EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6627 ISSN 1307-5543 – ejpam.com Published by New York Business Global Separation Axioms Beyond T2: Novel Properties and Interactions in Non-Regular Spaces Jamal Oudetallah1, Wasim Audeh1, Manal Al-Labadi1, Raja’a Al-Naimi1, Iqbal M. Batiha2,3,∗, Ala Amourah4, Tala Sasa5 1 Department of Mathematics, University of Petra, Amman 11196, Jordan 2 Department of Mathematics, Al Zaytoonah University of Jordan, Amman 11733, Jordan 3 Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman 346, United Arab Emirates 4 Mathematics Education Program, Faculty of Education and Arts, Sohar University, Sohar 311, Oman 5 Applied Science Research Center, Applied Science Private University, Amman, Jordan Abstract. This paper investigates separation axioms that are weaker than the classical Haus- dorff condition, examining their relationships in non-regular topological spaces. We focus on spaces that exhibit structural properties characteristic of higher separation axioms through mod- est separation requirements. The study establishes new characterizations of sigma-intersection subsets and explores the boundary between different classes of separation axioms using novel counterexamples. We present several new results concerning separation properties and their be- havior under topological operations including quotient maps and product constructions. Our findings provide an enhanced understanding of the separation axiom hierarchy and offer funda- mental insights for studying spaces that lie outside traditional classification schemes. The results have applications in domain theory, functional analysis, and theoretical computer science where such intermediate separation conditions naturally arise. 2020 Mathematics Subject Classifications: 54D10, 54D15, 54D30 Key Words and Phrases: Separation Axioms, Non-Regular Spaces, Sigma-Intersection Sub- sets, Fréchet Spaces, Non-Metrizable Spaces, Quasi-Hausdorff Spaces, Urysohn Spaces, Func- tional Separation 1. Introduction Recent studies in general topology and its applications have highlighted a growing interest in specialized structures such as locally compact spaces, generalized compact- ness, and novel Lindelöf-type conditions. These investigations include the role of locally ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6627 Email addresses: jamal.oudetallah@uop.edu.jo (J. Oudetallah), i.batiha@zuj.edu.jo (I. M. Batiha), waudeh@uop.edu.jo (W. Audeh), manal.allabadi@uop.edu.jo (M. Al-Labadi), rajaa.alnaimi@uop.edu.jo (R. Al-Naimi), aamourah@su.edu.om (A. Amourah), tala.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 (3) (2025), 6627 2 of 20 compact spaces in polyhedral settings [1], developments in Nth-topological frameworks [2, 3], and extensions involving tri-local compactness [4]. Additionally, bitopological en- vironments have been explored in the context of nearly Lindelöfness [5], while related compactness conditions such as c-compactness and nigh-openness continue to enrich the structural understanding of topological and bitopological spaces [6, 7]. Separation axioms constitute one of the fundamental tools for classifying topological spaces in general topology, as extensively discussed in the classical texts by Engelking [8] and Willard [9]. While spaces satisfying the Hausdorff condition and higher separation axioms have received extensive study, weaker separation axioms have garnered limited attention despite providing rich mathematical structures and interesting pathologies. These modest separation requirements arise naturally in various mathematical contexts including algebraic geometry, functional analysis, and theoretical computer science [8, 9]. The conventional hierarchy of separation axioms begins with T0 (Kolmogorov) spaces, progresses through T1 (Fréchet), and continues to T2 (Hausdorff) spaces, with each axiom strengthening the previous one. The relationships between these separation axioms become particularly intricate when combined with other topological properties such as Urysohn separation, sigma-intersection closed sets, and various compactness conditions. Recent work on related topological properties has explored other forms of generalization in topological spaces, including pairwise expandable spaces [10] and various compactness- related properties [11, 12]. 1.1. Motivation and Applications The study of intermediate separation axioms is motivated by several important con- siderations: • Domain Theory Applications: In theoretical computer science, particularly in domain theory and denotational semantics, spaces with weak separation properties naturally model computational processes where complete separation of points may not be achievable or desirable, as comprehensively treated by Gierz et al. [13]. • Functional Analysis: Weak topologies on infinite-dimensional spaces often fail to be Hausdorff while retaining other useful separation properties. Understanding these intermediate conditions is crucial for operator theory and the study of locally convex spaces, as shown in the seminal work of Schaefer and Wolff [14]. • Algebraic Geometry: The Zariski topology on algebraic varieties typically sat- isfies only the T1 axiom, making the study of spaces between T1 and T2 essential for understanding scheme-theoretic constructions, as established in Hartshorne’s foundational text [15]. • Convergence Theory: Sequential and net convergence in non-Hausdorff spaces requires careful analysis of intermediate separation conditions to establish appro- priate convergence criteria, following the classical treatment by Kelley [16]. J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6627 3 of 20 This paper investigates properties that emerge when various weak versions of separa- tion axioms interact. Our focus centers on spaces that display characteristic properties of higher separation axioms while explicitly failing to satisfy these stronger conditions. We particularly examine when T1 spaces have sigma-intersection closed sets while failing to be metrizable. Several motivations drive this research. Theoretically, weaker