EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 5578 ISSN 1307-5543 – ejpam.com Published by New York Business Global Some Types of Tri-Lindelöfness Spaces Jamal Oudetallah1, Rehab Alharbi2,∗, Iqbal M. Batiha3,4, Salsabiela Rawashdeh5, Ala Amourah6,7 1 Department of Mathematics, University of Petra, Amman, 11196, Jordan 2 Department of Mathematics, Jazan University, Jazan 2097, Saudi Arabia 3 Department of Mathematics, Al-Zaytoonah University of Jordan, Amman 11733, Jordan 4 Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman, UAE 5 Department of Mathematics, Irbid National University, Irbid 2600, Jordan 6 Mathematics Education Program, Faculty of Education and Arts, Sohar University, Sohar 3111, Oman 7 Applied Science Research Center, Applied Science Private University, Amman, Jordan Abstract. In this study, we investigate the Lindelöf property in the context of three topologies, introducing the concept of tri-Lindelöf spaces. Additionally, we analyze the characteristics of these spaces in relation to traditional Lindelöf spaces. Several theoretical results are presented and proven, extending various well-known theorems on Lindelöf spaces to the setting of three topologies. Furthermore, illustrative examples are provided to support and clarify the findings. 2020 Mathematics Subject Classifications: 54A05, 54D20 Key Words and Phrases: Tri-Topological Spaces, Lindelöf Spaces, Tri-Lindelöf Spaces, Tri- Metalindelöf Spaces, Compactness 1. Introduction Topology plays a crucial role in the analysis of differential equations, particularly in function spaces, continuity, and stability. Studies on Volterra integro-differential equations and fractional differential equations rely on topological concepts such as compactness and convergence to ensure solution existence and uniqueness [1, 2]. Additionally, understand- ing perturbation effects, like white noise, involves topological stability, reinforcing the connection between topology and applied mathematics [3]. Topology is a branch of mathematics that studies the properties of spaces that re- main unchanged under continuous deformations such as stretching or bending. It focuses on concepts like continuity, connectedness, and compactness, providing a fundamental ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5578 Email addresses: dr jamal@inu.edu.jo (J. Oudetallah), ralharbi@jazanu.edu.sa (R. Alharbi), i.batiha@zuj.edu.jo (I. M. Batiha), sabeelar27@gmail.com (S. Rawashdeh), aamourah@su.edu.om (A. Amourah) 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 (2) (2025), 5578 2 of 19 framework for analyzing geometric and abstract structures [4–10]. One of the key areas in set-theoretic topology focuses on developing and exploring the relationships between different classes of topological spaces that lie between Lindelöf spaces. In this context, the class of Lindelöf spaces plays a crucial role, as it naturally occupies an intermediate position between these classes. A Lindelöf space is defined as a topological space (Q, ϱ) in which every open cover of Q has a countable subcover. Extending this concept, a tri- topological space (Q, ϱ1, ϱ2, ϱ3) is called a tri-Lindelöf space if every ϱi-open cover of Q has a tripartite countable subcover of Q, where i = 1, 2, 3. A notable characteristic of these classes of spaces is that several important separation axioms, such as normality and the Hausdorff condition, coincide within these classes. This property makes them highly significant both theoretically and practically, whether addressing problems from a purely topological perspective or applications in other areas of mathematics. A set Q is called a tri-topological space if each point in Q has a fundamental system of almost open neighborhoods. It is worth noting that the concept of almost open sets in topological groups was first studied by Ghosh and Lahiri. The notion of a tri- topological group, which is the tri-topologized version of a topological group, has also been discussed in earlier works. The concept of a Lindelöf space in a topological space (Q, ϱ) was introduced by [11]. Recent research [12–14] has further explored and expanded upon these ideas. This paper investigates the concept of tri-Lindelöf and tri-metalindelöf spaces, presenting associated conclusions. In the next section, we introduce the fundamental concepts of topological spaces and tri-topological spaces, along with key notions in tri-topological spaces, such as open and closed sets, derived sets, closure sets, interior and exterior sets, and