EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 5802 ISSN 1307-5543 – ejpam.com Published by New York Business Global σ-compact Spaces in Nth-Topological Space Jamal Oudetallah1, Ala Amourah2,3,∗, Iqbal M. Batiha4,5, Jamal Salah6,∗, Mutaz Shatnawi7 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 Applied Science Research Center, Applied Science Private University, Amman, Jordan 4 Department of Mathematics, Al Zaytoonah University of Jordan, Amman 11733, Jordan 5 Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman 346, UAE 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 and Information Technology, Irbid National University, Irbid 21110, Jordan Abstract. The principal aims of this paper include the introduction of the concept of σ-compact Spaces in nth-topological spaces, the definition of compactness, discussion of various generalizations encompassing compactness in nth-topological space, the properties of σ-compact Spaces. Further- more, our study extends to the analysis of the notion of countable compactness in nth-topological spaces. Our investigation extends to various generalizations of these spaces. 2020 Mathematics Subject Classifications: 30C45 Key Words and Phrases: Tri-topological spaces, Compactness, Sigma compact spaces, Separa- tion axioms 1. Introduction The study of nth-topological spaces arises as a natural extension of the concepts found in bi-topological and tri-topological spaces, building upon the foundation of single topo- logical spaces. These extensions aim to provide a more comprehensive framework for understanding complex structures that involve multiple interrelated topologies. In this paper, we define an nth-topological space as a non-empty set K equipped with nth distinct topologies, denoted as η1, η2, ..., ηn, forming the structure (K, η1, η2, ..., ηn). ∗Corresponding author. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5802 Email addresses: jamal.oudetallah@uop.edu.jo (J. Oudetallah), AAmourah@su.edu.om (A. Amourah), i.batiha@zuj.edu.jo (I. M. Batiha), damous73@yahoo.com (J. Salah), m.shatnawi@inu.edu.jo (M. Shatnawi) 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), 5802 2 of 14 The idea of nth-topological spaces began gaining attention in the early 21st century, specifically around the year 2000, as researchers sought to expand the theoretical un- derstanding and practical applications of topological concepts to more intricate systems. Central to this study is the notion of nth-compactness, which generalizes the classical concept of compactness by requiring the existence of a finite subcover under the nth- topological framework. This generalization has proven to be a powerful tool in extending well-established results to broader contexts. nth-topological spaces have shown particular promise in their application to compact and metacompact spaces, serving as a bridge between classical topology and more spe- cialized fields. The exploration of nth-compactness has provided new insights into the behavior of finite subcovers and their interactions across multiple topologies. By expand- ing upon existing concepts such as nth-metacompactness, researchers have developed a wealth of generalized theorems and illustrative examples. This paper builds on these advancements, presenting several significant examples and discussing key results that extend the foundational theorems of topology to nth-topological spaces. The works of [J. Oudetallah], [Jamal Oudetallah (2021)], [Jamal Oudetallah and M Al-Hawari (2018)], and [Jamal Oudetallah, Mohammad M. Rousa (2021)] are particu- larly noteworthy in this regard, as they provide a robust theoretical underpinning for the concepts discussed here. These contributions underline the versatility and applicability of nth-topological spaces in modern topology, making them a crucial area of study for further exploration and development. 