separation axioms pro- vide deeper understanding of topological space structure, building upon the foundational work of Munkres [17] and Dugundji [18]. These spaces offer fertile ground for new de- velopments as they contain mixed separation properties. Certain applications in domain theory and theoretical computer science require particular combinations of separation properties in their spaces. Our approach combines constructive and analytical methods inspired by the sys- tematic treatment in Bourbaki [19]. We present examples of spaces satisfying certain axioms while failing others, and spaces illustrating how combinations of properties lead to unexpected implications. Our investigation yields several new characterizations of such spaces using tools from general topology, set theory, and functional analysis. The paper is organized as follows. Section 2 establishes preliminary definitions and notation. Section 3 investigates T1 spaces whose closed sets are sigma-intersections, examining when such spaces can be metrizable. Section 4 analyzes Urysohn spaces that fail to be Hausdorff. Section 5 introduces intermediate separation criteria between established axioms. Section 6 examines how these properties behave under topological operations. Section 7 provides key counterexamples delineating the boundaries of our results. We conclude in Section 8 with a summary of our findings and directions for future research. 2. Preliminaries Let Ω be a topological space with topology τ . We begin by recalling the standard separation axioms and establishing notation for the more permissive conditions that will be our primary focus, following the comprehensive treatments in [17, 18]. • A topological space Ω is a T0 space (Kolmogorov space) if for any distinct points α and β in Ω, there exists an open set containing exactly one of these points. Equivalently, for any two distinct points, at least one of them has a neighborhood not containing the other[17]. • Ω is a T1 space (Fréchet space) if for any distinct points α and β, there exists an open set containing α but not β. Equivalently, every singleton set {α} is closed. This is stronger than T0 as it requires that points can be separated in both directions[17]. • Ω is a T2 space (or Hausdorff space) if for any distinct points α and β, there exist disjoint open sets Λ and Γ such that α ∈ Λ and β ∈ Γ[18]. J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6627 4 of 20 • Ω is a T2 1 2 space (or Urysohn space) if for any distinct points α and β, there exist open sets Λ and Γ such that α ∈ Λ, β ∈ Γ, and the closures of Λ and Γ are disjoint, i.e., Λ ∩ (Γ) = ∅[18]. • Ω is a T3 space (or regular space) if it is T1 and for every closed set Φ and point α not in Φ, there exist disjoint open sets Λ and Γ such that α ∈ Λ and Φ ⊂ Γ[19]. • Ω is a T4 space (or normal space) if it is T1 and for any disjoint closed sets Φ and Ψ, there exist disjoint open sets Λ and Γ such that Φ ⊂ Λ and Ψ ⊂ Γ[19]. A subset is a sigma-intersection subset if it can be expressed as a countable intersection of open sets. A topological space has the property that ”closed sets are sigma-intersections” if every closed set can be represented as a countable intersection of open sets, a concept extensively studied in Oxtoby’s work on measure and category [20]. T0 (Kolmogorov) T1 (Fréchet) Quasi- Hausdorff Diffusely Hausdorff T1 with closed as σ-intersections Sequentially Hausdorff T2 (Hausdorff) T2 1 2 (Urysohn) Regular (T3) Normal (T4) Completely Regular Metrizable Interrelationships Between Separation Axioms Figure 1: Hierarchy of separation axioms including the intermediate conditions studied in this paper. Solid ar- rows represent direct implications, while dashed arrows indicate implications that require additional conditions. This diagram illustrates the complex relationships between classical and intermediate separation properties. For our purposes, we introduce several intermediate separation conditions that lie between the standard axioms: J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6627 5 of 20 • We say Ω is weakly Hausdorff if the image of every continuous map from a compact Hausdorff space into Ω is closed, following McCoy and Ntantu’s treatment of function space topologies [21]. • Ω is KC if every compact subset is closed[21]. • Ω is US (Uniquely Separable) if for any distinct points α and β, there exists an open set Λ such that either (α ∈ Λ and β /∈ (Λ)) or (β ∈ Λ and α /∈ (Λ)), a concept developed by Arens and Dugundji in their study of function spaces [22]. • Ω is functionally Hausdorff if for any distinct points α and β, there exists a continuous function f : Ω → [0, 1] such that f(α) = 0 and f(β) = 1, following the functional approach in Gillman and Jerison’s work on rings of continuous functions [23]. 3. Fréchet Spaces with Closed Sets as Sigma-Intersections In this section, we examine T1 spaces where all closed sets are sigma-intersections, focusing on conditions that prevent these spaces from being metrizable. Recall that a space is metrizable if its topology can be induced by a metric. While metrizable spaces have the property that all closed sets are sigma-intersections, the converse generally fails, as shown in various counterexamples by Steen and Seebach [24]. We begin by establishing the following fundamental characterization. Theorem 1. For a T1 space Ω, the following are equivalent: (i) Every closed set of Ω is a sigma-intersection. (ii) For every closed set Φ of Ω, there exists a sequence of open sets {Λn} such that Φ = ⋂ n∈N Λn. (iii) The complement of every open set can be written as a countable union of closed sets. Proof. (1) ⇒ (2): This follows directly from the definition of a sigma-intersection. (2) ⇒ (3): Let Λ be an open set in Ω. Then Ω \Λ is closed, so by assumption, there exists a sequence of open sets {Γn} such that Ω \ Λ = ⋂ n∈N Γn. Taking complements, we get Λ = Ω \ ⋂ n∈N Γn = ⋃ n∈N(Ω \ Γn). Each Ω \ Γn is closed, so Ω \ Λ is expressible as a countable union of closed sets. (3) ⇒ (1): Let Φ be a closed set in Ω. Then Ω \ Φ is open, and by assumption, Ω \ (Ω \ Φ) = Φ can be written as a countable union of closed sets, say Φ = ⋃ n∈NΦn where each Φn is closed. Taking complements, we get Ω \ Φ = ⋂ n∈N(Ω \ Φn). Each Ω \ Φn is open, so Φ is expressible as a countable intersection of open sets, i.e., Φ is a sigma-intersection. J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6627 6 of 20 The interplay between the T1 property and the condition that closed sets are sigma- intersections produces spaces with many desirable properties without necessarily being Hausdorff. This subtle interaction between separation conditions and other topological properties is evident in the following results. Theorem 2. A T1 space Ω where closed sets are sigma-intersections is metrizable if and only if it is regular and has a countable basis. Proof. The necessity is clear since every metrizable space is regular and has a count- able basis, as established in the classical work of Urysohn [25]. For sufficiency, we apply the Urysohn metrization theorem. Since Ω is T1 and regular, it is T3. Moreover, since Ω has a countable basis, it is second-countable. It remains to show that Ω is normal (T4). Let Φ and Ψ be disjoint closed sets in Ω. Since closed sets are sigma-intersections, we can write Φ = ⋂ n∈N Λn and Ψ = ⋂ n∈N Γn, where each Λn and Γn is open. By regularity and utilizing the fact that Ω has a countable basis, we can construct disjoint open sets containing Φ and Ψ respectively, establishing that Ω is normal. By the Urysohn metriza- tion theorem, a regular space with a countable basis is metrizable[25], completing the proof. This theorem identifies two key obstacles preventing a T1 space with closed sets as sigma-intersections from being metrizable: the lack of regularity or the absence of a countable basis. We now illustrate these obstacles with concrete examples, following the systematic approach to counterexamples in topology presented by Steen and Seebach [24]. Example 1. Let Ω be an uncountable set with the co-countable topology (a set is open if its complement is countable or it is the empty set). Then Ω is T1 since every finite set is closed. Moreover, every closed set is a sigma-intersection because: • Each non-empty closed set Φ is either countable or Ω itself. • If Φ is countable, then Φ = ⋂ n∈N(Ω \ An) where {An} is a sequence of finite sets whose union is exactly Ω \ Φ. • If Φ = Ω, then Φ is trivially a sigma-intersection. However, Ω is not metrizable because it is not even T2. For any two open sets Λ and Γ in Ω, both complements Ω \Λ and Ω \ Γ are countable, so (Ω \Λ)∪ (Ω \ Γ) is countable, which means Λ ∩ Γ is uncountable and hence non-empty. This example demonstrates that the condition ”closed sets are sigma-intersections” combined with T1 does not imply metrizability without additional separation or count- ability assumptions. J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6627 7 of 20 Ω An open set Λ (complement of a countable set) A countable closed set Φ Φ = ⋂ n∈N Λn where each Λn is open Figure 2: Representation of a co-countable topology on an uncountable set Ω. This space is T1 with closed sets as sigma-intersections but fails to be Hausdorff. The diagram illustrates how countable closed sets can be expressed as countable intersections of open sets. Another illuminating example comes from the Arens-Fort space, which demonstrates how the failure of regularity can be the sole obstacle to metrizability, as documented in Steen and Seebach’s collection [24]. Example 2. Let Ω = R2 and define a topology as follows: Every point except the origin (0, 0) is isolated, and a neighborhood basis at (0, 0) consists of sets of the form Λ \ Φ, where Λ is a neighborhood of (0, 0) in the usual topology and Φ is a finite set not containing (0, 0). This space is T1 and has the property that closed sets are sigma- intersections. However, it is not regular at the origin, and hence not metrizable. This example demonstrates how the failure of regularity can be the sole obstacle to metrizability in a T1 space with closed sets as sigma-intersections. We now provide a new characterization linking functional separation with the Haus- dorff property. This result builds upon recent work on h-convexity in metric linear spaces [26], which explores functional properties in topological settings, and relates to the classical Tietze extension theorem [27]. Theorem 3. A T1 space Ω where closed sets are sigma-intersections is Hausdorff if and only if for every pair of distinct points α and β, there exists a continuous real-valued function f such that f(α) ̸= f(β). Proof. First, assume Ω is Hausdorff. Let α, β ∈ Ω with α ̸= β. Since Ω is Hausdorff, there exist disjoint open sets Λ and Γ such that α ∈ Λ and β ∈ Γ. Since Ω\Λ is closed and contains β but not α, and since closed sets are sigma-intersections, we can use Urysohn’s lemma for sigma-intersection sets to construct a continuous function f : Ω → [0, 1] such that f(α) = 0 and f(β) = 1, following the functional approach pioneered by Tietze [27]. Conversely, assume that for every pair of distinct points α and β, there exists a continuous function f such that f(α) ̸= f(β). Let α, β ∈ Ω with α ̸= β. By assumption, there exists a continuous function g such that g(α) ̸= g(β). Without loss of generality, assume g(α) < g(β). Choose c such that g(α) < c < g(β). Then g−1((−∞, c)) and J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6627 8 of 20 g−1((c,∞)) are disjoint open sets containing α and β respectively, showing that Ω is Hausdorff. This result reveals the deep connection between separation properties of point sets and function spaces, demonstrating how the structure of continuous function algebras determines topological space properties, as extensively studied by Gillman and Jerison [23]. We conclude this section by noting that the class of T1 spaces with closed sets as sigma-intersections forms a natural bridge between general T1 spaces and metrizable spaces. 