separation axioms. We then discuss the Lindelöf property in tri-topological spaces, analyze its characteristics, and examine its applicability to other spaces. Additionally, we review well-known definitions that will be utilized throughout this work. Finally, we introduce and explore tripartite metalindelöfness spaces, highlighting their structural properties and significance. The terms ϱu, ϱdis, ϱcof , and ϱcoc represent the usual topology, discrete topology, cofinite topology, and cocountable topology, respectively. The concept of bitopological spaces is represented as Q = (Q, ϱ1, ϱ2), where ϱ1 and ϱ2 are two distinct topologies on Q. This aligns with prior research on bitopological spaces, where each topology satisfies a set of axioms. In [15], the notions of pairwise Hausdorff, pairwise regular, and pairwise normal spaces were discussed using well-established results such as Tietze extension theorems. The primary objective of this paper is to introduce and investigate a new class of tripartite Lindelöf spaces, termed tripartite metalindelöf spaces. Tri-topological spaces are sets equipped with three distinct topologies, denoted as: Q = (Q, ϱ1, ϱ2, ϱ3), where ϱ1, ϱ2, and ϱ3 are topologies on Q. These spaces exhibit structural variations that correspond to well-established properties in classical topology. J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5578 3 of 19 2. Preliminaries Before presenting the main results of this study, it is essential to establish the funda- mental concepts related to tri-topological spaces. In this section, we review key definitions and properties that will serve as the foundation for our discussion. These include funda- mental notions of topological spaces, tri-topological spaces, open and closed sets, derived sets, closure and interior sets, and separation axioms. Additionally, we introduce the Lindelöf property in tri-topological spaces, which ex- tends the classical Lindelöf concept to settings with multiple topologies. The definitions and results presented here will be instrumental in proving subsequent theorems and ex- ploring the structure of tri-Lindelöf and tri-metalindelöf spaces. Definition 1. [16] Let Q ̸= ∅ and let ϱ be a collection of subsets of Q, denoted by ϱ ⊆ P (Q) = {O | O ⊆ Q}. The collection ϱ is called a topological space on Q if it satisfies the following conditions: (i) ∅, Q ∈ ϱ (the empty set and the whole space are in ϱ). (ii) ϱ is closed under finite intersections. (iii) ϱ is closed under arbitrary unions. Definition 2. [13] Let Q ̸= ∅ and let ϱi ⊆ P (Q) for i = 1, 2, 3. We say that (Q, ϱ1, ϱ2, ϱ3) is a tri-topological space if each ϱi is a topology on Q, for all i = 1, 2, 3. Example 1. The space (Q, ϱ1, ϱ2, ϱ3) a tri-topological space. To see this, consider Q = {x, y, z}, with the following collections of subsets: ϱ1 = {∅, Q, {x}} ⊆ P (Q), ϱ2 = {∅, Q, {x}, {y}, {x, y}} ⊆ P (Q), ϱ3 = {∅, Q, {y}, {z}, {y, z}} ⊆ P (Q). Since each ϱi satisfies the conditions of a topology for i = 1, 2, 3, it follows that (Q, ϱ1, ϱ2, ϱ3) is a tri-topological space. However, consider the collection ϱ = {∅, Q, {x}, {y}}. This does not form a topology on Q because it does not satisfy all the conditions required for a topology. Definition 3. [15] Let (Q, ϱ1, ϱ2, ϱ3) be a tri-topological space and let O ⊆ Q. Then: (i) O is called a ϱi-open set if O ∈ ϱi for some i ∈ {1, 2, 3}. (ii) O is called a ϱi-closed set if its complement, Oc, belongs to ϱi for some i ∈ {1, 2, 3}. (iii) O is called a ϱi-clopen set if both O and Oc belong to ϱi for some i ∈ {1, 2, 3}. J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5578 4 of 19 Definition 4. [14] Let (Q, ϱ1, ϱ2, ϱ3) be a tri-topological space, where Q ̸= ∅ and O ⊆ Q. A point q ∈ Q is called a tri-limit point of O if for every ϱi-open set uq containing q, we have: uq ∩ (O \ {q}) ̸= ∅, for all i ∈ {1, 2, 3}. The set of all tri-limit points in a tri-topological space is called the tri-derived set, denoted by: O′ = {q | q is a tri-limit point of O}. Lemma 1. [13](Properties of the Tripartite Derived Set) Let (Q, ϱ1, ϱ2, ϱ3) be a tri-topological space, and let O,B ⊆ Q. Then: (i) ∅′ = ∅. (ii) If O ⊆ B, then O′ ⊆ B′. (iii) (O ∪B)′ = O′ ∪B′. (iv) (O ∩B)′ ⊆ O′ ∩B′. Proof. (i) By contradiction: Assume that ∅′ ̸= ∅. Then, there exists z ∈ ∅′, meaning that for all ϱi-open sets uz, we have: uz ∩ (∅ − {z}) ̸= ∅. However, since (∅ − {z}) = ∅, we obtain: uz ∩ ∅ ≠ ∅. This contradicts the fact that the intersection of any set with the empty set is always empty. Therefore, ∅′ = ∅. Definition 5. [16] Let (Q, ϱ1, ϱ2, ϱ3) be a tri-topological space, where Q ̸= ∅ and O ⊆ Q. The tri-closure set of O is denoted by: O = O ∪O′. Lemma 2. [5] (Properties of the Tripartite Closure Set) Let (Q, ϱ1, ϱ2, ϱ3) be a tri-topological space, and let O,B ⊆ Q. Then: (i) ∅ = ∅ and Q = Q. (ii) O ∪B = O ∪B and O ∩B ⊆ O ∩B. (iii) O is a ϱi-closed set. (iv) O = O if and only if O is a ϱi-closed set. (v) q ∈ O if and only if for every ϱi-open set uq such that q ∈ uq, we have uq ∩O ̸= ∅. J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5578 5 of 19 Proof. (ii) Let O,B ⊆ Q. Then: O ∪B = (O ∪B) ∪ (O ∪B)′ = (O ∪B) ∪ (O′ ∪B′) = (O ∪O′) ∪ (B ∪B′) = O ∪B. Similarly, for the intersection property: O ∩B = (O ∩B) ∪ (O ∩B)′ ⊆ (O ∩B) ∪ (O′ ∩B′) ⊆ (O ∪O′) ∩ (B ∪B′) = O ∩B. Definition 6. [8] Let (Q, ϱ1, ϱ2, ϱ3) be a tri-topological space, where Q ̸= ∅, and let O ⊆ Q. A point q ∈ O is said to be a tri-interior point of O if there exists at least one neighborhood N(q, ε) of q such that: N(q, ε) ⊆ O. The set of all tri-interior points of O is called the tri-interior set, denoted by: O◦ ≡ INT(O) = (Oc) c . Lemma 3. [6] (Properties of the Tripartite Interior Set) Let (Q, ϱ1, ϱ2, ϱ3) be a tri-topological space, and let O,B ⊆ Q. Then: (i) ∅◦ = ∅ and Q◦ = Q. (ii) (O ∩B)◦ = O◦ ∩B◦ and O◦ ∪B◦ ⊆ (O ∪B)◦. (iii) O◦ is a ϱi-open set. (iv) n ∈ O◦ if and only if there exists a ϱi-open set un such that n ∈ un ⊆ O. Definition 7. [7] Let (Q, ϱ1, ϱ2, ϱ3) be a tri-topological space, where Q ̸= ∅, and let O ⊆ Q. A point q is said to be a tri-exterior point of O if there exists at least one neighborhood N(q, ε) of q such that: N(q, ε) ∩O = ∅. The set of all tri-exterior points of O is called the tri-exterior set, denoted by: EX(O) = Int(Oc) = O c . Lemma 4. [14] (Properties of the Tripartite Exterior Set) Let (Q, ϱ1, ϱ2, ϱ3) be a tri-topological space, and let O,B ⊆ Q. Then: (i) EX(∅) = Q and EX(Q) = ∅. (ii) If O ⊆ B, then EX(B) ⊆ EX(O). (iii) EX(O) is a ϱi-open set. (iv) e ∈ EX(O) if and only if there exists a ϱi-open set ue such that e ∈ ue ⊆ Oc. J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5578 6 of 19 Proof. (iii) Since EX(O) = Int(Oc), we have: EX(O) = (Oc) c . But since Occ = O, it follows that: EX(O) = O c . By the definition of the tri-closure set, we know that O is a ϱi-closed set. Since the complement of a closed set is an open set, it follows that EX(O) is a ϱi-open set. As a result, the interior set is also a ϱi-open set. Definition 8. [12] Let (Q, ϱ1, ϱ2, ϱ3) be a tri-topological space, where Q ̸= ∅, and let O ⊆ Q. A point q is said to be a tri-boundary point of O if every neighborhood N(q, ε) of q satisfies: N(q, ε) ∩O ̸= ∅ and N(q, ε) ∩Oc ̸= ∅. The set of all tri-boundary points is called the tri-boundary set, denoted by: Bd(O) = O −O◦ = O ∩Oc. Lemma 5. [1] (Properties of the Tripartite Boundary Set) Let (Q, ϱ1, ϱ2, ϱ3) be a tri-topological space, and let O,B ⊆ Q. Then: (i) Bd(∅) = Bd(Q) = ∅. (ii) Bd(O) is a ϱi-closed set. (iii) b ∈ Bd(O) if and only if for every ϱi-open set ub containing b, we have: ub ∩O ̸= ∅ and ub ∩Oc ̸= ∅. Proof. (iii) Let s ∈ Bd(O), and let us be a ϱi-open set such that s ∈ us. Then, by definition: s ∈ O ∩Oc. This implies that: s ∈ O and s ∈ Oc. By the definition of closure, we have: s ∈ O ∪O′ and s ∈ Oc ∪ (Oc)′. This means that: (s ∈ O or s ∈ O′) and (s ∈ Oc or s ∈ (Oc)′). Thus, we conclude that: us ∩ (O \ {s}) ̸= ∅ and us ∩Oc ̸= ∅. J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5578 7 of 19 Since s ⊆ us, we obtain: us ∩O ̸= ∅ and us ∩Oc ̸= ∅. Hence, s satisfies the condition of a boundary point, completing the proof. Definition 9. [17] A topological space (Q, ϱ) is called a T0-space if for all distinct points q, s ∈ Q (q ̸= s), there exists an open set uq such that: q ∈ uq and s /∈ uq, or there exists an open set vs such that: s ∈ vs and q /∈ vs. Definition 10. [8] A tri-topological space (Q, ϱ1, ϱ2, ϱ3) is called a tri-T0-space if for all distinct points q, s ∈ Q (q ̸= s), there exists a ϱi-open set uq such that: q ∈ uq and s /∈ uq, or there exists a ϱj-open set vs such that: s ∈ vs and q /∈ vs, where i ̸= j and i, j ∈ {1, 2, 3}. Theorem 1. Let (Q, ϱ1, ϱ2, ϱ3) be a tri-topological space. Then, the following statements are equivalent: (i) Q is a tri-T0-space. (ii) For all q ̸= s, we have q /∈ {s} or s /∈ {q}. (iii) For all q ̸= s, we have {q} ≠ {s}. Proof. (i) ⇒ (ii): Let q ̸= s. Since Q is a tri-T0-space, there exists a ϱi-open set uq such that q ∈ uq and s /∈ uq, or there exists a ϱj-open set vs such that s ∈ vs and q /∈ vs, where i, j ∈ {1, 2, 3}. This implies that: q ∈ uq and uq ∩ {s} = ∅, or s ∈ vs and vs ∩ {q} = ∅. Therefore, we conclude that q /∈ {s} or s /∈ {q}. (ii) ⇒ (iii): Suppose q ̸= s. If q /∈ {s} and q ∈ {q}, then we must have {q} ̸= {s}. Similarly, if s /∈ {q} and s ∈ {s}, then again {q} ≠ {s}. Thus, condition (iii) holds. (iii) ⇒ (i): Suppose q ̸= s and we are given that {q} ≠ {s}. Since q ∈ {q} and s ∈ {s}, we must have: q /∈ Q− {q} = vs, where vs is a ϱi-open set in Q, since {q} is a ϱi-closed set. Furthermore, s ∈ Q−{q} = vs, where i ∈ {1, 2, 3}. Therefore, Q is a tri-T0-space. J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5578 8 of 19 Definition 11. [16] A tri-topological space (Q, ϱ1, ϱ2, ϱ3) is called a tri-T1-space if for all distinct points q, s ∈ Q (q ̸= s), there exists a ϱi-open set uq such that: q ∈ uq and s /∈ uq, and there exists a ϱj-open set vs such that: s ∈ vs and q /∈ vs, where i ̸= j and i, j ∈ {1, 2, 3}. Definition 12. [12] A tri-topological space (Q, ϱ1, ϱ2, ϱ3) is called a tri-T2-space if for all distinct points q, s ∈ Q (q ̸= s), there exists a ϱi-open set uq such that: q ∈ uq, and there exists a ϱj-open set vs such that: s ∈ vs and uq ∩ vs = ∅, where i ̸= j and i, j ∈ {1, 2, 3}. Definition 13. [1] A tri-topological space (Q, ϱ1, ϱ2, ϱ3) is called a tri-T2 1 2 -space if for all distinct points q, s ∈ Q (q ̸= s), there exist ϱi-closed sets Oq and Bs such that: q ∈ Oq, s ∈ Bs, and Oq ∩Bs = ∅, for some i ∈ {1, 2, 3}. Definition 14. [3] A tri-topological space (Q, ϱ1, ϱ2, ϱ3) is called a tri-regular space if for every point q /∈ O, where O is a ϱi-closed set, there exist a ϱi-open set uq and a ϱj-open set vO such that: q ∈ uq, O ⊆ vO, and uq ∩ vO = ∅, where i ̸= j and i, j ∈ {1, 2, 3}. Theorem 2. A space (Q, ϱ1, ϱ2, ϱ3) is a tri-regular space if and only if for all q ∈ uq, where uq is a ϱi-open set, there exists a ϱi-open set wq such that: q ∈ wq ⊆ wq ⊆ uq. Proof. (⇒) Let q ∈ uq. Since q /∈ ucq, and u c q is a ϱi-closed set, we denote ucq = O. By the definition of a tri-regular space, there exist ϱi-open sets wq and vO such that: q ∈ wq, O ⊆ vO, and wq ∩ vO = ∅. Clearly, we have wq ⊆ wq. To complete the proof, we need to show that wq ⊆ uq. Since wq ∩ vO = ∅, we can conclude that wq ⊆ vcO. Taking closures, we obtain: wq ⊆ vcO = vcO. J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5578 9 of 19 But since Oc = vcO and O ⊆ vO, we have: Oc ⊆ vcO. Therefore, vcO ⊆ uq, which implies: wq ⊆ uq. Thus, the condition holds. (⇐) Suppose q /∈ O and O is a ϱi-closed set. Then, q ∈ Oc, which is a ϱi-open set. By the given condition, there exists a ϱi-open set wq such that: q ∈ wq ⊆ wq ⊆ Oc. Now, we have two conditions: q ∈ wq and O ⊆ wq c. Since both wq and wq c are ϱi-open sets, it is sufficient to show: wq ∩ wq c = ∅. Suppose not. Then there exists some z ∈ wq ∩ wq c, which implies: z ∈ wq and z ∈ wq c. Since z ∈ wq c, it follows that z /∈ wq, contradicting z ∈ wq. Thus, our assumption is false, and we conclude: wq ∩ wq c = ∅. By these conditions, we establish that (Q, ϱ1, ϱ2, ϱ3) is a tri-regular space. Definition 15. [13] A topological space (Q, ϱ) is called a T3-space if it is both a T1-space and a regular space. Definition 16. [13] A tri-topological space (Q, ϱ1, ϱ2, ϱ3) is called a tri-T3-space if it is both a tri-T1-space and a tri-regular space. Definition 17. [15] A tri-topological space (Q, ϱ1, ϱ2, ϱ3) is called a tri-normal space if for every two disjoint ϱi-closed sets O and B, there exist ϱi-open sets uO and vB such that: O ⊆ uO, B ⊆ vB, and uO ∩ vB = ∅. Definition 18. [16] A tri-topological space (Q, ϱ1, ϱ2, ϱ3) is called a tri-T4-space if it is both a tri-T1-space and a tri-normal space. Theorem 3. [8] If (Q, ϱ1, ϱ2, ϱ3) is a tri-Tk-space, then it is also a tri-Tk−1-space. Theorem 4. If a space (Q, ϱ1, ϱ2, ϱ3) is a tri-T4-space, then it is also a tri-T3-space. J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5578 10 of 19 Proof. Let (Q, ϱ1, ϱ2, ϱ3) be a tri-T4-space. By definition, a tri-T4-space is both a tri-T1-space and a tri-normal space. That is, for all two disjoint ϱi-closed sets O and B, there exist ϱi-open sets uO and vB such that: O ⊆ uO, B ⊆ vB, and uO ∩ vB = ∅. (i) Now, let b ∈ B. Then, from (i), we have b ∈ vB. Since O ∩ B = ∅, it follows that b /∈ O. (ii) From (i) and (ii), we conclude that (Q, ϱ1, ϱ2, ϱ3) is a tri-regular space. Since a tri-T4-space is also a tri-T1-space, and we have shown it is tri-regular, we conclude that (Q, ϱ1, ϱ2, ϱ3) is a tri-T3-space. Definition 19. [13] Let (Q, ϱ) be a topological space, and let W = {Oα | α ∈ λ,Oα ⊆ Q} be a collection of