EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 7147 ISSN 1307-5543 – ejpam.com Published by New York Business Global Complex Neutrosophic Soft Topology with Multivariate Analysis: A Unified Framework for Signal-Template Relationships Maha Mohammed Saeed1, Raed Hatamleh2, Hamza Ali Abujabal3, Aqeedat Hussain4, Arif Mehmood4, M. I. Elashiry5, Abdelhalim Hasnaoui5, Alaa M. Abd El-latif5,∗ 1 Department of Mathematics, Faculty of Sciences, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia 2 Department of Mathematics, Faculty of Science, Jadara University, P.O. Box 733, Irbid 21110, Jordan 3 Department of Mathematics, King Abdulaziz University, P.O. Box 80003, Jeddah 21589, Saudi Arabia 4 Department of Mathematics, Institute of Numerical Sciences, Gomal University, Dera Ismail Khan 29050, KPK, Pakistan 5 Department of Mathematics, College of Science, Northern Border University, Arar 91431, Saudi Arabia Abstract. A novel theoretical method based on the topological formulation of complex neutrosophic soft sets (CNSS) is presented in this work. We go on to provide a thorough one-value complex neutrosophic soft topology, defining the most important topological properties such as interior, closure, and other related theoretical concerns, in order to give a strong mathematical foundation. An example is then provided to illustrate this framework’s analytical capabilities. To ensure a thorough empirical validation of this paradigm, we employ a multifaceted approach that combines PCA, clustering, and manifold learning. Everyone agreed that a clear target hierarchy should be established, even though we employed a variety of analytical methodologies. In this hierarchy, T1 was consistently described as the most stable and representative target, T2 as a transitional outlier, and T3 as the most divergent target. In its most basic form, multivariate analysis and topological theory offer a versatile framework for clarifying the connections between templates and signals while avoiding the drawbacks of over-methodologization. 2020 Mathematics Subject Classifications: 03E72, 06F25, 62H25, 54A10 Key Words and Phrases: Complex neutrosophic soft sets, complex neutrosophic soft topology, cotan- gent similarity measures, data visualization techniques 1. Introduction Garg and Rani studied the aggregation operators of Complex Interval-Valued Intuitionistic Fuzzy Sets [2], whereas Al-Qudah and Hassan studied the functions of Complex Multi-Fuzzy Sets [1]. In the event that there are several possible complex evaluations, Garg et al. later presented Complex Hesitant Fuzzy Sets, which include additional distance measures [3]. Singh constructed a concept lattice using Complex Vague Sets to illustrate the versatility of these ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.7147 Email addresses: mmmohammed@kau.edu.sa (M. M. Saeed), raed@jadara.edu.jo (R. Hatamleh), haabujabal@kau.edu.sa (H. A. Abujabal), aqeedathussain04@gmail.com (A. Hussain), mehdaniyal@gmail.com (A. Mehmood), mustafa.elashiry@nbu.edu.sa (M. I. Elashiry), abdllhalim.hasanawa@nbu.edu.sa (A. Hasnaoui), alaa.ali@nbu.edu.sa (A. M. Abd El-latif) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7147 2 of 41 ideas [4]. Scientists have expanded their conceptual richness and applicability based on the basic frameworks of complicated neutrosophic sets. Its ability to capture both the positive and negative memberships of truth and indeterminacy and falsity in the complex plane makes it a significant extension, first presented by RaBroumi et al. as Bipolar Complex Neutrosophic Sets [5]. As a result, it is a more effective tool for use in decision-making problems where opposing factors are present. Al-Quran and Alkhazaleh studied the significance of the relations between complicated neutro- sophic sets [7], and Quek et al. [6] constructed the sets in the theoretical framework of graph theory. Linguistic solutions to interval complicated neutrosophic sets were provided by Dat et al. [8] to bridge the gap between the numerical and linguistic representations, making them more useful in human-based decision-making scenarios. Molodtsov introduced Soft Set Theory, another similarly potent paradigm for handling uncertainty, at the same time [9]. This gave a totally new, parameterized manner of treating imprecision that was not subject to the limits of the old-fashioned membership functions. The combination of pre-existing fuzzy models and soft sets quickly became a productive area of study. Maji et al. made the initial attempt at this synergy when they launched fuzzy soft sets [10] and intuitionistic fuzzy soft sets [11], which parameterized soft sets but with fuzzy and intuitionistic fuzzy set capability, respectively. This trend of hybridization continued to embrace increasingly intricate forms of uncertainty. The researchers that had to deal with uncertain membership information devel- oped Interval-Valued Fuzzy Soft Sets [12] and Interval-Valued