EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6067 ISSN 1307-5543 – ejpam.com Published by New York Business Global G-Compact Spaces Characterized by the Intersection of Countable Neighborhoods Mutaz Shatnawi1, Jamal Oudetallah2, Anwar Bataihah3, Ala Amourah4,5,∗, Abdullah Alsoboh6,∗, Tala Sasa7 1 Department of Mathematics, Faculty of Science and Information Technology, Irbid National University, Irbid 21110, Jordan 2 Department of Mathematics, University of Petra, Amman 11196, Jordan 3 Department of Mathematics, Faculty of Science, Jadara University, Irbid 21110, Jordan 4 Mathematics Education Program, Faculty of Education and Arts, Sohar University, Sohar 311, Oman 5 Jadara Research Center, Jadara University, Irbid, Jordan 6 College of Applied and Health Sciences, A’Sharqiyah University, Post Box No. 42, Post Code No. 400, Ibra, Sultanate of Oman 7 Department of Mathematics, Faculty of Science, Applied Science Private University, Amman, Jordan Abstract. In this research, we introduce and analyze the concepts ofG-compactness, G-Lindelöfness, and G-countably compactness within the framework of topological spaces, which are characterized by more rigorous conditions compared to those governing compact and Lindelöf spaces. We for- mulate a series of theorems and present a variety of examples to clarify the relationships among G-compactness, G-Lindelöfness, compactness, and Lindelöfness. Additionally, we define the G- separation axioms utilizing Gδ sets and explore the interrelations among these concepts. 2020 Mathematics Subject Classifications: 54D30, 54E99, 54D10 Key Words and Phrases: Compact space, Lindelöf space, countably compact space, G-compact space, G-Lindelöf space, G-countably compact, separation axioms, G-separation axioms. 1. Introduction and Preliminary The notion of compactness in mathematics pertains to a characteristic of topological spaces that generalizes the idea of closed and bounded subsets found in Euclidean space. ∗Corresponding author. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6067 Email addresses: m.shatnawi@inu.edu.jo (M. Shatnawi), jamal.oudetallah@uop.edu.jo (J. Oudetallah), a.bataihah@jadara.edu.jo (A. Bataihah), AAmourah@su.edu.om (A. Amourah), abdullah.alsoboh@asu.edu.om (A. Alsoboh), t sasa@asu.edu.jo (T. Sasa) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. Shatnawi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6067 2 of 16 A topological space is deemed compact if every open cover of that space admits a finite subcover, indicating that from any collection of open sets that collectively cover the space, a finite selection of these sets can also serve as a cover. This concept is rooted in the Heine-Borel theorem, which defines compact subsets of Euclidean space as those that are both closed and bounded. The formalization of compactness within topology emerged alongside the establishment of topology as a separate mathematical discipline during the late 19th and early 20th centuries. This era was marked by significant contributions from mathematicians such as Henri Poincaré, who laid down essential principles of topology, and Felix Hausdorff, who formulated the axiomatic basis for general topological spaces. Compactness is integral to numerous branches of mathematics, including analysis and functional analysis. Notably, continuous functions defined on compact spaces exhibit significant properties, such as the ability to achieve maximum and minimum values and the potential for approximation by polynomial or Fourier series, as articulated in the Stone-Weierstrass theorem. The exploration and generalization of this concept continue to be a focal point in contemporary mathematical research, encompassing fields such as infinite-dimensional topology and set-theoretic topology see [1, 2]. Metric spaces play a crucial role in the field of topology, as they offer a framework for establishing a topology on a given set. A topology consists of a collection of open sets that adhere to specific axioms. In the context of a metric space, a subset is deemed open if, for every point within that subset, there exists a radius such that all points located within that radius are also included in the subset. This