppose I ⊆ J . Let A = {Uα : α ∈ Λ} be an open cover of X. Since (X, τ, I) is sβ-I-paracompact, by definition, there exists an sβ-I-locally finite refinement A′ of A consisting of strong β-I-open sets such that: X − ∪{V : V ∈ A′} ∈ I. Because I ⊆ J , it follows that: X − ∪{V : V ∈ A′} ∈ J . Thus, (X, τ,J ) is sβ-J -paracompact. Lemma 6. If an open cover A = {Uλ : λ ∈ Λ} of an ideal topological space (X, τ, I) has an sβ-I-locally finite strong β-I-open refinement B such that X−∪{V : V ∈ B} ∈ I, then there exists a precise sβ-I-locally finite strong β-I-open refinement C = {Hλ : λ ∈ Λ} of A such that X − ∪{Hλ : λ ∈ Λ} ∈ I. Proof. A similar technique is employed as in the proof of Lemma 1.3 in [29]. Definition 10. An ideal topological space (X, τ, I) is sβ-I-regular if for any closed subset F of X and x ̸∈ F , there exist disjoint strong β-I-open sets U and V such that x ∈ U and F − V ∈ I. Theorem 11. If (X, τ, I) is sβ-I-paracompact and Hausdorff, then (X, τ, I) is sβ-I- regular. Proof. Let F be a closed subset of X, and let x ̸∈ F . For each y ∈ F , since X is Hausdorff, there exist disjoint open sets Vx and Oxy such that x ∈ Vx and y ∈ Oxy. This implies that y ̸∈ cl(Vx). Now, consider the family A = {Oxy : y ∈ F} ∪ {X − F}, which is an open cover of X. By assumption, there exists an sβ-I-locally finite strong β-I-open refinement B = {Hxy : y ∈ F} ∪ {W} such that: Hxy ⊆ Oxy for each y ∈ F , W ⊆ X − F , and X − (∪{Hxy : y ∈ F} ∪ {W}) ∈ I. C. Boonpok, P. Raktaow, A. Sama-Ae / Eur. J. Pure Appl. Math, 18 (3) (2025), 6297 14 of 23 Next, let us define the sets V = ∪{Hxy : y ∈ F} and U = X−{sβ clI(∪(Hxy)) : y ∈ F}. We assert that U and V are disjoint strong β-I-open subsets of X. Given that Hxy ⊆ Oxy and sβ clI(Hxy) ⊆ cl(Hxy), it follows that sβ clI(Hxy) ⊆ cl(Oxy). As x /∈ cl(Oxy), it consequently follows that x /∈ sβ clI(Hxy), and thus x ∈ U . Furthermore, we observe that F − V = F − ∪{Hxy : y ∈ F} ⊆ X − (∪{Hxy : y ∈ F} ∪W ) ∈ I. Thus, U and V are indeed disjoint sβ-I-open sets, satisfying x ∈ U and F − V ∈ I. This confirms that the space (X, τ, I) is sβ-I-regular. Theorem 12. Let (X, τ, I) be an ideal topological space. The following statements are equivalent: (1) For every closed subset F of X and every x ̸∈ F , there exist disjoint strong β-I-open sets U and V such that x ∈ U and F − V ∈ I. (2) For every open subset G of X and every x ∈ G, there exists a strong β-I-open set U such that x ∈ U and sβ clI(U)−G ∈ I. Proof. (1) ⇒ (2): Let G be an open set and x ∈ G. Then X − G is closed, and since x /∈ X −G, by assumption, there exist disjoint strong β-I-open sets U and V such that x ∈ U and (X − G) − V ∈ I. Since U and V are disjoint, by Theorem 8, we have sβ clI(U) ⊆ X − V . Therefore, sβ clI(U) ∩ (X −G) ⊆ (X −G)− V . Hence, we conclude that sβ clI(U) ∩ (X −G) = sβ clI(U)−G ∈ I. (2) ⇒ (1): Let F be a closed set and x /∈ F . This implies that X − F is open, and x ∈ X − F . By assumption, there exists a strong β-I-open set U such that x ∈ U and sβ clI(U)−(X−F ) ∈ I. Thus, we define V = X−sβ clI(U), which is a strong β-I-open set. Since U and V are disjoint, we have F−V = F−(X−sβ clI(U)) = sβ clI(U)−(X−F ) ∈ I. By Theorem 11 and Theorem 12, we have the following colollary. Corollary 