subsets of Q. Then: (i) W is called a cover of Q if and only if:⋃ α∈λ Oα = Q. (ii) W is called an open cover of Q if and only if W is a cover and each Oα is an open set, where α ∈ λ. (iii) W is called a closed cover of Q if and only if W is a cover and each Oα is a closed set, where α ∈ λ. (iv) A collection C = {Bγ | γ ∈ Γ} is a subcover of W if and only if C ⊆ W and⋃ γ∈ΓBγ = Q. A space (Q, ϱ) is called compact if every open cover of Q has a finite subcover. Definition 20. [6] Let (Q, ϱ1, ϱ2, ϱ3) be a tri-topological space, and let W = {Oα | α ∈ λ,Oα ⊆ Q} be a collection of subsets of Q. Then: (i) W is called a tri-cover of Q if and only if:⋃ α∈λ Oα = Q. (ii) W is called a tri-open cover of Q if and only if W is a tri-cover and each Oα is a ϱi-open set, where α ∈ λ and i ∈ {1, 2, 3}. J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5578 11 of 19 (iii) W is called a tri-closed cover of Q if and only if W is a tri-cover and each Oα is a ϱi-closed set, where α ∈ λ and i ∈ {1, 2, 3}. (iv) A collection C = {Bγ | γ ∈ Γ} is called a tripartite subcover of W if and only if: (a) C ⊆W , (b) ⋃ γ∈ΓBγ = Q. A space (Q, ϱ1, ϱ2, ϱ3) is called a tripartite compact space if every tri-open cover of Q has a finite tripartite subcover. Definition 21. [7] Let (Q, ϱ1, ϱ2, ϱ3) be a tri-topological space. The space Q is said to be tripartite locally compact if each point of Q has a tri-open neighborhood whose tri-closure is tri-compact. Note: Every tri-compact space is tri-locally compact. 3. Tri-Lindelöf and Nearly Tri-Lindelöf Spaces In this section, we discuss the concept of Lindelöf spaces in classical topological spaces, as well as their extension to tri-topological spaces. Additionally, we introduce the notion of nearly Lindelöf spaces in tri-topological spaces and explore their fundamental properties and related theorems. Definition 22. [14] Let (Q, ϱ) be a topological space. It is called a Lindelöf space if every open cover of Q has a countable subcover. Definition 23. Let (Q, ϱ1, ϱ2, ϱ3) be a tri-topological space. It is called a tri-Lindelöf space if every ϱi-open cover of Q has a tripartite countable subcover. Note: A set A is said to be countable if either it is finite or if there exists an injective function from A into the set of natural numbers N. Definition 24. Let (Q, ϱ1, ϱ2, ϱ3) be a tri-topological space, and let C̃ be a cover of Q. We say that C̃ is a tri-open cover if: C̃ ⊆ 3⋃ i=1 υi. Remark 1. A tri-topological space (Q, ϱ1, ϱ2, ϱ3) is a second-countable base with respect to ϱ1, ϱ2, and ϱ3. Definition 25. A tri-topological space (Q, ϱ1, ϱ2, ϱ3) is called a tri-S-Lindelöf space if and only if it is a Lindelöf space, a pairwise Lindelöf space, and a tripartite Lindelöf space. Theorem 5. If a tri-topological space Q is a second-countable space, then it is a tripartite Lindelöf space. J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5578 12 of 19 Proof. Since Q is second-countable, each topology ϱ1, ϱ2, ϱ3 has a countable basis. Given any open cover in each topology, a countable subcover exists by definition of second- countability. Hence, Q is Lindelöf in each topology, making it a tripartite Lindelöf space. Remark 2. Every tripartite compact space is a tripartite Lindelöf space, but the converse need not be true. Theorem 6. Let (R, ϱu, ϱu, ϱu) be a tri-topological space. Then, (R, ϱu, ϱu, ϱu) is a tri- Lindelöf space but not a tri-compact space. Proof. Since the set O = {(q, s) | q, s ∈ Q} is a countable base with respect to ϱi of R, for i = 1, 2, 3, we conclude that R is a second- countable space. By the previous theorem, this implies that (R, ϱu, ϱu, ϱu) is a tri-Lindelöf space. However, the set (q, s) for all q < s in Q is not closed, meaning that (R, ϱu, ϱu, ϱu) is not compact. Therefore, (R, ϱu, ϱu, ϱu) is a tri-Lindelöf space but not a tri-compact space. Corollary 1. Every second-countable topological space is a tripartite topological space (X,ϑ1, ϑ2, ϑ3). Thus, X is a tri-S-Lindelöf space. Proof. Clearly by above theorem Remark 3. A compact subset of a tri-T2-space is tri-closed, but a tri-Lindelöf subset of a tri-T2-space need not be tri-closed. Definition 26. A space (Q, ϱ1, ϱ2, ϱ3) is called a tripartite space if and only if the countable intersection of tri-open sets is open. Theorem 7. If L is a Lindelöf subset of a tri-T2-space Q, then for each m /∈ L, we can separate m and L into two disjoint tri-open sets in Q. Proof. Since Q is a tri-T2-space, for each x ∈ L, there exist disjoint tri-open sets Ux and Vx such that x ∈ Ux and m ∈ Vx. The collection {Ux | x ∈ L} forms an open cover of L, which has a countable subcover {Uxi | i ∈ N} because L is Lindelöf. Let U = ⋃ i∈N Uxi and V = ⋂ i∈N Vxi . Then U and V are disjoint tri-open sets separating L and m, proving the theorem. Definition 27. [17] Suppose (Q, ϱ) is a topological space. A subset Y ⊆ Q is called nearly open if: Y = (Y )◦ in ϱ. Definition 28. Suppose (Q, ϱ1, ϱ2, ϱ3) is a tri-topological space. A subset Y ⊆ Q is called a tripartite nearly open set if: (Y )◦ is open in ϱ1, (Y )◦ is open in ϱ2, and (Y )◦ is open in ϱ3. J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5578 13 of 19 Definition 29. A collection Ũ = {Uα | α ∈ ∆} is called a nearly open cover of the tri-topological space (Q, ϱ1, ϱ2, ϱ3) if: (i) Each Uα is a tri-nearly open set for all α ∈ ∆. (ii) ⋃ α∈∆ Uα = Q. Definition 30. Suppose (Q, ϱ1, ϱ2, ϱ3) is a tri-topological space, and let Ũ = {Uα | α ∈ ∆} be a tri-nearly open cover of Q. A collection Sγ = {Uαγ | γ ∈ Γ} is called a tri-nearly subcover of Q if: ⋃ γ∈Γ Uαγ = Q. Definition 31. [11] Suppose (Q, ϱ) is a topological space. We say that Q is a nearly compact space if every nearly open cover of Q has a finite nearly subcover. Definition 32. Suppose (Q, ϱ1, ϱ2, ϱ3) is a tri-topological space. We say that Q is a tri- nearly compact space if every tri-nearly open cover of Q has a finite tri-nearly subcover. Definition 33. [12] Suppose (Q, ϱ) is a topological space. The space Q is called a nearly Lindelöf space if every nearly open cover of Q has a countable nearly subcover. Definition 34. Suppose (Q, ϱ1, ϱ2, ϱ3) is a tri-topological space. The space Q is called a tri-nearly Lindelöf space if every tri-nearly open cover of Q has a tripartite countable nearly subcover. Corollary 2. Every tri-nearly compact space is a tri-nearly Lindelöf space, but the con- verse need not be true. Proof. Let Q be a tri-nearly compact space. By definition, every tri-nearly open cover of Q has a finite ϱj-subcover of Q. Thus, for each tri-nearly open cover on Q, there exists a countable ϱj-subcover of Q, for all i ̸= j, where i, j ∈ {1, 2, 3}. Therefore, Q is a tri-nearly Lindelöf space. However, the converse need not be true. For example, the space (R, ϱu1, ϱu2, ϱu3) is a tri-nearly Lindelöf space but is not a tri-nearly compact space. Theorem 8. A tri-nearly Lindelöf space is preserved under an onto tri-continuous func- tion. Proof. Let i ̸= j where i, j ∈ {1, 2, 3}. Let F : (Q, ϱ1, ϱ2, ϱ3) → (Y, σ1, σ2, σ3) be a surjective continuous function, and suppose that Q is a tri-nearly Lindelöf space. We aim to show that Y is also a tri-nearly Lindelöf space. Assume Ũ = {Uα | α ∈ ϑ} is a nearly ϑi-open cover of Y , meaning that each Uα is a tri-open set for all α ∈ ϑ. Since F is continuous, it follows that F−1(Uα) is a tri-open set in Q for each α ∈ ϑ. Additionally, since F is surjective, we obtain: F−1(Ũ) = {F−1(Uα) | α ∈ ϑ} J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5578 14 of 19 which forms a tri-open cover of Q. Since Q is tri-nearly Lindelöf, there exists a countable ϑi-subcover {F−1(Uα) | α ∈ σ} where σ ⊆ ϑ and |σ| ≤ ℵ0. Thus, we have: Q ⊆ ⋃ α∈σ F−1(Uα). Since F is onto, it follows that: Y = F (Q) ⊆ F (⋃ α∈σ F−1(Uα) ) ⊆ ⋃ α∈σ Uα. Hence, Ũ has a countable ϑi-subcover of Y , proving that Y is a tri-nearly Lindelöf space. Remark 4. A compact subset of a tri-nearly T2-space is closed, but a tri-nearly Lindelöf subset of a tri-nearly T2-space need not be closed. Definition 35. A space (Q, ϱ1, ϱ2, ϱ3) is called a tripartite nearly tri-space if the countable intersection of nearly open sets is open. 4. Metalindelöfness spaces in Tri-topological Spaces In this section, we explore the concept of locally tri-metalindelöfness in tri-topological spaces and examine various fundamental properties of these spaces. Definition 36. A tri-topological space (Q, ϱ1, ϱ2, ϱ3) is called a tri-metalindelöf space if every tri-open cover of (Q, ϱ1, ϱ2, ϱ3) has a point-countable parallel refinement. Theorem 9. A countable tri-metalindelöf space is a tri-compact space. Proof. Let Q be a countable tri-metalindelöf space, and let U be an arbitrary tri- open cover of Q. Since Q is tri-metalindelöf, there exists a point-countable refinement V such that each point in Q is contained in at most countably many sets of V. Since Q is countable, V itself must be countable. By the Lindelöf property of V, there exists a countable subcover of U , proving that Q is tri-compact. Theorem 10. A separable tri-metalindelöf space (Q, ϱ1, ϱ2, ϱ3) is tri-Lindelöf. Proof. Since Q is separable, it has a countable