EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6513 ISSN 1307-5543 – ejpam.com Published by New York Business Global Novel Results On D-Soft Compact Spaces Jamal Oudetallah1, Ahmad Almalkawi2, Rahmeh Alrababah3, Ala Amourah4,5,∗, Abdullah Alsoboh6,∗, Khaled Al Mashrafi6, Tala Sasa7 1 Department of Mathematics, University of Petra, Amman, 11196, Jordan. 2 Modern College of Business and Science, Muscat, Sultanate of Oman. 3 Ajloun National University, College of Sciences, Dept. of Math, Jordan. 4 Mathematics Education Program, Faculty of Education and Arts, Sohar University, Sohar 311, Oman. 5 Jadara Research Center, Jadara University, Irbid 21110, Jordan 6 College of Applied and Health Sciences, A’Sharqiyah University, Post Box No. 42, Post Code No. 400, Ibra, Sultanate of Oman 7 Department of Mathematics, Faculty of Science, Applied Science Private University, Am- man, Jordan. Abstract. This paper introduces and investigates D-soft compact spaces, a novel generalization of compactness in soft topological spaces using D-soft covers. We establish fundamental properties, characterizations, and relationships between D-soft compactness and other forms of soft compact- ness. Key results include the hereditary nature of D-soft compactness under specific conditions, the equivalence between D-soft and soft compactness in soft locally indiscrete spaces, and preservation under continuous mappings. The study provides new perspectives on compactness conditions in soft topology with potential applications in decision-making and uncertainty modeling. Several examples and counterexamples illustrate the theoretical developments, demonstrating that D-soft compactness is generally stronger than soft compactness. 2020 Mathematics Subject Classifications: AMS 54D30, 54E99, 54D10 Key Words and Phrases: Soft sets, D-Soft set, soft cover, D-soft cover, Soft set theory, Soft topology, Soft compact space, D-soft compact space. Notation Throughout this paper, we use the following notation: ∗Corresponding author. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6513 Email addresses: jamal.oudetallah@uop.edu.jo (J. Oudetallah), ahmad.abdelqader@mcbs.edu.om (A. Almalkawi), rrr.ra2020r@gmail.com (R. Alrababah), AAmourah@su.edu.om (A. Amourah), abdullah.alsoboh@asu.edu.om (A. Alsoboh), khaled.almashrafi@asu.edu.om (K. Al Mashrafi), t sasa@asu.edu.jo (T. Sasa) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6513 2 of 14 • S denotes a universe set • A denotes a set of parameters • P (S) denotes the power set of S • (T,E) denotes a soft set where T : E → P (S) and E ⊆ A • (S,Υ, A) denotes a soft topological space • (U,A) denotes a soft open set when clear from context • ϕ denotes the null soft set 1. Introduction The theory of soft sets, introduced by Molodtsov [1] in 1999, has emerged as a powerful mathematical tool for handling uncertainty and vagueness in various fields including engi- neering, medicine, economics, and environmental sciences. Unlike traditional approaches such as probability theory, fuzzy sets, and rough sets, soft set theory provides a pa- rameterized family of subsets of the universe, offering greater flexibility in dealing with uncertainty. Following Molodtsov’s pioneering work, numerous researchers have contributed to the development and application of soft set theory. Maji et al. [2] introduced basic operations on soft sets, while Çağman and Enginoğlu [3, 4] developed soft set theory for decision- making and soft matrix theory. The algebraic structures of soft sets were studied by Aktaş and Çağman [5], leading to various applications in different mathematical structures. Abbas et al. [6] further generalized operations in soft set theory via relaxed conditions on parameters. Recent developments in soft set applications include the work of Al-Sharqi et al. [7] on interval complex neutrosophic soft sets, Gulistan et al. [8] on neutrosophic cubic soft matrices for decision making, Khan and Zhu [9] on parameter reduction algorithms, and Muhiuddin et al. [10] on generalized ideals based on fuzzy soft set theory. Ulucay [11] introduced soft representation of soft groups, while practical applications were explored by Xiao [12] in medical diagnosis using hybrid fuzzy soft sets, Voskoglou [13] in decision making with TFNs and soft sets, and Zhang et al. [14] with N-soft rough sets. The introduction of soft topology by Shabir and Naz [15] in 2011 marked a significant milestone in extending topological concepts to the soft set framework. This development opened new avenues for research in generalized topological spaces, building upon classical topology foundations established by Willard [16] and Engelking [17]. Subsequently, various topological properties have been investigated in the soft setting, including the work of Aygünoğlu and Aygün [18] on notes about soft topological spaces and Arockiarani and Selvi [19] on soft slightly πg-continuous functions. Compactness, being one of the most fundamental concepts in topology, has naturally attracted attention in soft topology. The notion of soft compactness was introduced and J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6513 3 of 14 studied in the context of soft topological spaces as an extension of classical compactness. However, despite these efforts, the theory of soft compactness remains relatively underde- veloped compared to its classical counterpart, particularly regarding covering properties and their applications. The concept of D-sets in classical topology, introduced by Tong [20], provides a gen- eralization of open sets that has proven useful in studying various topological properties. Qoqazeh et al. [21] investigated D-compact topological spaces, while Mustafa and Qo- qazeh [22] studied supra D-sets and associated separation axioms. The soft version of D-sets was introduced by Mahmood [23], leading to new perspectives on soft topological structures and weak soft dñ-sets. Oudetallah et al. have made significant contributions to the theory of D-sets and related concepts. Oudetallah et al. [24] investigated D-metacompactness in topological spaces, establishing fundamental properties and relationships with other covering prop- erties. This work was extended to r-compactness in both topological and bitopological spaces [25], providing new insights into generalized compactness conditions. Further in- vestigations by Oudetallah [26] on nearly metacompact spaces in bitopological settings, and Oudetallah and AL-Hawari [27] on other generalizations of pairwise expandable spaces, have enriched the theory of covering properties in generalized topological spaces. Recent advances (2023-2025) in soft topological spaces have focused on refining com- pactness conditions and exploring new types of coverings. Notable contributions include the work of Al-shami et al. [28] on compactness and connectedness via soft somewhat open sets, Mhemdi [29] on novel types of soft compact and connected spaces inspired by soft q-sets, and Alqahtani and Ameen [30] on soft nodec spaces. These studies have re- vealed the rich structure of soft topological spaces and motivated further investigations into specialized forms of compactness. More recent developments include Alghamdi et al. [31] on soft submaximal and soft door spaces, Saleh and Salih [32] on c-continuity and c-compact spaces via soft sets, Al- Mufarrij and Al-Ghour [33] on regular-closed functions between soft topological spaces, and Alshammari et al. [34] on r-fuzzy soft δ-open sets with applications. These works demonstrate the continuing evolution and diversification of soft topology research. Beyond topological considerations, the mathematical structures underlying our work have connections to geometric and functional analytic concepts. Oudetallah and Abualigah [35] investigated h-convexity in metric linear spaces, which shares structural similarities with compactness properties in terms of covering and separation conditions. Oudetal- lah [36] recently studied novel results on near Lindelöfness in topological spaces, which provides relevant insights for extending covering properties to soft settings. Recent developments in approximation spaces and their applications have shown promis- ing connections to soft topological methods. Studies on κ-neighborhoods [38], innova- tive rough set approaches for medical diagnosis of COVID-19 variants using novel initial- neighborhood systems [37], accurate diagnosis for COVID-19 variants using nearly initial- rough sets [38], enhancing rheumatic fever analysis via tritopological approximation spaces [39], and exploring β-basic rough sets and their applications in medicine [40] have demon- strated the practical relevance of these theoretical frameworks in real-world medical