1. An ideal topological space (X, τ, I) is sβ-I-paracompact and Hausdorff if and only if for any open set G ⊆ X and for every point x ∈ G, there exists a strong β-I-open set U such that x ∈ U and sβ clI(U)−G ∈ I. Theorem 13. If an ideal topological space (X, τ, I) is sβ-I-paracompact and regular, then every open cover of X has an sβ-I-locally finite I-cover refinement of strong β-I-closed sets. Proof. Let A be an open cover of X. By the regularity of X, for each x ∈ X and Ux ∈ A containing x, there exists an open set Gx such that x ∈ Gx and cl(Gx) ⊆ Ux. Thus, the family A1 = {Gx : x ∈ X} is an open cover of X. As X is sβ-I-paracompact, the cover A1 has an sβ-I-locally finite refinement B1 = {Vλ : λ ∈ Λ} consisting of strong β-I-open sets such that X − ∪{Vλ : λ ∈ Λ} ∈ I. Since Vλ ⊆ sβ clI(Vλ) for each λ, and I is an ideal, we conclude that X−∪{sβ clI(Vλ) : λ ∈ Λ} ∈ I. By Theorem 9, the collection C. Boonpok, P. Raktaow, A. Sama-Ae / Eur. J. Pure Appl. Math, 18 (3) (2025), 6297 15 of 23 B = {sβ clI(Vλ) : Vλ ∈ B1} is sβ-I-locally finite. Since B1 refines A1, for each λ ∈ Λ, there exists some Gx ∈ A1 such that Vλ ⊆ Gx. Therefore, we have: sβ clI(Vλ) ⊆ cl(Vλ) ⊆ cl(Gx). Consequently, sβ clI(Vλ) ⊆ Ux. This shows that B refines A. Thus, the collection B = {sβ clI(Vλ) : Vλ ∈ B1} is an sβ-I-locally finite I-cover refinement of strong β-I-closed sets. Theorem 14. If an ideal topological space (X, τ, I) is Hausdorff and A is an sβ-I- paracompact subset of X, then A is a closed set in (X, τ∗). Proof. Let x ∈ X − A. Since X is Hausdorff, for each y ∈ A, there exists an open set Gy ∈ τ such that y ∈ Gy and x ̸∈ Gy. Thus, the family U = {Gy : y ∈ A} forms an open cover of A. Since A is an sβ-I-paracompact subset of X, the cover U has an sβ-I-locally finite strong β-I-open refinement V = {Vα : α ∈ Λ} such that A− ∪{Vα : α ∈ Λ} ∈ I. Since x ̸∈ cl(Vα) for all α ∈ Λ, it follows that x ̸∈ ∪{cl(Vα) : α ∈ Λ}. Moreover, since the locally finite family is closure-preserving, we have: ∪{cl(Vα) : α ∈ Λ} = cl (∪{Vα : α ∈ Λ}) , so that x ̸∈ cl (∪{Vα : α ∈ Λ}). Let G = X − cl (∪{Vα : α ∈ Λ}) and J = A− cl (∪{Vα : α ∈ Λ}). We know that G ∈ τ and J ⊆ A − ∪{Vα : α ∈ Λ} ∈ I, and additionally, (G − J) ∩ A = ∅. Thus, x ̸∈ A∗, confirming that A∗ ⊆ A. Theorem 15. Let A and B be subsets of an ideal topological space (X, τ, I). If A and B are sβ-I-paracompact subsets of X, then A ∪B is also an sβ-I-paracompact subset of X Proof. Let A = {Uλ : λ ∈ Λ} be an open cover of A ∪ B. This cover A also serves as an open cover for both A and B. By assumption, there exist sβ-I-locally finite strong β-I-open families B = {Vα : α ∈ Λ1} for A and C = {Wµ : µ ∈ Λ2} for B, which refine A such that: A− ∪{Vα : α ∈ Λ1} ∈ I, B − ∪{Wµ : µ ∈ Λ2} ∈ I. This implies that: A ⊆ ∪{Vα : α ∈ Λ1} ∪ I1, B ⊆ ∪{Wµ : µ ∈ Λ2} ∪ I2, where I1, I2 ∈ I. Therefore, we have: A ∪B ⊆ ∪({Vα : α ∈ Λ1} ∪ {Wµ : µ ∈ Λ2}) ∪ (I1 ∪ I2). It follows that: A ∪B − ∪{Vα ∪Wµ : α ∈ Λ1, µ ∈ Λ2} ⊆ I1 ∪ I2 ∈ I. We see that the collection D = {Vα ∪ Wµ : α ∈ Λ1, µ ∈ Λ2} of strong β-I-open sets is sβ-I-locally finite and refines A. Consequently, A ∪ B is an sβ-I-paracompact subset of X. C. Boonpok, P. Raktaow, A. Sama-Ae / Eur. J. Pure Appl. Math, 18 (3) (2025), 6297 16 of 