EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 5764 ISSN 1307-5543 – ejpam.com Published by New York Business Global Some Types of Tri-Locally Compactness Spaces Jamal Oudetallah1, Ala Amourah2,∗, Sultan Alsaadi2, Iqbal M. Batiha3, Jamal Salah4,∗, Tala Sasa5 1 Department of Mathematics, University of Petra, Amman, 11196, Jordan 2 Mathematics Education Program, Faculty of Education and Arts, Sohar University, Sohar 311, Oman 3 Department of Mathematics, Al Zaytoonah University of Jordan, Amman 11733, Jordan. 4 College of Applied and Health Sciences, A’Sharqiyah University, Post Box No. 42, Post Code No. 400 Ibra, Sultanate of Oman 5 Department of Mathematics, Faculty of Science, Applied Science Private University, Amman, Jordan Abstract. Three topologies, or tri-locally compact spaces, will be examined in this study in order to examine the locally compactness spaces attribute. Furthermore, these spaces’ characteristics will be analyzed in light of locally limited spaces. Several well-known theorems about locally compact spaces have been expanded to apply to three topologies, and many theoretical results have been proposed and verified. The results are supported by illustrative instances. 2020 Mathematics Subject Classifications: 47B38 Key Words and Phrases: Tri-locally compact spaces, Tri-topological spaces, Locally compact- ness, Metacompactness 1. Introduction The study of the connections between different classes of topological spaces located between countably paracompact spaces is one of the main areas of set theoretic topology [1–4]. Since it naturally lies between these classes, the class of locally compact spaces is important in this context. According to Dugundji (1966), a locally compact space is a topological space (X,ϑ) in which each point a ∈ X has a neighborhood that is also contained within a compact space. Similarly, a tri-topological space (X,ϑ1, ϑ2, ϑ3) is called a tri-locally metacompact space if every point a ∈ X has a neighborhood that is contained within a tri-compact area. ∗Corresponding author. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5764 Email addresses: jamal.oudetallah@uop.edu.jo (J. Oudetallah), AAmourah@su.edu.om (A. Amourah),alsaad99@hotmail.com (S. Alsaadi), i.batiha@zuj.edu.jo (I. M. Batiha), damous73@yahoo.com (J. Salah), 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 (2) (2025), 5764 2 of 11 One important feature of such spaces is that several important separation axioms, such as normality and collection-wise Hausdorff, agree for them. There are significant theoretical and practical implications to problems that come from other areas of mathe- matics or from a strictly topological standpoint. If there is a fundamental system of nearly open neighborhoods for every point in a set X, then X is a tri-topological space. Keep in mind that the first people to examine almost open sets in a topological group were Ghosh and Lahiri [5]. The concept of a tri-topological group, or the tri-topologized form of a topological group, has previously been discussed in earlier research. Since all of the spaces examined in this study are assumed to be nonempty and T0 spaces, any two open neighborhoods of a meet to generate another open neighborhood of a for any point a in the space. The concept of a locally compact space in topological space (X,ϑ) was first proposed by