ap- J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6513 4 of 14 plications. This study aims to introduce and investigate D-soft compact spaces, a new class of soft topological spaces that generalizes both soft compactness and D-compactness. By introducing D-soft covers—a novel type of covering in soft topology—we provide a fresh perspective on compactness conditions. Our approach builds upon the extensive work on D-sets and related concepts by Oudetallah and collaborators [24–27, 35, 36], adapting these ideas to the soft topological setting. This addresses existing gaps in the literature and offers a more nuanced understanding of covering properties in soft topological spaces. The significance of this research is multifaceted. Theoretically, it extends the foun- dational understanding of compactness by introducing adjustable D-soft parameters that can be tailored to different topological structures. Practically, it provides tools for model- ing uncertainty in complex systems where traditional compactness conditions may be too restrictive. The framework developed here has potential applications in decision-support systems, pattern recognition, and data analysis where soft computing approaches are ben- eficial, as demonstrated by recent applications in medical diagnosis [37–40]. The main contributions of this paper are: (i) Introduction of D-soft covers and D-soft compact spaces with their fundamental properties (ii) Establishment of relationships between D-soft compactness and existing forms of soft compactness (iii) Characterization theorems for D-soft compact spaces under various conditions (iv) Investigation of hereditary properties and preservation under mappings (v) Illustrative examples demonstrating the distinctions between different types of soft compactness The organization of the paper is as follows: In Section 2, we review essential concepts from soft set theory and soft topology, providing the foundation for our work. In Section 3, we present the main definitions of D-soft covers and D-soft compactness concepts; we then develop the theory through several theorems and results, supported by illustrative figures and examples. Section 4 concludes the paper with a summary of our findings and directions for future research. 2. Basic Principles and Primary Notions In this section, we present the main concepts and results that we employed in our study in the theory of soft sets in soft topological spaces. Definition 1. [1] Let S be a universe set and A be a set of parameters. Suppose that the power set of S is represented by P (S) and E ⊂ A. A pair (T,E) is called a soft set over S if T : E −→ P (S); for any e ∈ E, we have T (e) ∈ P (S). J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6513 5 of 14 Definition 2. [15] A soft topological space is a triplet (S,Υ, A) where S is a nonempty set and Υ is a collection of soft sets over S with parameter set A such that Υ satisfies the following axioms: (i) Both S and ∅ belong to Υ. (ii) Any union of soft sets in Υ belongs to Υ. (iii) Any finite intersection of soft sets in Υ belongs to Υ. The soft topological space is denoted by (S,Υ, A) where A is a parameter set and such a collection, Υ, over S is called a soft topology. The elements of Υ are called soft open sets, and their complements are called soft closed sets. Table 1: Comparison between Classical and Soft Topological Concepts Classical Topology Soft Topology Set X Soft set (F,A) Open set U Soft open set (U,A) Topology τ Soft topology Υ Open cover Soft open cover Compact space Soft compact space D-set D-soft set D-cover D-soft cover D-compact D-soft compact Definition 3. [18] The soft-set (T,A) is said to be covered by a family σ of soft-sets if (T,A) ⊆ ∪{(Ti, A); (Ti, A) ∈ σ; i ∈ I}, where σ is referred to as a soft-open cover if all of its elements are soft-open sets. Definition 4. [3] A soft topological space (S,Υ, A) is called soft-compact if every soft-open cover has a finite soft-subcover. This generalizes the classical notion of compactness to the soft set setting. Definition 5. [16, 17] A topological space (S,Υ) is compact if every open cover of S has a finite subcover. This means that whenever S = ⋃ i∈I Ui where each Ui is open, there exists a finite subset J ⊆ I such that S = ⋃ i∈J Ui. Definition 6. [20] Let (S,Υ) be a topological space. A subset S1 ⊆ S is called a D-set if there exist two open