23 Theorem 16. Let (X, τ, I) be an ideal topological space. If A is an sβ-I-paracompact subset of X and B is a closed subset of X, then A∩B is also an sβ-I-paracompact subset of X. Proof. Let A = {Uλ : λ ∈ Λ} be an open cover of A ∩ B. Since X − B is open in X, the collection A1 = {Uλ : λ ∈ Λ}∪{X−B} forms an open cover of A. By assumption and Lemma 6, A1 has a precise sβ-I-locally finite strong β-I-open refinement B = {Vλ : λ ∈ Λ} ∪ {V } such that: Vλ ⊆ Uλ for all λ ∈ Λ, V ⊆ X −B, A− (∪{Vλ : λ ∈ Λ} ∪ {V }) ∈ I. Now, observe that: A∩B−∪{Vλ : λ ∈ Λ} = A∩B− (∪{Vλ : λ ∈ Λ}∪{V }) ⊆ A− (∪{Vλ : λ ∈ Λ}∪{V }) ∈ I. Thus, we have A∩B−∪{Vλ : λ ∈ Λ} ∈ I. It follows that the collection B1 = {Vλ : λ ∈ Λ}, consisting of strong β-I-open sets, is sβ-I-locally finite and refines A. Therefore, A ∩ B is an sβ-I-paracompact subset of X. As a consequence of Theorem 16, we obtain the following corollaries. Corollary 2. If A is a closed subset of an sβ-I-paracompact space (X, τ, I), then A is also an sβ-I-paracompact subset of X. Corollary 3. Let A and B be closed subsets of an sβ-I-paracompact space of (X, τ, I), then A ∪B is also an sβ-I-paracompact subset of X Corollary 4. If A is an sβ-I-paracompact subset of X and B is an open set contained A, then A−B is an sβ-I-paracompact subset of X. Lemma 7. [19] 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 17. Let A and B be subsets of an ideal topological space (X, τ, I). If A is an sβ-IB-paracompact subset of B and B is a subset of X whose intersection with any strong β-I-open is again strong β-I-open, then A is an sβ-I-paracompact subset of X. Proof. Let A = {Uα : α ∈ Λ} be an open cover of A inX. Then, UA = {Uα∩B : α ∈ Λ} is an open cover of A in B. Since A is an sβ-IB-paracompact subset of B, the collection UA has a precise sβ-IB-locally finite strong β-I-open refinement VA = {Vα ∩ B : α ∈ Λ} such that Vα ⊆ Uα for all α ∈ Λ, and A − ∪{Vα ∩ B : α ∈ Λ} ∈ IB. Since Vα is a strong β-I-open subset of X for all α ∈ Λ, the collection B = {Vα : α ∈ Λ} of strong β-I-open sets of X is sβ-I-locally finite and refines A. Now, we have A− ∪{Vα : α ∈ Λ} ⊆ A− ∪{Vα ∩B : α ∈ Λ} ∈ IB ⊆ I. Therefore, A is an sβ-I-paracompact subset of X. C. Boonpok, P. Raktaow, A. Sama-Ae / Eur. J. Pure Appl. Math, 18 (3) (2025), 6297 17 of 23 4. Preservation of Strong β-I-Paracompactness In this part, we will show that the property of sβ-I-paracompactness is maintained under specific circumstances. We begin by introducing the following definition. Definition 11. Let (X, τ, I) and (Y, τ ′,J ) be ideal topological spaces, and let f : X → Y be a function. (1) The function f is said to be sβ-I-open if the image of every strong β-I-open set in X is a strong β-J -open set in Y . (2) The function f is said to be sβ-I-closed if the image of every strong β-I-closed set in X is a strong β-J -closed set in Y . (3) The function f is said to be sβ-I-irresolute if the preimage of every strong β-J -open set in Y is a strong β-I-open set in X. Observe that f−1(J ) forms an ideal on X if f : X → Y is a function and Y is a topological space endowed with an ideal J . Furthermore, if f is surjective and X has an ideal I, it follows that f(I) forms an ideal in Y . We now present the characteristics