EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6497 ISSN 1307-5543 – ejpam.com Published by New York Business Global Almost, Weakly, and Nearly Lindelöf Ideal Topological Spaces Eman Almuhur1,∗, Manal Al-labadi2, Enoch Suleiman3, Nazneen Khan4, Mohammad Esmael Samei5 1 Department of Mathematics, Applied Science Private University, Amman, Jordan 2 Department of Mathematics, University of Petra, Amman, Jordan 3 Department of Mathematics, Federal University Gashua, Yobe State, Nigeria 4 Department of Mathematics,Taibah University, Madina Munawwara, Saudi Arabia 5 Department of Mathematics, Faculty of Science, Bu-Ali Sina University, Hamedan, Iran Abstract. The concepts of almost, weakly and nearly Lindelöf closed ideal topological spaces are introduced in this work. We examine their subspaces and the connection between the subspaces and their topological characteristics and explain how countable covers affect Almost Lindelöf spaces and concentrate on the significance of these covers. These covers are made up of countable subfamilies whose closures cover the ideal spaces. Definitions, claims, characterizations, and observations pertaining to the recently presented concepts of almost and weakly Lindelöf ideal topological spaces are stated, examined, and discussed. Additionally, the connections among various ideal topological spaces are analyzed and explored. We provide examples of the consequences of these novel ideal spaces. 2020 Mathematics Subject Classifications: 54A05, 54A10 Key Words and Phrases: Almost Lindelöf, Weakly Lindelöf, Nearly Lindelöf, Ideal spaces 1. Introduction The ideal J on the (X, τ) is a non-empty collection of ℘(X) subsets satisfying: i) If Z ∈ J and W ⊂ Z, then W ∈ J ; ii) If Z ∈ J and W ∈ J , then Z ∪W ∈ J . Hamlett and Jankovic [1, 2] created the first generalization of several important features of general topology via topological ideals in ideal topological spaces. Features like separation axioms, decomposition of continuity, connectedness, compactness, and resolvability have all been generalized using the concept of ideals [3, 4]. Historically, there have been two ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6497 Email addresses: e almuhur@asu.edu.jo (E. Almuhur), manal.allabadi@uop.edu.jo (M. Al-labadi), enochsuleiman@gmail.com (E. Suleiman), nkkhan@taibahu.edu.sa (N. Khan), mesamei@basu.ac.ir (M.E. Samei) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) E. Almuhur et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6497 2 of 11 primary developments in the application of ideals in general topology. The study of topological spaces’ local properties, which can be extended to their global properties, is the focus of the first line [5, 6]. The compatibility of an ideal with a topology is the main idea in these studies. An example of an immediate corollary of the σ-Extension Theorem is the well-known Banach Category Theorem [5]. The second-line works employ ideals to generalize some topological space features, like the separation axioms [7] and compactness [8]. In 1996, Santoro and Cammaroto [9] presented the concept of weakly Lindelöf spaces (SL). The well-known Lindelöf property naturally deteriorates into the weak Lindelöf property. The fact that we only need the countable set Ŵ to cover a dense subset of the space X naturally weakens the Lindelöf criterion. In particular, if there exists a countable Ŵ Û such that ∪Ŵ is dense in X for each open cover Û of X, then X is a weakly SL [10]. The traditional idea of SLs in topology is generalized by the concept of almost SLs. An almost SL is a topological space where a little weaker requirement is satisfied: we just need that each open cover has a ”countable refinement” that covers the space, rather than that each open cover has a countable subcover. Definition 1. The space X is almost Lindelöf if for each family of open subsets{ Uα : α ∈ Λ } , covering it, there exists { U ′ n : n ∈ N } a countable collection of open sets: i) There exists α ∈ Λ : U ′ n ⊂ Uα; ii) X = ∪n∈NU ′ n. Remark 1. Important differences between almost Lindelöf and Lindelöf spaces a) Countable subcovers are directly extracted from the original open cover in a SL; b) Before extracting a countable subcover, we decrease the sets (while keeping them open) in an almost SL, allowing us to modify the original open cover. As a result, all SLs are almost Lindelöf, albeit this is not always the case. Moreover, compact spaces are Lindelöf and so, they are almost Lindelöf. ([0, 1], τu) and (R, τu) are Lindelöf and hence almost SLs. Examine R that has the lower limit topology, which is produced by intervals of the [a, b) type a, b ∈ R. Because some open covers cannot have a countable subcover extracted, this space is not Lindelöf. Nonetheless, it is almost Lindelöf since a countable subfamily spanning R may be created using the closures of the intervals [a, b]. If each open cover in the space X has a countable subcollection whose union is dense in X, then the space is weakly Lindelöf. This trait is stronger than separability but weaker than the Lindelöf property. Understanding how compactness, separability, and covering qualities interact in topology requires an understanding of weakly SLs. The study of functional analysis and general topology naturally leads to weakly SLs. They offer a helpful compromise between the Lindelöf characteristic and separability. They are E. Almuhur et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6497 3 of 11 especially pertinent in areas like measure theory and function space analysis where density arguments are crucial. Comparison with SLs: If a countable subcover exists, its union is the entire space (and hence dense), making every SL, weakly Lindelöf. The opposite isn’t true, though. Weakly Lindelöf areas that are not Lindelöf do exist [11]. Example 1. i) Weakly Lindelöf is any separable space. This is due to the fact that a countable subcollection of an open cover whose union is dense may always be constructed from a countable dense subset. ii) With the conventional topology, the real line R is both Lindelöf and weakly Lindelöf. iii) The true line with the lower limit topology, the Sorgenfrey line, is not Lindelöf but faintly so. Remark 2. Relation to Further Topological Features: a) Separability: While the opposite is not true, any separable space is weakly Lindelöf. For instance, the one-point compactification of a discrete uncountable space is weakly Lindelöf but not separable; b) First Countability: Neither first countability nor weak Lindelöfness are a prerequisite for the other; c) Normality: A weakly SL that is normal is not always Lindelöf. The Moore (Niemytzki) plane, for instance, is normal and weakly Lindelöf but not Lindelöf. Typically, the extent e(X) of X represents the smallest cardinal number κ (The small- est cardinal number in set theory is represented by the formula κ = ℵ◦ = |N| [12]) for which the cardinality of each discrete closed subset of X is less than or equal κ. If A ⊂ X, then |A| denotes the cardinality of A. Definitely, ω is the first infinite cardinal number, ω1is the first uncountable cardinal number and c represents the cardinality of R. Additionally, Cammaroto and Santoro [9] created an example (see [9, Example 3.11]) demonstrating that there is a Tychonoff weakly SL that is not nearly Lindelöf. A generalization or relaxation of the concept is a nearly SL, which is frequently described in terms of specific “almost” requirements pertaining to Lindelöfness. The exact concept of nearly SLs depends on the context and the specific generalization being studied. 