sets U1 and U2 in Υ such that U1 ̸= S and S1 = U1 − U2. We say that S1 is the D-set generated by U1 and U2. Definition 7. [23] A soft subset (F,A) of a soft topological space (S,Υ, A) is called a D-soft set if there exist two soft open sets (U,A) and (V,A) in Υ such that U ̸= S and F ⊂ (U,A)− (V,A). J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6513 6 of 14 Remark 1. [23] Every soft-open set is a D-soft set. However, as the following example shows, the converse of Remark 2.1 is not true in general. Definition 8. [21] Let (S,Υ) be a topological space. A cover U = {Uγ : γ ∈ Γ} of S is called a D-cover if every Uγ is a D-set for all γ ∈ Γ. Definition 9. [21] A topological space (S,Υ) is called D-compact if every D-cover has a finite subcover. Definition 10. [22] The topological space (S,Υ) is called locally indiscrete if every open set in Υ is a clopen set. Definition 11. [19] A soft topological space (S,Υ, A) is called soft locally indiscrete if every soft open set in (Υ, A) is a soft clopen set. 3. Main Results The notion of D-soft compactness in soft topological spaces is presented in this section along with some of its properties and its relations with other spaces. Our approach builds upon the extensive work on D-compactness and related concepts in classical topology [24, 25], adapting these ideas to the soft topological framework. Example 1. Let S = {4, 5, 7} and A = {a1, a2} such that a1 =“even number”, a2 =“odd numbers”. Suppose that T (a1) = {4}, T (a2) = {5, 7}, then the collection (T,E) = {(a1, {4}), (a2, {5, 7})} is a soft set over S. Example 2. Let S = N, A = {a} and Υ = {S, ϕ, (U,A)} be a soft topology on S, where the soft set (U,A) is defined by U(a) = {1}. Then (U,A) is a D-soft set but not a soft open set. To see this, take the soft open sets (W,A) and (V,A) where W (a) = S and V (a) = S − {1}. Then (U,A) = (W,A) − (V,A), showing it is a D-soft set. However, (U,A) /∈ Υ, so it is not soft open. Definition 12. A soft cover (U,A) = {(Uγ , A) : γ ∈ Γ} of a soft topological space (S,Υ, A) is called a D-soft cover if (Uγ , A) is a D-soft set for every γ ∈ Γ. Every soft-open cover is clearly a D-soft cover, but the converse is not true in general. For example, in the soft topological space (R,Υcof , A = {a}), the collection (U,A) = {(a, {n}) : n ∈ R} is a D-soft cover that is not a soft open cover. Definition 13. The soft topological space (S,Υ, A) is called D-soft compact if every D-soft cover of S has a finite soft-subcover. J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6513 7 of 14 Figure 1: The space of D-soft compact. Figure 1 shows the D-soft compact space for which (U,A) = {(Uγ , A) : γ ∈ Γ} is D-soft cover of the soft topological space (S,Υ, A) possesses a finite soft-subcovers {(Uγ1 , A1), (Uγ2 , A2), . . . , (Uγn , An)}, in which each (Uγ , A) is a D-soft set for all γ ∈ Γ. Example 3. Consider the following examples of D-soft compact and non-D-soft compact spaces: • Let S = {1, 2, 3}, A = {a1, a2, a3}, and Υ = {ϕ, S, (U1, A), (U2, A)} where (U1, A) is defined by U1(ai) = {1, 2} for all i, and (U2, A) is defined by U2(ai) = {2, 3} for all i. Then (S,Υ, A) is D-soft compact because S is finite. • Let A = {a} and (R,Υ, A) be the soft usual topological space. Then (R,Υ, A) is not D-soft compact. To see this, consider the D-soft cover (U,A) = {((n− 1, n+1), A) : n ∈ Z}. This cover has no finite subcover, hence (R,Υ, A) is not D-soft compact. Lemma 1. In a soft locally indiscrete space (S,Υ, A), the union of any collection of D-soft sets is a D-soft set. Proof. Let {(Fi, A) : i ∈ I} be a collection of D-soft sets in a soft locally indiscrete space. For each i, there exist soft open sets (Ui, A) and (Vi, A) such that Ui ̸= S and (Fi, A) ⊆ (Ui, A)− (Vi, A). Since the space is soft locally indiscrete, all soft open sets are soft clopen. Let (U,A) = ⋃ i∈I(Ui, A) and (V,A) = ⋃ i∈I(Vi, A). Then (U,A) and (V,A) are soft open (hence soft clopen), and ⋃ i∈I(Fi, A) ⊆ (U,A) − (V,A). Since at least one Ui ̸= S, we have U ̸= S (as S is soft clopen). Therefore, ⋃ i∈I(Fi, A) is a D-soft set. Theorem 1. For every soft topology on S, the space (S,Υ, A) is D-soft compact if S is a nonempty finite set. Proof. Suppose S = {s1, s2, . . . , sn} is a finite set and (U,A) = {(Uγ , A) : γ ∈ Γ} is a D-soft cover of S. Since (U,A) covers S, for each si ∈ S there exists some γi ∈ Γ such that si ∈ (Uγi , A). Therefore, the collection (U,A)∗ = {(Uγ1 , A), (Uγ2 , A), . . . , (Uγn , A)} is a finite soft subcover of (U,A) for S. Thus, (S,Υ, A) is a D-soft compact space. J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6513 8 of 14 Corollary 1. In a soft locally indiscrete space, every D-soft set is soft clopen. Proof. Let (U,A) be a D-soft set in a soft locally indiscrete space. Then (U,A) = (W,A)− (Z,A) for some soft open sets (W,A) and (Z,A). Since the space is soft locally indiscrete, both (W,A) and (Z,A) are soft clopen. Therefore, (U,A) is the difference of two soft clopen sets, which makes it soft clopen. Corollary 2. In a soft locally indiscrete space, the union of any collection of D-soft sets is a D-soft set. Proof. This follows directly from Lemma 3.1. Theorem 2. Every D-soft compact space is soft compact. Proof. Let (S,Υ, A) be a D-soft compact space and let (U,A) = {(Uγ , A) : γ ∈ Γ} be any soft open cover of S. Since every soft open set is a D-soft set (Remark 2.1), (U,A) is also a D-soft cover. By D-soft compactness, (U,A) has a finite soft subcover. Therefore, (S,Υ, A) is soft compact. Figure 2: The relation of D-soft compact and soft compact spaces. The primary connection between D-soft compact and soft compact spaces is depicted in Figure 2. Any D-soft compact space should be soft compact, according to the important claim that (U,A) = {(Uγ , A) : γ ∈ Γ} is a D-soft cover of the soft topological space (S,Υ, A) with a finite soft-subcover, and (V,B) = {(Vγ , B) : γ ∈ Γ} is a soft-cover of the soft topological space (T, θ,A). We demonstrate in the following example that the opposite of Theorem 3.2 does not necessarily have to be true. Example 4. The soft-cofinite topological space (R,Υ, A), where A = {a1, a2}, is soft- compact but not a D-soft compact. In a soft topological space (R,Υ, A), any soft-set of the form (a1,R − {y}) or (a2,R − {y}); y ∈ R is a soft-open set. Let (W,A) = (a1,R − {y1}) and (Z,A) = (a2,R − {y2}) stand now. In such case, (U,A) = (W,A) − (Z,A) = J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6513 9 of 14 (a1, {y2}) is an inaccessible D-soft set. With no finite soft-subcover, the collection (U,A) = (ai=1,2, {y}) : y ∈ R} is a D-soft cover of (R,Υ, A). If (U,A) = {(ai=1,2, {y}) : y ∈ R} has a finite soft-subcover {(ai=1,2, {y1}), (ai=1,2, {y2}), . . . , (ai=1,2, {yn})}, then R ⊆⋃n k=1(ai=1,2, {yk}), i.e., R is a finite soft-set, which is contradictory. In the next example, we show that the contrapositive of Theorem 3.2. is true in general. Example 5. Since A is an arbitrary parameter, the soft topological space (R,Υl.r, A) is not soft-compact and thus not D-soft compact. In the next example presents that the converse of Theorem 3.2. may be true with extra conditions. Theorem 3. If the soft compact space (S,Υ, A) is a soft locally indiscrete, then it is D-soft compact. Proof. The collection (Uγ , A) is a soft-clopen set for every γ ∈ Γ. Let (U,A) = {(Uγ , A) : γ ∈ Γ} be a D-soft cover of (S,Υ, A). (U,A) is a soft-open cover of (S,Υ, A), in this case. Now, (U,A) has a finite soft-subcover since the space (S,Υ, A) is soft compact. Thus, we have what is needed. Figure 3: The relation of D-soft compact and soft compact spaces with soft locally indiscreet condition. Figure 3 illustrates the complex relationship between D-soft compact and soft compact spaces with addition condition. It presents a curial assertion that any D-soft compact space should be soft compact, while the opposite is true if the soft compact space satisfies the soft locally indiscreet condition, such that (U,A) = {(Uγ , A) : γ ∈ Γ} is D-soft cover of the soft topological space (S,Υ, A) has a finite soft-subcover, and (V,B) = {(Vγ , B) : γ ∈ Γ} is soft-cover of the soft topological space (T, θ,A) has a finite-subcover. Example 6. (1) Let S = R, A = {a} and Υind be a soft indiscrete topology on S. Then the soft space (S,Υind, A) is D-soft compact, because it is locally indiscrete and soft compact. (2) Let S = R, A = {a1, a2, a3, a4, a5, a6} and Υ = {ϕ,R,R − {7}, {7}} be soft topology on S. Then the space (S,Υ, A) is soft locally indiscrete and soft compact, so it is D-soft compact. J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6513 10 of 14 Theorem 4. Let (S,Υs, As) be a soft topological space and Q ⊆ S, then (Q,Υq, Aq) is D-soft compact iff every each soft-cover of Q by D-soft sets in S has a finite