Levine [6]. These subjects were examined in greater detail in more recent research [7–10].... The notions of tri-locally compact and tri-locally metacompact in tri-topological spaces, as well as related findings, are examined in this work. In tri-topological spaces, we present the notion of tri-locally compactness, analyze its properties, and apply it to different spaces. We go over common definitions that will be used in the parts that follow. Typically, ϑu, ϑdis, ϑcof , and ϑcoc represent discrete, co-finite, and co-countable topolo- gies, respectively. X = (X,ϑ1, ϑ2) is a representation of the concept of bitopological spaces, where ϑ1, ϑ2 are two topologies on X. This is related to earlier research on bitopological spaces, where a topology is a collection of points that satisfy a set of axioms. Kim’s pa- per [5] described pairwise Hausdorff, pairwise regular, and pairwise normal spaces using a set of standard results known as the Tietze extension. Bitopological space study was further explored in [11, 12]. According to [7, 8], bitopological space can expand, nearly ex- pand, and feebly expand. The primary goal of this paper is to introduce and investigate a new kind of tripartite compact space: the tripartite locally compact space. Tri-topological spaces are sets containing three topologies, where ϑ1, ϑ2, and ϑ3 are topologies on X. They are represented as X = (X,ϑ1, ϑ2, ϑ3). Tri-topological space variations match well-known topological space features. 2. Preliminaries We will illustrate some of the fundamental concepts of tri-topological space in this part, including paracompactness, dense sets, and compact space. Definition 1. [1] ϑ ⊂ P(X) = {A : A ⊆ X} is a collection of subsets of X, where X is a non-empty set. If ϑ satisfies the following requirements, it is considered a topology on X: (i) ∅, X ∈ ϑ (ii) We have A ∩B ∈ ϑ for every A,B ∈ ϑ. (iii) ⋃ α∈λAα ∈ ϑ if E = {Aα : α ∈ λ,Aα ∈ ϑ} is any collection of sets in ϑ Definition 2. [10]. Let X be a non-empty set, and for i = 1, 2, 3, ϑi ⊂ P(X). (X,ϑ1, ϑ2, ϑ3) is a tri-topological space if ϑi is a topology on X for all i = 1, 2, 3. J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5764 3 of 11 Example 1. Assume X = {a, b, c} so that they (i) ϑ1 = {∅, X, {a}} ⊂ P(X) (ii) ϑ2 = {∅, X, {a}, {b}, {a, b}} ⊂ P(X) (iii) ϑ3 = {∅, X, {b}, {c}, {b, c}} ⊂ P(X) For i = 1, 2, 3, ϑi satisfies the requirements of a topological space. A tri-topological space is thus (X,ϑ1, ϑ2, ϑ3). For example, {a} ∪ {b} = {a, b} ̸∈ ϑ, therefore the space ϑ = {∅, X, {a}, {b}} is not a topological space. Definition 3. [11] Presuming that (X,ϑ1, ϑ2, ϑ3) is a tri-topological space and that A ⊂ X (i) If A ∈ ϑi for some i = 1, 2, 3, then A is a ϑi-open set. (ii) If Ac ∈ ϑi for some i = 1, 2, 3, then A is a ϑi-closed set. (iii) If A and Ac are both in ϑi for some i = 1, 2, 3, then A is referred to as a ϑi-clopen set. Definition 4. [7] Given a tri-topological space (X,ϑ1, ϑ2, ϑ3), X ̸= ∅, and A as a subset of X, a ∈ X is a tri-limit point of A if, for any ϑi-open set ua containing a, ua∩(A−{a}) ̸= ∅.. A′ = {a : a is a tri-limit point of A} is the representation of the tri-derived set, which is the set of all tri-limit points. Theorem 1. [8] If (X,ϑ1, ϑ2, ϑ3) and A,B ⊂ X are tri-topological spaces, then: (i) ∅′ = ∅ (ii) (A ∪B)′ = A′ ∪B′ (iii) (A ∩B)′ ⊂ A′ ∩B′ (iv) If A ⊂ B, then A′ ⊂ B′ Definition 