2. Main results Definition 2. A topological space X is an almost Lindelöf ideal space modulo J if each Ü = { Uα : α ∈ Λ } , an open cover of X admits countable subfamily: X = ∪n∈NCl ( Uαn ) [13]. Definition 3. The space X is an almost ideal space modulo J if every almost Lindelöf subset of X modulo J is closed. E. Almuhur et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6497 4 of 11 Theorem 1. A topological space X is almost SL modulo J if and only if Ĥ = { Hα : α ∈ Λ } , the family of closed subsets of X : ∩α∈ΛHα = ∅ admits the countable subfamily ∩ n∈N Hαn = ∅. Proof. Let Ĥ = { Hα : α ∈ Λ } be a family consisting of closed subsets of X : ∩α∈ΛHα = ∅. So, { X \ Hα : α ∈ Λ } covers X. Since X is almost Lindelöf modulo J , there exists{ α1, α2, . . . } a countable subfamily: X = ∪n∈NCl ( X \Hαn ) . Hence, ∩ n∈N int ( Hαn ) = ∅. On the other hand, if Ü = { Uα : α ∈ Λ } forms an open cover of X modulo J . Therefore,{ X \ Uα : α ∈ Λ } forms a family of closed subsets of X : ∩α∈Λ ( X \ Uα ) = ∅. Thus, there is { α1, α2, . . . } a countable subfamily: ∩ n∈N ( X \ Uαn ) = ∅, i.e X = ∪n∈NCl ( Uαn ) . Theorem 2. If X is an almost regular SL modulo J , then it is Lindelöf. Proof. Suppose that Ũ = { Uα : α ∈ Λ } forms an open cover of X for each Uα a regularly open subset. But X is almost regular Lindelöf modulo J , then for every a ∈ X, there exists αa ∈ Λ : a ∈ Uαa. For the regularly open a-neighborhood Vαa , a ∈ Vαa ⊂ Cl(Vαa) ⊂ Uαa . For X, being a nearly L-closed, there is { a1, a2, . . . } a countable subfamily : X = ∪ n∈N Cl(Vαa). Thus, Ṽ = { Vαa : α ∈ Λ } covers X regularly because X is almost modulo J . Therefore, X is Lindelöf modulo J . Corollary 1. X is an almost SL modulo J if and only if it is nearly Lindelöf. Theorem 3. X is an almost SL modulo J if and only if for every family M̃ = { Mα : α ∈ Λ } of closed subsets of X modulo J , there is Ũ = { Uα : α ∈ Λ } a family of open subsets with ∩ α∈Λ Cl(Uα) = ∅ : Mα ⊂ Uα, and there exists { α1, α2, . . . } a countable subfamily such that ∩n∈N int(Mαn) = ∅. E. Almuhur et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6497 5 of 11 Proof. Direct part: Assume that M̃ = { Mα : α ∈ Λ } is a family of closed subsets of X modulo J . If Ũ = { Uα : α ∈ Λ } is a family of open subsets with ∩α∈ΛCl(Uα∈Λ) = ∅, then Mα ∈ Ũ . Since X is regular modulo J , then Mα ⊂ Uα ⊂ Cl(Uα). So, X \ Uα ⊂ X \Mα, and X = ∪α∈Λ ( X \Mα ) , because of being an almost regular space modulo J . But X is a SL modulo J , there exists a countable subset { α1, α2, . . . } : X = ∪ n∈N Cl ( X \Mαn ) = ∪ n∈N ( X \ int(Mαn) ) = X \ ∩ n∈N int(Mαn). Thus, ∩n∈N int(Mαn) = ∅. Converse part: If the family Ũ = { Uα : α ∈ Λ } comprises regular open subsets of X modulo J with the empty set ∩n∈N int(Mαn). Then, for every α ∈ Λ, there is Mα ⊂ X : Mα ⊂ Uα where Mα is regular and closed modulo J with the empty set ∪n∈N int(Mαn). Thus, ∩ n∈N Cl(X \Mαn) = ∩ n∈N ( X \ int(Mαn) ) = X \ ∩ n∈N int(Mαn) = X \ X = ∅. Hence, there exists { α1, α2, . . . } : ∩n∈N int(X \ Uαn) is empty. Consequently, X \ ∪ n∈N Cl(Uαn) = ∅, too and X = ∪n∈NCl(Uαn). Definition 4. X is a hereditarily almost SL modulo J if each of its subspaces is almost SL modulo J . Remark 3. If X is a hereditarily almost SL modulo J , then X is countable discrete if and only if X is hereditarily Lindelöf. Definition 5. X is an almost SL modulo J if each almost regular Lindelöf subset of X modulo J is closed. Definition 6. If X is an almost regular space modulo J , then a subspace Y is almost regular relative to X modulo J ′ if for each Ŏ = { Oα : α ∈ Λ } the collection of open subsets of X modulo J :⊂ ∪α∈ΛOα, there exists Kα ⊂ X a regularly closed subset moulo J : Kα ⊂ Oα and Y ⊂ ∪α∈Λ int(Kα). In addition, there exists { αn : n ∈ N } the countable subset of X modulo J : Y ⊂ ∪n∈NCl(Oα). Remark 4. If X is almost regular modulo J , then X SL modulo J . The remark above is not true for the Urysohn space [14]. E. Almuhur et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6497 6 of 11 Example 2. Let ℵ = { nα : α < ω1 } , ℵ̀ = { mi : i ∈ ω } and Y = {( nα,mi )} α<ω1 for each i ∈ ω. If X = Y ∪ ℵ ∪ { n } for each n /∈ Y ∪ ℵ, then X modulo J is topologized as