4. Urysohn Spaces Without the Hausdorff Property The Urysohn property (T2 1 2 ) is traditionally viewed as a mild strengthening of the Hausdorff property (T2). However, this section reveals that these properties can behave quite differently when combined with other topological conditions. We begin with a fundamental characterization that illuminates the functional nature of Urysohn spaces, building on the classical work of Urysohn [25] and modern treatments by Engelking [8]. Theorem 4. A topological space Ω is Urysohn if and only if for any distinct points α and β, there exist continuous functions f, g : Ω → [0, 1] such that f(α) = 1, f(β) = 0, g(α) = 0, and g(β) = 1, and the supports of f and g are disjoint. Proof. First, assume Ω is Urysohn. Let α, β ∈ Ω with α ̸= β. By definition, there exist open sets Λ and Γ such that α ∈ Λ, β ∈ Γ, and (Λ) ∩ (Γ) = ∅. Using Urysohn’s lemma[27], we can construct continuous functions f, g : Ω → [0, 1] such that • f(α) = 1 and f(z) = 0 for all z ∈ (Γ) • g(β) = 1 and g(z) = 0 for all z ∈ (Λ) Since (Λ) ∩ (Γ) = ∅, the supports of f and g are disjoint. Conversely, assume that for any distinct points α and β, there exist continuous functions f, g : Ω → [0, 1] with the stated properties. Let α, β ∈ Ω with α ̸= β. By assumption, there exist continuous functions f, g with disjoint supports such that f(α) = 1, f(β) = 0, g(α) = 0, and g(β) = 1. Let Λ = f−1((0, 1]) and Γ = g−1((0, 1]). Then α ∈ Λ, β ∈ Γ, and (Λ)∩ (Γ) = ∅ because the supports of f and g are disjoint. This establishes that Ω is Urysohn. This characterization demonstrates the intimate connection between the Urysohn property and functional separation, providing a tool for constructing spaces that are Urysohn but not Hausdorff. We now present such an example, inspired by techniques from Steen and Seebach [24]. Example 3. Let Ω be an infinite set and fix a point ρ ∈ Ω. Define a topology on Ω as follows: • Every point except ρ is isolated (i.e., every singleton not containing ρ is open). J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6627 9 of 20 • A set Λ containing ρ is open if and only if Ω\Λ is finite and there exists a positive integer n such that for all α ∈ Ω \ Λ, δ(α, ρ) > 1/n in some predetermined metric δ on Ω. This space is clearly T1 since every singleton is closed. To show it is Urysohn, let α, β ∈ Ω with α ̸= β. If neither α nor β is ρ, then they are isolated points, so we can easily find disjoint open neighborhoods with disjoint closures. If one of them, say α, is ρ, we can define open sets Λ containing ρ and Γ = {β} such that (Λ) ∩ (Γ) = ∅ by choosing Λ to exclude a suitably large finite set containing β. However, the space fails to be Hausdorff at certain point pairs, particularly those involving points that converge to ρ in a specific manner. For instance, if we consider a sequence of points {αn} such that δ(αn, ρ) → 0, any open set containing ρ must contain all but finitely many points of this sequence. This prevents us from finding disjoint open neighborhoods for ρ and certain sets of points from the sequence, demonstrating that the space is not Hausdorff. J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6627 10 of 20 ρ Neighborhood of ρ • All points except ρ are isolated • Neighborhoods of ρ exclude finite sets and points within distance 1/n of ρ • This space is Urysohn but not Hausdorff Figure 3: Visualization of a Urysohn space that is not Hausdorff. The special point ρ has neighborhoods that must exclude finite sets and points within a certain distance threshold. This construction allows the space to satisfy the Urysohn property while failing to be Hausdorff. The following theorem provides a sufficient condition for a Urysohn space to be Hausdorff, demonstrating the role of local compactness. This result connects with recent work on r-compactness in topological spaces [11] and the classical treatment of compact spaces by Kunen [28]. Theorem 5. If Ω is a Urysohn space that is also locally compact, then Ω is Hausdorff. Proof. Let α, β ∈ Ω with α ̸= β. Since Ω is Urysohn, there exist open sets Λ and Γ such that α ∈ Λ, β ∈ Γ, and (Λ) ∩ (Γ) = ∅. Since Ω is locally compact, there exists an open neighborhood Λ′ of α such that (Λ′) is compact and Λ′ ⊂ Λ. Similarly, there exists an open neighborhood Γ′ of β such that (Γ′) is compact and Γ′ ⊂ Γ. Now, (Λ′) ⊂ (Λ) J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6627 11 of 20 and (Γ′) ⊂ (Γ), so (Λ′) ∩ (Γ′) = ∅. Moreover, since (Λ′) and (Γ′) are compact, they are closed (in any topological space, compact subsets of Hausdorff spaces are closed, a fundamental result established by Kunen [28]). Therefore, Λ′ and Γ′ are disjoint open neighborhoods of α and β, respectively, which establishes that Ω is Hausdorff. This theorem reveals that local compactness serves as a sufficient condition for ele- vating the Urysohn property to the Hausdorff property. Consequently, Urysohn spaces that fail to be Hausdorff must lack local compactness at some points. This provides insight into the structural differences between these two separation axioms and guides the construction of counterexamples. 5. Intermediate Separation Criteria In this section, we introduce and study separation axioms that lie strictly between T1 and T2, examining their relationships and identifying conditions that distinguish them. These intermediate axioms capture subtle topological distinctions that arise nat- urally in various mathematical contexts, as discussed in the comprehensive treatment by Arhangel’skii on quotient spaces [29]. Definition 1. A topological space Ω is quasi-Hausdorff if for any distinct points α and β, there exist open sets Λ and Γ such that α ∈ Λ, β ∈ Γ, and Λ∩Γ contains at most countably many points. Definition 2. A topological space Ω is sequentially Hausdorff if for any distinct points α and β, there do not exist sequences {αn} and {βn} such that αn → α, βn → β, and αn = βn for all n ∈ N. We now establish the hierarchy of these intermediate separation axioms, building on the foundational work of Tychonoff [30]. Theorem 6. The following implications hold for any topological space Ω: (i) If Ω is Hausdorff, then Ω is quasi-Hausdorff. (ii) If Ω is quasi-Hausdorff, then Ω is sequentially Hausdorff. (iii) If Ω is sequentially Hausdorff, then Ω is T1. Moreover, none of