2. Literature review The concept of a σ space in topological space (K, η) was presented by [7]. Recent research [2], [3], [6] has delved deeper into these areas. This paper explores the concept of nth-σ presents associated conclusions. Then, we introduce the concept of topological space, nth-topological space and some important concept in nth-topological space like: open and closed sets, derived set, closure set, interior and exterior sets, separation axioms, etc... . Then we talk about the concept of σ space in nth-topological spaces, discuss its features, and apply it to other spaces. We examine well-known definitions that will be applied in the sequel. The terms ηu, ηdis, ηcof and ηcoc represent the ordinary or usual topology, discrete topology, co-finite topology, and co-countable topology, respectively. The concept of tri- topological spaces can be represented as K = (K, η1, η2, η3) where η1, η2 and η3 are topolo- gies on K and the concept of nth-topological space (K, η1, η2, ..., ηn) where η1, η2 ... η3 are topologies on K . This is connected to prior research on nth-topological spaces, where each topology is a set of points that meet a set of axioms. [4] explained Hausdorff,regular and normal spaces in nth topological spaces using a set of standard results known as Tietze extintion. The primary goal of this paper is to introduce and investigate a novel sort of nth-σ space and nth-topological spaces are sets containing n topologies. J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5802 3 of 14 3. Preliminaries Definition 1.1 [3]: Let K ̸= ϕ, η ⊂ P(K) = {A:A ⊆ K}, then η is called topology on K if the following conditions are satisfied : i) ϕ , K ∈ η. ii) Closed under intersection. iii) The union of any collection of sets in η is also in η. Definition 2.1[1] : Let K be a non empty set , ηi ⊂ P(K) = {A:A ⊆ K}, where i=1,2,3,...,n . we say that (K , η1 , η2, ... ,ηn) is nth-topological space if ηi is topology on K , for all i=1,2,3,...,n . Example consider K = {1,2,3} η1 = {ϕ, K , {1}} ⊂ P(K) η2 = {ϕ , K, {1} , {2} , {1,2}} ⊂ P(K) η3 = {ϕ , K, {2} , {3} , {2,3}} ⊂ P(K) ηi’s satisfies the condition of topological space , i=1,2,3 so, ( K, η1 , η2 , η3 ) is tri- topological space. But, for example η = { ϕ, 1} is not topological space because it is not contains K. Definition 3.1 : Let ( K , η1 , η2 , ..., ηn ) be a nth-topological space . E ⊂ K, then: i) E is called nth-Open set , If E ∈ ηi for some i=1,2,3. ii) E is called nth-cloced set , If Ec ∈ ηi for some i=1,2,3. iii) E is called nth-clopen set , If E and Ec are both in ηi for some i=1,2,3. Example Let K = {x, y, z}, η1 = {ϕ,K, {x}}, η2 = {ϕ,K, {y}}, η3 = {ϕ,K, {z}} . The sets : ϕ,K, {x}, {y}, {z} are nth-open sets in K. The sets : ϕ,K, {y, z}, {x, z}, {x, y} are nth-closed sets in K. The sets : ϕ ,K are nth-clopen sets in K. Definition 4.1[5] : Let (K,η1,η2,...,ηn) be a nth-topological space , K ̸= ϕ and A is subset of K ,then q ∈ K is called nth-Limit point of A If for all ud nth-open set such that ud ∩ (A− {d}) ̸= ϕ . The set of all nth-limit points is called nth-derived set and it is denoted by A′ ={d : d is nth-limit point of A }. Properties of derived set: Let (K,η1,η2,..., ηn) be a nth-topological space and let W,M ⊂ K, then: i) the derived set of ϕ is ϕ. ii) If W subset of M, then W’ subset of M’. iii) The derived set of union of W and M equal the union of the derived set of W and the derived set of M. iv) The derived set of intersection of W and M equal the intersection of the derived set of J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5802 4 of 14 W and the derived set of M. Example letK ={x, y, z, q} , η1={ϕ,K, {x}, {x, y}}, η2 = {ϕ,K, {x}}, η3 = {ϕ,K, {x}, {x, z}} and A = {x, z}. Now y ∈ K and {x,y} nth-open set in η1 and y ∈ A we have {x,y} ∩ (A− {y}) ={x} ̸= ϕ then y is a nth-limit point of A. and the same argement we get also x and z are limit point, and A′= {x,y,z}. Definitio 5.1 : Let (K, η1, η2, ..., ηn) be a nth-topological space , K ̸= ϕ and W is subset of K , then the nth-clousre set is denoted by W = W ∪ W ′ Note that : Z is nth-closure set. Properties of closure set: let (K,η1,η2,...,ηn) be a nth-topological space and W is subset of K. then : i) W is nth-closed set. ii) W C is nth- open set. iii) w ∈ W if and only if for all ηi-open set uw and w ∈ uw, we have uw ∩ W ̸= ϕ. iv) w /∈ W if and only if for all ηi-open set uw (i= 1,2,3,...,n) and w ∈ uw, we have uw ∩ W = ϕ. Definition 6.1 : Let (K,η1,η2,...,ηn) be a nth-topological space , K ̸= ϕ and W ⊂ D, then a piont d ∈ W is said to be nth-Interior piont of W if there exist at least one neighborhood of d (N(d,ε)) such that N(d,ε) ⊆ W. The set of all nth-interior piont is called the nth-Interior set and it is denoted by A◦ ≡ INT(W) =AC C . Note That : A◦ is nth-open set. Properties of interior set: Let (K,η1,η2,...,ηn) be a nth-topological space and let X,Y ⊂ K ,then: (i) ϕ◦ = ϕ and K◦ = K. (ii) (X ∩ Y )◦ = X◦ ∩Y ◦ and X◦ ∪Y ◦ ⊂ (X ∪ Y )◦. (iii) X◦ is ηi-open set. (iv) n ∈ X◦ if and only if there exist ηi-open set un such that n ∈ un ⊂ X. Definition 7.1 : Let (K,η1,η2,...,ηn) be a nth-topological space , K ̸= ϕ and W is subset of K, then the point d is said to be nth-Exterior point of W, If there exist at least one neighborhood of d such that N(d,ε) ∩ W = ϕ The set of all nth-Exterior point is called nth-Exterior set and it is denoted by EX(W)=Int(W c)= W C Note That :Ex(W)is nth-close set . Properties of exterior set: Let (K,η1,η2,...,ηn) be a nth-topological space and let W,M ⊂ K ,then: J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5802 5 of 14 (i) The exterior set of ϕ is K and the exterior set of K is ϕ. (ii) if W ⊂ M, then EX(M) ⊂ EX(W). (iii) EX(W) is ηi-open set, i=1,2,...,n. (iv) m ∈ EX(w) if and only if there exsit ηi-open set um such that m ∈ uK ⊂ W c. Proof: (iii) since EX(W) = INT(W c), then EX(W) = W ccc ,but W cc = W thus, EX(W ) = W c and by definition of nth-closure set we have W is ηi-closed set so,the complement of ηi-closed set is ηi-open set, therefor EX(W) is ηi-open set, i=1,2,...,n. Definition 8.1 : Let (K,η1,η2,...,ηn) be a nth-topological space , K ̸= ϕ and W is sub- set of K , then the point d is said to be nth-Boundary point of W, If every neighborhood of d satisfy that N(d,ε) ∩ A ̸= ϕ and N(d,ε) ∩ Ac ̸= ϕ. The set of all nth-boundary point is called nth-Boundary set , and it is denoted by Bd(W) = W −W ◦ = W ∩W c. Note That : Bd (W) is nth-Closed set. Properties of boundary set: Let (K,η1,η2,...,ηn) be a nth-topological space and let X,Y ⊂ K ,then: (i) The boundary set of the empty set and K equal ϕ. (ii) Bd(X) is ηi-closed set, i=1,2,...,n. (iii) y ∈ Bd(X) if and only if for all ηi-open set uy such that y ∈ uy we have uy ∩ X ̸= ϕ and uy ∩ Xc ̸= ϕ. Proof: (iii) Let y ∈ Bd(X) and uy be a ηi-open set such that y uy ,then y ∈ (X∩XC) ,if and only if b ∈ X and y ∈ Xc ,if and only if y ∈ (X ∪ X ′) and y ∈ Xc ∪ (Xc)′ ,if and only if ( y ∈ X or y ∈ X’ ) and (y ∈ Xc or y ∈ (Xc)′ ,if and only if y ∈ X’ and y ∈ Xc ,if and only if uy ∩ (X/{y}) ̸= ϕ and y ∩ Xc ̸= ϕ ,but y ⊂ uy ,so we have uy ∩ X ̸= ϕ and uy ∩ Xc ̸= ϕ . Definition 9.1 : A nth-topological space (K,η1,η2,...,ηn) is T◦-space, If for all c ̸= d in K , there exists ηi-open set uc such that c ∈ uc and d /∈ uc or there exists ηi-open set vd such that c /∈ vd and d ∈ vd, for all i= 1,2,...,n. Theorem 1 : Let (K,η1,η2,...,ηn) be a topological space, then the following are equiv- alent: i) K is nth-T0-space . ii) For all m ̸= n in K, we