EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6375 ISSN 1307-5543 – ejpam.com Published by New York Business Global An Exploration of Compactness and Separation Axioms in Generalized Primal Topological Spaces Muhammad Shahbaz1, Tayyab Kamran1, Mariam Imtiaz2, Umar Ishtiaq3, Mohammad Akram4,∗, Ioan-Lucian Popa5,6 1 Department of Mathematics, Quaid-I-Azam University Islamabad, Pakistan 2 Department of Mathematics, The Islamia University of Bahawalpur, Bahawalnagar Campus, Pakistan 3 Office of Research, Innovation and Commercialization, University of Management and Technology, Lahore 54770, Pakistan 4 Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah 42351, Saudi Arabia 5 Department of Computing, Mathematics and Electronics, “1 Decembrie 1918” University of Alba Iulia, 510009 Alba Iulia, Romania 6 Faculty of Mathematics and Computer Science, Transilvania University of Brasov, Iuliu Maniu Street 50, 500091 Brasov, Romania Abstract. The research explores S∗ g -compactness together with S∗ g -connectedness in generalized primal topological spaces to enhance theoretical knowledge of non-classical topological systems. This paper provides an extensive analysis of these two concepts to show their characteristics and potential applications. The research examines the relationship dynamics between T0, T1, and T2 separation axioms and these concepts throughout their expanded theoretical framework. Examining S∗ g -compactness and S∗ g -connectedness independently provides an advanced under- standing of generalized primal spaces, although they diverge from standard separation properties. This study simultaneously supports theoretical research of these domains while building essential foundations for math investigations in this field. 2020 Mathematics Subject Classifications: 54A05, 54A10 Key Words and Phrases: Generalized topological space, Hausdorff space, compactness, primal topological space 1. Introduction Topology, as a core branch of modern mathematics, offers powerful tools for understand- ing ideas such as convergence, continuity, and separation axioms in different types of ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6375 Email addresses: mshahbaz@math.qau.edu.pk (M. Shahbaz), tkamran@qau.edu.pk (T. Kamran), mariamimtiaz122@gmail.com (M. Imtiaz), umarishtiaq000@gmail.com (U. Ishtiaq), akramkhan_20@rediffmail.com (M. Akram), lucian.popa@uab.ro (I. L. Popa) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. Shahbaz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6375 2 of 22 spaces. The theoretical and practical world bases its work on compactness and connect- edness—influential concepts which lead topology to its most important advancements. The recent study conducted by Shahbaz et al. [1, 2] investigates various continuity, compactness, and connectedness frameworks that generate novel insights which define modern topological research. Their research on generalized topological spaces leads the way toward investigating particular cases of spaces especially within generalized primal topological spaces. Topological ideas face increasing interest from scholars to rethink traditional interpreta- tions through abstract flexible frameworks within the last few years. Generalized primal topological spaces emerged because of heightened interest and created new ways to un- derstand familiar concepts. Historical fundamental principles about open and closed sets form the basis of developing these spaces. The generalized closed sets theory of Levine from 1970 provided innova- tive methods for examining covering properties and separation even though his original research investigated generalized topology but these principles align with contemporary primal environment research. Császár developed a set of theoretical generalized open sets [3] that built the essential framework for open set analysis for non-traditional math- ematical spaces. Maki, Balachandran and Devi [4], Maki, Rao, and Gani [5], and Navalagi along with Page [6], have made essential scholarly contributions to the field through their semi- generalized closed sets and their connected structures of semi-open and pre-open sets. These fundamental theoretical foundations have set the base for analyzing analogous features within primal spaces with generalization. The field of generalized primal topology continues to evolve through Choquet’s initial research on grills and filters from 1961 which remains influential in current investiga- tions. Structural analysis of primal spaces received clearer explanations through research conducted by Acharjee, Özkoç, and Issaka [7] and Özkoç and Köstel [8]. Saadi and Malki [9, 10] developed our knowledge of these spaces through their examinations of different open set categories. A combination of above-mentioned aspects results in the adaptabil- ity of primal spaces being demonstrated also through extension to the soft structures [11]. Besides these, new operator-based solutions have been suggested to reinforce the primal topology framework [12]. The most significant advancement emerged through Missier and Jesti’s research which proposed S∗ g -open sets [13]. The research initiated by Missier and Jesti [13] created a whole family of connected functions for improved set operation analysis in generalized primal spaces. The initial concept of µ-compactness created by Thomas and John [14] started in general settings before it helped researchers understand compactness definitions in specific spaces. The investigation examines basic space differentiation criteria known as T0, T1 and T2 separation axioms because they serve as fundamental tools to distinguish space types. M. Shahbaz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6375 3 of 22 These axioms introduced by Urysohn [15] then improved by Freudenthal and Est [16] continue to play an important role in contemporary topological classifying systems. Stone [17] proved that any topological space can become a T0 space through the process of point merging to merge indistinguishable points. Meanwhile Youngs [18] examined separation conditions between T0 and T1. The current research established S∗ g -compact and S∗ g -connected spaces as its main focus inside generalized primal topological worlds. The paper studies how spaces change un- der the influence of T0, T1 and T2 separation axioms in their internal structure. Thus this piece develops the concepts of S∗ g -T0, S∗ g -T1 and S∗ g -T2 spaces to demonstrate their alignment with S∗ g -set characteristics. The study explores theoretical growth through its results which establish foundations for future research in generalized primal topology. 