Intuitionistic Fuzzy Soft Sets [13]. Moreover, neutrosophic theory was adopted with the introduction of Neutrosophic Soft Sets by Maji [14] and Interval-Valued Neutrosophic Soft Sets by Deli [17]. These two unconnected con- cepts describe the truth, indeterminacy, and falsity of parameters separately. Abu Qamar and Hassan’s [18] introduction of the Q-Neutrosophic Soft Set, which demonstrates the contemporary and dynamic intersection of complex, neutrosophic, and soft set theories to handle the complexity of uncertainty in data, is one of the even more extended forms of the area that are now being explored. More sophisticated and specialized models have been pro- duced as a result of the ongoing hybridization trend. By studying the algebraic structures of Q-neutrosophic soft fields, Abu Qamar et al. developed the theoretical foundation of the study and further formalized the work on Q-Neutrosophic Soft Sets [19].The model’s ability to progress from pure set theory to standard algebraic frameworks was a sign of its maturity. The actual merger of soft set theory and the multifaceted fuzzy model was another noteworthy par- allel comparable development. It resulted in the Complex Fuzzy Soft Set [20], a paradigm that leverages the two-dimensional information store of complex membership degrees and the pa- rameterization of soft sets. Soon after, this hybrid was expanded to handle greater imprecision, giving rise to the Interval-Valued Complex Fuzzy Soft Set [21]. Recent advances have proposed even more powerful hybrid frameworks that can simulate complicated and higher-order uncer- tainty, pushing this integration to even greater heights. As an example, Fujita [22] developed strong models that can handle data represented by multi-dimensional membership degree by incorporating a hyper-fuzzy environment into the already outstanding VIKOR and DEMATEL approaches. In the same vein, Gul [23] proposed a bipolar fuzzy rough set model fused with the VIKOR approach, which offers a more detailed model on the capacity to manage conflict- ing criteria and is useful in reflecting the positive and negative preferences of decision-makers in a d-covering context. Alongside these advancements, the study of more complex algebraic structures has produced a solid theoretical foundation for the use of these methods in signifi- cant fields. The work of Abdalla et al. [24] on Interval-Valued Fermatean Neutrosophic Super Hyper Soft Sets is an illustration of this trend. It describes how complex forms of uncertainty in real-world contexts like health care, where the information is indeterminate, incomplete, and may come from multiple sources, can be represented by sophisticated set-theoretic models. M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7147 3 of 41 1.1. Literature Review The recent study by AlAkram et al. [25], which suggested Complex Fermatean Fuzzy N-Soft Sets, is one example of how novel this is at the moment. This model is a significant amalgam of many potent paradigms: These three cutting-edge ideas are combined in this hybrid model to provide an unparalleled capacity for handling multi-level, complicated, and ambiguous data while making decisions. Its creation demonstrates the field’s constant drive to create increas- ingly expressive and adaptable mathematical tools in order to represent the complexity of the real world. By establishing the Complex Bipolar-Valued Neutrosophic Soft Set [26], Al-Quran et al. advanced the research of bipolarity on the complex neutrosophic framework and improved the model’s ability to handle contradictory data dimensions. Akram et al. produced Complex Neutrosophic N-Soft Sets [27], which provide a multi-level graded structure of truth, indeter- minacy, and falsity, by combining the influential idea of N-soft sets with neutrosophic theory in complex space. Formally describing the complex neutrosophic soft expert relation [28], which provides a mathematical foundation for comparing and connecting the many expert evalua- tions in this intricate framework, was a significant theoretical and practical leap. A theory of Interval-Valued Complex Neutrosophic Soft Sets [29] and related relations [31] also extended this relational paradigm to address imprecision. When these models were eventually articulated in complex space [30], it was a necessary generalization, and their representational power was significantly increased. Effective comparison tools are essential to all of these advancements. Similarity measurements, which are essential for applications like pattern recognition and medi- cal diagnosis, have been the subject of extensive research as a result. Numerous structures have been developed, including Lattice Ordered Multi-Fuzzy Soft Sets [33], Single and Multi-Valued Neutrosophic Hypersoft Sets [32], and m-polar-interval valued Neutrosophic Soft Sets [34]. The real-world effects of such progressively abstract theoretical constructions are demonstrated