dense subset D. Given any tri-open cover U of Q, each point of D is contained in at most countably many sets of a point- countable refinement V due to the tri-metalindelöf property. Since D is countable, we can extract a countable subcollection V ′ covering D. The closure of D, which is Q, is covered by V ′, ensuring a countable subcover of U . Thus, Q is tri-Lindelöf. J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5578 15 of 19 Definition 37. A tri-topological space (Q, ϱ1, ϱ2, ϱ3) is called tripartite countably met- alindelöf if every countable tri-open cover of (Q, ϱ1, ϱ2, ϱ3) has a point-countable parallel refinement. Theorem 11. The tri-topological space (Q, ϱ1, ϱ2, ϱ3) is a tri-metalindelöf space. It is also tripartite countably metacompact. Proof. Since (Q, ϱ1, ϱ2, ϱ3) is tri-metalindelöf, every open cover of Q has a point- countable open refinement. To show that it is also tripartite countably metacompact, let U be a countable open cover of Q. By the tri-metalindelöf property, there exists a point- countable refinement V of U , meaning each point of Q is contained in at most countably many sets of V. Since V is countable, we can extract a countable subcover from U , proving that Q is tripartite countably metacompact. Theorem 12. Every tri-Lindelöf tripartite countably metacompact space (Q, ϱ1, ϱ2, ϱ3) is a tri-metalindelöf space. Proof. Let Ũ = {Uα | α ∈ ∆} be a tri-open cover of Q. Since Q is tri-Lindelöf, there exists a tripartite countable subcover, say à = {Aαi}∞i=1. Furthermore, since Q is tripartite countably metacompact, the subcover à has a point- countable parallel refinement Ǧ of Ũ . Hence, (Q, ϱ1, ϱ2, ϱ3) is a tri-metalindelöf space. Theorem 13. Every tri-metalindelöf countably metacompact space (Q, ϱ1, ϱ2, ϱ3) is a tri- metacompact space. Proof. Let (Q, ϱ1, ϱ2, ϱ3) be a tri-metalindelöf countably metacompact space. By the tri-metalindelöf property, for each i ∈ {1, 2, 3}, for any open cover Ui of Q, there exists a countable subcover Uc i ⊆ Ui. By countable metacompactness, for each i ∈ {1, 2, 3}, for the subcover Uc i , there exists a countable subcover Vi ⊆ Uc i such that for each point x ∈ Q, there exists a neighborhood Nx of x that intersects only finitely many sets of Vi. Therefore, (Q, ϱ1, ϱ2, ϱ3) is tri-metacompact. Example 2. The tri-topological space (R, ϱdis, ϱdis, ϱdis) is tri-metalindelöf since ϱdis(i) forms a tri-open cover V = {{x} | x ∈ R} of R. It is also tripartite countably metacompact. Clearly, (R, ϱdis, ϱdis, ϱdis) is a tri- metalindelöf space. Theorem 14. Every tripartite countably metacompact topological space (Q, ϱ1, ϱ2, ϱ3) is tri-compact. J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5578 16 of 19 Proof. Let (Q, ϱ1, ϱ2, ϱ3) be a tripartite countably metacompact topological space. By the definition of countable metacompactness, every countable open cover of Q has a finite subcover. We are required to show that Q is tri-compact, which means that every open cover of Q has a finite subcover with respect to the topology ϱ1, ϱ2, ϱ3. Since Q is countably metacompact, for any countable collection of open sets {Ui}i∈N that cover Q, there exists a finite subcollection {Ui1 , Ui2 , . . . , Uik} such that their union covers Q. For each of the topologies ϱ1, ϱ2, ϱ3, we have the same result since the topologies are compatible with the metacompactness condition. Therefore, we can find finite subcovers for each of the topologies, implying that Q is tri-compact. Hence, the space (Q, ϱ1, ϱ2, ϱ3) is tri-compact. Theorem 15. The product of a tri-compact space Q and a tri-metalindelöf space Y is tri-metalindelöf, where (Q, ϱ1, ϱ2, ϱ3) and (Y, σ1, σ2, σ3) are tri-topological spaces. Proof. Let f : Q× Y → Y be the tri-projection function, defined by f(q, s) = s, such that (q, s) ∈ Q × Y . Then, f : Q × Y → Y is a tri-perfect function. Since Y is tri-metalindelöf, it follows that Q× Y is also tri-metalindelöf. Theorem 16. Let f : (Q, ϱ1, ϱ2, ϱ3) → (Y, σ1, σ2, σ3) be a continuous, tri-closed, and onto function. Then, if Q is tri-metalindelöf, then Y is also tri-metalindelöf. Proof. Let à = {Uα | α ∈ ∆} ∪ {Vβ | β ∈ Γ} be any tri-open cover of Y , where {Uα | α ∈ ∆} consists of tripartite σ-open members of Ã. Since f is a continuous and onto function, the set Ũ = {f−1(Uα) | α ∈ ∆} ∪ {f−1(Vβ) | β ∈ Γ} is a tri-open cover of Q. Given that Q is tri-metalindelöf, there exists a point-countable tri-open parallel refinement of Ũ , denoted by Ũ∗ = {f−1(U∗ α) | α ∈ ∆} ∪ {f−1(V ∗ β ) | β ∈ Γ}. Thus, Ã∗ = {U∗ α | α ∈ ∆} ∪ {V ∗ β | β ∈ Γ} is