have m /∈ {n} or n /∈ {m}. iii) For all m ̸= n in K, we have {m} ̸= {n}. Proof: ((i) implies (ii)) Let m ̸= n, then there exist ηi-open set such that m ∈ um and n /∈ u −m or there exist ηj-open set vn such that m ∈ vn and m /∈ vn where i=1,2,...,n. so, we have m ∈ um and um ∩ {n} = ϕ or n in vn and vn ∩ {m} = ϕ. thus, m /∈ {n} or n /∈ {m}. ((ii) implies (iii)) J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5802 6 of 14 Let m ̸= n, then if m /∈ {n} and m ∈ {m}, then we have {m} ≠ {n}. Additionally, if n /∈ {n} and n ∈ {n}, then we have {m} ̸= {n}. ((iii) implies (i)) Let m ̸= n and by given {m} ≠ {n}, but m ∈ {m} and n ∈ {n}, then m /∈ K − {m} = vn which is ηi open set in K since {m} is ηi-closed set in K and n ∈ K − {m} = vn where i=1,2,...,n. thus, K is nth-T0-space. Definition 10.1 : A nth-topological space (K,η1,η2,...,ηn) is T1-Space, If for all c ̸= d in K, there exists ηi-open set uc such that c ∈ uc and d /∈ uc and there exists ηi-open set vd such that c /∈ vd and d ∈ vd, for all i= 1,2,...,n. Definition 11.1 : A nth-topological space (K,η1,η2,...,ηn) is T2-Space, If for all c ̸= d in K, there exists ηi-open set uc and there exists ηi-open set vd such that c ∈ uc and d ∈ vd and uc ∩ vd . = θ, for all i= 1,2,...,n. Definition 12.1 : A nth-topological space (K,η1,η2,...,ηn) is T2 1 2 -Space, If for all c ̸= d in K there exists ηi-closed set AC and there exists ηi-closed set Bd in K, such that c ∈ Ac, d ∈ Bd and Ac ∩ Bd = θ, for all i= 1,2,...,n. Definition 13.1 : A nth-topological space (K,η1,η2,...,ηn) is nth-regular space if for all c /∈ A and A nth-closed set in K, there exist ηi-open set uc and ηi-open set VA in K such that c ∈ uc, A ⊂ VA and uc ∩ VA . = θ Theorem 2 : A space (K,η1,η2,...,ηn) is nth-regular space if and only if for all m ∈ um ,where um is ηi-open set ,there exist ηi-open set wm such that m∈wm⊂wm⊂um. Proof: (→) Let m ∈ um, then m /∈ um c ,but um c is ηi-closed set, then we can say that um c=M. so, by definition of nth-regular space, there exist ηi-open set wm and vM such that a ∈ wm ,M ⊂ vM and wm ∩ vM =ϕ ,but clearly wm⊂wm . It is enough to show wm ⊂ um, now wm ∩ vM=ϕ, then we can say that wm ⊂ vM c, then wm ⊂ vMc = vM c . so, we have wm ⊂ vM c ,but um c= M ⊂ vM ,then um c ⊂ vM ,then vM c ⊂ um ,thus wm ⊂ um. (←) Let m /∈ M and M is ηi-open set, then m ∈ M c and M c is ηi-open set, then by given there exist ηi-open set wm such that m ∈ wm ⊂ wm ⊂ M c. now we have two givens, m ∈ wm and M ⊂ wm c ,where wm and wm c is ηi-open sets, i=1,2,...,n. ..... (*). it is enough to show that wm ∩ wm c = ϕ ,suppose not ,then there exist y such that y ∈ (wm ∩ wm c) ,that is implies y ∈ w and y ∈ wm c, then y ∈ wm and y /∈ w and y ∈ wm ′ ,then we have y ∈ ( wm ∩ wm c ) that is contradiction. J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5802 7 of 14 so ,wm ∩ wm c = ϕ ..... (**) By (*) and (**) we have, a space (K,η1,η2,...,ηn) is n th-regular space. Definition 14.1 : A nth-topological space (K,η1,η2,...,ηn) is T3-Space if K is nth- regular space and T1-Space. Definition 15.1 : A nth-topological space (K,η1,η2,...,ηn) is nth-normal apace if for all M, W be a nth- closed set in K and M ∩ W = θ, there exists ηi-open set uM and ηi-open set VW in M . such that M ⊂ uM , W ⊂ VB and uA ∩ VB . = θ. Definition 16.1 : A nth- topological space (K,η1,η2,...,ηn) is T4-Space if K is nth normal space and T1-Space. Definition 17.1 : A nth-topological space (K,η1,η2,...,ηn) and let A subset of K. the complement of nth-α open set is called nth-α closed set. 4. Nth-compact space and some type of it This section includes several important concepts in nth-topological spaces. Definition 2.1 [1] : Let (K, η) be a topological space and Q = {Wα : α ∈ λ,Wα ⊂ K} is called : i) cover of K if and only if ⋃ α∈λ Wα = K. ii) open cover of K if and only if Q is cover and Wα is open set ,where α ∈ λ. iii) closed cover of K if and only if Q is cover and Wα is closed set ,where α ∈ λ. iv) C = {Rγ : γ ∈ Γ} is a subcover of Q if and only if : (i) C ⊂ Q (ii) ⋃ γ∈Γ Rγ = K A space (K, η) is called compact space, if every open cover of K has a finite subcover. Example [7] (R, η) is not compact. Proof: by contradiction , assume that (R, ηu) is compact ,so every open cover of R has a finite subcover, but Q = {(−n, n) : n = 1, 2, 3, ...