2. Preliminaries This section provides some findings and definitions from the literature in order to clarify the main part. Definition 2.1. [19] An empty set does not exist yet the set V ̸= ∅. A collection Gτ ⊆ 2V satisfies the criteria to be considered a generalized topology (GT) on V if it contains the empty set and all possible unions of non-empty subclasses within Gτ are also contained in Gτ . The pair made up of (V, Gτ ) represents a GTS (Generalized Topological Space). Remark 2.1. [3] Each member of the set Gτ is identified as open in the space. Any set E present in the context of (V, Gτ ) forms an essential part of our consideration. It becomes a closed set whenever the complement of E relative to the set V is open. The closure of a set E receives the notation Clg(E) through intersection of every closed set which contains E. Interior of a set known as Intg(E) consist of every open subset found within E. Definition 2.2. [3] According to GTS a ψ operator maps elements x from V to sets in 22 V and it fulfills the condition where x ∈ F whenever F belongs to the image of x. According to Definition a generalized neighbourhood of point x in set V refers to the element F ∈ ψ(x). The collection of every generalized neighbourhood that exists within V gets symbolized by Ψ(V). Definition 2.3. [4–6, 20] (i) Consider a GTS (V, Gτ ). If a set E in a GTS has an open container set F that satisfies the condition F ⊆ E ⊆ Cl(F) or if E ⊆ Cl(Int(E)) then it is known as generalized Gτ -semi-open. M. Shahbaz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6375 4 of 22 (ii) The complement of a generalized Gτ -semi-open set results in a generalized Gτ -semi- closed set. All generalized Gτ -semi-open sets within (V, Gτ ) form the set known as SO(V,Gτ ). (iii) In a generalized Gτ -semi-open setting the overall collection of such sets contained within E forms the generalized semi-interior which is denoted as sgInt(E). (iv) A generalized semi-closure consists of the intersection between all generalized semi- closed sets of V that contain E. This set bears the notation sgCl(E). Definition 2.4. [14] If (V, Gτ ) has a finite subcover for every open cover, then a gener- alized topological space (V, Gτ ) is called Gτ -compact. Definition 2.5. [1] Assume (V Gτ 1) and (Z, Gτ 2) as GTS. A mapping j : (V,Gτ 1) → (Z,Gτ 2) is classified as Gτ -S∗ g -irresolute when the preimage of every Gτ -S∗ g -open set in (Z,Gτ 2) is a Gτ -S∗ g -open set in (V, Gτ 1). Main Results 3. Gpt-S∗ g -Compact Space in GPTS This particular section evolves to generalized primal topological spaces. The extension of the previous ideas is emphasized in this part, which also uses the Kuratowski closure operator to examine closure features in primal topological spaces. Definition 3.1. [21] Assume that V ̸= ∅. A grill on V is a family G ⊆ 2V if the following criteria are met: (i) ∅ is not a member of G. (ii) For D, E ⊆ V having D ⊆ E implies E ∈ G if D ∈ G. (iii) For D, E ⊆ V, then D ∪ E ∈ G, whenever D ∈ G or E ∈ G. Definition 3.2. [8] Assume that V ̸= ∅. A primal on V is a collection P of 2V if the following criteria are met: (i) V ̸∈ P. (ii) If D ∈ P and E ⊆ D then E ∈ P. (iii) If D ∩ E ∈ P, then D ∈ P or E ∈ P. A pair (V, Gτ ) with a primal P on V is termed generalized primal topological space (GPTS) symbolized as (V, Gτ , P). The members of (V, Gτ , P) are known as Gpt-open sets, and their complements are considered Gpt-closed sets. Definition 3.3. [9] Assume A ⊆ V. Let an operator (.)◦ : 2V → 2V in GPTS is defined as A ◦ (V, Gτ , P) = {x ∈ V : A c ∪ Oc ∈ P, ∀ O ∈ ψ(x)} where O is generalized primal neighbourhood of x in V and the collection of all generalized neighbourhood of V is termed as Ψ(V). M. Shahbaz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6375 5 of 22 Definition 3.4. Let A ⊆ V in GPTS. The generalized Kuratowski closure operator Cl◦ is defined as Cl◦(A) = A∪ A◦, with the condition: Cl◦(A ∪B) ⊇ Cl◦(A) ∪ Cl◦(B). Remark 3.1. The operator Cl◦ satisfies the following properties: (i) Extensivity: A ⊆ Cl◦(A). (ii) Monotonicity: If A ⊆ B, then Cl◦(A) ⊆ Cl◦(B). (iii) Idempotency: Cl◦(Cl◦(A)) = Cl◦(A). (iv) Generalized Subset Union Property: Cl◦(A ∪B) ⊇ Cl◦(A) ∪ Cl◦(B). In standard topological spaces, Cl◦ reduces to the classical Kuratowski closure operator. Proof. 1. Extensivity By definition, the generalized Kuratowski closure operator Cl◦(A) consists of all points in A. All generalized limit points of A, i.e., points where every generalized primal neigh- bourhood intersects A. Since every point in A is trivially in its closure, thus A ⊆ Cl◦(A). 2. Monotonicity Suppose A ⊆ B. Any generalized primal neighbourhood of a point x that intersects A also intersects B. Hence, any generalized limit point of A must also be a generalized limit point of B, implying Cl◦(A) ⊆ Cl◦(B). 3. Idempotency Expanding Cl◦(Cl◦(A)). Applying the closure operator twice, Cl◦(Cl◦(A)) = Cl◦(A∪A◦). Using the definition of closure again. By the definition of the closure operator Cl◦(A∪A◦) = (A∪A◦)∪(A∪A◦)◦. Since closure always includes the interior, (A ∪ A◦)◦ = A◦. By substituting this, we get Cl◦(Cl◦(A)) = (A ∪ A◦) ∪ A◦. Since A◦ is already included in Cl◦(A), implies Cl◦(Cl◦(A)) = Cl◦(A). 