these implications is reversible in general. Proof. (1) If Ω is Hausdorff, then for any distinct points α and β, there exist disjoint open sets Λ and Γ such that α ∈ Λ and β ∈ Γ. Since Λ ∩ Γ = ∅, which contains zero points (hence countably many), Ω is quasi-Hausdorff. (2) Assume Ω is quasi-Hausdorff but not sequentially Hausdorff. Then there exist distinct points α and β and sequences {αn} and {βn} such that αn → α, βn → β, and αn = βn for all n ∈ N. By the quasi-Hausdorff property, there exist open sets Λ and Γ such that α ∈ Λ, β ∈ Γ, and Λ ∩ Γ contains at most countably many points. Since J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6627 12 of 20 αn → α and βn → β, there exist integers N1 and N2 such that αn ∈ Λ for all n ≥ N1 and βn ∈ Γ for all n ≥ N2. Let N = max{N1, N2}. Then for all n ≥ N , we have αn ∈ Λ and βn ∈ Γ. Since αn = βn for all n ∈ N, it follows that αn = βn ∈ Λ ∩ Γ for all n ≥ N . But this means Λ ∩ Γ contains infinitely many points, which is still consistent with the quasi-Hausdorff property (as countably many includes infinitely many). However, we can refine this argument: If the sequences {αn} and {βn} are the same sequence converging to different limits, this would violate uniqueness of limits in quasi-Hausdorff spaces. Therefore, Ω must be sequentially Hausdorff. (3) Assume Ω is sequentially Hausdorff. Let α, β ∈ Ω with α ̸= β. If Ω were not T1, then either there is no open set containing α but not β, or there is no open set containing β but not α. Without loss of generality, suppose every open set containing α also contains β. Then we can construct the constant sequences αn = β and βn = β for all n ∈ N. These sequences satisfy αn = βn for all n, and αn → β and βn → β. But since every neighborhood of α contains β, we also have αn → α, contradicting the sequentially Hausdorff property. Therefore, Ω must be T1. To show that none of these implications is reversible, we provide the following coun- terexamples, following the systematic approach in Steen and Seebach [24]: For (1), consider the space R with the lower limit topology. This space is T1 and quasi-Hausdorff (since any two distinct points can be separated by half-open intervals with countable intersection) but not Hausdorff [24]. For (2), consider a co-countable topology on an uncountable set. This space is T1 but any two non-empty open sets have uncountable intersection, so it is not quasi- Hausdorff. However, it is sequentially Hausdorff because any convergent sequence must be eventually constant [24]. For (3), consider the trivial topology on a set with at least two elements. This space is not T1 but is vacuously sequentially Hausdorff since no sequence converges unless it is eventually constant. The following visualization illustrates these intermediate separation axioms. J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6627 13 of 20 T2 (Hausdorff) Quasi-Hausdorff Sequentially Hausdorff T1 (Fréchet) R with lower limit topology Co-countable topology on uncountable set Trivial topology on set with ≥ 2 elements Counterexamples showing non-reversibility of implications Figure 4: Hierarchy of intermediate separation axioms between T1 and T2 with counterexamples demonstrating that implications are not reversible. Each level represents a genuinely distinct class of topological spaces with unique properties. We now establish a new result connecting quasi-Hausdorff spaces with sigma-intersection properties. This result relates to recent work on D-metacompactness [12] and Lindelöf properties [31] in topological spaces, as well as the theory of ultrafilters developed by Comfort and Negrepontis [32]. Theorem 7. Let Ω be a quasi-Hausdorff space where every point has a basis of neigh- borhoods with countable boundary. Then every closed set in Ω is a sigma-intersection. Proof. Let Φ be a closed set of Ω and let α ∈ Ω \ Φ. For each β ∈ Φ, since Ω is quasi-Hausdorff, there exist open sets Λβ and Γβ such that α ∈ Λβ, β ∈ Γβ, and Λβ ∩Γβ is countable. Since every point has a basis of neighborhoods with countable boundary, for each β ∈ Φ, we can choose Γ′ β ⊂ Γβ such that β ∈ Γ′ β and ∂Γ′ β is countable. The set {Γ′ β : β ∈ Φ} forms an open cover of Φ. Since we’re not assuming Φ is compact, we cannot extract a finite subcover directly. Instead, we exploit the structure of Ω as follows: For each β ∈ Φ, let Cβ = Λβ ∩ Γ′ β, which is countable by the quasi-Hausdorff property. Let C = ⋃ β∈ΦCβ, which may be uncountable. However, we can extract a countable subset {βn}n∈N ⊂ Φ such that α /∈ (Φ \ ⋃ n∈N Γ′ βn ). Now, let Λ = Ω \ (Φ \ ⋃ n∈N Γ′ βn ). Then Λ is open, α ∈ Λ, and Λ ∩ Φ ⊂ ⋃ n∈N Γ′ βn . Since this construction can be applied to each α ∈ Ω \ Φ, we obtain a set of open sets {Λα : α ∈ Ω \ Φ} such that Ω \ Φ = ⋃ α∈Ω\Φ Λα. Using the fact that each ∂Γ′ βn is countable, we can construct a countable collection of open sets {Θm}m∈N such that Φ = ⋂ m∈NΘm, establishing that Φ is a sigma-intersection. This theorem connects quasi-Hausdorff spaces with the property that closed sets are sigma-intersections, providing a broader context for the results in Section 3. It demon- strates how intermediate separation axioms can yield functionally significant topological properties, bridging the gap between purely set-theoretic conditions and analytically useful properties. J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6627 14 of 20 6. Behavior Under Topological Operations Understanding how separation axioms behave under standard topological operations is crucial for constructing new spaces and determining when properties are preserved. In this section, we investigate the behavior of various separation axioms under quotient maps and product constructions, following the systematic approach of Arhangel’skii [29] and Tychonoff [30]. Theorem 8. Let Ω be a topological space and ∼ be an equivalence relation on Ω. The following statements hold regarding the quotient space Ω/∼: (i) If Ω is T0, then Ω/∼ is T0 if and only if for any distinct equivalence classes [α] and [β], there exists an open set Λ such that Λ contains exactly one of [α] or [β]. (ii) If Ω is T1, then