} is open cover of K because ∞⋃ n=1 (−n, n)=R and (-n,n) is open set,so E has a finite subcover say C={(−n1, n1), (−n2, n2).(−n3, n3), . . . , (−nm, nm)} ,then m⋃ i=1 (−ni, ni) = R ,then (a, b) = R where a = min{−ni} i=1,...,m and b = max{ni} i=1,...,m ,then R=(a,b) ⊂ [a, b]⇒ R ⊂ [a,b] ≡ bounded set so, R is bounded set and that is contradicted. ∴ (R, ηu) is not compact space. J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5802 8 of 14 Definition 2.2 : Let (K,η1,η2,...,ηn) be an nth-topological space and let W subset of K is called an nth-α∗-open set in k, if W subset of η1 int∗(η2CL(η3int ∗W ). Example let (K,η1,η2,...,ηn) be an nth-topological space where k = {1,2,3} , η1={ϕ,K,{1}}, η2={ϕ,K,{1},{1,2}}, η3= {ϕ,K,{1},{1,3}} then the nth-α∗-open set are {ϕ,{1},{1,2},{1,3},K}. Theorem 1 : If W is nth open set, then W is nth α∗-open set. Proof: W is nth-open set ⇒ W ⊂ η1 int(η2int(η3intA) ⇒ W ⊂ η1 int(η2CL(η3intA) ⇒ W ⊂ η1 int∗(η2CL(η3int ∗A). ⇒ W is nth α∗-open set. Note: The converse is not true. Example: Let (K,η1,η2,...,ηn) be an nth-topological space where K = {1,2,3} , η1 = {ϕ,K, {1}}, η2 = {ϕ,K, {1}, {1, 2}}, η3 = {ϕ,K, {1}, {1, 3}}, then: nth α∗-open set are {ϕ,{1},{1,2},{1,3},K}. nth α∗-closed set are {ϕ,{2},{3},{2,3},K}. W={1,2} is nth α∗-open set. But, W={1,2} is not nth open set. Definition 3.2 : Let (K,η1,η2,...,ηn) be an nth-topological space and let W subset of K is called an nth-α∗-closed set in K , if W ⊃ η1 int∗(η2CL(η3int ∗A). Theorem 2: Every nth-closed set is nth α∗-closed set. Proof: W is nth-closed set implies w subset of η1 cl η2 cl η3 cl W implies W c ⊂ η1 int η2 int η3 int W c implies W c ⊂ η1 int∗η2 int η3 int∗ W c implies W c ⊂ η1 int∗ η2 CLη3 int∗ W c implies W c is nth α∗-open implies W is nth α∗-closed set. Definition 4.2 : Let (K,η1,η2,...,ηn) be an nth-topological space and U = {Wα : α ∈ λ,Wα ⊂ K} is called : i) nth-cover of K if and only if ⋃ α∈λ Wα = K. ii) nth-open cover of K if and only if U is nth-cover and Wα is ηi-open set ,where α ∈ λ , i=1,2,...,n. iii) nth-closed cover of K if and only if U is nth-cover and Wα is ηi-closed set, where α ∈ λ, i=1,2,...,n. J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5802 9 of 14 iv) E = {Mγ : γ ∈ Γ} is a nthpartite subcover of U if and only if : (a) E ⊂ U (b) ⋃ γ∈Γ Mγ = K A space (K, η1, η2, ..., ηn) is called nth-compact space, if every nth-open cover of K has a finite nth subcover. Example : The nth-topological space (R, ηu1 , ηu2 ,..., ηun) is not n th-compact space. Proof: by contradiction , assume that (R, η1, η2, ..., ηn) is nth-compact, so every nth-open cover of R has a finite nth subcover ,but E = {(−n, n) : n = 1, 2, 3, ...} is nth-open cover of K because ∞⋃ n=1 (−n, n)=R and (-n,n) is ηui-open set ,i=1,2,...,n ,so E has a finite nth-subcover say C={(−n1, n1), (−n2, n2).(−n3, n3), ..., (−nm, nm)} ,then m⋃ i=1 (−ni, ni) = R ,then (x, y) = R where x = min{−ni} i=1,...,m and y = max{ni} i=1,...,k ,then R=(x,y) ⊂ [x, y] ⇒ R⊂ [x,y]≡ bounded set so, R is bounded set and that is contradicted.