4. Generalized Finite Union Property Consider x ∈ Cl◦(A)∪Cl◦(B). This means x is either in Cl◦(A) or Cl◦(B), so every gen- eralized primal neighbourhood of x intersects either A or B. Therefore, every generalized primal neighbourhood of x intersects A∪B, implying Cl◦(A∪B) ⊇ Cl◦(A)∪Cl◦(B). Definition 3.5. (i) Assume (V, Gτ , P) as a GPTS. If there exists an open set F in V such that F ⊆ E ⊆ Cl◦ (F) or equivalently if E ⊆ Cl◦(Int(E)), then the subset E of (V, Gτ , P) is known as generalized primal semi-open [10]. (ii) The generalized primal semi-closed set exists as the complement of generalized pri- mal semi-open sets. All generalized primal semi-open sets make up the collection known as Gpt-SO in (V, Gτ , P) [10]. M. Shahbaz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6375 6 of 22 (iii) The union of all Gpt-SO sets of V contained in E is called Gpt-semi−interior of E (briefly Gpt-sInt(E). (iv) The intersection of all Gpt-SC sets of V containing E is Gpt-semi-closure of E (briefly Gpt-sCl(E)). Definition 3.6. [1] (i) Consider (V, Gτ , P) be a GPTS. If Gpt-sCl(E) ⊆ F whenever E ⊆ F and F is Gpt- semi−open in V, then the subset E of V is known as generalized primal semi- generalized closed (briefly Gpt- Sg-closed). (ii) If Cl(E) ⊆ F whenever E ⊆ F and F is Gpt- semi−open in V, then E as a subset of a space V is generalized primal semi−star generalized closed (briefly Gpt-S∗-closed). (iii) The complement of Gpt- Sg-closed set (Gpt-S∗g-closed set) is generalized primal semi- generalized open (generalized primal semi-star generalized open). It is represented by Gpt-Sg-open (Gpt-S∗g-open) appropriately. (iv) The generalized primal semi-generalized interior (briefly Gpt-sInt∗(E)) of E is in- dicated as the union of all Gpt-Sg-open sets of V contained in E. (v) The intersection of all Gpt- Sg−closed sets in V containing E when E is a subset of V is called generalized primal semi−generalized closure (briefly Gpt-sCl∗(E)) of E. Definition 3.7. Assume a GPTS (V, Gτ , P) and a subset E of (V, Gτ , P), a collection { Eαi : i ∈ ⋏ } of Gpt-S∗ g -open set in Gpt is referred to as Gpt-S∗ g -open cover of E if E ⊂ ∪ i ∈ ⋏ Eαi . Definition 3.8. If every Gpt-S∗ g -open cover of (V, Gτ , P) has a finite subcover, then the GPTS is called Gpt-S∗ g -compact. Definition 3.9. The subset E of GPTS is named as Gpt-S∗ g -compact relative of V if there exists ⋏o of ⋏ as a finite subset which satisfies E ⊂ ∪ { Zi : i ∈ ⋏o } for each { Zi : i ∈ ⋏ } consisting of Gpt-S∗ g -open subset of V such that E ⊂ ∪ { Zi : i ∈ ⋏ }. Definition 3.10. Assume E as a subset of GPTS. A subset E of V is named Gpt-S∗ g - compact when it maintains this property as a subspace of V. Theorem 3.1. (i) Every Gpt-S∗ g -compact space is Gpt-compact. (ii) The property of being Gpt-semi-compact implies Gpt-S∗ g -compactness. Proof. (i) Let U = {ui : i ∈ Λ, ui ∈ Gpt} be an Gpt-open cover of X, so X = ⋃ i∈Λ ui. Because every Gpt-open set is Gpt-S∗ g , U is also a Gpt-S∗ g -open cover of X. If X is Gpt- S∗ g -compact, then by definition every Gpt-S∗ g -open cover of X has a finite subcover, hence there exist a finite subset Λ0 = {i1, . . . , in} ⊆ Λ such that X = ⋃n k=1 uik . Therefore X admits a finite subcover of the Gpt-open cover. M. Shahbaz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6375 7 of 22 (ii) Since every Gpt-S∗ g -open set is a Gpt-semi-open set, the result follows similarly to part (i). Example 3.1. Let V be the set of all bounded spherical regions in 3-dimensional Eu- clidean space: V = {S ⊆ R3 | S is a bounded spherical region}. where S is defined as: S = {(l,m, n) ∈ R3 | √ (l − a)2 + (m− b)2 + (n− c)2 ≤ r}, with r > 0 as the radius of the spherical region and a, b, c ∈ R. A bounded spherical region is a subset of R3 consisting of points within a sphere of finite radius. Gτ = {E ⊆ V \ Sr1 | Sr1 = {S ∈ V | radius(S) = r1}}, where r1 > 0 is a fixed radius, and Sr1 is the set of all spherical regions in V with radius exactly r1, forms a generalized topology on V. We now prove that Gτ satisfies the axioms of a generalized topology. A collection Gτ forms a generalized topology if it satisfies the following conditions: Condition 1: ∅ ∈ Gτ The empty set trivially belongs to Gτ since there is no restriction preventing ∅ from being included. Condition 2: Arbitrary Unions of Elements of Gτ Remain in Gτ Let {Eαi}i∈I be a family of sets in Gτ , meaning each Eαi ⊆ V \ Sr. Consider their union: E = ⋃ i∈I Vi. Since each Eαi excludes all spherical regions of radius exactly r, their union E must also exclude all such regions. That is, E ⊆ V \ Sr. Thus, Sα satisfies the definition of Gτ , ensuring that arbitrary unions remain in Gτ . Since Gτ satisfies both required conditions, so it forms a generalized topology on V and define a collection Gpt ⊆ 2V as: Gpt = {E ⊆ V | all spherical region in E have radius strictly less than r, where r > 0}. To determine whether E is Gpt-semi-compact, consider a family {Zi : i ∈ ⋏} of Gpt-S∗ g - open sets satisfying E ⊆ ⋃ {Zi : i ∈ ⋏}. Within this covering, a finite subcover can always be extracted. Hence, E is Gpt-S∗ g -semi-compact. Next, let {Zi : i ∈ ⋏} be a collection of Gpt-S∗ g -open sets such that E ⊆ ⋃ {Zi : i ∈ ⋏}. Each point in E must be contained within at least one Zi, and thus a finite subset ⋏o ⊆ ⋏ is sufficient to cover E. Consequently, E is Gpt-S∗ g -compact. M. Shahbaz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6375 8 of 22 All open covers from Gpt naturally have finite subcovers in the context of E. Thus, E is Gpt-compact. Theorem 3.2. A Gpt-S∗ g -compact space makes every of its Gpt-S∗ g -closed subsets compact relative to (V, Gτ , P). Proof. Consider E as a Gpt-S∗ g -closed subset of Gpt-S∗ g -compact space (V, Gτ , P) implies Ec is Gpt-S∗ g -open in (V, Gτ , P). The collection { Zi : i ∈ ⋏ } serves as a cover of the set E since each member of the collection is an element from the family of Gpt-S∗ g -open subsets of V so that E ⊂ ∪ { Zi : i ∈ ⋏ } implies Ec ⊂ ∪ { Zi : i ∈ ⋏ } = V. Therefore, (V, Gτ , P) is Gpt-S∗ g -compact then there exists a finite subset Eo of E so that E ⊂ Ec ∪ { Zi : i ∈ ⋏ } = V. Then, E ⊂ ∪ { Zi : i ∈ ⋏ } and therefore E is Gpt-S∗ g -compact relative to V. Example 3.2. Assume V = R and (V, Gτ , P) be defined as follows: U ∈ Gτ if and only if either U = ∅ or 1 ∈ U , see Example 10 in [22]. Let Gpt be defined on R as follows: U ∈ Gpt if and only if 1 /∈ U . Then, (V, Gτ , P) is a generalized primal topology. Now, consider the subset N ⊂ R. Let S is index set and {Vη}η∈S be a Gpt-S∗ g -open cover of N such that Vη ̸= ∅ for every η ∈ S . This implies that: N ⊆ ⋃ η∈S Vη. Let S0 = {Vi}ni=1 ⊆ {Vη}η∈S . Then, for any x ∈ N \ ⋃n i=1 Vi, it must follow that R \ [ N \ ⋃n i=1 Vi ] /∈ Gpt. Thus, there exists a finite subcover S0 that covers N, proving that N is Gpt-S∗ g -compact relative to (V, Gτ , P). Theorem 3.3. Consider a Gpt-S∗ g -continuous surjective map f : (V, Gτ 1, Pα) → (Z, Gτ 2, Pβ) from V to Z. If (V, Gτ 1, Pα) is Gpt-S∗ g -compact, then (Z, Gτ 2, Pβ) is Gpt-compact. Proof. Consider { Eαi : i ∈ ⋏ }, an open cover of Z. As f is Gpt-S∗ g -continuous implies { f−1 (Eαi) : i ∈ ⋏ } is a Gpt-S∗ g -open cover of V. Furthermore, there exists a finite Gpt-subcover { f−1(Eα1), f−1(Eα2), f−1(Eα3),..., f−1(Eαn)} as V is Gpt-S∗ g -compact. The surjectiveness of V implies { Eα1 , Eα2 , Eα3 ,..., Eαn} is a finite Gpt-subcover of Z implies Z is Gpt-compact. Theorem 3.4. Consider a Gpt-S∗ g -irresolute surjective map f : (V, Gτ 1 , Pα) → (Z, Gτ 2, Pβ) from GPTS V into a GPTS Z. If (V, Gτ 1, Pα) is Gpt-S∗ g -compact, then (Z, Gτ 2, Pβ) is Gpt-S∗ g -compact. Proof. Let { Eαi : i ∈ ⋏ }, a Gpt-S∗ g -open cover of Z. As f is Gpt-S∗ g -irresolute implies { f−1 (Eαi) : i ∈ ⋏ } is a Gpt-S∗ g -open cover of V. Furthermore, there exists a finite Gpt-subcover { f−1(Eα1), f−1(Eα2), f−1(Eα3),..., f−1(Eαn)} as V is Gpt-S∗ g -compact. Now, f is onto implies { Eα1 , Eα2 , Eα3 ,..., Eαn} is a finite Gpt-subcover of Z implies Z is Gpt- S∗ g -compact. M. Shahbaz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6375 9 of 22 Theorem 3.5. Let f: (V, Gτ 1, Pα) → (Z, Gτ 2, Pβ) a Gpt-S∗ g -irresolute map and a subset D of (V, Gτ 1, Pα) is Gpt-S∗ g -compact relative to V, then the image f(D) is Gpt-S∗ g -compact relative to Z. Proof. Any collection of Gpt-S∗ g -open subsets {Eαi : i ∈ ⋏} of Z containing the image of D under the mapping f becomes part of the union function. Then, D ⊂ ∪ { f−1(Eαi) : i ∈ ⋏ } holds. D is Gpt-S∗ g -compact relative to V by hypothesis, then there exists ⋏o a finite subset of ⋏ such that D ⊂ ∪ { f−1(Eαi) : i ∈ ⋏o } implies f(D) ⊂ ∪ { Eαi : i ∈ ⋏o }. Thus, f(D) is Gpt-S∗ g -compact relative to Z. Theorem 3.6. If a surjective map f: (V, Gτ 1, Pα) → (Z, Gτ 2 , Pβ) is strongly Gpt-S∗ g - continuous and (V, Gτ 1, Pα) is Gpt-compact, then (Z, Gτ 2, Pβ) is Gpt-S∗ g -compact. Proof. Let a Gpt-S∗ g -open cover { Eαi : i ∈ ⋏ } of Z. As f exhibits strong Gpt-S∗ g - continuity it implies that { f−1 (Eαi) : i ∈ ⋏ } creates an open cover of V. Since V has the Gpt-compactness property it contains the finite subcover { f−1(Eα1), f−1(Eα2), f−1(Eα3),..., f−1(Eαn)}. The surjectiveness of map f leads to the finite subcover { Eα1 , Eα2 , Eα3 ,..., Eαn} of collection Z which makes Z Gpt-S∗ g -compact. Example 3.3. Let V be the set of all bounded spherical regions in R3, where a spherical region S is defined as: S = {(l,m, n) ∈ R3 | √ (l − a)2 + (m− b)2 + (n− c)2 ≤ r}, with r > 0 as the radius of the spherical region and a, b, c ∈ R, τg1 = {E ⊆ V | E ⊆ V \ {S | radius(S) = r} for some fixed r > 0} (see example 3.1) and define a collection Pα ⊆ 2V as: Pα = {E ⊆ V | all spherical region in E have radius strictly less than r, where r > 0}. Let Z be the set of all bounded circular regions in R2, where a circular region C is defined as: C = {(l,m) ∈ R2 | √ (l − a)2 + (m− b)2 ≤ r1}, with (a, b) is the center of the circular region r1 > 0 as the radius of the circular region and a, b ∈ R, τg2 = {E ⊆ V | E ⊆ V \ {C | radius(C) = r1} for some fixed r1 > 0} is generalized topology, similarly step (see example 3.1) and Pβ be a primal on Z such that Pβ = {E ⊆ V | E does not contain any point on the boundary }. A surjective map f : V → Z from a spherical region to a circular region is defined as: h(l,m, n) = ( r r1 l, r r1 m ) . M. Shahbaz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6375 10 of 22 This map is surjective, as every element of Z has a pre-image in V. Assume that f is strongly Gpt-S∗ g -continuous, meaning it preserves the primal structure and satisfies the required continuity. Since V is Gpt-compact, and f is a surjective map that is strongly Gpt-S∗ g -continuous. Thus, Z inherits the Gpt-S∗ g -compactness. Theorem 3.7. If a surjective map f: (V, Gτ 1 , Pα) → (Z, Gτ 2, Pβ) is perfectly Gpt-S∗ g - continuous and (V, Gτ 1, Pα) is Gpt-compact, then (Z, Gτ 2, Pβ) is Gpt-S∗ g -compact. Proof. As every perfectly Gpt-S∗ g -continuous is strongly Gpt-S∗ g -continuous. Theorem 3.6 leads to the obtained result. 3.1. Gpt-S∗ g -Connected Space in GPTS Definition 3.11. Assume a GPTS and two disjoint non-empty open sets D and E in V, then (V, Gτ , P) is known as Gpt-disconnection if D ∪ E = V. Definition 3.12. A GPTS, (V, Gτ , P) is known as Gpt-connected space if (V, Gτ , P) has no Gpt-disconnection. Definition 3.13. The condition for a space to become Gpt-S∗ g -connected exists when two non-empty Gpt-S∗ g -open sets D and E in (V, Gτ , P) cannot be disjoint as D ∪ E = V. Otherwise called Gpt-S∗ g -disconnected space if D ∪ E = V. Example 3.4. Assuming V = { j1, k1, l1 }, Gτ = { ∅, {j1, k1} } and P = { ∅, {j1}, {l1}, {j1, l1} }. In this GPTS (V, Gτ , P)O = { ∅, {j1, k1} } and (Gτ , P)S∗ gO = { ∅, {j1, k1}, V }. There does not exists two disjoint non-empty Gpt-S∗ g -open sets D and E in (V, Gτ , P) such that D ∪ E = V implies (V, Gτ , P) is Gpt-S∗ g -connected space. Theorem 3.8. A space which satisfies Gpt-S∗ g -connectedness contains the condition of Gpt-connectedness. Proof. Assume a Gpt-S∗ g -connected space (V, Gτ , P). Consider (V, Gτ , P) is not a Gpt- connected space, then there exist nonempty open subsets D and E in V such that their union equals V itself. Every open set holds the status of being both Gpt-S∗ g -open and Gpt-open in the defined topology. The pair of open sets D and E belongs to the class of Gpt-S∗ g -open sets since V = D ∪ E. The finding of two nonempty open sets in V leads to a contradiction when applying the definition of Gpt-S∗ g -connected space. A space represented by (V, Gτ , P) serves as a connected space under the Gpt structure. Remark 3.2. An illustration disproves the invalidity of the converse statement derived from above. Example 3.5. Assuming V = {j1, k1, l1}, Gτ = { ∅, {j1}, {k1}, {j1, k1} } and P = { ∅, {j1}, {l1}, {j1, l1} }. In this GPTS, (V, Gτ , P)O = { ∅, {j1}, {k1}, {j1, k1}} and (Gτ , P)S∗ gO = { ∅, {j1}, {k1}, {j1, k1}, {j1, l1}, {k1, l1}}. There does not exists two disjoint non-empty open sets D and E in (V, Gτ , P) such that D ∪ E = V implies (V, Gτ , P) M. Shahbaz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6375 11 of 22 is Gpt-S∗ g -connected space. But two disjoint non-empty Gτ -S∗ g -open sets {j1} and {k1, l1} exists such that {j1} ∪ {k1, l1} = V implies (V, Gτ , P) is not Gpt-S∗ g -connected space. Theorem 3.9. Assume a GPTS, then the subsequent are equivalent. i. V