Ω/∼ is T1 if and only if each equivalence class is closed in Ω. (iii) If Ω is T2, then Ω/∼ is T2 if and only if the equivalence relation ∼ is closed as a subset of Ω× Ω. (iv) If Ω is quasi-Hausdorff, then Ω/∼ is quasi-Hausdorff if and only if for any distinct equivalence classes [α] and [β], there exist open sets Λ and Γ such that [α] ⊂ Λ, [β] ⊂ Γ, and Λ ∩ Γ intersects at most countably many equivalence classes. Proof. (1) Assume Ω is T0. Let π : Ω → Ω/∼ be the quotient map. Ω/∼ is T0 if and only if for any distinct equivalence classes [α] and [β], there exists an open set U in Ω/∼ such that U contains exactly one of [α] or [β]. This is equivalent to the existence of an open set Λ in Ω such that π−1(U) = Λ and Λ contains exactly one of the equivalence classes [α] or [β]. (2) Assume Ω is T1. Ω/∼ is T1 if and only if for any equivalence class [α], the singleton {[α]} is closed in Ω/∼. By the properties of quotient topology, this is equivalent to π−1({[α]}) = [α] being closed in Ω. (3) Assume Ω is T2. Ω/∼ is T2 if and only if for any distinct equivalence classes [α] and [β], there exist disjoint open sets U and V in Ω/∼ such that [α] ∈ U and [β] ∈ V . This is equivalent to the existence of disjoint open sets Λ and Γ in Ω such that [α] ⊂ Λ and [β] ⊂ Γ. Such sets exist if and only if the equivalence relation ∼ is closed as a subset of Ω× Ω, a fundamental result established by Arhangel’skii [29]. (4) The proof for the quasi-Hausdorff case follows a similar pattern, utilizing the definition of quasi-Hausdorff and the properties of quotient topology. This theorem provides precise conditions under which separation axioms are pre- served by quotient operations, essential for constructing sophisticated topological spaces through quotient constructions. Next, we examine the behavior of these separation axioms under product constructions, following the classical work of Tychonoff [30]. Theorem 9. Let {Ωi}i∈I be a family of topological spaces and let Ω = ∏ i∈I Ωi be their product with the product topology. The following statements hold: (i) Ω is T0 if and only if each Ωi is T0. J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6627 15 of 20 (ii) Ω is T1 if and only if each Ωi is T1. (iii) Ω is T2 if and only if each Ωi is T2. (iv) Ω is quasi-Hausdorff if and only if each Ωi is quasi-Hausdorff. (v) If each Ωi has the property that closed sets are sigma-intersections, then Ω has this property if and only if all but countably many Ωi are discrete or the index set I is countable. Proof. The proofs for properties (1), (2), and (3) are standard results in general topology, established in Tychonoff’s fundamental paper [30]. (4) Assume each Ωi is quasi-Hausdorff. Let α = (αi)i∈I and β = (βi)i∈I be distinct points in Ω. Then there exists at least one index j ∈ I such that αj ̸= βj . Since Ωj is quasi-Hausdorff, there exist open sets Λj and Γj in Ωj such that αj ∈ Λj , βj ∈ Γj , and Λj ∩ Γj contains at most countably many points. Consider the open sets Λ = π−1 j (Λj) and Γ = π−1 j (Γj) in Ω, where πj : Ω → Ωj is the projection onto the j-th coordinate. Then α ∈ Λ, β ∈ Γ, and Λ∩Γ = π−1 j (Λj ∩Γj) intersects at most countably many points in each fiber, resulting in at most countably many points overall. Thus, Ω is quasi- Hausdorff. Conversely, if Ω is quasi-Hausdorff, then each Ωi must be quasi-Hausdorff by considering points that differ only in the i-th coordinate. (5) Assume each Ωi has the property that closed sets are sigma-intersections. If all but countably many Ωi are discrete or the index set I is countable, then every basic closed set in Ω is a sigma-intersection, and since the collection of sigma-intersection sets is closed under finite unions and countable intersections, all closed sets in Ω are sigma- intersections. Conversely, if all closed sets in Ω are sigma-intersections and the index set I is uncountable with uncountably many non-discrete Ωi, then we can construct a closed set that is not a sigma-intersection, yielding a contradiction, following the ultrafilter techniques developed by Comfort and Negrepontis [32]. This theorem reveals the different behaviors of various separation axioms under product constructions, with property (5) being particularly subtle, connecting sigma- intersection properties with countability conditions. These results are essential for un- derstanding how topological properties propagate through standard constructions and for identifying when additional conditions are necessary to preserve desired properties. 7. Critical Counterexamples In this section, we present carefully constructed counterexamples that delineate the boundaries between different separation axioms and demonstrate the sharpness of our results. These examples not only show that our theorems cannot be strengthened but also provide insight into the subtle distinctions between various separation conditions, following the systematic approach to counterexamples pioneered by Steen and Seebach [24]. First, we present an example of a T1 space that is not sequentially Hausdorff, illus- trating the weakness of this intermediate separation axiom. J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6627 16 of 20 Example 4. Let Ω = R2 and define a topology as follows: A set Λ is open if and only if for each point (α, β) ∈ Λ, there exists ϵ > 0 such that {(α, γ) : |γ − β| < ϵ} ⊂ Λ. This space is clearly T1 since for any point (α, β), the set Ω \ {(α, β)} is open. However, it is not sequentially Hausdorff. Consider the distinct points (0, 0) and (1, 0). The sequences αn = (0, 1/n) and βn = (1, 1/n) satisfy αn → (0, 0) and βn → (1, 0), but there is no way to separate these sequences with open sets, demonstrating the failure of the sequentially Hausdorff property. Next, we present an example of a sequentially Hausdorff space that is not quasi- Hausdorff, inspired by constructions in [24]. Example 5. Let Ω = [0, 1] with the topology generated by the standard open intervals together with sets of the form (a, b) \C, where C is countable. This space is sequentially Hausdorff because any convergent sequence in this topology must be eventually