5. [9] A = A ∪A′ is the representation of the tri-closure set if (X,ϑ1, ϑ2, ϑ3) is a tri-topological space, X ̸= ∅, and A is a subset of X. Theorem 2. [8] Let (X,ϑ1, ϑ2, ϑ3) be a tri-topological space and A,B ⊂ X. Then: (i) X = X and ∅ = ∅ (ii) A ∪B = A ∪B (iii) A ∩B ⊂ A ∩B (iv) We have Ua ∩A ̸= ∅ for every point a ∈ A and every ϑi-open set Ua containing a. (v) A is a ϑi-closed set for each i ∈ {1, 2, 3} if and only if A = A. J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5764 4 of 11 Definition 6. [4] allow X ̸= ∅, A be a subset of X, and allow (X,ϑ1, ϑ2, ϑ3) be a tri- topological space. If there is at least one ϑi-neighborhood N(a, ϑi) of a for some i ∈ {1, 2, 3} such that N(a, ϑi) ⊆ A, then a point a ∈ A is called a tri-interior point of A. A◦ or INT(A) indicate the tri-interior of A, which is the set of all tri-interior points of A. A◦ = INT(A) = (Ac)c is another way to express this, in which Ac is the tri-closure of the complement of A. Theorem 3. [7] The following characteristics are true given a tri-topological space (X,ϑ1, ϑ2, ϑ3) and A,B ⊂ X: (i) X◦ = X and ∅◦ = ∅. (ii) (A ∩B)◦ = A◦ ∩B◦ and A◦ ∪B◦ ⊂ (A ∪B)◦. (iii) A◦ is a ϑi-open set for each i ∈ {1, 2, 3} if and only if there exists a ϑi-open set Un such that n ∈ Un ⊂ A for each i ∈ {1, 2, 3}. Definition 7. [8] Since X ̸= ϕ and A is a subset of X, let (X,ϑ1, ϑ2, ϑ3) be a tri-topological space. If there is at least one neighborhood of a such that N(a, ε) ∩ A = ϕ, then a is a tri-exterior point of A. The tri-exterior set, which is the set of all tri-exterior points, is represented by the equation EX(A) = Int(Ac) = A C . Theorem 4. If we define EX(A) = Int(Ac) (the interior of the complement of A) given a tri-topological space (X,ϑ1, ϑ2, ϑ3) and A,B ⊂ X, then: (i) EX(∅) = X and EX(X) = ∅. If and only if there is a ϑi-open set Ue such that e ∈ Ue ⊂ Ac, then for all e ∈ X, e ∈ EX(A) EX(B) ⊂ EX(A) if A ⊂ B. Proof. (i) EX(∅) = Int(∅c) = Int(X) = X and EX(X) = Int(Xc) = Int(∅) = ∅. (ii) If and only if there is a ϑi-open set Ue such that e ∈ Ue ⊂ Ac, then by definition of interior, e ∈ Int(Ac) = EX(A). Assume that A ⊂ B. Bc ⊂ Ac, then. Set inclusion is maintained by taking the interior, so Int(Bc) ⊂ Int(Ac). EX(B) ⊂ EX(A), so. Definition 8. [9] Let (X,ϑ1, ϑ2, ϑ3) be a Tri-topological space, and let X ̸= ∅, A be a subset of X. If every neighborhood N(a, ϑi) of a (with respect to each topology ϑi, i = 1, 2, 3) fulfills both N(a, ϑi)∩A ̸= ∅ and N(a, ϑi)∩Ac ̸= ∅, then the point a ∈ X is a tri-boundary point of A. The collection of all tri-boundary points of A is its tri-boundary, represented by Bd(A). It may be written as follows: Bd(A) = A ∩ Ac = A− A◦, where A is the tri-closure of A and A◦ is the tri-interior of A. Theorem 5. Given A,B ⊂ X and a tri-topological space (X,ϑ1, ϑ2, ϑ3), (i) Bd(∅) = Bd(X) = ∅ (ii) Bd(A) is a ϑi-closed set J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5764 5 of 11 (iii) b ∈ Bd(A) if and only if for all ϑi-open sets ub containing b, we have ub ∩A ̸= ∅ and ub ∩Ac ̸= ∅ Proof. We only prove (iii) here. If b ∈ Bd(A) and ub is a ϑi-open set containing b, then: b ∈ Bd(A) = A ∩Ac if and only if b ∈ A ∧ b ∈ Ac if and only if b ∈ (A ∪A′) ∧ b ∈ Ac ∪ (Ac)′ if and only if (b ∈ A ∨ b ∈ A′) ∧ (b ∈ Ac ∨ b ∈ (Ac)′) if and only if b ∈ A′ ∧ b ∈ (Ac)′ if and only if ub ∩ (A/{b}) ̸= ∅ ∧ ub ∩ (Ac/{b}) ̸= ∅ But we have b ⊂ ub, so we obtain