∴ (R, ηu1 , ηu2 , ..., ηun) is not nth-compact space. Theorem 3 : Let (K, η) be a topological space and U ⊂ k,then U is compact space if and only if U is closed set and bounded set. Theorem 4 : Let (K,η1,η2,...,ηn) be a nth topological space and U ⊂ K,then U is nth-compact space if and only if U is ηi-closed set and nth-bounded set. Proof: Suppose M ⊂ R is nth-compact. For each m ∈ M , consider the open interval (m − 1,m + 1) = Wm. Each Wm is ηi-open in R, so {Wm : m ∈ M} forms an nth-open cover of M . Since M is nth-compact, there exist finitely many points m1,m2, . . . ,mn ∈M such that M ⊆ ⋃n i=1Wmi . Let Q = max(m1,m2, . . . ,mn) and e = min(m1,m2, . . . ,mn). Then M ⊆ ⋃n i=1Wmi ⊆ [q − 1, q + 1], so Y is nth-bounded. Since M is nth-compact in the Euclidean space R (denoted as T2 space), Y is ηi-closed set. Conversely, suppose Y is ηi-closed and nth-bounded in R. If Y is nth-bounded, then Y ⊂ [x, y] for some x < y in R. Since Y is ηi-closed in the nth-compact subset [x, y], then Y is nth-compact set. Therefore, Y is ηi-closed and nth-bounded if and only if M is nth-compact. Definition 5.2 [6] : Let (K, η) be a topological space, and let L be a family of subsets of K. We say that L has a finite intersection property (f.i.p) if and only if the intersection of any finite number of members of L is non-empty. Definition 6.2 : Let (K,η1,η2,...,ηn) be a nth-topological space, and let L be a family of subsets of K. We say that L has a finite intersection property (f.i.p) if and only if the J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5802 10 of 14 intersection of any finite number of members of L is non-empty. Theorem 5 : Let (K,η1,η2,...,ηn) be a nth-topological space. Then K is nth-compact if and only if every family of nth-closed subsets of K with the finite intersection property (f.i.p) has a non-empty intersection. Proof: Suppose K is nth-compact space. If there exists a family of nth-closed subsets of K, say {Wα : α ∈ λ}, with f.i.p such that ⋂ α∈λWα = ∅, then ⋃ α∈λ(K \Wα) = K \ ⋂ α∈λWα = K \ ∅ = K. Since Wα is ηi-closed set in K for all α ∈ λ, K \Wα is ηi-open set in K for all α ∈ λ. Therefore, N = {K \ Wα : α ∈ λ} is an nth-open cover of K. By the compactness of K, N has a finite subcover of K, say {K \Wαi : i = 1, 2, . . . , n}. Thus, K = ⋃n i=1(K \Wαi) = K \ ⋂n i=1Wαi . This contradicts ⋂ α∈λWα = ∅, proving that every family of nth-closed subsets of K with f.i.p has a non-empty intersection. Conversely, suppose every family of nth-closed subsets of K with f.i.p has a non-empty intersection. If K is not nth-compact, then there exists an nth-open cover of K, say {wα : α ∈ λ}. Since wα is nth-open for all α ∈ λ, {K \wα : α ∈ λ} is a family of nth-closed subsets of K. Claim: {K \ wα : α ∈ λ} has f.i.p. If not, there exist w1, w2, . . . , wn such that ⋂n i=1(K \ wi) = ∅, hence ⋃n i=1wi = K. This implies {wi : i = 1, 2, . . . , n} is a finite subcover of K, which is a contradiction. Therefore, {K \ wα : α ∈ λ} has f.i.p. By assumption, ⋂ α∈λwα ̸= ∅. So, ∅ ≠ K \ ⋂ α∈λwα = ⋃ α∈λ(K \ (K \ wα)) = ⋃ α∈λwα, which is a contradiction. Hence, K must be compact. Theorem 6 : Every nth-closed subset of a nth-compact space is nth-compact. Proof: Suppose that T is a nth-closed subset in a compact space K. Let E = {tα : α ∈ λ} be an nth-open cover of T . Then K = T ∪ (K - T) = ⋃ α∈λ tα∪ (K - T) is an nth-open cover of K. Since K is nth-compact, T ∪ (K − T ) can be reduced to a nth finite subcover. Say nth-finite subcover Theorem 7 [4] : Let W be a compact subset in a nth-T2-space K. Then for all n /∈W there exists an open set Un containing n such that W ∩ Un = ∅ Theorem 8 : Let W be a nth-compact subset in a nth-T2-space K. Then for all n /∈W there exists a ηi-open set Un containing n such that W ∩ Un = ∅, i=1,2,...,n. Definition 7.2 : Let (K,η1,η2,...,ηn) be a nth-topological space ,then a set G in (K,η1,η2,...,ηn) is called nth-Dense set If G = K. On another hand, if G is dense in (K,η1,η2,...,ηn),then for all ηi-open set u we have u ∩ S ̸= ϕ. J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5802 