soft-subcover. Proof. =⇒) Let (Q,Υq, Aq) be D-soft compact and (U,As) = {(Uγ , As) : γ ∈ Γ} be D-soft cover of Q by D-soft sets in S. Now, for all γ ∈ Γ we have that the collection (U∗ γ , A ∗ q) = (Uγ , As) ∩ Q is a D-soft sets in Q. Then (U∗, A∗ q) = {(U∗ γ , A ∗ q) : γ ∈ Γ} is D-soft cover of Q by D-soft sets in Q. Since the space (Q,Υq, Aq) is a is D-soft com- pact, then (U∗, A∗ q) has a finite soft-subcover {(U∗ γ1 , A ∗ q1), (U ∗ γ2 , A ∗ q2), . . . , (U ∗ γn , A ∗ qn)} for Q. Hence we get that the family {(Uγ1 , As1), (Uγ2 , As2), . . . , (Uγn , Asn)} is a finite soft-subcover of (U,As) in S for Q, where (U∗ γi , A ∗ qi) = (Uγi , Asi) ∩Q for all i = 1, 2, . . . , n and γ ∈ Γ. Hence we have the required. ⇐=) Let Q have a finite soft-subcover for every D-soft cover of Q by D-soft sets in S. It is assumed that (V,Aq) = {(Vγ , Aq) : γ ∈ Γ} is a D-soft cover of Q by D- soft sets in Q. Then, for any γ ∈ Γ, there are D-soft sets (Uγ , As) in S such that (Vγ , Aq) = (Uγ , As)∩Q. As we now know, (U,As) = {(Uγ , As) : γ ∈ Γ} is the D-soft cover of Q by D-soft sets in S. then by assumption the collection (U,As) has a finite-subcover {(Uγ1 , As1), (Uγ2 , As2), . . . , (Uγn , Asn)}. Hence we get that the family {(Vγ1 , Aq1), (Vγ2 , Aq2), . . . , (Vγn , Aqn)} is a finite soft-subcover of (V,Aq) for Q, because Vγi ⊆ Uγi, for all γ ∈ Γ and i = 1, 2, . . . , n. Hence we have the required. Figure 4: The soft subspace topology of D-soft compact space. A key distinction between the soft topological space and its soft subspace is depicted in Figure 4, in which the soft topological subspace (Q,Υq, Aq) of the D-soft compact space (S,Υs, As) must be D-soft compact space such that (U,A) is D-soft cover of the soft topological subspace (Q,Υq, Aq) has a finite soft-subcover, for any D-soft sets in the space (S,Υs, As). Corollary 3. If (Q,Υq, Aq) is a D-soft compact space, then any soft-open cover of Q by soft-open sets in S has a finite soft-subcover. Theorem 5. Let (S,Υ1, A1) and (S,Υ2, A2) be two soft topological spaces. If (S,Υ2, A2) is D-soft compact and (Υ1, A1) ⊆ (Υ2, A2), then (S,Υ1, A1) is a D-soft compact space. J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6513 11 of 14 Proof. Let (U,A1) = {(Uγ , A1) : γ ∈ Γ} be a D-soft cover of (S,Υ1, A1). Then we get that (U,A1) is also D-soft cover of (S,Υ2, A2), because (Υ1, A1) ⊆ (Υ2, A2), so it has finite soft-subcover, Hence we have the required that the space (S,Υ1, A1) is D-soft compact. Theorem 6. All of the D-soft compact subspaces are closed soft spaces of D-soft compact spaces. Proof. Assume that S is a D-soft compact space and Q ⊂ S is a closed set. Let (U,A) = ((Aγ , A) : γ ∈ Γ). Given a D-soft cover of Q by D-soft sets of S. When S is D soft compact (which we denote by writing Sc as a D soft cover), so is its collection (U,A)∩{S−Q}, and we will have a finite soft sub cover of that family, or, we have a D soft cover to the problem, that is (U,A)∗, of finite size. Now the collection (U,A)∗ − {S −Q} is a finite soft subcover of (U,A) for Q. Thus, we have what is needed. 4. Conclusion In this paper, we have introduced and investigated the concept of D-soft compact spaces, providing a new perspective on compactness in soft topological spaces. Using the novel concept of D-soft cover, we have established a framework that generalizes classical compactness while addressing limitations in existing soft topology literature. Our main contributions include: • The introduction of D-soft covers as a new type of covering in soft topology • The definition and characterization of D-soft compact spaces • Proof that D-soft compactness implies soft compactness, with the converse holding in soft locally indiscrete spaces • Investigation of hereditary properties and behavior under continuous mappings • Illustrative examples distinguishing D-soft compactness from soft compactness The results obtained show that D-soft compact spaces not only enhance the theoretical foundation of soft set theory but also open new avenues for applications in fields requir- ing uncertainty modeling, such as computer science and decision-support systems. The framework developed here provides an efficient and innovative contribution to the evolving field of soft topology and encourages further research into the structural characteristics and interrelationships of soft compact spaces. The study of D-soft