ub ∩A ̸= ∅ and ub ∩Ac ̸= ∅. Definition 9. [7] A tri-topological space (X,ϑ1, ϑ2, ϑ3) functions as a tri-T0-space when it contains either a ϑi-open set ua containing a but excluding b or it possesses a ϑj-open set vb with element b inside but element a kept outside from vb where i ̸= j and i, j = 1, 2, 3 for every pair of distinct elements a Definition 10. [8] If, for each of the two distinct elements a and b in X, there exists ϑi-open set ua such that a ∈ ua and b /∈ ua, or ϑj-open set vb such that b ∈ vb and a /∈ vb, where i ̸= j and i, j = 1, 2, 3, then (X,ϑ1, ϑ2, ϑ3) is a tri− T0-space. Theorem 6. Let (X,ϑ1, ϑ2, ϑ3) be a tri-topological space . Then thefollowing statements are equivalent: (i) X represents a tri-T0-space (ii) For any two distinct elements a and b, a /∈ {b} or b /∈ {a} (iii) If a and b are two distinct elements, we have {a} ≠ {b} Proof. (i) ⇒ (ii): Let a ̸= b be distinct elements of X, and suppose X is a tri-T0-space. By definition of a tri-T0-space, there exists a ϑi-open set Ua such that a ∈ Ua and b /∈ Ua for some i ∈ {1, 2, 3}, or a ϑj-open set Vb such that b ∈ Vb and a /∈ Vb for some j ∈ {1, 2, 3}. In the first case, since a ∈ Ua and Ua ∩ {b} = ∅, this implies b /∈ {a}. In the second case, since b ∈ Vb and Vb ∩ {a} = ∅, this implies a /∈ {b}. As a result, a /∈ {b} or b /∈ {a}. (ii) ⇒ (iii): For distinct elements a and b, we have a /∈ {b} or b /∈ {a}. Without loss of generality, assume that a /∈ {b}. Since a ∈ {a}, we have {a} ≠ {b}. (iii) ⇒ (i): Assume a and b are distinct elements such that {a} ≠ {b}. Then, there exists either c ∈ {a} \ {b} or d ∈ {b} \ {a}. Without loss of generality, assume b /∈ {a} (and obviously b ∈ {b}). Since {a} is a ϑi-closed set in X for each i ∈ {1, 2, 3}, the set X \ {a} = Vb is ϑi-open in X for each i. We have b ∈ Vb and a /∈ Vb. As a result, X is a tri-T0-space. J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5764 6 of 11 Definition 11. [7] (X,ϑ1, ϑ2, ϑ3) is a tri-T1-space if, for each of the two distinct elements a and b in X, there exists ϑi-open set ua such that a ∈ ua, b /∈ ua, and b ∈ vb such that b ∈ vb and a /∈ vb, where i ̸= j such that i, j = 1, 2, 3. Definition 12. [9] (X,ϑ1, ϑ2, ϑ3) is a tri-T2-space if, for each of the two distinct elements a and b in X, there exists ϑi-open set ua such that a ∈ ua and ϑj-open set vb such that b ∈ vb and ua ∩ vb = ϕ, where i ̸= j such that i, j = 1, 2, 3. Definition 13. [7] For i = 1, 2, 3, a topological space (X,ϑ1, ϑ2, ϑ3) is a tri-T2 1 2 -space if, for each of the two distinct elements a and b in X, there exists a ϑi-closed set Aa, b ∈ Bb, and Aa ∩Bb = ϕ. Definition 14. [8] (X,ϑ1, ϑ2, ϑ3) is a tri-regular space. If A is a ϑi-closed set and a ∈ ua, A ⊂ vA, and ua ∩ vA = ϕ, then i ̸= j, for i, j = 1, 2, 3. Theorem 7. A tri-topological space (X,ϑ1, ϑ2, ϑ3) is a tri-regular space if and only if, for every point a ∈ X and ϑi-open set ua containing a, there exists a ϑi-open set wa such that a ∈ wa ⊂ wa ⊂ ua. Proof. (⇒) If a ∈ ua, then a /∈ uca. Since uca is a ϑi-closed set and (X,ϑ1, ϑ2, ϑ3) is a tri-regular space, there exist ϑi-open sets wa and vuc a such that a ∈ wa, uca ⊂ vuc a , and wa ∩ vuc a = ∅. Thus, wa ⊂ vcuc a . Since vuc a is ϑi-open, v c uc a is ϑi-closed, which means wa ⊂ vcuc a . Moreover, uca ⊂ vuc a implies vcuc a ⊂ ua. Consequently, a ∈ wa ⊂ wa ⊂ vcuc a ⊂ ua. (⇐) Let a ∈ X and F be a