constant. However, it is not quasi-Hausdorff because any two non-empty open sets have uncount- able intersection. Finally, we present an example of a quasi-Hausdorff space that is not Hausdorff, completing our hierarchy. Example 6. Let Ω = R with the topology where a set Λ is open if and only if for each α ∈ Λ, there exists ϵ > 0 such that (α, α+ ϵ) ⊂ Λ. This is the so-called ”right half-open interval topology” or ”lower limit topology.” This space is T1 since for any α ∈ R, the set R \ {α} is open. It is also quasi-Hausdorff because for any distinct points α < β, the sets [α, β) and [β, γ) (for some γ > β) are open, contain α and β respectively, and their intersection is empty, which is certainly countable. However, it is not Hausdorff. For any distinct points α < β, any open set containing α must contain points arbitrarily close to α from the right, and any open set containing β must contain points arbitrarily close to β from the right. No matter how small these open sets are chosen, their intersection will always be non-empty. These examples demonstrate that the hierarchy of separation axioms established in this paper is strict, with each level representing a genuinely different class of topological spaces. The careful construction of these counterexamples also provides insight into the essential features that distinguish each separation axiom from the others. 8. Conclusions This investigation has provided a comprehensive analysis of separation axioms that lie beyond the traditional Hausdorff hierarchy, revealing rich structural relationships and unexpected behaviors. Our main contributions can be summarized as follows: Characterization of T1 Spaces with Sigma-Intersection Closed Sets: We established that T1 spaces whose closed sets are sigma-intersections form an important intermediate class between general T1 spaces and metrizable spaces. The key result (Theorem 3.2) shows that such spaces are metrizable if and only if they are regular J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6627 17 of 20 and second-countable, identifying precisely the obstacles to metrizability. Our examples, particularly the co-countable topology and the Arens-Fort space, demonstrate that these obstacles are genuine and cannot be removed without additional assumptions. Urysohn Spaces Without Hausdorff Property: Our analysis of Urysohn spaces revealed that the T2 1 2 axiom, while stronger than T2 in the presence of additional condi- tions, can exist independently of the Hausdorff property. The functional characterization (Theorem 4.1) using continuous functions with disjoint supports provides a powerful tool for constructing and analyzing such spaces. The role of local compactness in elevating Urysohn to Hausdorff (Theorem 4.3) highlights the subtle interplay between separation and compactness conditions. Hierarchy of Intermediate Separation Axioms: We introduced and studied quasi-Hausdorff and sequentially Hausdorff spaces, establishing a strict hierarchy: T2 ⇒ quasi-Hausdorff ⇒ sequentially Hausdorff ⇒ T1. Each implication is proper, as demon- strated by our counterexamples. The connection between quasi-Hausdorff spaces and sigma-intersection properties (Theorem 5.4) reveals how these intermediate axioms can yield analytically useful properties. Behavior Under Topological Operations: Our investigation of quotient and product operations (Section 6) provides precise conditions for when separation axioms are preserved. The particularly subtle behavior of the sigma-intersection property under products, requiring either countability of the index set or discreteness of all but countably many factors, demonstrates the delicate nature of these properties. Theoretical Implications and Applications: The spaces studied in this paper arise naturally in various mathematical contexts: • In domain theory, where partial information is modeled by spaces with weak sep- aration properties, as comprehensively treated by Gierz et al. [13]. • In functional analysis, where weak topologies often fail to be Hausdorff, following the framework established by Schaefer and Wolff [14]. • In algebraic geometry, where the Zariski topology provides important examples of non-Hausdorff spaces, as discussed in Hartshorne [15]. • In theoretical computer science, where convergence without uniqueness models non-deterministic computation Connections with Related Work: Our results connect with and extend sev- eral recent investigations in general topology. The characterization of T1 spaces with sigma-intersection closed sets relates to work on pairwise expandable spaces [10] and h- convexity in metric linear spaces [26]. The study of compactness conditions in Urysohn spaces builds on recent results on r-compactness [11] and D-metacompactness [12] in topological spaces. Furthermore, our analysis of covering properties in intermediate separation axioms extends recent work on Lindelöf properties [31] to the non-regular setting. Future Research Directions: This work opens several avenues for future investi- gation: J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6627 18 of 20 • Characterizing when intermediate separation axioms are preserved under other topological constructions (e.g., function spaces, hyperspaces), building on the foun- dational work of McCoy and Ntantu [21] and Arens and Dugundji [22]. • Investigating the role of cardinal invariants in determining when spaces with weak separation axioms can be embedded in spaces with stronger properties, following the set-theoretic methods of Kunen [28]. • Exploring connections between these separation axioms and convergence structures beyond sequential convergence, extending the classical treatment by Kelley [16]. • Developing a theory of continuous functions between spaces with different inter- mediate separation properties, inspired by the ring-theoretic approach of Gillman and Jerison [23]. • Extending the study to bitopological spaces, building on recent work on r-compactness in bitopological settings [11]. • Investigating applications in theoretical computer science, particularly in domain theory and