of a function mapped between two ideal topological spaces, where one space reflects the same properties as the other. To begin, we introduce the notion of sβ-I-compactness and state a lemma that will be employed in proving Theorem 18. Definition 12. An ideal topological space (X, τ, I) is said to be sβ-I-compact if ev- ery cover A of strong β-I-open subsets of X has a finite subcover, i.e., there exist sets A1, A2, . . . , An ∈ A such that X ⊆ A1 ∪A2 ∪ · · · ∪An. Lemma 8. Let (X, τ, I) and (Y, τ ′,J ) be ideal topological spaces, and f : X → Y be surjective. Then f is strong β-I-closed if and only if for every y ∈ Y and for every strong β-I-open set U in X containing {f−1(y)}, there exists a strong β-J -open set V containing y such that f−1(V ) ⊆ U . Proof. Let y ∈ Y and let U be a strong β-I-open subset of X such that {f−1(y)} ⊆ U . Define the set V = Y − f(X − U), which is strong β-J -open. Clearly, y ∈ V and f−1(V ) ⊆ U . Therefore, the necessity is established. Now, let F be a strong β-I-closed subset of X, and let y ∈ Y − f(F ). This implies that {f−1(y)} ⊆ X −F . By assumption, there exists a strong β-J -open set Vy containing y such that f−1(Vy) ⊆ X − F , and consequently, y ∈ Vy ⊆ Y − f(F ). Therefore, the set Y − f(F ) = ∪{Vy : y ∈ Y } is a strong β-J -open set. Hence, f(F ) is a strong β-J -closed set in Y . Theorem 18. Let (X, τ, I) and (Y, τ ′,J ) be ideal topological spaces. Suppose that f : X → Y satisfies the following statements: C. Boonpok, P. Raktaow, A. Sama-Ae / Eur. J. Pure Appl. Math, 18 (3) (2025), 6297 18 of 23 (1) f is continuous; (2) f is sβ-I-open; (3) f is sβ-I-closed; (4) f is surjective with {f−1(y)} being sβ-I-compact for every y ∈ Y ; and (5) f(I) ⊆ J . If X is sβ-I-paracompact, then Y is sβ-J -paracompact. Proof. Let A = {Uλ : λ ∈ Λ} be an open cover of Y . This implies that B = {f−1(Uλ) : λ ∈ Λ} is an open cover of X. Since X is sβ-I-paracompact, the collection B has a precise sβ-I-locally finite refinement C = {Vλ : λ ∈ Λ}, where each Vλ is strong β-I-open and X − ∪λ∈ΛVλ ∈ I. Since f is sβ-I-open, the family f(C) = {f(Vλ) : λ ∈ Λ} consists of strong β-J -open sets and is a refinement of A. Moreover, we have Y − ∪λ∈Λf(Vλ) ∈ J . Next, we check that f(C) is sβ-J -locally finite. Let y ∈ Y . Since C is sβ-I-locally finite, for each x ∈ {f−1(y)}, there exists a strong β-I-open set Gx containing x such that Gx intersects at most finitely many elements of C. Because {f−1(y)} is sβ-I-compact, and the family {Gx : f(x) = y} forms a strong β-I-open cover of {f−1(y)}, there exists a finite subcover {Hy} such that {f−1(y)} ⊆ ∪Hy, and ∪Hy intersects at most finitely many elements of C. As f is sβ-I-closed, applying Lemma 8, there exists a strong β-J -open set Wy containing y such that f−1(Wy) ⊆ ∪Hy. Thus, f−1(Wy) intersects at most finitely many elements of C, which implies that Wy intersects at most finitely many elements of f(C). Therefore, f(C) is sβ-J -locally finite in Y . Consequently, (Y, τ ′,J ) is sβ-J -paracompact. The next theorem characterizes a function from an sβ-I-paracompact ideal topological space (X, τ, I) to a topological space (Y, τ ′), ensuring that Y retains the same structural properties as X. Theorem 19. Let (X, τ, I) be