ϑi-closed set such that a /∈ F . Then a ∈ F c, and F c is a ϑi-open set containing a. By our hypothesis, there exists a ϑi-open set wa such that a ∈ wa ⊂ wa ⊂ F c. This implies a ∈ wa and F ⊂ (wa) c. Since wa is ϑi-closed, (wa) c is ϑi-open. We also have wa ∩ (wa) c = ∅. Therefore, (X,ϑ1, ϑ2, ϑ3) is a tri-regular space. Definition 15. [8] A tri-topological space (X,ϑ1, ϑ2, ϑ3) is a tri-T3-space if it is both a tri-T1-space and tri-regular. Definition 16. [7] A tri-topological space (X,ϑ1, ϑ2, ϑ3) is a tri-normal space if for any two disjoint ϑi-closed sets A and B, there exist ϑi-open sets uA and vB such that A ⊂ uA, B ⊂ vB, and uA ∩ vB = ∅. Theorem 8. A tri-topological space (X,ϑ1, ϑ2, ϑ3) is a tri-normal space if and only if for every ϑi-closed set F and ϑi-open set U containing F , there exists a ϑi-open set V such that F ⊂ V ⊂ V ⊂ U . Proof. (⇒) Let F be a ϑi-closed set and U be a ϑi-open set containing F . Then U c is a ϑi-closed set and F ∩ U c = ∅. Since (X,ϑ1, ϑ2, ϑ3) is tri-normal, there exist ϑi-open sets V and W such that F ⊂ V , U c ⊂ W , and V ∩W = ∅. Thus, V ⊂ W c. Since W is ϑi-open, W c is ϑi-closed, which means V ⊂ W c. Additionally, U c ⊂ W implies W c ⊂ U . Therefore, F ⊂ V ⊂ V ⊂ W c ⊂ U . (⇐) Let F and G be disjoint ϑi-closed sets. Then F ⊂ Gc and Gc is a ϑi-open set. By our hypothesis, there exists a ϑi-open set V such that F ⊂ V ⊂ V ⊂ Gc. This implies F ⊂ V and G ⊂ (V )c. Since V is ϑi-closed, (V )c is ϑi-open. We also have V ∩ (V )c = ∅. Therefore, (X,ϑ1, ϑ2, ϑ3) is tri-normal. J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5764 7 of 11 Definition 17. [7] A tri-topological space (X,ϑ1, ϑ2, ϑ3) is a tri-T4-space if it is both a tri-T1-space and tri-normal. Theorem 9. [8] It is tri-Tk−1-space if (X,ϑ1, ϑ2, ϑ3) is tri-Tk-space. Theorem 10. [9] A space is tri-T3-space if (X,ϑ1, ϑ2, ϑ3) is tri-T4-space. Proof. By considering (X,ϑ1, ϑ2, ϑ3) to be tri-T4-space, it is tri-T1-space and tri-normal space. This suggests that for any of the two disjoint ϑi-closed sets A and B, there are ϑi- open sets uA and vB such that A ⊂ uA and B ⊂ vB with uA ∩ vB = ϕ. (1) Let b ∈ B now, followed by b ∈ vB. On the other hand, A ∩B = ϕ. Thus, b /∈ Aisobtained. (2) Thus, (X,ϑ1, ϑ2, ϑ3) is a tri-regular space according to (1) and (2). Thus, (X,ϑ1, ϑ2, ϑ3) is obtained. (X,ϑ1, ϑ2, ϑ3) is tri-T3-space since is tri-T1-space and tri-regular space. Theorem 11. [8] A space is tri-normal if (X,ϑ1, ϑ2, ϑ3) is tri-completely normal. Proof. Consider the space (X,ϑ1, ϑ2, ϑ3) to be tri-completely normal. By definition, there is a ϑi-continuous function fi : X → [0, 1] for any two disjoint ϑi-closed sets A and B. such that fi(B) = {1}, fi(A) = {0} (3) We now define the inverse images of the open intervals (0, 12) and (12 , 1) under fi, yielding two ϑi-open sets uA and vB such that A ⊂ uA, B ⊂ vB, uA ∩ vB = ∅. (4) The tri-normality of (X,ϑ1, ϑ2, ϑ3) may be inferred from (3) and (4). Definition 18. Given a tri-topological space (X,ϑ1, ϑ2, ϑ3), a set D in (X,ϑ1, ϑ2, ϑ3) is referred to as a tri-dense set if D = X. Conversely, for every ϑi-open set u, we obtain u ∩D ̸= ϕ if D is dense in (X,ϑ1, ϑ2, ϑ3). Definition 19. Let (X,ϑ1, ϑ2, ϑ3) and (Y, σ1, σ2, σ3) be tri-topological spaces. If f(u) = v, where u is ϑi-open set and v is σi-open set, then f : (X,ϑ1, ϑ2, ϑ3) is a tri-open function. J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5764 