follows: Each y ∈ Y is an isolated point. The neighborhood of nα ∈ ℵ has the form: Unα(i) = { nα } ∪ { (nα,mi) } α<ω1 , ∀i ∈ ω. The neighborhood of n is Un(α) = { nα } ∪ { (nγ ,mi) : γ > α } α<ω1 , ∀i ∈ ω. Now, X is a not regular Urysohn space modulo J because n cannot be separated from the uncountable discrete closed set ℵ. Hence, X is not a SL modulo J . If U covers X modulo J , so there exists Un ∈ U for each n ∈ Un and there is γ < ω1 : Un(γ) ⊂ Un. Now,{ nα : α > γ } ∪ { n } ∪ { (nγ ,mi) : γ > α } α<ω1 ⊂ Un, ∀ i ∈ ω, and X \ Un is countable. So there is Ŵ ⊂ U countable such that X \ Un ⊂ ∪Ŵ . If Õ = Ŵ ∪ { Un } , then Õ ⊂ U is a countable subfamily. Theorem 4. If X is almost regular modulo J , and Y is an almost regular subspace of X modulo J ′, then Y is almost regular relative to X modulo J ′. Proof. If Ŏ = { Oα : α ∈ Λ } forms an open cover of Y modulo J ′, then there exists Fα ⊂ X regularly closed subset Fα ⊂ Oα, Y ⊂ ∪α∈Λ int(Fα), and there is { αn : n ∈ N } the countable subset of X modulo J : Y ⊂ ∪n∈NCl(Oα). If Uα = int(Fα) ∩ Y, Wα = Oα ∩ Y, both are open subsets of Y: Cl(Uα) is regularly closed subset of Y modulo J ′. So, Cl(Uα) ⊂ Fα ∩ Y ⊂ Oα ∩ Y, ∀α ∈ Λ. Now, X = ∩α∈ΛUα and Uα ⊂ int(Cl ( Uα) ) , hence Y = ∪α∈Λ int ( Cl(Uα) ) . Since Y is almost regular L-closed modulo J ′, ∃ { αn : n ∈ N } : X = ∪ n∈N Cl(Wαn). But Cl(Wαn) ⊂ Cl(Uα), so Y = ∪n∈NCl(Oαn). As a consequence, Y is almost regular subspace relative to X modulo J ′. Corollary 2. If X is an almost regular space modulo J and each of its proper regularly closed subset is almost regular L-closed modulo J , then X is almost regular modulo J . Theorem 5. If X is almost regular modulo J , then each of its proper clopen subsets is almost regular relative to X modulo J . E. Almuhur et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6497 7 of 11 Proof. If K ⊂ X and Ü = { Uα : α ∈ Λ } covers K : ∀α ∈ Λ, there is Aα regularly closed and Aα ⊂ Uα. Since K ⊂ ∪ α∈Λ int(Aα), there exists { α1, α2, . . . } a countable subset such that K ⊂ ∪n∈NCl(Uαn). But,{ Uα } α∈Λ ∪ (X \K), covers X regularly, and X is almost regular L-closed modulo J . Hence, X = ∪n∈N ( X \K ) . Definition 7. The space X modulo J is weakly Lindelöf if for each cover Õ = { Oβ : β ∈ Γ } , of its subsets, there is { α1, α2, . . . } a countable subset such that X = Cl ( ∪n∈N (Oβn) ) . Definition 8. The space X modulo J is weakly regular if for each of its covers by the regular open subsets { Uβ : β ∈ Γ } , there exists { α1, α2, . . . } a countable subset: X = Cl(∪n∈N ( Uβn) ) . Theorem 6. The space X modulo J is weakly if and only if for the family K ∼ = { Kβ : β ∈ Γ } , of closed subsets: ∩β∈Γ(Kβ) is empty, there exists { α1, α2, . . . } a countable subset such that int ( ∩n∈N (Kβn) ) is empty. Proof. Let K ∼ = { Kβ : β ∈ Γ } consists of closed subsets of X modulo J : ∩β∈Γ(Kβ). Now, X = ∪β∈Γ(X \ Kβ), there exists { α1, α2, . . . } a countable subset such that X = Cl(X \Kβn). Hence, X \ Cl(X \Kβn) = ∅. Thus, int ( X \ ∪ n∈N (X \Kβn) ) = int ( ∩ n∈N (Kβn) ) , is empty. Theorem 7. There is a Tychonoff weakly SL modulo J that is not almost Lindelöf. Proof. If Y is a discrete space modulo J : |Y| = ω1, X = ( βY × (ω + 1) ) − ( (βY \ Y)× { ω }) , is a subspace of βY×(ω+1) modulo J , and Ü is an open cover of X modulo J . βY×ω is a σ-compact dense in X modulo J , there exists Ŵ a countable subset of Ü : βY×ω ⊂ ∪Ŵ . E. Almuhur et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6497 8 of 11 Hence, X = ∪Ŵ and so X is a weakly SL modulo J . Since |Y| = ω1, Y can be enumerated by { yγ : γ < ω1 } and for every n ∈ ω. Let Uγ = { yγ } × (ω + 1) and Wn = βY × { n } . If Ü = { Uγ : γ < ω1 } ∪ { Wn : n ∈ ω } , then ∪Ŵ = ∪ { w : w ∈ Ŵ } . If γ◦ = sup { γ : Uγ ∈ Ŵ } , then γ◦ < ω1 because Ŵ is countable. Considering γ′ > γ◦, then( yγ′ , ω ) /∈ { w : w ∈ Ŵ } . But (yγ′ , ω) ∈ Uγ′ ⊂ Ü , hence X is not an almost SL modulo J . Theorem 8. For each infinite cardinal