an ideal topological space and (Y, τ ′) a topological space. Suppose that f : X → Y satisfies the following statements: (1) f is sβ-I-irresolute; (2) f is continuous; C. Boonpok, P. Raktaow, A. Sama-Ae / Eur. J. Pure Appl. Math, 18 (3) (2025), 6297 19 of 23 (3) f is sβ-I-open; (4) f is surjective; and (5) f(V) is sβ-f(I)-locally finite in Y for every sβ-I-locally finite V in X. If (X, τ, I) is sβ-I-paracompact, then (Y, τ ′, f(I)) is sβ-f(I)-paracompact. Proof. Let A = {Uλ : λ ∈ Λ} be an open cover of Y . This implies that B = {f−1(Uλ) : λ ∈ Λ} forms an open cover of X. Since X is sβ-I-paracompact, the collection B has a precise sβ-I-locally finite strong β-I-open refinement C = {Vλ : λ ∈ Λ} such that X − ∪λ∈ΛVλ ∈ I. Since Y − ∪λ∈Λf(Vλ) ⊆ f (X − ∪λ∈ΛVλ) , and f (X − ∪λ∈ΛVλ) ∈ f(I), it follows that Y − ∪λ∈Λf(Vλ) ∈ f(I). Because f is surjective, f(I) is an ideal in Y . By assumption, f(C) = {f(Vλ) : λ ∈ Λ} is a precise sβ-f(I)-open refinement of strong β-f(I)-open sets in Y . Now, we proceed to verify that f(C) refines A. For each f(Vλ) ∈ f(C), we have Vλ ∈ C, and there exists Uλ ∈ A such that Vλ ⊆ f−1(Uλ), as C refines B. This implies that f(Vλ) ⊆ f(f−1(Uλ)) ⊆ Uλ. Consequently, (Y, τ ′, f(I)) is sβ-f(I)-paracompact. The following theorem outlines conditions under which a function from a topological space X to an sβ-J -paracompact ideal topological space Y ensures that X shares the same properties as Y . Theorem 20. Let (X, τ) be a topological space and (Y, τ ′,J ) an ideal topological space. Suppose that f : X → Y satisfies the following statements: (1) f is open; (2) f is sβ-f−1(J )-irresolute; and (3) f is bijective. If (Y, τ ′,J ) is sβ-J -paracompact, then (X, τ, f−1(J )) is sβ-f−1(J )-paracompact. Proof. Let A = {Uλ : λ ∈ Λ} be an open cover of X. Since f is open, the collection f(A) = {f(Uλ) : λ ∈ Λ} is an open cover of Y . By hypothesis, f(A) has a precise sβ-J -locally finite strong β-J -open refinement B = {Vλ : λ ∈ Λ} such that Y − ∪λ∈ΛVλ ∈ J . C. Boonpok, P. Raktaow, A. Sama-Ae / Eur. J. Pure Appl. Math, 18 (3) (2025), 6297 20 of 23 This implies that Y − ∪λ∈ΛVλ = J for some J ∈ J , which means f−1(Y )− ∪λ∈Λf −1(Vλ) = f−1(Y )− f−1(∪λ∈ΛVλ) = f−1(J). Hence, X − ∪λ∈Λf −1(Vλ) ∈ f−1(J ). Let I = f−1(J ). Since f is sβ-I-irresolute, the collection C = {f−1(Vλ) : λ ∈ Λ} forms an sβ-I-locally finite collection of strong β-I-open sets. For each f−1(Vλ) ∈ C, since Vλ ∈ B, there exists Uλ ∈ A such that Vλ ⊆ f(Uλ) as B refines f(A). Thus, f−1(Vλ) ⊆ f−1(f(Uλ)) = Uλ. The refinement of A by C is then asserted. Therefore, (X, τ, I) is shown to be sβ-I- paracompact. 5. Conclusions Conclusions: This paper introduces and investigates the concept of strong β-I-submaximal ideal topological spaces, which generalizes submaximality in the context of ideal topology. We have demonstrated that the following statements are equivalent: • (X, τ, I) possesses the property of strong β-I-submaximality. • For every subset A of X, the set sβ clI(A)−A is strong β-I-closed. • Each subset of X is locally strong β-I-closed. • Every subset of X qualifies as a B-sI set. • Any strong β-I-dense subset of X is a B-sI set. • Every subset of X is co-locally strong β-I-closed. • If a subset A of X satisfies