8 of 11 3. Tri-locally compact spaces We then introduce the idea of tri-locally compact spaces and establish some of their most important properties in this section. Definition 20. A subset A of a tri-topological space (X,ϑ1, ϑ2, ϑ3) is tri-compact if every cover of A by ϑi-open sets (for i = 1, 2, 3) has a finite subcover. Definition 21. A tri-topological space (X,ϑ1, ϑ2, ϑ3) is tri-locally compact if for every point a ∈ X, there exists a ϑi-open set Ua containing a such that Ua is tri-compact. Theorem 12. A tri-topological space (X,ϑ1, ϑ2, ϑ3) is tri-locally compact if and only if for every point a ∈ X and every ϑi-open set U containing a, there exists a ϑi-open set V such that a ∈ V ⊂ V ⊂ U and V is tri-compact. Proof. (⇒) Assume (X,ϑ1, ϑ2, ϑ3) is tri-locally compact. Let a ∈ X and U be a ϑi- open set containing a. By definition, there exists a ϑj-open set Wa containing a such that Wa is tri-compact. Let V = Wa ∩ U , which is a ϑi-open set containing a. Then V ⊂ Wa ∩ U ⊂ Wa ∩ U . Since V is a closed subset of the tri-compact set Wa, V is tri-compact. Therefore, a ∈ V ⊂ V ⊂ U and V is tri-compact. (⇐) This direction follows directly from the definition of tri-locally compact spaces. Theorem 13. Let (X,ϑ1, ϑ2, ϑ3) be a tri-T2-space. If X is tri-locally compact, then for every tri-compact set K and every ϑi-open set U containing K, there exists a ϑi-open set V such that K ⊂ V ⊂ V ⊂ U and V is tri-compact. Proof. Let K be a tri-compact set and U be a ϑi-open set containing K. For each a ∈ K, there exists a ϑi-open set Va such that a ∈ Va ⊂ Va ⊂ U and Va is tri-compact. The collection {Va : a ∈ K} forms an open cover of K. Since K is tri-compact, there exists a finite subset {a1, a2, . . . , an} ⊂ K such that K ⊂ ⋃n j=1 Vaj . Let V = ⋃n j=1 Vaj . Then V is a ϑi-open set and K ⊂ V ⊂ V ⊂ ⋃n j=1 Vaj ⊂ ⋃n j=1 Vaj ⊂ U . Since V ⊂ ⋃n j=1 Vaj and each Vaj is tri-compact, their finite union ⋃n j=1 Vaj is tri- compact. Therefore, V is tri-compact as a closed subset of a tri-compact set. Theorem 14. Let (X,ϑ1, ϑ2, ϑ3) be a tri-topological space. If X is tri-locally compact and tri-T2, then for any tri-compact set K and any closed set F such that K ∩ F = ∅, there exist ϑi-open sets U and V such that K ⊂ U , F ⊂ V , and U ∩ V = ∅. Proof. Let K be a tri-compact set and F be a closed set such that K ∩ F = ∅. For each a ∈ K, we have a /∈ F , which means a ∈ F c. Since X is tri-T2, for each a ∈ K and b ∈ F , there exist ϑi-open sets Ua and Vb such that a ∈ Ua, b ∈ Vb, and Ua ∩ Vb = ∅. For each a ∈ K, the collection {Vb : b ∈ F} forms an open cover of F . Since X is tri- locally compact, there exists a finite subset {b1, b2, . . . , bna} ⊂ F such that F ⊂ ⋃na j=1 Vbj . Let Va = ⋃na j=1 Vbj . Then Va is a ϑi-open set containing F . J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5764 9 of 11 Now, let Wa = X \ Va. Then Wa is a ϑi-closed set and a ∈ Wa. By the previous theorem, there exists a ϑi-open set Ua such that a ∈ Ua ⊂ Ua ⊂ Wa and Ua is tri-compact. The collection {Ua : a ∈ K} forms an open cover of K. Since K is tri-compact, there exists a finite subset {a1, a2, . . . , am} ⊂ K such that K ⊂ ⋃m j=1 Uaj . Let U = ⋃m j=1 Uaj and V = ⋂m j=1 Vaj . Then U is a ϑi-open set containing K, and V is a ϑi-open set containing F . Furthermore, U ∩ V = ∅ because Uaj ∩ Vaj = ∅ for each j = 1, 2, . . . ,m. Theorem 