κ, there is a Tychonoff weakly SL, X modulo J with e(X) > κ. Although there are other methods for defining nearly Lindelöf ideal spaces, weakening the Lindelöf condition is a popular strategy. The following are two common definitions. i) Dense Subsets Definition: The ideal space X is nearly Lindelöf if, for each open cover Û of X modulo J , there is a dense subset D ⊆ X such that a countable subcollection of Û may cover D. ii) Definition via Nearly Covers: If there exists a countable subcollection Õ ⊆ Û where the union of the sets in Õ is dense in X modulo J for each open cover Û of X, then the topological space X is nearly SL modulo J . Since any open cover has a countable subcover if X is Lindelöf ideal space, this trivially satisfies the nearly Lindelöf condition, making any Lindelöf ideal space nearly Lindelöf [15]. According to some definitions, any ideal space that has a countable dense subset— such as separable spaces — is nearly Lindelöf because the dense subset is frequently covered by a countable subcollection of any open cover. Furthermore, if and only if a discrete ideal space is countable, it is Lindelöf. Depending on the definition, anything that is uncountable may nevertheless meet certain weakened kinds of Lindelöfness even though it is not Lindelöf [16]. The study of covering qualities and their generalizations in general topology inevitably leads to the emergence of nearly Lindelöf ideal spaces. They are especially helpful in situations when density-based covering or a weaker type of countability is still preferred but Lindelöfness is an excessively strong condition [17, 18]. E. Almuhur et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6497 9 of 11 Applications I) Functional Analysis: The study of function spaces and their topological character- istics involves the use of nearly L-closed ideal SLs. II) Set-Theoretic Topology: Cardinal invariants and combinatorial principles are exam- ined in relation to these spaces. III) Generalized Metric Spaces: Nearly Lindelöf ideal spaces serve as a link between more generic ideal topological spaces and metric-like spaces. Theorem 9. If X is Lindelöf ideal and A ⊆ X, then the following are equivalent: a) A is an almost Lindelöf subset modulo J . b) A is weakly Lindelöf subset modulo J . Proof. (a ⇒ b) Clear. (b ⇒a) If X is not a SL modulo J and Õ covers X. Then, for every O ∈ Õ, { O × {0, 1} } covers A. Hence, A is not a weakly Lindelöf subset modulo J because all points of X× { 1 } are isolated in A. 3. Conclusion This paper introduces the notions of virtually, weakly, and almost Lindelöf closed ideal topological spaces. In addition to explaining how countable covers impact Almost Lindelöf spaces and focusing on their importance, we also look at their subspaces and the relationship between the subspaces and their topological properties. The ideal spaces are covered by the closures of countable subfamilies that make up these covers. We begin, analyze, and discuss definitions, assertions, characterizations, and observations related to the recently introduced notions of almost and weakly Lindelöf ideal topological spaces. The relationships between different ideal topological spaces are also examined and studied. We illustrate the implications of these new ideal environments. Declarations Availability of Data and Materials Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study. Competing interests The authors declare that they have no competing interests. Conflict of interest No conflicts of interest are disclosed by the authors. Funding No particular grant for this study was provided by funding agencies from the commercial, nonprofit sectors, or public. E. Almuhur et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6497 10 of 11 Authors’ contributions All authors have equally contributed, read, and approved the final manuscript. References [1] T. R. 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