relationship enables metric spaces to exemplify topological spaces, facilitating the investigation of concepts such as continuity and convergence see [3]. For more on metric theory we refer [4–14] and references therein. Separation axioms in topology represent a collection of criteria utilized to differentiate among various types of topological spaces, particularly in terms of the ability to separate distinct points and sets through neighborhoods. The emergence of these axioms coin- cided with the maturation of topology as a specialized area of mathematics, significantly influenced by the work of mathematicians such as Felix Hausdorff. In his seminal 1914 publication, ”Grundzüge der Mengenlehre,” Hausdorff articulated the separation property, which became a cornerstone of contemporary topology by establishing axioms applicable to general spaces. This property, referred to as the Hausdorff condition, stipulates that for any two distinct points within a space, there exist disjoint open sets that separate them, thereby classifying the space as a Hausdorff space, or (T2) space. The development of separation axioms is closely linked to the overall progression of topology, which originated in the 19th century, with key contributions from mathematicians such as Henri Poincaré, who played a pivotal role in defining topology as a distinct discipline through his 1895 work, ”Analysis Situs.” The advancement of topology involved the investigation of vari- ous spatial properties, including compactness and convergence, which were systematically formalized by mathematicians like Maurice Fréchet and Pavel Alexandrov through ax- iomatic methods. Today, separation axioms are integral to topological research, offering a structured approach to understanding the interactions between points and sets within a topological space. They are essential for differentiating among various types of spaces and for establishing numerous theorems within the field of topology see [1, 2]. M. Shatnawi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6067 3 of 16 Various forms of compactness are examined in numerous research articles, as illus- trated in references [15–18]. In [19] the authors study topological spaces on symbolic m-plithogenic intervals, while [20] extends this to neutrosophic and refined neutrosophic real intervals, both using partial order relations. In this document, we represent the set of natural numbers as and the set of real numbers as . Definition 1 (Metric Space [3]). A metric space (X, d) consists of a set X together with a function d : X ×X → R satisfying: (i) d(x, y) ≥ 0 for all x, y ∈ X, and d(x, y) = 0 if and only if x = y. (ii) d(x, y) = d(y, x) for all x, y ∈ X (symmetry). (iii) d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X (triangle inequality). Metric spaces provide a framework for establishing topology on a given set, where we consider a subset open if, for every point within that subset, there exists a radius such that all points within that radius also belong to the subset. Definition 2 (Separation Axioms [1], [2]). Separation axioms in topology serve as crite- ria differentiating various topological spaces, particularly regarding the ability to separate distinct points and sets through neighborhoods. The main separation axioms include: (i) T0 (Kolmogorov): For any two distinct points, at least one has a neighborhood not containing the other. (ii) T1 (Fréchet): For any two distinct points, each has a neighborhood not containing the other. (iii) T2 (Hausdorff): For any two distinct points, disjoint neighborhoods exist containing them separately. (iv) T3 (Regular): A T1 space where for any point and any closed set not containing it, disjoint neighborhoods exist containing them separately. (v) T4 (Normal): A T1 space where for any two disjoint closed sets, disjoint neighbor- hoods exist containing them separately. These axioms form a hierarchy where T4 ⇒ T3 ⇒ T2 ⇒ T1 ⇒ T0. Definition 3 (Compactness [1]). A topological space (X, τ) demonstrates compactness if every open cover has a finite subcover. Equivalently, a space exhibits compactness if every collection of closed sets with the finite intersection property has a non-empty intersection. A space