15. Let (X,ϑ1, ϑ2, ϑ3) be a tri-T2-space. If X is tri-locally compact, then for any tri-compact set K and any ϑi-open set U containing K, there exists a ϑi-open set V with compact closure such that K ⊂ V ⊂ V ⊂ U . Proof. Let K be a tri-compact set and U be a ϑi-open set containing K. For each a ∈ K, by tri-local compactness, there exists a ϑi-open set Va such that a ∈ Va ⊂ Va ⊂ U and Va is tri-compact. The collection {Va : a ∈ K} forms an open cover of K. Since K is tri-compact, there exists a finite subset {a1, a2, . . . , an} ⊂ K such that K ⊂ ⋃n j=1 Vaj . Let V = ⋃n j=1 Vaj . Then V is a ϑi-open set and K ⊂ V ⊂ V ⊂ ⋃n j=1 Vaj ⊂ U . Since each Vaj is tri-compact, their finite union ⋃n j=1 Vaj is tri-compact. Therefore, V is tri-compact as a closed subset of a tri-compact set. Theorem 16. Let (X,ϑ1, ϑ2, ϑ3) be a tri-topological space. If X is tri-locally compact and tri-T3, then X is tri-regular. Proof. Let a ∈ X and F be a ϑi-closed set such that a /∈ F . Since X is tri-T3, it is tri-T1, which means {a} is a tri-compact set. By the previous theorem, there exist ϑi-open sets U and V such that {a} ⊂ U , F ⊂ V , and U ∩ V = ∅. Therefore, X is tri-regular. Theorem 17. Let (X,ϑ1, ϑ2, ϑ3) be a tri-topological space. If X is tri-locally compact and tri-T2, then for any two disjoint tri-compact sets K1 and K2, there exist ϑi-open sets U1 and U2 such that K1 ⊂ U1, K2 ⊂ U2, and U1 ∩ U2 = ∅. Proof. Let K1 and K2 be disjoint tri-compact sets. For each a ∈ K1 and b ∈ K2, since X is tri-T2, there exist ϑi-open sets Ua and Vb such that a ∈ Ua, b ∈ Vb, and Ua ∩ Vb = ∅. For each a ∈ K1, the collection {Vb : b ∈ K2} forms an open cover of K2. Since K2 is tri-compact, there exists a finite subset {b1, b2, . . . , bna} ⊂ K2 such that K2 ⊂ ⋃na j=1 Vbj . Let Va = ⋃na j=1 Vbj . Then Va is a ϑi-open set containing K2 and Ua ∩ Va = ∅. The collection {Ua : a ∈ K1} forms an open cover of K1. Since K1 is tri-compact, there exists a finite subset {a1, a2, . . . , am} ⊂ K1 such that K1 ⊂ ⋃m j=1 Uaj . Let U1 = ⋃m j=1 Uaj and U2 = ⋂m j=1 Vaj . Thus K1 is an element of U1 which is a ϑi open set and similarly for K2 and U2. Furthermore, U1 ∩ U2 = ∅ because Uaj ∩ Vaj = ∅ for each j = 1, 2, . . . ,m. J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5764 10 of 11 4. Tri-locally metacompact spaces In this part, we raise the idea of metacompactness to tri topological space and discuss the connection of tri locally compact and tri locally metacompact. Definition 22. A family U of subsets of a tri-topological space (X,ϑ1, ϑ2, ϑ3) is called point-finite if each point of X belongs to at most finitely many members of U . Definition 23. A tri-topological space (X,ϑ1, ϑ2, ϑ3) is tri-metacompact if every open cover of X has a point-finite open refinement. Definition 24. A tri-topological space (X,ϑ1, ϑ2, ϑ3) is tri-locally metacompact if for every point a ∈ X, there exists a ϑi-open set Ua containing a such that Ua is tri-metacompact. Theorem 18. Every tri-locally compact space is tri-locally metacompact. Proof. Let (X,ϑ1, ϑ2, ϑ3) be a tri-locally compact space. For every point a ∈ X, there exists a ϑi-open set Ua containing a such that Ua is tri-compact. Since every tri-compact space is tri-metacompact, Ua is tri-metacompact. Therefore, (X,ϑ1, ϑ2, ϑ3) is tri-locally metacompact. Theorem 19. Let (X,ϑ1, ϑ2, ϑ3) be a tri-topological space. If X is