compact spaces opens several promising research directions: (i) Extension to other covering properties: Future studies could focus on ex- panding D-soft compactness into more complex soft topological frameworks, such as D-soft Lindelofness and D-soft metacompactness, building upon the work on near Lindelofness [36] and D-metacompactness [24]. J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6513 12 of 14 (ii) Bitopological generalizations: The concepts developed here could be extended to soft bitopological spaces, following the approach used for nearly metacompact spaces [26] and pairwise expandable spaces [27]. (iii) Connections with approximation spaces: Building on the work of Nawar et al. (2022) on θβ-approximation spaces and El-Bably et al. (2025) on their theoretical refinements, there is potential for integrating D-soft compactness with ideal-related topological frameworks. These connections could enhance both theoretical under- standing and practical applications. (iv) Applications in decision-making: The framework developed here could be ap- plied to real-world problems in medical diagnosis, pattern recognition, and data analysis, following the successful applications of rough set approaches demonstrated in recent literature. (v) Relationships with fuzzy and rough set theories: Investigating how D-soft compact spaces connect with fuzzy topological spaces and rough set models could yield new hybrid approaches for handling uncertainty. (vi) Geometric connections: The relationship between D-soft compactness and con- vexity properties in soft settings, inspired by work on h-convexity [35], presents an intriguing avenue for future research. We emphasize that any extensions should maintain fidelity to the original formulations, particularly regarding the θβ-concepts established by Nawar et al. (2022) and refined by El-Bably et al. (2025), as these provide the accurate foundation for ideal-related topological structures. We hope that this study will encourage researchers to explore this field and discover additional connections with other areas. Acknowledgements We appreciate the reviewer’s useful suggestions, which helped make the paper more successful. References [1] Dmitriy Molodtsov. Soft set theory—first results. Computers & Mathematics with Applications, 37(4-5):19–31, 1999. [2] Pradip Kumar Maji, Ranjit Biswas, and A. Ranjan Roy. Soft set theory. Computers & Mathematics with Applications, 45(4-5):555–562, 2003. [3] Naim Çağman and Serdar Enginoğlu. Soft set theory and uni–int decision making. European Journal of Operational Research, 207(2):848–855, 2010. [4] Naim Çağman and Serdar Enginoğlu. Soft matrix theory and its decision making. Computers & Mathematics with Applications, 59(10):3308–3314, 2010. J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6513 13 of 14 [5] Hacı Aktaş and Naim Çağman. Soft sets and soft groups. Information Sciences, 177(13):2726–2735, 2007. [6] Mujahid Abbas, Muhammad Irfan Ali, and Salvador Romaguera. Generalized oper- ations in soft set theory via relaxed conditions on parameters. Filomat, 31(19):5955– 5964, 2017. [7] Faisal Al-Sharqi, Abd Ghafur Ahmad, and Ashraf Al-Quran. Mapping on inter- val complex neutrosophic soft sets. International Journal of Neutrosophic Science, 19(4):77–85, 2022. [8] Muhammad Gulistan, Ismat Beg, and Naveed Yaqoob. A new approach in decision making problems under the environment of neutrosophic cubic soft matrices. Journal of Intelligent & Fuzzy Systems, 36(1):295–307, 2019. [9] Abid Khan and Yuanguo Zhu. New algorithms for parameter reduction of intuition- istic fuzzy soft sets. Computational and Applied Mathematics, 39(3):232, 2020. [10] G. Muhiuddin, Deena Al-Kadi, K. P. Shum, and Abdulaziz Mohammed Alanazi. Generalized ideals of bck/bci-algebras based on fuzzy soft set theory. Advances in Fuzzy Systems, 2021(1):8869931, 2021. [11] Vakkas Ulucay. Soft representation of soft groups. New Trends in Mathematical Sciences, 4(2):23–29, 2016. [12] Fuyuan Xiao. A hybrid fuzzy soft sets decision making method in medical diagnosis. IEEE Access, 6:25300–25312, 2018. [13] Michael Gr. Voskoglou. A hybrid model for decision making utilizing tfns and soft sets as tools. Equations, 2:65–69, 2022. [14] Di Zhang, Pi-Yu Li, and Shuang An. N-soft rough sets and its applications. Journal of Intelligent & Fuzzy Systems, 40(1):565–573, 2021. [15] Muhammad