a point-countable tri-open parallel refinement of Ã. Therefore, Y is tri-metalindelöf. Lemma 6. Let f : (Q, ϱ1, ϱ2, ϱ3) → (Y, σ1, σ2, σ3) be a continuous and onto function. If à = {Aα | α ∈ ∆} is a point-countable family of subsets of Q, then {f(Aα) | α ∈ ∆} is a point-countable family of subsets of Y . J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5578 17 of 19 Proof. Let f : (Q, ϱ1, ϱ2, ϱ3) → (Y, σ1, σ2, σ3) be a continuous and onto function. Suppose à = {Aα | α ∈ ∆} is a point-countable family of subsets of Q, i.e., for each point x ∈ Q, the set {α ∈ ∆ | x ∈ Aα} is countable. We aim to prove that {f(Aα) | α ∈ ∆} is a point-countable family of subsets of Y . Consider any point y ∈ Y . Since f is onto, there exists some x ∈ Q such that f(x) = y. Since à = {Aα | α ∈ ∆} is a point-countable family of subsets of Q, the set {α ∈ ∆ | x ∈ Aα} is countable. Thus, the set {α ∈ ∆ | f(x) ∈ f(Aα)} is also countable, since for any α ∈ ∆, we have f(x) ∈ f(Aα) if and only if x ∈ Aα. Therefore, {f(Aα) | α ∈ ∆} is a point-countable family of subsets of Y . Q.E.D. Definition 38. A space (Q, ϱ1, ϱ2, ϱ3) is called a tri-metalindelöf space if every tri-open cover of Q has a tri-open locally countable refinement. Definition 39. A subset ψ of a space (Q, ϱ1, ϱ2, ϱ3) is said to be tri-paralindelöf relative to Q if every tri-open cover of ψ by members of ϱ has a tripartite locally countable parallel refinement in Q by members of ϱ. Corollary 3. Every tri-paralindelöf space is tri-metalindelöf. Proof. Clearly by above lemma Theorem 17. Every tri-closed subspace of a tri-metalindelöf space (Q, ϱ1, ϱ2, ϱ3) is tri- metalindelöf. Proof. Let X be a tri-closed subspace of the tri-metalindelöf space (Q, ϱ1, ϱ2, ϱ3). To show that X is tri-metalindelöf, we need to prove that for every open cover U = {Uα} of X, there exists a countable subcover V = {Vβ} such that X ⊆ ⋃ β Vβ, and each Vβ is a tri-open set. Since X is a subspace of Q, the open cover U of X is also an open cover of Q. Since (Q, ϱ1, ϱ2, ϱ3) is tri-metalindelöf, there exists a countable subcover W = {Wγ} of Q, where each Wγ is a tri-open set, such that Q = ⋃ γ Wγ . Now, because X is tri-closed in Q, each set Wγ ∩X is tri-open in X. Therefore, the collection V = {Wγ ∩X} forms a countable subcover of X and each Wγ ∩X is tri-open in X. Hence, X is tri-metalindelöf. Theorem 18. Every tri-metalindelöf subset of a tri-T2 locally indiscrete space (Q, ϱ1, ϱ2, ϱ3) is tri-closed. Proof. Let f : (Q, ϱ1, ϱ2, ϱ3) → (Y, σ1, σ2, σ3) be a bijective and continuous map. Suppose that (Y, σ1, σ2, σ3) is a tri-T2 and tripartite locally indiscrete space, and that (Q, ϱ1, ϱ2, ϱ3) is tri-metalindelöf. Then, f is a tri-homomorphism. It suffices to show that f is tri-closed. Let A be a proper tri-closed subset of Q. Since A is a tri-metalindelöf subset of Q, and f : (Q, ϱ1, ϱ2, ϱ3) → (Y, σ1, σ2, σ3) is continuous, it follows that f(A) J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5578 18 of 19 is a tri-metalindelöf subset of Y . Thus, f(A) is a tri-closed subset of Y . Therefore, f : (Q, ϱ1, ϱ2, ϱ3) → (Y, σ1, σ2, σ3) is a tri-closed function. Hence, the result follows. Corollary 4. Let f : (Q, ϱ1, ϱ2, ϱ3) → (Y, σ1, σ2, σ3) be a tripartite bijective continuous map. If (Y, σ1, σ2, σ3) is a tri-T2 and tripartite locally indiscrete space, and (Q, ϱ1, ϱ2, ϱ3) is tri-paralindelöf, then f is a tri-homomorphism. Proof. Clearly by above theorem 5. Conclusion and Future Directions In this study, we have introduced and explored the concepts of Lindelöfness, met- alindelöfness, and nearly Lindelöfness in the setting of tri-topological spaces. By extend- ing classical topological properties to this generalized framework, we established several theoretical results and analyzed their behavior. Various separation axioms were exam- ined, revealing intricate relationships between tri-compactness, tri-Lindelöfness, and tri- metalindelöfness. The findings contribute to the broader understanding of general topol- ogy, offering new insights into the structure and properties of tri-topological spaces. Future research can extend these findings in several directions. One promising avenue is the investigation of finer separation axioms such as tri-regularity and tri-normality, as well as their implications in different tri-topological settings. Another important direction is the extension of these concepts to fuzzy tri-topological spaces, which could provide useful applications in both theoretical and applied mathematics. 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