by the practical effects of such measurements, particularly in sensitive fields like medical diagnos- tics [34], which completes the circle between the formulation of mathematical issues and their practical resolution. Since the proliferation of these sophisticated hybrid forms has necessitated the development of robust methods for comparing and contrasting these forms, recent studies have included research on similarity metrics and entropy. They are crucial in applications like as pattern recognition, medical diagnosis, and multi-criteria decision-making, where the ability to measure the degree of similarity or ambiguity across different sets is crucial. Basic similarity measures between neutrosophic and interval neutrosophic sets were founded by Broumi and Smarandache [38] and Ye [39], respectively. These were soon generalized to hybrid structures, as was the case with similarity measures of Interval Valued Neutrosophic Soft Sets [40]. Medical usages are also a dominant factor and Liu et al. [35] utilize the Euclidean distance to neutro- sophic sets to diagnose, which Zulqarnain et al.[34] apply to more advanced m-Polar features. The comparison techniques grew more intricate as the models became more intricate. The re- searchers employed Lattice Ordered Multi-Fuzzy Soft Sets [33], Possibility Intuitionistic Fuzzy Hyper-soft Sets [37], and T-Spherical Fuzzy Sets [36] as special measures. The same idea holds true in the complex realm, where Complex Multi-Fuzzy Soft Sets [42] have been subjected to entropy and similarity measures in order to handle the dimensional expansion. Additionally, equivalent entropy and distance metrics have been added to generalization to structures like Q-Neutrosophic Soft Sets [41], ensuring that even the most generalized models have the math- ematical resources needed to be practical. To put it simply, the creation of each new hybrid model is now entangled with the creation of the corresponding entropy and similarity metrics. This ensures that the theoretical power of such complex structures can be appropriately applied to the solution of complex real-world problems where comparison and uncertainty quantification are the best qualities. In order to address the issue of multi-criteria group decision-making, Xu et al. focused on distance measures of Interval Complex Neutrosophic Sets [44], while Selvachan- dran et al. developed a similarity measure on Complex Vague Soft Sets and demonstrated its applicability in pattern recognition [43]. These metrics of comparison have been developed and M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7147 4 of 41 applied extensively on a variety of complex hybrid models. Neutrosophic set and its character- istics were studied in [45]. A fundamental idea in the field is brought to light by this consistent correlation between the new set-theoretic models and specially designed similarity and distance measures: the practice drives and validates the theory’s progress. Lastly, our ability to mathematically describe un- certainty has grown exponentially since the original fuzzy sets proposed by Zadeh [46] evolved into the complex, neutrosophic, and soft hybrid models of today. This has been called a cycle of innovation, whereby a new theoretical framework is created by identifying a representational deficiency, then new components are added, and finally, real-world applications of similarity measures and aggregation operators are added to deal with the real world. The result is a multifaceted and varied ecosystem of mathematical tools that can simulate the gradation, am- biguity, cyclicality, and intrinsic contradictions of complex systems, enabling robots to reason more like humans. Since the principles of these frameworks have been extended beyond set theory to abstract algebraic and mathematical structures, the final step of the evolutionary process demonstrates how theoretically profound they are. This development was supported by the introduction of Interval Neutrosophic Sets [48] and is based on Smarandache’s early work on Neutrosophy [47], which provides the logical and philosophical underpinnings of neutrosophic logic. Fuzzy, intuitionistic, and neutrosophic notions can now be applied to basic algebraic fields thanks to this groundbreaking discovery. The operations of Intuitionistic and Interval-Valued Intuitionistic Fuzzy Soft Sets [49] have been more extensively generalized to distance measure- ments, which gives them an algebraic basis. Concurrently, the set-theoretic models have been expanded to include more complex models that can process multi-dimensional attribute values, as the Plithagenic Hyper-soft Set [50]. Most significantly, a robust research thread that builds algebraic systems on top of the ex- isting uncertainty-based foundations has been produced. These include studies of structures like Intuitionistic Anti Fuzzy Normal Subrings over Normed Rings [52] and the Neutrosophic Multiplication