sβ IntI(A) = ∅, then it is strong β-I-closed. • All strong β-I-codense subsets of X are strong β-I-closed. • Every subset A of X for which sβ IntI(A) = ∅ holds is both strong β-I-closed and strong β-I-discrete. • For any subset A of X, the set sβ clI(A) − A is both strong β-I-closed and strong β-I-discrete. • Each strong β-I-codense subset of X is strong β-I-closed and strong β-I-discrete. C. Boonpok, P. Raktaow, A. Sama-Ae / Eur. J. Pure Appl. Math, 18 (3) (2025), 6297 21 of 23 This study explores various characterizations of sβ-I-paracompactness in ideal topo- logical spaces, highlighting it as a stronger variant of β-paracompactness. It is established that every sβ-I-paracompact space is necessarily I-β-paracompact, and in the case of Hausdorff spaces, sβ-I-paracompactness implies sβ-I-regularity. Moreover, it is shown that the union of two sβ-I-paracompact subsets remains sβ-I- paracompact, and the intersection of an sβ-I-paracompact subset with a closed set also retains the property. The work further demonstrates the preservation of sβ-I-paracompactness under cer- tain mappings. Specifically, if a map f : X → Y is continuous, sβ-I-open, sβ-I-closed, and surjective, with each fiber {f−1(y)} being sβ-I-compact, and if f(I) ⊆ J , then sβ-I-paracompactness of X implies the same for Y . Similarly, if f is sβ-I-irresolute, continuous, sβ-I-open, and surjective, and for ev- ery sβ-I-locally finite family V, the image f(V) is sβ-J -locally finite, then the sβ-I- paracompactness of X ensures that Y is sβ-J -paracompact. Finally, if f is open, sβ-I-irresolute, and bijective, and Y is sβ-J -paracompact, then it follows that X is also sβ-I-paracompact. Applications: Strong β-I-submaximality and paracompactness contribute significantly to fields such as digital topology, computational geometry, and data analysis by utilizing generalized open sets and locally finite covers. These properties broaden the scope of con- tinuity results within ideal spaces and find relevance in areas like fuzzy and soft topology, effectively linking abstract topological theory with computational approaches to support better approximation techniques and structural interpretations. Future Works: Potential research avenues include extending the concepts to fuzzy and neutrosophic ideals, exploring their impact on separation axioms and product topolo- gies, investigating connections with measure theory, analyzing relationships among vari- ous generalized open sets, and creating algorithms for detecting these properties in finite spaces—thereby enriching both the theoretical framework and real-world applications. Acknowledgements We sincerely appreciate all those who contributed to our research. Their direction, collaboration, and assistance have been essential to the success of this project. This research was supported by the National Science, Research, and Innovation Fund (NSRF) and Prince of Songkla University (Ref. No. SAT6801325S). References [1] E. D. Khalimsky, R. Kopperman, and P. R. Meyer. Computer graphics and connected topologies on finite ordered sets. Topology and its Applications, 36:1–17, 1990. [2] T. Y. Kong, R. Kopperman, and P. R. Meyer. 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