is Gpt-S∗ g -connected. ii. ∅ and V are only Gpt-S∗ g -open set and Gpt-S∗ g -closed set in V. iii. Any Gpt-S∗ g -continuous mapping f: (V, Gτ 1 , Pα) → (Z, Gτ 2 , Pβ) is a constant map where (Z, Gτ 2 , Pβ) at least two-point discrete space. Proof. (i) then (ii): Let a Gpt-S∗ g -open set and Gpt-S∗ g -closed set E in (V, Gτ , P) implies Ec is also Gpt-S∗ g -open set and Gpt-S∗ g -closed set in (V, Gτ , P). Then, E ∪ Ec = V as E and Ec are both disjoint Gpt-S∗ g -open sets which implies contradiction. As (V, Gτ , P) is Gpt-S∗ g -connected space. Hence, V is ∅ or V. (ii) then (i): Assume E and D as disjoint Gpt-S∗ g -open sets in (V, Gτ , P) and E ∪ D = V. Since Ec = D, then E is Gpt-S∗ g -open set (Gpt-S∗ g -closed set). Then, by hypothesis, E is ∅ or V, this contradicts. Hence, (V, Gτ , P) is Gpt-S∗ g -connected space. (ii) then (iii): A mapping f: (V, Gτ 1 , Pα) → (Z, Gτ 2 , Pβ) that is both Gpt-S∗ g -continuous and constant functions to at least two-point discrete space works as an assumption. The pre-image of any set point under f satisfies both Gpt-S∗ g -open and Gpt-S∗ g -closed properties throughout the elements of Z. The domain set V equals the union of these pre-image sets f−1 ({x}), while each set point x belongs to Z making the pre-images both Gpt-S∗ g -open and Gpt-S∗ g -closed in V. The failure for f to be a proper mapping occurs when each inverse image set f−1 ({x}) equals either empty set or the entire space V for all elements x in Z. Among all points x in Z there exists at least one where the pre-image f−1 ({x}) consists of V so f functions as a constant. (iii) then (ii): Asume E is Gpt-S∗ g -open and Gpt-S∗ g -closed in GPTS where E is non-empty set. Consider Gpt-S∗ g -continuous mapping f: (V, Gτ 1 , Pα) → (Z, Gτ 2 , Pβ) defined as f (E) = {x} and f (Ec) = {y}, where x, y ∈ Z and x ̸= y. By hypothesis, f is constant map and E = V. Theorem 3.10. Let f: (V, Gτ 1, Pα) → (Z, Gτ 2, Pβ) is a surjective, Gpt-S∗ g -continuous mapping and (V, Gτ 1, Pα) is Gpt-S∗ g -connected space then (Z, Gτ 2, Pβ) is Gpt-connected space. Proof. The space (Z, Gτ 2 , Pβ) fails to be a Gpt-connected space. The space contains two non-empty disjoint open sets E and D which exist within Z. Such as E ∪ D = Z. The fact that f demonstrates Gpt-S∗ g -continuity creates a relationship between f−1(E) ∪ f−1(D) = V and f−1(E), f−1(D) which implies a contradiction. Thus, V is Gpt-S∗ g -connected. Both conditions demonstrate that the space Z is Gpt-connected. M. Shahbaz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6375 12 of 22 Theorem 3.11. Let f: (V, Gτ 1, Pα) → (Z, Gτ 2, Pβ) is a surjective, Gpt-S∗ g -irresolute function and (V, Gτ 1, Pα) is Gpt-S∗ g -connected space, then (Z, Gτ 2, Pβ) is Gpt-connected space. Proof. Assume (Z, Gτ 2 , Pβ) is Gpt-S∗ g -connected space. Let E and D be two non-empty Gpt-S∗ g -open sets in Z. Such as E ∪ D = Z. As f is surjective, Gpt-S∗ g -continuous so f−1(E) ∪ f−1(D) = V and f−1(E), f−1(D) is Gpt-S∗ g -open disjoint subsets in V. this implies a contradiction. So, V is Gpt-S∗ g -connected. Therefore, Z is also Gpt-S∗ g -connected space. Theorem 3.12. Suppose f: (V, Gτ 1, Pα) → (Z, Gτ 2 , Pβ) is a strongly Gpt-S∗ g -continuous function and (V, Gτ 1, Pα) is Gpt-connected space, then its image is Gpt-S∗ g -connected space. Proof. Consider f: (V, Gτ 1 , Pα) → (Z, Gτ 2 , Pβ) is a strongly Gpt-S∗ g -continuous map and (V, Gτ 1 , Pα) is Gpt-connected space. Let (Z, Gτ 2 , Pβ) is not Gpt-S∗ g -connected space for Gpt-S∗ g -open sets E and D in Z. Such as E ∪ D = Z. As f is strongly Gpt-S∗ g -continuous so f−1(E) ∪ f−1(D) = V and f−1(E), f−1(D) is open disjoint sets in V. This implies a contradiction. Thus, V is Gpt-connected. Hence, Z is Gpt-S∗ g -connected space. 3.2. Gpt-S∗ g -Separation Axioms Definition 3.14. A GPTS (V, Gτ , P) is called Gpt-S∗ gTc space if every Gpt-S∗ g -closed is closed. Definition 3.15. Let E be a subset of V which is Gpt-T0 space when every explicit point r, s of V satisfies conditions: either s /∈ Mα and r ∈ Mα or r /∈ Mα, s ∈ Mα, where Mα is an Gpt-open set of V. Definition 3.16. Let Mα ⊆ V. E is Gpt-T1 space if for every explicit point r and s of V, s /∈ Mα, r ∈ Mα and r /∈ Nα, s ∈ Nα, where Mα and Nα are Gpt-open sets of V. Definition 3.17. Let E ⊆ V. E is known as Gpt-T2 space if for every explicit point r and s of V, s /∈ Mα, r ∈ Mα and r /∈ Nα, s ∈ Nα, where Mα and Nα are disjoint Gpt-open sets of V. Definition 3.18. The function j : V → Z performs as an Gpt-S∗ g -continuous operator. A function is Gpt-S∗ g -continuous when its inverse images is Gpt-S∗ g -open in (V, Gτ 1, Pα) for every open set in (Z, Gτ 2 , Pβ). 3.2.1. Gpt-S∗ g -T0, Gpt-S∗ g -T1, Gpt-S∗ g -T2 Spaces Definition 3.19. A subset E of V is called Gpt-S∗ g -T0 space if for any two different points r and s satisfies either s /∈ Mα and r ∈ Mα or r /∈ Mα and s∈ Mα, where Mα is Gpt-S∗ g−open set. M. Shahbaz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6375 13 of 22 Definition 3.20. A subset E of V is called Gpt-S∗ g -T1 space if for any two distinct point r and s of V, s /∈ Mα, r ∈ Mα and r /∈ Nα, s ∈ Nα, where Mα and Nα are Gpt-S∗ g -open sets of V. Definition 3.21. Let E ⊆ V. E is called Gpt-S∗ g -T2 (Gpt-S∗ g -Housdorff) space if there exist two disjoint Gpt-S∗ g -open sets Mα and Nα for any two different points r and s of V, s /∈ Mα, r ∈ Mα and r /∈ Nα, s ∈ Nα. Theorem 3.13. i. If V is Gpt-T0, then V is Gpt-S∗ g -T0. ii. If V is Gpt-T1, then V is Gpt-S∗ g -T0 and Gpt-S∗ g -T1. iii. If V is Gpt-T2, then V is Gpt-S∗ g -T2. iv. If V is Gpt-S∗ g -T2, then V is Gpt-S∗ g -T0. v. If V is Gpt-S∗ g -T2, then V is Gpt-S∗ g -T1. Proof. i) Given, V is Gpt-T0 space. An open set F exists which covers every pair of points r, s belonging to V such that s /∈ F while r ∈ F or r /∈ F but s ∈ F. The family set F belongs to the collection Gpt-S∗ gO(V) while satisfying two conditions: s /∈ F and r ∈ F or r /∈ F and s ∈ F. Thus, V is Gpt-S∗ g -T0 space. The demonstration of all proposed points ii), iii), iv), and v) is also possible in the same way. Remark 3.3. An illustration disproves the invalidity of the converse statement derived from above. Example 3.6. Let V = {j1, k1, l1} and Gτ = { ∅, V, {l1}, {j1, k1} }, P = { ∅, {j1}, {l1}, {j1, l1} }. In (V, Gτ , P)O = { ∅, V, {l1}, {j1, k1} } and Gpt-S∗ gO = P(V). Hence, (V, Gτ , P) is ◦ Gpt-S∗ g -T0 but not Gpt-T0. No open set exists with s /∈ Mα, r ∈ Mα or r /∈ Mα, s ∈ Mα, where Mα is an open set of V for explicit points r and s of V. ◦ Gpt-S∗ g -T1 space but not Gpt-T1 space. No two open sets Mα and Nα exist with s /∈ Mα, r ∈ Mα and r /∈ Nα, s ∈ Nα for any explicit points s and r of V. ◦ Gpt-S∗ g -T2 space but not Gpt-T2 space. No two distinct open sets Mα and Nα exist with s /∈ Mα, r ∈ Mα and r /∈ Nα, s ∈ Nα for any explicit points s and r of V. Theorem 3.14. If f is bijective, strongly Gpt-S∗ g -open and V is Gpt-S∗ g -T0, then Z is Gpt-S∗ g -T0 space. Proof. Take r2 and s2 of Z with r2 ̸= s2. By hypothesis, r2 = f(rα) and s2 = f(s1) where rα and s1 are the explicit points of V. By hypothesis, Mα ∈ Gpt-S∗ gO(V) with rα ∈ Mα and s1 /∈ Mα. Therefore, f(rα) ∈ f(Mα) and f(s1) /∈ f(Mα). f(Mα) ∈ Gpt-S∗ gO(Z) as V is strongly Gpt-S∗ g -open. Thus, f(Mα) is Gpt-S∗ g -open set in Z with r2 ∈ f(Mα) and s2 /∈ f(Mα). So, Z is a Gpt-S∗ g -T0 space. M. Shahbaz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6375 14 of 22 Figure 1: Illustrates the above theorems and considerations. Theorem 3.15. If f is Gpt-S∗ g -irresolute, injective, and Z is Gpt-S∗ g -T0, then X is Gpt- S∗ g -T0 space. Proof. Let rα and s1 distinct points of V. Since f is injective implies f(s1) ̸= f(rα). As Z is Gpt-S∗ g -T0 there exists F ∈ Gpt-S∗ gO(Z) such that f(rα) ∈ F, f(s1) /∈ F or exists Fβ ∈ Gpt-S∗ gO(Z) such that f(s1) ∈ Fβ , f(rα) /∈ Fβ with f(s1) ̸= f(rα). As f is Gpt-S∗ g -irresolute then f−1( F) ∈ Gpt-S∗ gO(V), there exists f−1(rα) ∈ F, f−1(s1) /∈ F or f−1(Fβ) ∈ Gpt-S∗ gO(V) implies f−1(s1) ∈ Fβ , f−1(rα) /∈ Fβ . Hence, V is Gpt-S∗ g -T0 space. Theorem 3.16. Assume V is Gpt-S∗ g -T1 iff s1 ∈ V singleton {s1} ∈ Gpt-S∗ gC(V). Proof. Assume V is Gpt-S∗ g -T1, rα ∈ V. Then, s1 ∈ V - {rα} implies rα ̸= s1 ∈ V. But V is Gpt-S∗ g -T1 space implies there exists F, Fβ ∈ Gpt-S∗ gO(V) implies rα /∈ F, s1 ∈ Fβ ⊆ (V - {rα}). Also s1 ∈ Fβ ⊆ (V - {rα}) implies (V - {rα}) ∈ Gpt-S∗ gO(V). Thus, {rα} is Gpt-S∗ g -closed. Conversely, Consider the distinct elements rα ̸= s1 ∈ V where sets {rα} and {s1} form Gpt-S∗ g -closed sets and their complement {rα}c represents an Gpt-S∗ g -open subset. Certainly, {rα} /∈ {rα}c and {s1} ∈ {rα}c. Similarly {s1}c is Gpt-S∗ g -open, {s1} /∈ {s1}c and {rα} ∈ {s1}c. Thus, V is Gpt-S∗ g -T1 space. M. Shahbaz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6375 15 of 22 Theorem 3.17. Assume f : V → Z. Then the subsequent results hold: i) If f is injective, Gpt-S∗ g -continuous, Z is Gpt-T1, then V is Gpt-S∗ g -T1. ii) If f is injective, Gpt-S∗ g -continuous, Z is Gpt-T2, then V is Gpt-S∗ g -T2. iii) If f is injective, Gpt-S∗ g -irresolute, Z is Gpt-S∗ g -T2, then V is Gpt-S∗ g -T2. Proof. i) Assume rα ̸= s1, where rα, s1 ∈ V, then f(rα) = r2 and f(s1) = s2. Additionally, f(rα) ̸= f(s1). As (Z, Gτ 2 , Pβ) Gpt-T1 implies r2 ∈ Mα, s2 /∈ Mα and s2 ∈ Nα, r2 /∈ Nα. Then rα ∈ f−1(Mα), rα /∈ f−1(Nα) and s1 ∈ f−1(Nα), s1 /∈ f−1(Mα). According to the definition of Gpt-S∗ g -continuity, f−1(Mα) and f−1(Nα) ∈ Gpt-S∗ gO(V). For rα ̸= s1, rα, s1 ∈ V implies rα ∈ f−1(Mα), rα /∈ f−1(Nα) and s1 ∈ f−1(Nα), s1 /∈ f−1(Mα). Thus, (V, Gτ 1 , Pα) is Gpt-S∗ g -T1 space. In the same way, ii) and iii) can be proven. Theorem 3.18. The subsequent statements are equivalent. (i) V is Gpt-S∗ g -T2. (ii) If rα ∈ V, then rα ̸= s1, there exists Uα containing rα and s1 /∈ Gpt-S∗ gCl(Uα). Proof. (1) implies (2) Take rα ∈ V and s1 ∈ V with rα ̸= s1 there exists disjoint set Uα and V ∈ Gpt-S∗ gO(V) such that rα ∈ Uα and s1 ∈ V. Then, rα ∈ Uα ⊆ Vc and Vc ∈ Gpt-S∗ gC(V) and s1 /∈ Vc implies s1 /∈ Gpt-S∗ gCl(Uα). (2) implies (1) Consider rα ∈ V and s1 ∈ V with rα ̸= s1 implies there exists Gpt-S∗ g -open Uα containing rα such that s1 /∈ Gpt-S∗ gCl(Uα) implies s1 ∈ (V - (Gpt-S∗ gCl(Uα))). (V - (Gpt-S∗ gCl(Uα))) ∈ Gpt-S∗ gO(V) and rα /∈ (V - (Gpt-S∗ gCl(Uα))). Furthermore, Uα ∩ (V - (Gpt-S∗ gCl(Uα))) = ∅. So V is Gpt-S∗ g -T2. 3.2.2. Gpt-S∗ g -Regular Space Definition 3.22. If for all F ∈ Gpt-S∗ gC(V) and r /∈ F, there exists disjoint open sets E and D such that F ⊆ E, r ∈ D. Theorem 3.19. The condition of being Gpt-S∗ g -regular implies Gpt-regularity. Proof. Let V be a Gpt-S∗ g -regular space. Take F ∈ Gpt-S∗ gC(V) and r /∈ F. As V is Gpt-S∗ g -regular, there exists a pair of disjoint open sets E and D such that F ⊆ E, r ∈ D. Hence, V is a Gpt-regular. Example 3.7. Let V be the set of all bounded spherical regions in R3, where a spherical region S is defined as: S = {(x, y, z) ∈ R3 | √ (x− a)2 + (y − b)2 + (z − c)2 ≤ r}, M. Shahbaz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6375 16 of 22 with r > 0 as the radius of the spherical region and a, b, c ∈ R, τg = {E ⊆ V | E ⊆ V \ {S | radius(S) = r} for some fixed r > 0} (see example 3.1) and define a collection Gpt ⊆ 2V as: Gpt = {E ⊆ V | all spherical region in E have radius strictly less than r, where r > 0}. Let x ∈ V and C ⊆ V be a Gpt-S∗ g -closed set such that x /∈ C. Since C is Gpt-S∗ g -closed, its complement V\C is Gpt-S∗ g -open in V. Let an Gpt-S∗ g -open set U such that x ∈ U and U ∩C = ∅. Therefore, (V,Gτ ,Gpt), satisfies the condition for Gpt-S∗ g -regularity. Now, we check if V is Gpt-regular: Let x ∈ V and C ⊂ V be a closed set such that x /∈ C. Since every closed set is Gpt-S∗ g - closed set. By construction, such a set U exists and satisfies the Gpt-regularity condition which implies (V,Gτ ,Gpt) is Gpt-regular. Thus, (V,Gτ ,Gpt) is both Gpt-S∗ g -regular and Gpt-regular. Remark 3.4. Every Gpt-regular is a not Gpt-S∗ g -regular space. Example 3.8. Let V = {j1, k1, l1} and Gτ = { ∅, V, {l1}, {j1, k1} }, P = { ∅, {j1}, {l1}, {j1, l1} }. Hence, (V, Gτ , P) is Gpt-regular but not Gpt-S∗ g -regular space. For {k1} ∈ Gpt-S∗ gC(V) and r /∈ {k1}, there does not exists disjoint open sets E and D such that {k1} ⊆ E, r ∈ D. Theorem 3.20. Every Gpt-regular with Gpt-S∗ gTc space is Gpt-S∗ g -regular. Proof. Under the condition that V is Gpt-regular and Gpt-S∗ gTc. Take a set F which be- longs to Gpt-S∗ gC(V) along with an element r belonging to both V and non-corresponding to F. Because V functions as a Gpt-S∗ gTc space, its concluding that F is closed yet r belongs to the exterior of F. Since V possesses the property of Gpt-regularity a pair