Module [51]. The work on Polish Groups [53], conditions on P. P. Rings [54], Classical Artinian Modules [55], Fractional Ideals [56], and structures in Topological Groups [57] all point to a parallel study program in pure algebra that is extended by this subject. This indicates that in addition to advancing these uncertainty theories, the mathematical com- munity is also contributing to the advancement of the classical algebraic structures to which they are applied. In conclusion, it is important to note that the current state of amazing in- tellectual synergy is a direct result of the fuzzy sets of Zadeh. Beginning with the very simple idea of graded membership, it evolved into a whole ecosystem of models to address uncertainty through phases of integration and generalization to complex planes, neutrosophic logic, and parameterized soft sets. This evolution typically follows a positive feedback loop: theoretical innovation results in new hybrid models, which are then given useful tools like comparable mea- sures to be applied, and their ideas are abstracted to enhance fundamental fields like abstract algebra. This leads to a dynamic and intricate area of mathematics that ultimately enhances our ability to reason, model, and navigate a world that is fundamentally complex and uncertain. The thorough examination of the structures, like the Neutrosophic Multiplication Module [51] and Intuitionistic Anti Fuzzy Normal Sub-rings over Normed Rings [52], continues this robust line of inquiry that seeks to create algebraic systems with uncertainty-driven foundations. As evidenced by work on Classical Artinian Modules [58], Polish Groups [53], conditions on P.P. Rings [54], Fractional Ideals [56], and the study of new types of mappings in Topological spaces with Strongly Closed Graphs [59], this line of inquiry is part of a larger, parallel research pro- gram in pure algebra and topology. The investigation of supra open soft sets and its various decompositions and applications, as well as soft set theory, are both superseded by supra soft topology, according to the sources provided. Fundamental results on decompositions of supra soft sets, together with the concept of soft continuity in this context, were provided by El-Sheikh and Abd Ellatif’s basic contribu- M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7147 5 of 41 tions [60]. Supra soft strongly generalized closed sets [62], supra soft strongly extra generalized closed sets [63], and supra soft strongly semi∗ generalized closed sets [65] are only a few of the specific classes of soft sets that are now being studied in this field. Furthermore, the ideas have been crucial in defining and researching different forms of soft connectedness [67] as well as related soft separation axioms [61][64][66]. The research of supra soft compactness [69] and interactions with soft ideals [68] have also contributed to the development of the theory. The creation of complicated frameworks that can manage uncertainty, indeterminacy, and in- tricate relationship systems has greatly influenced recent developments in computational math- ematics and logic. Among these, neutrosophic theory has shown itself to be an effective tool for simulating real-world issues. It generalizes fuzzy and intuitionistic fuzzy sets by adding a truth-membership, indeterminacy-membership, and falsity-membership function. Its wide range of uses, from resolving complicated differential equations [70] to examining structural characteristics in algebra and graph theory [71, 72], demonstrate this. Simultaneously, research into hybrid systems that combine fuzzy and neutrosophic sets with other mathematical fields continues to provide new discoveries. For example, recent research has established complex tangent trigonometric methods for advanced fuzzy sets [73] and extended complex cubic intu- itionistic fuzzy sets to algebraic structures [74]. Additionally, the analysis of minimal units in fuzzy neutrosophic rings [76] and the foundational work on topological spaces using symbolic n-plithogenic intervals [75] show the growing frontier of this field, offering a solid mathematical foundation for addressing problems with inherent ambiguity and multifaceted information. 