shows countable compactness if every countable open cover has a finite subcover. Definition 4 (Lindelöf Space [2]). A topological space (X, τ) qualifies as Lindelöf if every open cover has a countable subcover. This property ranks weaker than compactness but stronger than separability for many spaces. M. Shatnawi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6067 4 of 16 Definition 5 (Gδ Set [1]). In a topological space (X, τ), a Gδ set is any set expressible as a countable intersection of open sets. That is, G ⊆ X constitutes a Gδ set if and only if G = ∞⋂ n=1 Un where each Un belongs to τ . Definition 6 (Fσ Set [2]). In a topological space (X, τ), an Fσ set is any set expressible as a countable union of closed sets. That is, F ⊆ X constitutes an Fσ set if and only if F = ∞⋃ n=1 Cn where each Cn is closed in X. Definition 7 (Paracompact Space [1]). A topological space (X, τ) achieves paracompact- ness if every open cover of X has a locally finite open refinement. This property generalizes compactness and holds particular importance in differential geometry and analysis. Definition 8 (Metacompact Space [17]). A topological space (X, τ) demonstrates meta- compactness if every open cover of X has a point-finite open refinement. This property ranks weaker than paracompactness but stronger than the Lindelöf property in many cases. Topologists have examined various compactness forms, including sequential compact- ness, countable compactness, and pseudo-compactness. Each notion captures different aspects of the intuitive ”boundedness” idea in topological spaces, as illustrated in refer- ences [15], [16], [17], and [18]. The interplay between countability, compactness, and separation properties forms a rich research area in topology. Particularly, the study of Gδ sets and Fσ sets provides insight into the fine structure of topological spaces and has applications in descriptive set theory and analysis [3]. 2. G-compact and G-Lindelöf spaces Definition 9. Let (X, τ) be a topological space. Then the collection˜= {Gα : α ∈ ∆} is called a G-cover of X provided that X = ⋃ α∈∆ Gα where, Gα is a Gδ set in X for all α ∈ ∆. Definition 10. The topologica space (X, τ) is called G-compact space if every G -cover has a finite subcover. Theorem 1. Every G-compact space is compact. M. Shatnawi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6067 5 of 16 Proof. Let (X, τ) be a G-compact space and let Ũ = {Uα : α ∈ ∆} be an open cover of X. Since every open set Uα is countable intersection of itself, Uα is a Gδ set. Hence Ũ is a G-cover of X. But X is G-compact space, so there is a finite subcover of Ũ that covers X and therefore, X is compact. Here, it should be mentioned that the converse of Theorem 1 need not be true. To show this we have the following example: Example 1. The cofinite topology on , (R, τcof ) is compact but not G-compact space. Proof. It is known that (, τcof ) is compact. However, (R, τcof ) is not G-compact space. To show that let˜= {Gk : k∈ N}, where Gk = ∞⋂ i=k (R\ {i}) = (R\N) ∪ {1, 2, . . . k − 1} . Then˜is a G-cover of R which has no finite subcover. Assume by contrary that R = n⋃ j=1 Gkj , where kj−1 < kj for j = 2, 3, · · · , n. Then n⋃ j=1 Gkj = n⋃ j=1 ∞⋂ i=kj (R\ {i}) = (R\N) ∪ {1, 2, . . . , kn − 1} = Gkn which is a contradiction. Theorem 2. Every finite topological space is G-compact space. Proof. Let (X, τ) be a topological space such that X is finite set, we can write X as X = {x1, x2, . . . xn}. Let˜= {Gα : α ∈ ∆} be a G -cover of X, that means X = n⋃ i=1 {xi} = ⋃ α∈∆ Gα. So, for all 1 ≤ i ≤ n, xi ∈ Gαi, for some αi ∈ ∆. Hence, {Gα1, Gα2, . . . , Gαn} is a finite subcover of˜for X. Therefore, (X, τ) is G-compact space. Theorem 3. If (X, τ) represents a G-compact space and (Y, τ ′) demonstrates homeomor- phism with X, then Y exhibits G-compactness. Proof. Let f : X → Y establish a homeomorphism, and consider a G-cover˜= {Gα : α ∈ ∆} of Y . For each Gα, we express Gα = ⋂∞ n=1 Uα,n where each Uα,n belongs to τ ′. Since f creates a homeomorphism, f−1(Gα) = f−1( ⋂∞ n=1 Uα,n) = ⋂∞ n=1 f −1(Uα,n). As each f−1(Uα,n) belongs to τ , f−1(Gα) forms a Gδ set in X. M. Shatnawi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6067 6 of 16 