Shabir and Munazza Naz. On soft topological spaces. Computers & Mathematics with Applications, 61(7):1786–1799, 2011. [16] Stephen Willard. General Topology. Addison Wesley Pub. Co, Reading, MA, 1970. [17] Ryszard Engelking. General Topology, volume 529. Heldermann, Berlin, 1989. MR1039321 (91c:54001). [18] Abdülkadir Aygünoğlu and Halis Aygün. Some notes on soft topological spaces. Neural Computing and Applications, 21(Suppl 1):113–119, 2012. [19] I. Arockiarani and A. Selvi. On soft slightly πg-continuous functions. Journal of Progressive Research in Mathematics, 3(2):168–174, 2015. [20] J. C. Tong. A separation axiom between t0 and t1. Annales de la Société Scien- tifique de Bruxelles, Séries 1: Sciences Mathématiques, Astronomiques et Physiques, 96(2):85–90, 1982. [21] Hamza Qoqazeh, Yousef Al-Qudah, Mohammad Almousa, and Ali Jaradat. On d- compact topological spaces. Journal of Applied Mathematics & Informatics, 39(5- 6):883–894, 2021. [22] Jamal M. Mustafa and Hamzeh A. Qoqazeh. Supra d-sets and associated separation axioms. International Journal of Pure and Applied Mathematics, 80(5):657–663, 2012. [23] Sabiha I. Mahmood. On weak soft n-open sets and weak soft dn-sets in soft topological spaces. Al-Nahrain Journal of Science, 20(2):131–141, 2017. J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6513 14 of 14 [24] Jamal Oudetallah, Rehab Alharbi, Salsabiela Rawashdeh, and Ala Amourah. Lin- delöfness spaces in nth topological spaces. International Journal of Neutrosophic Science (IJNS), 25(3), 2025. [25] Ala Amourah, Jamal Oudetallah, Iqbal M. Batiha, Salsabiela Rawashdeh, Sultan Alsaadi, and Tala Sasa. Some types of tri-locally compactness spaces. European Journal of Pure and Applied Mathematics, 18(2):5764–5764, 2025. [26] Jamal Oudetallah, Rehab Alharbi, Iqbal Batiha, Salsabiela Rawashdeh, and Ala Amourah. Some types of tri-lindelöfness spaces. European Journal of Pure and Applied Mathematics, 18(2):5578–5578, 2025. [27] Ala Amourah, Jamal Oudetallah, Iqbal Batiha, Jamal Salah, and Mutaz Shatnawi. σ-compact spaces in nth-topological space. European Journal of Pure and Applied Mathematics, 18(2):5802–5802, 2025. [28] Tareq M. Al-shami, Abdelwaheb Mhemdi, Radwan Abu-Gdairi, and Mohammed E. El-Shafei. Compactness and connectedness via the class of soft somewhat open sets. AIMS Mathematics, 8(1):815–840, 2023. [29] Abdelwaheb Mhemdi. Novel types of soft compact and connected spaces inspired by soft q-sets. Filomat, 37(28):9617–9626, 2023. [30] Mesfer H. Alqahtani and Zanyar A. Ameen. Soft nodec spaces. AIMS Mathematics, 9(2):3289–3302, 2024. [31] Ohud F. Alghamdi, Mesfer H. Alqahtani, and Zanyar A. Ameen. On soft submaximal and soft door spaces. Contemporary Mathematics, pages 663–675, 2025. [32] Hind Y. Saleh and Areen A. Salih. c-continuity, c-compact and c-separation axioms via soft sets. Neutrosophic Sets and Systems, 73(1):51, 2024. [33] Jawaher Al-Mufarrij and Samer Al-Ghour. Regular-closed functions between soft topological spaces. International Journal of Neutrosophic Science (IJNS), 25(3), 2025. [34] Ibtesam Alshammari, Osama Taha, Mostafa El-Bably, and Islam Taha. On r-fuzzy soft δ-open sets with applications in fuzzy soft topological spaces. European Journal of Pure and Applied Mathematics, 18(1):5733–5733, 2025. [35] Jamal A. Oudetallah and Abualigah Laith. h-convexity in metric linear spaces. In- ternational Journal, 8(6), 2019. [36] Mikhail Tkachenko. Locally homeomorphic infinite lindelöf p-groups are homeomor- phic. Topology and its Applications, 355:109005, 2024. [37] Mostafa K. El-Bably, Rodyna A. Hosny, and Mostafa A. El-Gayar. Innovative rough set approaches using novel initial-neighborhood systems: Applications in medical diagnosis of covid-19 variants. Information Sciences, page 122044, 2025. [38] Radwan Abu-Gdairi and Mostafa K. El-Bably. The accurate diagnosis for covid-19 variants using nearly initial-rough sets. Heliyon, 10(10), 2024. [39] A. Nawar, R. Abu-Gdairi, M. El-Bably, and H. Atallah. Enhancing rheumatic fever analysis via tritopological approximation spaces for data reduction. Malaysian Jour- nal of Mathematical Sciences, 18(2):321–341, 2024. [40] Mostafa K. El-Bably, Radwan Abu-Gdairi, K. K. Fleifel, and Mostafa A. El-Gayar. Exploring β-basic rough sets and their applications in medicine. European Journal of Pure and Applied Mathematics, 17(4):3743–3771, 2024.