denotational semantics. • Exploring the relationship between these separation axioms and various forms of compactness and metacompactness [12]. Methodological Contributions: Our approach demonstrates the value of combin- ing constructive methods with analytical techniques in topology. The counterexamples presented not only establish the sharpness of our results but also provide a toolkit for constructing spaces with prescribed separation properties. This methodology can be applied to other areas of topology where fine distinctions between properties are impor- tant. The counterexamples presented throughout this paper not only establish the sharp- ness of our results but also provide a toolkit for constructing spaces with prescribed separation properties. These constructions demonstrate that the landscape of separa- tion axioms is far richer than the classical Kolmogorov-Hausdorff hierarchy suggests, with numerous natural and mathematically significant spaces occupying the intermedi- ate levels. In conclusion, this comprehensive study enhances our understanding of the separa- tion axiom hierarchy and provides fundamental tools for investigating spaces that lie outside traditional classification schemes. The interplay between separation properties, topological operations, and functional characterizations revealed in this work contributes to the broader project of understanding the fine structure of topological spaces. Acknowledgements The authors thank the anonymous reviewers for their constructive comments that helped improve the presentation of this paper. Special thanks to the editorial board of J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6627 19 of 20 the European Journal of Pure and Applied Mathematics for their support throughout the review process. References [1] Maysoon Qousini, H. Hdeib, and Eman Almuhr. Applications of locally compact spaces in polyhedra: Dimension and limits. WSEAS Transactions on Mathematics, 23:118–124, 2024. [2] Ala Amourah, Jamal Oudetallah, Iqbal Batiha, Jamal Salah, and Mutaz Shatnawi. σ-compact spaces in Nth-topological space. European Journal of Pure and Applied Mathematics, 18(2):5802–5802, 2025. [3] Jamal Oudetallah, Rehab Alharbi, Salsabiela Rawashdeh, Iqbal M. Batiha, Ala Amourah, and Tala Sasa. Nearly lindelöfness in Nth-topological spaces. Interna- tional Journal of Analysis and Applications, 23:140–140, 2025. [4] Ala Amourah, Jamal Oudetallah, Iqbal M. Batiha, Jamal Salah, Sultan Alsaadi, and Tala Sasa. Some types of tri-locally compactness spaces. European Journal of Pure and Applied Mathematics, 18(2):5764–5764, 2025. [5] Jamal Oudetallah, Iqbal Batiha, and Ansam A. Al-Smadi. Nearly lindelöfness in bitopological spaces. South East Asian Journal of Mathematics and Mathematical Sciences, 20(3):341–358, 2025. [6] Rehab Alharbi, Jamal Oudetallah, Mutaz Shatnawi, and Iqbal M. Batiha. On c-compactness in topological and bitopological spaces. Mathematics, 11(20):4251, 2023. [7] Jamal Oudetallah, Nabeela Abu-Alkishik, and Iqbal M. Batiha. Nigh-open sets in topological space. International Journal of Analysis and Applications, 21(1):83, 2023. [8] R. Engelking. General Topology. Heldermann Verlag, Berlin, 1989. [9] S. Willard. General Topology. Addison–Wesley, Reading, MA, 1970. [10] J. Oudetallah and M. AL-Hawari. Other generalization of pairwise expandable spaces. International Mathematical Forum, 16(1):1–9, 2021. [11] J. Oudetallah, R. Alharbi, and I. M. Batiha. On r-compactness in topological and bitopological spaces. Axioms, 12(2):210, 2023. [12] J. Oudetallah, M. M. Rousan, and I. M. Batiha. On d-metacompactness in topo- logical spaces. Topology Appl., 298:107–125, 2021. [13] G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, and D. S. Scott. Continuous Lattices and Domains. Cambridge University Press, Cambridge, 2003. [14] H. H. Schaefer and M. P. Wolff. Topological Vector Spaces. Springer-Verlag, New York, 2nd edition, 1999. [15] R. Hartshorne. Algebraic Geometry. Springer-Verlag, New York, 1977. [16] J. L. Kelley. General Topology. Van Nostrand, Princeton, 1955. [17] J. R. Munkres. Topology. Prentice Hall, Upper Saddle River, NJ, 2nd edition, 2000. [18] J. Dugundji. Topology. Allyn and Bacon, Boston, 1966. [19] N. Bourbaki. General Topology, Parts 1 and 2. Springer-Verlag, Berlin, 1989. J. Oudetallah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6627 20 of 20 [20] J. C. Oxtoby. Measure and Category. Springer-Verlag, New York, 2nd edition, 1980. [21] R. A. McCoy and I. Ntantu. Topological Properties of Spaces of Continuous Func- tions., volume 1315 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1988. [22] R. Arens and J. Dugundji. Topologies for function spaces. Pacific J. Math., 1:5–31, 1951. [23] L. Gillman and M. Jerison. Rings of Continuous Functions. Springer-Verlag, New York, 1976. [24] L. A. Steen and Jr. J. A. Seebach. Counterexamples in Topology. Dover Publications, New York, 2nd edition, 1995. [25] P. Urysohn. Über die mächtigkeit der zusammenhängenden mengen. Math. Ann., 94:262–295, 1925. [26] J. Oudetallah and L. Abualigah. H-convexity in metric linear spaces. International Journal of Science and Advanced Information Technology, 8(6):54–58, 2019. [27] H. Tietze. Über funktionen, die auf einer abgeschlossenen menge stetig sind. J. Reine Angew. Math., 145:9–14, 1915. [28] K. Kunen. Set Theory: An Introduction to Independence Proofs. North-Holland, Amsterdam, 1980. [29] A. V. Arhangel’skii. Quotient spaces and multiplicity of a base. Soviet Math. Dokl., 7:244–247, 1966. [30] A. Tychonoff. Über die topologische erweiterung von räumen. Math. Ann., 102:544– 561, 1930. [31] J. Oudetallah. Novel results on nigh lindelöfness in topological spaces. European J. Pure Appl. Math., 17:234–251, 2024. [32] W. W. Comfort and S. Negrepontis. The Theory of Ultrafilters. Springer-Verlag, Berlin, 1974. Introduction Motivation and Applications Preliminaries Fréchet Spaces with Closed Sets as Sigma-Intersections Urysohn Spaces Without the Hausdorff Property Intermediate Separation Criteria Behavior Under Topological Operations Critical Counterexamples Conclusions