Therefore, {f−1(Gα) : α ∈ ∆} constitutes a G-cover of X. Since X demonstrates G-compactness, we find a finite subcover {f−1(Gα1), f −1(Gα2), . . . , f −1(Gαn)}. It follows that {Gα1 , Gα2 , . . . , Gαn} forms a finite subcover of˜for Y , proving that Y exhibits G-compactness. Definition 11. The topological space (X, τ) is called G-Lindelöf space if every G -cover has a countable subcover. Theorem 4. Every G-compact space is G-Lindelöf space. Proof. The proof is obvious, because every finite subcover is countable. Here, it should be mentioned that the converse of Theorem 4 need not be true. To show that we have the following example: Example 2. Consider the cofinite topology on , (N, τcof ). To show that (N, τcof ) is a G-Lindelöf space, let˜= {Gα : α ∈ ∆} be a G -cover of N. Then for any n∈ N, there is Gαn ∈ G such that n ∈ Gαn. Hence {Gαn : n∈ N} is a countable subcover of˜ for N. Therefore, (N, τcof ) is a G-Lindelöf space. Now, to show that (N, τcof ) is not G-compact space, let n∈ N. Then {n} = ∞⋂ k = 1, k ̸= n (N\ {k}) is a Gδ set containing n. Hence, {{n} : n∈ N} forms a G -cover of N that has no finite subcover. Therefore, (N, τcof ) is not G-compact space. Definition 12. A space X is called G-countably compact if every countable G-cover has a finite subcover. Remark 1. Every G-countably compact space is countably compact, but the reverse is not true in general (see Example 2). Theorem 5. Every G-compact space exhibits G-countably compactness, but the converse generally fails. Proof. Let (X, τ) represent a G-compact space, and consider a countable G-cover˜= {Gn : n ∈ N} of X. Since X demonstrates G-compactness, every G-cover (including countable ones) admits a finite subcover. Therefore, X exhibits G-countably compactness. To show the converse fails generally, we examine the space (ω1, τ), where ω1 represents the first uncountable ordinal with the order topology. This space lacks G-compactness because the collection {{α} : α < ω1} creates a G-cover with no finite subcover, as each singleton forms a Gδ set. However, (ω1, τ) exhibits G-countable compactness because any countable cover of ω1 must include a set containing points arbitrarily close to ω1, and by the order topology’s nature, such a set would include a tail of the ordinals. M. Shatnawi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6067 7 of 16 Theorem 6. Every G-Lindelöf space is Lindelöf space. Proof. Let (X, τ) be a G-Lindelöf space and let Ũ = {Uα : α ∈ ∆} be an open cover of X. Since every open set Uα is countable intersection of itself, Uα is a Gδ set. Hence Ũ is a G –cover of X. But X is G-Lindelöf space, so there is a countable subcover of Ũ that covers X and therefore, X is Lindelöf. Here, it should be mentioned that the converse of Theorem 6 need not be true. To show that we have the following example: Example 3. Let (R, τu) be the real numbers with the usual topology. Then (R, τu) is Lindelöf but not G- Lindelöf space. Proof. Since (R, τu) is second countable, it is Lindelöf. Now, to show that (R, τu) is not G-Lindelöf space, let G̃ = {{x} : x∈ R} be a G-cover of R, one can verify that for any x∈ R, {x} = ∞⋂ n=1 ( x− 1 n , x+ 1 n ) . However, it is clear that G̃ has no countable subcover. Theorem 7. If X represents a G-Lindelöf space and Y constitutes a continuous image of X, then Y exhibits G-Lindelöfness. Proof. Let f : X → Y establish a continuous surjection, and consider a G-cover˜= {Gα : α ∈ ∆} of Y . For each Gα, we have Gα = ⋂∞ n=1 Uα,n where each Uα,n belongs to the topology of Y . Since f maintains continuity, each f−1(Uα,n) belongs to the topology of X, and f−1(Gα) = ⋂∞ n=1 f −1(Uα,n) forms a Gδ set in X. Thus, {f−1(Gα) : α ∈ ∆} creates a G-cover of X. Since X demonstrates G-Lindelöfness, we find a countable subcover {f−1(Gαi) : i ∈ N}. As f achieves surjectivity, {Gαi : i ∈ N} forms a countable subcover of˜for Y , proving that Y exhibits G-Lindelöfness. Remark 2. Compact spaces need not be G- Lindelöf spaces. To show that we have the following example: Example 4. The closed interval [0, 1] with the usual topology exhibits compactness but lacks G-Lindelöfness. Proof. We know that (by the Heine-Borel Theorem) I = [0, 1] demonstrates com- pactness because it fulfills