tri-locally metacompact and tri-T3, then for any tri-metacompact set M and any ϑi-open set U containing M , there exists a ϑi-open set V such that M ⊂ V ⊂ V ⊂ U and V is tri-metacompact. Proof. Let M be a tri-metacompact set and U be a ϑi-open set containing M . For each a ∈ M , by tri-local metacompactness, there exists a ϑi-open set Va such that a ∈ Va ⊂ Va ⊂ U and Va is tri-metacompact. The collection {Va : a ∈ M} forms an open cover of M . Since M is tri-metacompact, there exists a point-finite open refinement {Wα : α ∈ Λ} of {Va : a ∈ M}. For each α ∈ Λ, there exists aα ∈ M such that Wα ⊂ Vaα. Let V = ⋃ α∈ΛWα. Then V is a ϑi-open set and M ⊂ V ⊂ V ⊂ ⋃ α∈ΛWα ⊂⋃ α∈Λ Vaα ⊂ U . Since each Vaα is tri-metacompact and the collection {Wα : α ∈ Λ} is point-finite, V is tri-metacompact. 5. Conclusion In this paper, we have introduced and studied the concept of tri-locally compact spaces in the context of tri-topological spaces. Several fundamental properties and theorems regarding tri-locally compact spaces are proved and their connections with tri regular spaces and tri T3 spaces are established. For that, we have also shown that every tri-locally compact space is tri-locally metacompact, and then we have explored the connection between tri-locally compact spaces and tri-locally metacompact spaces. The behavior of tri-locally compact spaces under various continuous operations, such as continuous mappings, products, and quotients could be further researched. One could also J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5764 11 of 11 explore the relationship between tri-locally compact spaces and other types of tri-topological spaces in general, i.e. between tri-locally compact spaces and tri-paracompact spaces and tri-Lindelf spaces. References [1] Stephen Willard. General topology. Addison-Wesley, Reading, MA, 1970. [2] James R. Munkres. Topology. Prentice-Hall, Englewood Cliffs, NJ, 2 edition, 2000. [3] Jun-iti Nagata. Modern general topology. Elsevier, Amsterdam, 1985. [4] John L. Kelley. General topology. Graduate Texts in Mathematics, 27, 1955. [5] Yong Woon Kim. Pairwise compactness. Publicationes Mathematicae Debrecen, 15:87–90, 1968. [6] Norman Levine. Semi-open sets and semi-continuity in topological spaces. The Amer- ican Mathematical Monthly, 70(1):36–41, 1963. [7] Jamal Oudetallah, Rehab Alharbi, Salsabiela Rawashdeh, and Ala Amourah. Lin- delöfness spaces in N th topological spaces. International Journal of Neutrosophic Science, 25(3):206–216, 2025. [8] Rehab Alharbi, Jamal Oudetallah, Salsabiela Rawashdeh, and Ala Amourah. Some types of N th-locally compactness spaces. International Journal of Neutrosophic Sci- ence, 25(3):217–228, 2025. [9] Jamal Oudetallah, Mohammad M. Rousan, and Iqbal M. Batiha. On D- metacompactness in topological spaces. Journal of Applied Mathematics & Infor- matics, 39(5-6):919–926, 2021. [10] Jamal Oudetallah. Nearly expandability in bitopological spaces. Advances in Mathe- matics: Scientific Journal, 10(2):705–712, 2021. [11] N. Alharbi, H. Shukri, and M. S. M. Noorani. Some properties of pairwise β-irresolute and strongly β-irresolute bitopological mappings. Symmetry, 15(2):375, 2023. [12] S. Hnaif, M. Abu-Saleem, and W. Shatanawi. On results of relations between pair- wise almost s-regular and related bitopological spaces. Journal of Function Spaces, 2021:5580806, 2021.