of open sets named E and D exists with the property that F belongs to E while r be- longs to D and these open sets are disjoint. The space V meets the criteria for being a Gpt-S∗ g -regular. Theorem 3.21. If V is Gpt-S∗ g -regular, then it is Gpt-regular space. Proof. According to the fact, every closed set belongs to Gpt-S∗ gC(V). Theorem 3.22. The subsequent statements are equivalent: (i) V is Gpt-S∗ g -regular. (ii) For all r ∈ V and each Gpt-S∗ g -open neighbourhood Uα there exists open neighbour- hood Nα of V such that Cl◦ (Nα) ⊆ Uα. Proof. (1) implies (2) Assume Uα is Gpt-S∗ g -neighbourhood of r, there exists E ∈ Gpt- S∗ gO(V) such that r ∈ E ⊆ Uα. Now, Ec ∈ Gpt-S∗ gC(V) and r /∈ Ec. From (1), there exists Rα, Sα such that Ec ⊆ Rα, r ∈ Sα, Rα ∩ Sα = ∅. Thus Sα ⊆ Mα c. Now, Cl◦(Sα) ⊆ Cl◦(Rα c) = Ec and Ec ⊆ Rα implies Rα c ⊆ E ⊆ Uα. Thus Cl◦(Sα) ⊆ Uα. (2) implies (1) Consider Gpt-S∗ g -closed F in V and r /∈ F or r ∈ (F)c and V is Gpt-S∗ g -open implies (F)c is M. Shahbaz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6375 17 of 22 Gpt-S∗ g -neighbourhood of r. By hypothesis, there exists an open neighbourhood Nα such that r ∈ Nα, Cl◦ (Nα) ⊆ (F)c implies F ⊆ { V - Cl◦ (Nα) } and Nα ∩ { V - Cl◦(Nα) } = ∅. Thus, V is Gpt-S∗ g -regular. Theorem 3.23. Assume V is Gpt-S∗ g -regular iff for every E ∈ Gpt-S∗ gC(V) and point p ∈ (V - E) then r ∈ Uα, E ⊆ Nα and Cl◦(Nα) ∩ Cl◦(Uα) = ∅, where Uα and Nα are open sets. Proof. Given that V is Gpt-S∗ g -regular. Assume E ∈ Gpt-S∗ gC(V) and r /∈ E. Then, p ∈ Mα and E ⊆ Nα and Mα ∩ Nα = ∅, where Mα and Nα are open sets implies Mα ∩ Cl◦(Nα) = ∅. As V is Gpt-S∗ g -regular, p ∈ Rα and Cl◦(Nα) ⊆ Sα, Rα ∩ Nα = ∅ where Rα and Sα are open. Furthermore, Cl◦(Rα) ∩ Sα = ∅. Assume Uα = Mα ∩ Rα implies p ∈ Uα, E ⊆ Nα and Cl◦(Nα) ∩ Cl◦(Uα) = ∅ where Nα and Uα are open in V. On the other hand, consider Nα and Uα are open sets. p ∈ Uα, E ⊆ Nα and Cl◦(Nα) ∩ Cl◦(Uα) = ∅ for all E ∈ Gpt-S∗ gC(V) and p ∈ (V - E) implies p ∈ Uα, E ⊆ Nα and Uα ∩ Nα = ∅. Thus, V is Gpt-S∗ g -regular. Theorem 3.24. A subspace Z of Gpt-S∗ g -regular (Z, Gτ , P) is Gpt-S∗ g -regular. Proof. Obvious. Theorem 3.25. Assume f is bijective, Gpt-S∗ g -irresolute and open map from Gpt-S∗ g - regular V into Z, then Z is Gpt-S∗ g -regular. Proof. Let rα ∈ Z and F ∈ Gpt-S∗ gC(V) and rα /∈ F. Furthermore, f is Gpt-S∗ g -irresolute, then f−1(F) ∈ Gpt-S∗ gC(V). Now, assume rα = f(r) then f−1(rα) = r and r /∈ f−1(F). As V is Gpt-S∗ g -regular then there exists Rα and Sα such that r ∈ Rα and f−1(F) ⊆ Sα, Rα ∩ Sα = ∅. Since f is open and bijective implies rα ∈ f(Rα), F ⊆ f(Sα) and f(Rα ∩ Sα) = f(∅) = ∅. Then, Z is Gpt-S∗ g -regular. 3.2.3. Gpt-S∗ g -Normal Space Definition 3.23. Assume V is Gpt-S∗ gnormal if for each pair E, D ∈ Gpt-S∗ gC(V), there exists open sets Rα and Sα in V such that D ⊆ Rα and E ⊆ Sα. Theorem 3.26. Every Gpt-S∗ g -normal is Gpt-normal. Proof. As V is a Gpt-S∗ g -normal. Assume disjoint sets E and D in V. So E, D ∈ Gpt- S∗ gC(V). As V is Gpt-S∗ g -normal implies there exist a pair F, Hα such that D ⊆ F, E ⊆ Hα. Thus, V is Gpt-normal. Example 3.9. Consider V = {j1, k1, l1} and Gτ = { ∅, V, {k1}, {l1}, {k1, l1}, {j1, k1} }, P = { ∅, {j1}, {l1}, {j1, l1} }. Here, (V, Gτ , P) is Gpt-normal but not Gpt-S∗ g -normal space. For disjoint sets {j1}, {k1, l1} ∈ Gpt-S∗ gC(V), there does not exist open sets Rα and Sα in V. M. Shahbaz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6375 18 of 22 Theorem 3.27. If Z is Gpt-normal, Gpt-S∗ gTc space, then Z is Gpt-S∗ g -normal. Proof. Since Z is Gpt-normal. Consider disjoint set E, D ∈ Gpt-S∗ gC(Z). As Gpt-S∗ gTc space, then E and D are closed. Since Z is Gpt-normal, then there exist disjoint open sets Rα and Sα in Z such that E ⊆ Rα and D ⊆ Sα. Thus, Z is Gpt-S∗ g -normal. Theorem 3.28. Every Gpt-S∗ g -normal is Gpt-g-normal. Proof. As V is Gpt-S∗ g -normal. Assume disjoint set E, D ∈ Gpt-S∗ gC(Z) implies there exist a disjoint Rα, Sα such that E ⊆ Rα and D ⊆ Sα. Thus, V is Gpt-g-normal. Remark 3.5. Every Gpt-g-normal is not Gpt-S∗ g -normal. Example 3.10. Consider V = {j1, k1, l1} and Gτ = { ∅, V, {k1}, {l1}, {k1, l1}, {j1, l1} }, P = { ∅, {j1}, {l1}, {j1, l1} }. Here, (V, Gτ , P) is Gpt-g-normal but not Gpt-S∗ g -normal space. For disjoint sets {j1}, {k1, l1} ∈ Gpt-S∗ gC(V), there does not exist open sets Rα and Sα in V. Theorem 3.29. Every Gpt-S∗ g -normal is Gpt-w-normal. Proof. Similar to theorem 3.28. Remark 3.6. An illustration disproves the invalidity of the converse statement derived from above. Example 3.11. Consider V = {j1, k1, l1} and Gτ = { ∅, V, {k1}, {l1}, {k1, l1}, {j1, l1} }, P = { ∅, {j1}, {l1}, {j1, l1} }. Here, (V, Gτ , P) is Gpt-w-normal but not Gpt-S∗ g -normal space. For disjoint sets {j1}, {k1, l1} ∈ Gpt-S∗ gC(V), there does not exists open sets Rα and Sα in V. Theorem 3.30. If Z is Gpt-S∗ g -closed subspace of Gpt-S∗ g -normal V, then Z is Gpt-S∗ g - normal. Proof. Assume V is Gpt-S∗ g -normal and Z is Gpt-S∗ g -closed subspace. Consider a pair of disjoint sets E and D ∈ Gpt-S∗ gC(Z) implies there exists F, Hα ∈ V such that E ⊆ F and D ⊆ Hα implies F ∩ Z and Hα ∩ Z are open in Z. Furthermore, E ⊆ F and D ⊆ Hα implies E ∩ Z ⊆ Z ∩ F and Z ∩ D ⊆ Z ∩ Hα and (F ∩ Z) ∩ (Z ∩ Hα) = Z ∩ (F ∩ Hα) = ∅. Thus, Z is Gpt-S∗ g -normal. Theorem 3.31. The following conditions in (V, Gτ , P) are equivalent: 1) The space V is Gpt-S∗ g -normal. 2) For each E belonging to Gpt-S∗ gC(V), there exists an open set T1 such that E ⊆ T1 ⊆ Cl(T1) ⊆ T2 for some T2 in Gpt-S∗ gO(V) with E ⊆ T2. M. Shahbaz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6375 19 of 22 3) Given two disjoint sets E and D in Gpt-S∗ gC(V), there exists an open set T1 such that E ⊆ T1 and Cl(T1) ∩D = ∅. 