closedness and boundedness, as stated in [3]. However, I lacks G-Lindelöfness. We establish this by considering points x ∈ I in three cases: Case 1: If x = 0, then {0} = ∞⋂ n=1 [ 0, 1 n ) . M. Shatnawi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6067 8 of 16 Case 2: If x = 1 then {1} = ∞⋂ n=1 ( n+ 1 n+ 2 , 1] . Case 3: If x ∈ (0, 1), then we find an ε-neighborhood containing x with (x−ε, x+ε) ⊆ (0, 1). By the Archimedean property, we find Mx∈ N such that 1 Mx < ε, establishing( x− 1 n , x+ 1 n ) ⊆ (x− ε, x+ ε) ⊆ (0, 1) for all n ≥ Mx. Also, {x} = ∞⋂ n=Mx ( x− 1 n , x+ 1 n ) . In each case, {x} forms a Gδ set for any x ∈ I. Therefore, G̃ = {{x} : x ∈ I} creates a G-cover of I with no countable subcover since I demonstrates uncountability. This proves that I lacks G-Lindelöfness. Remark 3. G- Lindelöf spaces need not be compact spaces. To show that we have the following example: Example 5. The nested interval topology on (0, 1) exhibits G-Lindelöfness but lacks com- pactness. Proof. On the open interval X = (0, 1) The nested interval topology τ is defined by declaring all open sets of the form Vn = ( 0, 1− 1 n ) , for n = 2, 3, 4, . . ., together with ∅ andX. This topological space is G- Lindelöf but not compact space. To see this, let G̃ = {Gα : α ∈ ∆} be a G -cover of X, but any Gδ set in X belongs to {∅, X, Vn : n = 2, 3, 4, . . .}. Hence, G̃ is countable. So, we can choose G̃ itself as a countable subcover for X. Theorem 8. Every Fσ subspace of a G-compact space is G-compact. Proof. Let (X, τ) be a G-compact space and let F be an Fσ subspace of X. To show that F is G-compact, let G̃ = {Gα : α ∈ ∆} be a G -cover of F . Then G̃ ∪ {X\F} is a G -cover of X. As X is G-compact, there exists a finite subcover of G̃ ∪ {X\F}, say A = {Gα1, Gα2, . . . , Gαn}. This covers F by the fact that it covers X. Suppose X\F is an element of A. Then X\F may be removed from A, and the rest of A still covers F . Thus we have a finite subcover of G̃ which covers F . Hence F is G-compact. Theorem 9. If (X, τ) represents a G-compact space and U constitutes a Gδ set in X, then U with the subspace topology exhibits G-Lindelöfness. Proof. Let U form a Gδ set in the G-compact space (X, τ), and consider a G-cover˜= {Gα ∩ U : α ∈ ∆} of U , where each Gα represents a Gδ set in X. M. Shatnawi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6067 9 of 16 Since U forms a Gδ set, we write U = ⋂∞ n=1 Vn where each Vn belongs to τ . For each α ∈ ∆, we express the set Gα = ⋂∞ n=1Wα,n where each Wα,n belongs to τ . Consider the collection {Gα ∪ (X \ U) : α ∈ ∆}. Each set in this collection creates a Gδ set in X since: Gα ∪ (X \ U) = Gα ∪ (X \ ∞⋂ n=1 Vn) = Gα ∪ ∞⋃ n=1 (X \ Vn) This collection forms a G-cover of X. By G-compactness, it admits a finite subcover {Gα1 ∪ (X \ U), Gα2 ∪ (X \ U), ..., Gαk ∪ (X \ U)}. Therefore, {Gα1 ∩ U,Gα2 ∩ U, ..., Gαk ∩ U} constitutes a finite (and hence countable) subcover of˜for U , demonstrating that U exhibits G-Lindelöfness. Theorem 10. Let (X, τ) be a topological space. The space X is G-compact if and only if each family F of Fσ subsets of X with the finite intersection property has nonempty intersection. Proof. First suppose that (X, τ) is G-compact. If {Fα : α ∈ ∆} is a family of Fσ sets of X having empty intersection, then {X\Fα : α ∈ ∆} is a G -cover of X. By G- compactness, there is a finite subcover {X\Fα1 , X\Fα2 , . . . , X\Fαn } and then n⋂ i=1 Fαi = ∅. So {Fα : α ∈ ∆} does not have the finite intersection property which is a contradiction. Conversely, suppose that any family of Fσ sets of X with the finite intersection prop- erty has nonempty intersection and let G̃ = {Gα : α ∈ ∆} be a G -cover of X with no finite subcover. Then X\ (Gα1 ∪Gα2 ∪ . . . ∪Gαn) ̸= ∅ for each finite collection {Gα1, Gα2, . . . , Gαn} from G̃, in other words n⋂ i=1 (X\Gαi) ̸= ∅. Hence, we conclude that the collection {X\Gα : α ∈ ∆} is a family of Fσ sets of X that has the finite intersection property. Therefore, we get ⋂ α∈∆ (X\Gα) ̸= ∅, and hence, G̃ is not a cover for X which is a contradiction. Theorem 11. The continuous image of a G-compact space is G-compact. Proof. Suppose X is G-compact space and h is a continuous map of X onto Y . If G̃ = {Gα : α ∈ ∆} be a G -cover of Y , then { h−1 (Gα) : α ∈ ∆ } is a G -cover of X and by G-compactness, a finite subcover exists, say { h−1 (Gα1) , h−1 (Gα2) , . . . , h−1 (Gαn) } . Then, since h is onto, the sets Gα1 , Gα2 , . . . , Gαn covers Y . Thus Y is G-compact space. Remark 4. Paracompact spaces need not be G-compact spaces. To show that we have the following example: M. Shatnawi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6067 10 of 16 Example 6. Consider the Sorgenfrey line Rℓ (the real line with the lower limit topology). This space demonstrates paracompactness but lacks G-compactness. Proof. Researchers have established that Rℓ exhibits paracompactness, as shown in [2]. To prove it lacks G-compactness, we observe that for any x ∈ R, the singleton {x} forms a Gδ set since {x} = ∞⋂ n=1 [x, x+ 1 n ) Hence, the collection {{x} : x ∈ R} creates a G-cover of Rℓ that has no finite subcover. Therefore, Rℓ lacks G-compactness. Theorem 12. Let {Xα : α ∈ Λ} represent a family of topological spaces. If the product space X = ∏ α∈ΛXα exhibits G-compactness, then each Xα demonstrates G-compactness. Proof. Let X = ∏ α∈ΛXα exhibit G-compactness and fix β ∈ Λ. Consider a G-cover G̃ = {Gi : i ∈ I} of Xβ. For each Gi, define G̃i = π−1 β (Gi), where πβ denotes the projection map from X onto Xβ. Since πβ maintains continuity and Gi forms a Gδ set in Xβ, G̃i constitutes a Gδ set in X. The collection {G̃i : i ∈ I} creates a G-cover ofX. SinceX exhibits G-compactness, we find a finite subcover {G̃i1 , G̃i2 , . . . , G̃in}. Then {Gi1 , Gi2 , . . . , Gin} forms a finite subcover of G̃ for Xβ, proving that Xβ exhibits G-compactness. Theorem 13. Let (X, τ) represent a topological space. If X demonstrates metrizability and separability, then X exhibits G-Lindelöfness if and only if X demonstrates second countability. Proof. Suppose X constitutes a metrizable, separable space exhibiting G-Lindelöfness. Let {xn : n ∈ N} form a countable dense subset of X. For each pair of rational numbers p, q with p > 0, and each n ∈ N, define the open ball B(xn, p) = {x ∈ X : d(x, xn) < p}. The collection B = {B(xn, p) : n ∈ N, p ∈ Q+} demonstrates countability. We claim that B forms a base for τ . Consider any open set U ∈ τ and point x ∈ U . By the metric topology’s definition, we find ϵ > 0 such that B(x, ϵ) ⊂ U . Since {xn : n ∈ N} demonstrates density, some xm exists such that d(x, xm) < ϵ/3. Choose a rational number p such that d(x, xm) < p < ϵ/3. Then x ∈ B(xm, p) ⊂ B(x, ϵ) ⊂ U . This proves that B constitutes a countable base for X, establishing X’s second countability. Conversely, suppose X demonstrates metrizability and second countability. Let B form a countable base for τ . Every Gδ set G in X can be expressed as G = ⋂∞ n=1 Un where each Un ∈ τ . But we can express each Un as a union of elements from the base B. Since X demonstrates second countability, this means we find at most countably many different Gδ sets in X. Therefore, any G-cover of X contains at most countably many distinct elements, automatically making X G-Lindelöf. M. Shatnawi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6067 11 of 16 3. G- Separation Axioms Definition 13. A space X is called Tδ0 space if whenever x and y are distinct points in X, there is a Gδ set containing one and not the other. Definition 14. A space X is called Tδ1 space if whenever x and y are distinct points in X, there is a Gδ sets G1 and G2 such that x ∈ G1, y /∈ G1 and y ∈ G2, x /∈ G2. Evidently, every Tδ1 space is Tδ0 . Theorem 14. A space X is Tδ0 if and only if it is T0. Also, a space X is Tδ1 if and only if it is T1. Proof. It is easy to show that a space X is Tδ0 if and only if it is T0. Now, to show that a space X is Tδ1 if and only if it is T1 it is enough to show that if X is Tδ1 , then it is T1. Hence, let X be a Tδ1 space and let x, y ∈ X be two distinct elements, since X is Tδ1 there are two Gδ sets G1 and G2 such that x ∈ G1, y /∈ G1 and y ∈ G2, x /∈ G2. By the definition of Gδ sets assume G1 = ⋂∞ n=1G1n , G2 = ⋂∞ n=1G2n where G1n and G2n are open in X for n = 1, 2, · · · . Since x /∈ G2, so there is G2i for some i such that x /∈ G2i . Similarly, Since y /∈ G1, so there is G1k for some k such that y /∈ G1k but x ∈ G1k and y ∈ G2i . Therefore X is T1 space. Definition 15. A space X is called Tδ2 space if whenever x and y are distinct points in X, there are disjoint Gδ sets G1 and G2 such that x ∈ G1 and y ∈ G2. Evidently, every Tδ2 space is Tδ1 . Remark 5. Clearly every T2 space is Tδ2 space but the converse is not true in general. To show that we give the following example. Example 7. Consider the cofinite topology on , (N, τcof ). Then (N, τcof ) is not T2 space because is uncountable. To show that (N, τcof ) is Tδ2, let n, m be two distinct points in . Then, {m}, and {n} are two disjoint Gδ-sets. In fact {m} = ∞⋂ i=1 ((−{i}) ∪ {m}), {n} = ∞⋂ i=1 ((−{i}) ∪ {n}). Remark 6. Every Tδ2 space is Tδ1 but the converse is not true in general. M. Shatnawi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6067 12 of 16 Example 8. Consider with the cofinite topology. Then, (, τcof ) is Tδ1 which is not Tδ2. Clearly, (, τcof ) is T1 and hence, Tδ1. Now, assume to the contrary that (, τcof ) is Tδ2. Then for any a, b ∈ there are two Gδ sets say G1 = ∞⋂ i=1 Ui and G2 = ∞⋂ i=1 Vi, such that a ∈ G1, b ∈ G2 and G1 ∩G2 = ϕ. So, ( ∞⋂ i=1 Ui )⋂( ∞⋂ i=1 Vi ) = ϕ. =⇒ ∞⋂ i=1 ( Ui ⋂ Vi ) = ϕ. =⇒ ( ∞⋂ i=1 ( Ui ⋂ Vi ))c = . =⇒ ∞⋃ i=1 ( Ui ⋂ Vi )c = . =⇒ ∞⋃ i=1 ( Ui c ⋃ Vi c ) = . Since Ui c and Vi c are finite for each i, it follows that is countable, which a contradiction. Theorem 15. A space X exhibits Tδ2 if and only if the diagonal ∆ = {(x, x) : x ∈ X} forms a Gδ set in the product space X ×X. Proof. Suppose X represents a Tδ2 space. For each pair of distinct points x, y ∈ X, we find disjoint Gδ sets Gx and Gy containing x and y respectively. This means (x, y) ∈ Gx ×Gy and (Gx ×Gy) ∩∆ = ∅. Let Ux,y = Gx×Gy, which constitutes aGδ set inX×X. Then (X×X)\∆ = ⋃ x ̸=y Ux,y, making (X ×X) \∆ an Fσ set in X ×X. Therefore, ∆ forms a Gδ set in X ×X. Conversely, suppose ∆ constitutes a Gδ set in X ×X. Then (X ×X) \∆ represents an Fσ set, which we can write as ⋃∞ n=1 Fn where each Fn is closed in X ×X. For any distinct points x, y ∈ X, the pair (x, y) ∈ (X × X) \ ∆, so (x, y) ∈ Fn for some n. Since Fn is closed, (x, y) has an open neighborhood U ×V contained in Fn. Since (x, x) /∈ Fn, we must have x /∈ V or y /∈ U . Without loss of generality, assume x /∈ V . Let Gx = U and Gy = V . Then Gx and Gy form open sets containing x and y respectively, and Gx ∩ Gy = ∅. By taking countable intersections of such open sets for different n, we can construct disjoint Gδ sets containing x and y, establishing that X exhibits Tδ2 . M. Shatnawi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6067 13 of 16 4. Connections with Other Compactness Notions In this section, we explore relationships between G-compactness and other well-established compactness notions in topology. Theorem 16. Every G-compact space exhibits metacompactness. Proof. Let (X, τ) represent a G-compact space and consider an open cover U = {Uα : α ∈ ∆} ofX. Since every open set Uα equals a countable intersection of itself, it constitutes a Gδ set. Hence, U forms a G-cover of X. Since X demonstrates G-compactness, we find a finite subcover {Uα1 , Uα2 , . . . , Uαn} of U . This finite collection qualifies as point-finite, establishing that X exhibits metacompactness. Theorem 17. A space X demonstrates G-compactness if and only if every filter base on X consisting of Fσ sets has a cluster point. Proof. Suppose X exhibits G-compactness and consider a filter base F consisting of Fσ sets. If F lacks any cluster point, then for each x ∈ X, we find Fx ∈ F such that x /∈ Fx. This means x ∈ X \ Fx, which forms an open set. Since Fx constitutes an Fσ set, its closure Fx also forms an Fσ set (as the closure of an Fσ set remains an Fσ set in a regular space). Therefore, X \ Fx constitutes a Gδ set. The collection {X \ Fx : x ∈ X} creates a G-cover of X. By G-compactness, we find a finite subcover {X \ Fx1 , X \ Fx2 , . . . , X \ Fxn}. This implies that X = ⋃n i=1(X \ Fxi), or equivalently, ⋂n i=1 Fxi = ∅. But since F forms a filter base, the sets Fx1 , Fx2 , . . . , Fxn have non-empty intersection, which means⋂n i=1 Fxi ̸= ∅, creating a contradiction. Conversely, suppose every filter base consisting of Fσ sets has a cluster point, and consider a G-cover G̃ = {Gα : α ∈ ∆} ofX with no finite subcover. For any finite collection {Gα1 , Gα2 , . . . , Gαn} from G̃, we have ⋃n i=1Gαi ̸= X, which means ⋂n i=1(X \Gαi) ̸= ∅. Since each Gαi constitutes a Gδ set, each X \ Gαi forms an Fσ set. The collection {X \ Gα : α ∈ ∆} has the finite intersection property and consists of Fσ sets, so it generates a filter base of Fσ sets. By our assumption, this filter base has a cluster point x ∈ X. But xmust belong to some Gβ from the original G-cover, which means x ∈ X\(X\Gβ). This contradicts x being a cluster point for the filter base, as x has a neighborhood disjoint from X \ Gβ. Therefore, the original G-cover must have a finite subcover, establishing that X exhibits G-compactness. Theorem 18. Every paracompact Hausdorff space in which every closed set forms a Gδ set exhibits G-Lindelöfness. Proof. Let (X, τ) represent a paracompact Hausdorff space in which every closed set forms a Gδ set. Consider a G-cover G̃ = {Gα : α ∈ ∆} of X. Since X demonstrates paracompactness, we find a locally finite open refinement V = {Vβ : β ∈ Γ} of G̃. M. Shatnawi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6067 14 of 16 For each x ∈ X, we find a neighborhood Nx of x that intersects only finitely many members of V, say Vβ1 , Vβ2 , . . . , Vβnx . For each Vβi , we find a Gαi in G̃ such that Vβi ⊆ Gαi . Let G′ = {Gαi : i = 1, 2, . . . , nx, x ∈ X}. Since V demonstrates local finiteness in a paracompact space, G′ demonstrates countability. Therefore, G′ forms a countable subcover of G̃, establishing that X exhibits G-Lindelöfness. Future research We encourage scholars to explore G-compactness within metric spaces and investigate its behavior under various topological constructions, extending work in [3] and [4]. Fur- ther research could examine G-compactness in relation to filter convergence, nets, and ultrafilters, building on concepts from [15] and [16]. The interaction between G-properties and topological dimension theory also presents an interesting investigation avenue, as suggested by [18]. Another promising direction involves studying G-compactness in function spaces, par- ticularly regarding continuous functions, uniform convergence, and equicontinuity. The relationship between G-compactness and completeness in metric spaces could yield new insights into topological spaces’ structure. Additionally, developing more refined G-separation axioms based on Gδ sets could generate new set-theoretic topology insights, following approaches in [9] and [17]. The connections between G-compactness and descriptive set theory, particularly the classifica- tion of Borel sets and analytic sets, present rich exploration opportunities. Conclusion In this paper, we have established a new framework for studying topological spaces through Gδ sets. The introduced concepts of G-compactness, G-Lindelöfness, and G- countably compactness impose stronger requirements than their classical counterparts, creating finer distinctions among topological spaces. Our results demonstrate that while these G-properties imply their corresponding classical properties, the converse relation- ships generally fail, as our carefully constructed counterexamples illustrate. The G-separation axioms we’ve developed form a hierarchy paralleling the classical separation axioms, but with distinctive characteristics allowing classification of spaces indistinguishable under traditional separation properties. We’ve shown that certain spaces can exhibit Tδ2 without demonstrating T2, highlighting these new axioms utility. The relationships we’ve established with other compactness-like properties (metacom- pactness, paracompactness) further integrate our G-properties into the broader topologi- cal theory landscape. Furthermore, our results on G-compactness behavior under various topological operations, such as continuous maps, product spaces, and subspaces, provide tools for recognizing and applying these properties in diverse contexts. Our characterization of G-compactness in terms of filter bases consisting of Fσ sets connects this concept with classical filter-theoretic approaches to compactness. Similarly, M. Shatnawi et al. / Eur. J. Pure Appl. 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