11 of 14 Definition 8.2 : Let (K,η1,η2,...,ηn) and (Q,Λ1,Λ2, ...,Λn) are nth-topological space ,then the function h:(K,η1,η2,...,ηn) → (Q,Λ1,Λ2, ...,Λn) is called nth-open function if h(u) = v ,where u is ηi-open set and v is σi-open set. 5. σ-compact space in Nth-topological space Now, we will show some concept of σ-compact space in topological space, σ-compact space in nth-topological space and some theorems and their properties. Definition 1.3 [2] : Let (K, η) be a topological space ,then it is called σ-compact space if every open cover of K has a countable subcover of K.On the same time A topo- logical space K is called σ-compact if it can be expressed as a countable union of compact subspaces. Definition 2.3 : Let (K, η1, η2, ..., ηn) be a nth-topological space ,then it is called nth σ topological space if every ηi-open cover of K has a nth countable subcover of k. Theorem 1 : Any closed subspace of a nth σ-compact space is also nth σ-compact. Proof: Let (K, η1, η2, ..., ηn) be a nth σ-compact space, meaning K = ⋃∞ ni=1M(ni), where each M(ni) is compact in K. For any closed subspace Q ⊆ K, each Mn ∩ Q is compact (since compactness is preserved under closed subspaces). Thus, Q = ⋃∞ ni=1M(ni) ∩ F , which is a countable union of compact sets in Q, proving that Q is nth σ-compact, i=1,2,...,n. Theorem 2 : The continuous image of a nth σ-compact space is also nth σ-compact space. Proof: Let h : (K, η1, η2, ..., ηn) → (J,Λ1,Λ2,Λ3) be a continuous map and D = ⋃∞ ni=1 M(ni), where each M(ni) is compact. Since h(M(ni)) is compact (compactness preserved under continuous maps), we have h(K) = ⋃∞ ni=1 f(M(ni)), which is a countable union of compact sets in J. Hence, h(K) is nth σ-compact, for all i=1,2,...,n. Theorem 3 : The product of a nth σ-compact space with a nth compact space is nth σ-compact. Proof: Let K = ⋃∞ ni=1M(ni), where each M(ni) ⊂ K is nth compact, and let Y be nth compact. Then k × Y = ⋃∞ ni=1 (M(ni)× Y ), where each M(ni)× Y is nth compact in K ×Y (since J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5802 12 of 14 the product of nth compact spaces is nth compact). Therefore, k × Y is nth σ-compact. Theorem 5 : The countable union of nth σ-compact subspaces is nth σ-compact. Proof: Let K = ⋃∞ i=1Ki, where each Di is n th σ-compact. Then, for each i, we can write Ki =⋃∞ j=1 Sij , where Sij are nth compact. Thus,K = ⋃∞ i=1 ⋃∞ j=1 Sij , which is a nth countable union of nth compact sets, proving K is nth σ-compact. Theorem 6 : Every second-countable compact space is nth σ-compact space. proof: Since (K, η1, η2, ..., ηn) is nth topological space and K is second countable, there exists a countable base {Vi}∞i=1. Each compact subset Mi = Vi is compact in a second-countable space, and since K = ⋃∞ i=1Mi, K is nth σ-compact space. Theorem 7 : Every compact subspace of a nth σ-compact space is nth σ-compact space. proof: Let (K, η1, η2, ..., ηn) be nth σ-compact and M ⊂ K compact. Since K = ⋃∞ n=1M(ni) where each M(ni) is n th compact, M = ⋃∞ n=1 (M ∩M(ni)) , which is a countable union of compact sets. Thus, M is nth σ-compact space, i=1,2,...,n. Theorem 8 : If K is nth σ-compact space, then any quotient space R = K ∼ is nth σ-compact space. proof: Let K = ⋃∞ n=1 M(ni) with each M(ni) nth σ-compact space. Since the quotient map Q : K → R is continuous, Q(M(ni)) is nth σ-compact in R. Thus, R = ⋃∞ n=1Q(M(ni)), proving that R is nth σ-compact space, i=1,2,...,n. Theorem 9 : Every nth σ-compact space is separable if it is a metric space. proof: Let K be an nth σ-compact metric space. Since K can be expressed as a countable union of compact sets M(ni), each M(ni) is separable (nth compact subsets of metric spaces are separable). Therefore, we can find a countable dense subset T(ni) for each M(ni). The union T = ⋃∞ n=1 T(ni) is dense in k, showing that K is separable, for all i=1,2,...,n. Example: The real numbers R with the standard topology are nth σ-compact and separable (since the set of rational numbers Q is dense in R). J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5802 13 of 14 Theorem 10 : Any locally compact in nth σ-compact space is nth σ-compact space. proof: Let (K, η1, η2, ..., ηn) be nth topological space and K is nth σ-compact space and let K be a locally compact nth σ-compact space, and let K = ⋃∞ n=1M(ni), where each M(ni) is nth compact. For each point di ∈ K, there exists a neighborhood V(di) that is compact. Thus, we can cover K by a countable union of compact neighborhoods, confirming that K is nth σ-compact space. Example: The space Rn is locally compact and nth σ-compact space, as it can be covered by compact sets (closed balls) in a countable manner. Theorem 11 : The space C(K) of continuous functions on aN nth σ-compact space, K is nth σ-compact under the compact open topology. proof: Let K = ⋃∞ n=1M(ni), where each M(ni) is nth compact. The compact open topology is generated by sets of the form {h : M → R | h(M) ⊆ U}. Each nth compact set M(ni) gives rise to a countable family of compact sets C(M(ni)) in C(K). Thus, C(K) can be expressed as a countable union of compact sets, proving it is nth σ-compact. Theorem 12 : If K is nth σ-compact space and Q is a closed subset of K, then Q is nth σ-compact space. proof: Since K = ⋃∞ n=1M(ni) is n th σ-compact Space, the intersection M(ni) ∩ Q is compact for each ni. Therefore, Q = ⋃∞ n=1 (M(ni) ∩ Q), which is a countable union of compact sets. Thus, Q is nth σ-compact Space, for all i=1,2,...,n. Theorem 13 : Every nth σ-compact space can be expressed as a countable union of locally finite open covers. proof: Let K = ⋃∞ n=1M(ni) be nth σ-compact space. Each compact set M(ni) can be covered by a locally finite collection of open sets. The union of these open covers from all M(ni) remains locally finite. Thus, K can be expressed as a countable union of locally finite open covers, for all i=1,2,...,n. Example: The space R can be covered by intervals (n, n+ 1) for each n ∈ Z. Theorem 15 : A nth σ-compact Hausdorff space is second-countable. J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5802 14 of 14 proof: Suppose K is a nth σ-compact Hausdorff space that may be written as a countable union of compact sets. Each compact subset is second-countable, resulting in a countable base for the topology. The union of these countable bases produces a countable base for K, indicating that it is second-countable. Example: The space Rn is Hausdorff, nth σ-compact, and second-countable, meaning it may be covered by balls in a countable way. 6. Conclusion In this paper, we obtained some results related to σ-compact spaces, and applied this concept in nth-topological spaces. Several characteristics of these spaces and their interactions with other topologies are presented. Also, our study of σ-compact spaces solved some important mathematical problems in nth-topological spaces. Acknowledgments We very much appreciate everyone who made a contribution to this research project. Their help, guidance, and teamwork have been key to the completion of this research. References [1] Dugundji ;J. ,( 1966). Topology, Allyn and Bacon, Boston . [2] J. Oudetallah , On feebly pairwiese expandable space, J. Math. Comput. Sci. 11 (2021), No. 5, 6216-6225 [3] J. Oudetallah, Nearly Expandability in bitopological spaces, Advances in Mathemat- ics: Scientific Journal 10 (2021), 705-712. [4] J. Kelley, General topology, Van Nostrand Company, 1955.. kyungpook Math.J.,32, No. 2(1992), 273-284 [5] Kim,Y. W. (1968). Pairwise Compactness. Publ. Math. 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