4) For any two disjoint sets E, D in Gpt-S∗ gC(V), there exist open sets T1 and T2 such that E ⊆ T2, D ⊆ T1, and Cl(T2) ∩ Cl(T1) = ∅. Proof. (1) implies (2): Suppose E belongs to Gpt-S∗ gC(V) and T2 is an element of Gpt- S∗ gC(V) with E ⊆ T2. Since E and V−T2 are disjoint, the assumption of Gpt-S∗ g -normality guarantees open sets T1 and T3 such that E ⊆ T1 and V − T2 ⊆ T3 with T1 ∩ T3 = ∅. This ensures T1 ⊆ V− T3 and further Cl(T1) ⊆ V− T3 ⊆ T2, leading to Cl(T1) ⊆ T2. (2) implies (3): Given two disjoint sets E and D in Gpt-S∗ gC(V), we note that E ⊆ V−D. By (2), there exists an open set T1 such that E ⊆ T1 and Cl(T1) ⊆ V−D, ensuring that Cl(T1) ∩D = ∅. (3) implies (4): Given two disjoint sets E and D in Gpt-S∗ gC(V), (3) guarantees an open set T2 such that E ⊆ T2 and Cl(T2) ∩D = ∅. Since Cl(T2) is closed, applying (3) again ensures the existence of an open set T1 containing D such that Cl(T2) ∩ Cl(T1) = ∅. (4) implies (1): Suppose E and D are two disjoint sets in Gpt-S∗ gC(V). By (4), there exist disjoint open sets T1 and T2 such that E ⊆ T2 and D ⊆ T1. This confirms that V satisfies the definition of Gpt-S∗ g -normality. Theorem 3.32. Assume a mapping f : V → Z. If f is bijective open, Gpt-S∗ g -irresolute from Gpt-S∗ g -normal V onto Z, then Z is Gpt-S∗ g -normal. Proof. Assume disjoint sets E, D ∈ Gpt-S∗ gC(V). Since f is Gpt-S∗ g -irresolute then f−1(E) and f−1(D) are in Gpt-S∗ gC(V). As V is Gpt-S∗ g -normal implies f−1(E) ⊆ T2 and f−1(D) ⊆ T1 where T1 and T2 are open in V. Also, as f is bijective and open, f(T2) and f(T1) are open and E ⊆ f(T2), D ⊆ f(T1). Thus, Z is Gpt-S∗ g -normal. Theorem 3.33. The following statements hold equivalently in (V, Gτ , P): 1) V is Gpt-g-normal. 2) There exist disjoint open sets T1, T2 ∈ Gpt-S∗ gO(V) such that E ⊆ T2 and D ⊆ T1 for any disjoint sets E and D. Proof. (1) implies (2) Suppose that V is Gpt-g-normal, and let E and D be two disjoint subsets of V. By the assumption that (V, Gτ , P) is Gpt-g-normal, there exist disjoint Gpt- g-open sets T1 and T2 such that E ⊆ T2 and D ⊆ T1. Consequently, T1, T2 ∈ Gpt-S∗ gO(V), satisfying E ⊆ T2 and D ⊆ T1 while ensuring T1 ∩ T2 = ∅. (2) implies (1) consider that for any two disjoint Gpt-S∗ g -closed sets E, D ∈ Gpt-S∗ gC(V), there exist disjoint open sets T1 and T2 such that E ⊆ T2, D ⊆ T1, and T1 ∩ T2 = ∅ where T1, T2 ∈ Gpt-S∗ gO(V). Since E is contained in Gpt-gInt(T2) and D in Gpt-gInt(T1), and their interiors remain disjoint, it follows that (V, Gτ , P) is Gpt-g-normal. M. Shahbaz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6375 20 of 22 4. Methodology A theoretical framework demonstrates the examination of compactness as well as con- nectedness and separation axioms in generalized primal topological spaces. The research method includes the following main stages of development: ◦ Review of Relevant Literature: This research explores the developmental his- tory of compactness and connectedness and separation properties through extensive study of published literature across generalized contexts. The research focuses on reviewing studies that created fundamental educational frameworks to transfer the principles of S∗ g -compactness and S∗ g -connectedness to generalized primal topolog- ical spaces. ◦ Definition Development: The article introduces new definitions of S∗ g -compactness and S∗ g -connectedness under generalized primal topology. New definitions for gen- eralized primal structures are developed but go through extensive evaluation to confirm their alignment with core principles of primal topology. ◦ Analyzing the Separation Axioms: Generalized primal topology utilizes S∗ g - open sets to study the classical separation conditions T0, T1 and T2. This study establishes the S∗ g -T0, S∗ g -T1 together with the S∗ g -T2 conditions before performing their respective analyses. ◦ Establishing Theoretical Proofs: The formal verification process proves the various attributes defined in the proposed constructions. Mathematical proofs es- tablish internal validity while ensuring logical consistency of new concepts in order to provide respectable grounds for theoretical future research. ◦ Comparative Analysis: The paper evaluates the new developed theory by show- ing its distinctions and correspondences to traditional theories alongside demon- strating advantages for accepting generalized primal topological methods. ◦ Results Synthesis: The research findings transform into structural models for describing S∗ g -compact as well as S∗ g -connected spaces through the generalized pri- mal topological framework, together with separation axiom analysis. Furthermore, the work discusses what aspects the advancement means for developing topological theory. 5. Conclusions Researchers perform an extensive investigation into separation axioms together with con- nectedness and compactness phenomena in generalized primal topological spaces. This paper extends traditional concepts to the new framework, which increases our under- standing of S∗ g -compactness and S∗ g -connectedness at a theoretical level. The new def- initions specifically designed for generalized primal spaces combined with their formal support system enhances these concepts for application throughout nonclassical spaces. M. Shahbaz et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6375 21 of 22 The study introduces a new set of classification tools through S∗ g -T0, S∗ g -T1, and S∗ g -T2 separation axiom reinterpretations to analyze spaces based on their topological proper- ties. In their independent studies, S∗ g -compactness and S∗ g -connectedness do not strictly mirror traditional separation axioms, yet their examination broadens the fundamental understanding of generalized primal spaces. This research provides significant value to the expansion of generalized primal topology as a field of exploration. Author contributions Conceptualization, M.S., U.I. and I.L.P.; methodology, M.I. and I.L.P.; software, T.K., U.I. and I.L.P.; validation, M.A. and I.L.P.; formal analysis, U.I. and M.A.; investigation, M.A.; resources, T.K.; data curation, U.I.; writing—original draft preparation, M.S., M.I., and U.I.; writing—review and editing, T.K.; visualization, M.A. supervision, T.K. and I.L.P.; project administration, U.I. and I.L.P.; funding acquisition, M.A. All authors have read and agreed to the published version of the manuscript. Acknowledgements The authors extend their gratitude to the Deanship of Graduate Studies and Scientific Research of the Islamic University of Madinah for the support provided to the Post- Publication Program 4. Competing interests The authors declare that they have no competing interests. References [1] M. Shahbaz, T. Kamran, U. Ishtiaq, M. Imtiaz, I.L. Popa, and F.M. Maiz. Some new notions of continuity in generalized primal topological space. 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