1.2. Novelty and Motivation for Research For the first time, a one-value complex neutrosophic soft topology that strictly satisfies the fundamental features, notably the closure and the interior, is developed by combining the topological formulation with complex neutrosophic soft sets (CNSS). Such a study formalizes a topological basis that ensures theoretical coherence and generalizability, in contrast to the present neutrosophic or soft set methodologies, which are mostly algebraic or descriptive. The relationship between this abstract model and specific analytical methods, such as PCA, clustering, and manifold learning, is truly novel since it demonstrates how an entirely mathe- matical model can be applied directly to empirical classification and pattern recognition issues. Although it does not rely too much on any one approach, this interdisciplinary connection be- tween theory topology and real-world multivariate analysis ensures interpretability and strength. The limitations of the existing classification and similarity metrics to address ambiguity, insta- bility, and divergence in signal-template connections have been the driving force behind this investigation. When dealing with intermediate instances, such as T2, that do not fully fall into ei- ther the stability or the divergence category, conventional procedures are extremely challenging. This study aims to propose a topological framework that implements neutrosophic uncertainty handling in a way that is both theoretically rigorous and practically adaptive, generating con- sistent target hierarchies. One solid argument for the need for such a unified paradigm is the consensus that T1 is the most stable, T2 is intermediate and transitional, and T3 is divergent in different analytical lenses. The ultimate objective of the study, however, is to balance the practicality of the empirical data with the depth of concepts in order to ensure that complicated signal systems may be accurately and vividly interpreted while being open to analysis. 2. Preliminaries In this section, we will give some preliminary information for the present study. Definition 1. [45] A neutrosophic set A on the universe of discourse X is defined as: A = {⟨x, TA(x), IA(x), FA(x)⟩ : x ∈ X} where, T, IF : X −→]0, 1[ and 0 ≤ TA(x) + IA(x) + FA(x) ≤ 3. M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7147 6 of 41 Definition 2. [9] Let X be an initial universe, E be a set of all parameters and P (X) denotes the powerset of X. A pair of (F,E) is called a soft set over X, where F is a mapping given by F : E −→ P (X). In other words, the soft set is a parameterized family of subsets of the setX. For e ∈ E,F (e) may be considered as the set of e-elements of the soft set (F,E), or as the set of e-approximate elements of the soft set, i.e., (F,E) = {(e, F (e)) : e ∈ E,F : E −→ P (X)} Firstly, neutrosophic soft set defined by Maji [14] and later this concept has been modified by Deli and Broumi [15] as given below: Definition 3. Let X be an initial universe, E be a set of all parameters. Let P (X) denote the set of all neutrosophic sets of X. Then, a neutrosophic soft set (F̃ .E) over X is a set defined by a set valued function F̃ representing a mapping F̃ : E −→ P (X) where F̃ is called approximate function of neutrosophic soft set (F̃ , E). In other words, the neutrosophic soft set is a parametrized family of some elements of the set P (X) and therefore it can be written as a set of order pairs, (F̃ , E) = {(e, ⟨x, TF̃ (e)(x), IF̃ (e)(x), FF̃ (e)(x)⟩ : x ∈ X) : e ∈ E} where TF̃ (e)(x), TF̃ (e)(x), FF̃ (e)(x) ∈ [0, 1], respectively called the truth-membership, indeterminacy- membership, falsity-membership of F̃ (e). Since supremum of each T, I, F is1 so the inequality 0 ≤ TF̃ (e)(x) + IF̃ (e)(x) + FF̃ (e)(x) ≤ 3 is obvious. Definition 4. [16] Let (F̃ , E) be a neutrosophic soft sets over the universe set X. The com- plement of (F̃ , E) is denoted by (F̃ , E)c and is defined by: (F̃ , E)c = {(e, ⟨x, FF̃ (e)(x), 1− IF̃ (e)(x), TF̃ (e)(x)⟩ : x ∈ X) : e ∈ E} Obvious that, ((F̃ , E)c)c = (F̃ , E). Definition 5. [14] Let (F̃ , E) and (G̃, E) be two neutrosophic soft sets over the universe set X. (F̃ , E) is said to be neutrosophic soft subset of (G̃, E) if TF̃ (e)(x) ≤ TG̃(e)(x), IF̃ (e)(x) ≤ IG̃(e)(x), FF̃ (e)(x) ≥ FG̃(e)(x), for all e ∈ E, for all x ∈ X. It is denoted by (F̃ , E) ⊆ (G̃, E). (F̃ , E) is said to be neutrosophic soft set equal to (G̃, E) if (F̃ , E) is neutrosophic soft subset of (G̃, E) and (G̃, E) is neutrosophic soft subset of (F̃ , E). It is denoted by (F̃ , E) = (G̃, E). 3. Operations on Complex Single-Valued Neutrosophic Soft Sets In this section , the operations of union, intersection, difference, AND and OR on complex single-valued neutrosophic soft sets are defined. In addition to this, examples are given for better understanding these operations. Definition 6. Let (F̃1, E) and (F̃2, E) be two complex single-valued neutrosophic soft sets over the universe set X. Then their union is represented by (F̃1, E) ∪ (F̃2, E) = (F̃3, E) is defined by: (F̃3, E) = {( e, ⟨x, TF̃3(e) (x), IF̃3(e) (x), FF̃3(e) (x)⟩ : x ∈ X ) : e ∈ E } , Where TF̃3(e) (x) = max{TF̃1(e) (x), TF̃2(e) (x)} IF̃3(e) (x) = max{IF̃1(e) (x), IF̃2(e) (x)} FF̃3(e) (x) = min{FF̃1(e) (x), FF̃2(e) (x)} M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7147 7 of 41 Definition 7. Let (F̃1, E) and (F̃2, E) be two complex single-valued neutrosophic soft sets over the universe set X. Then their intersection is represented by (F̃1, E) ∩ (F̃2, E) = (F̃3, E) is defined by: (F̃3, E) = {( e, ⟨x, TF̃3(e) (x), IF̃3(e) (x), FF̃3(e) (x)⟩ : x ∈ X ) : e ∈ E } , Where TF̃3(e) (x) = min{TF̃1(e) (x), TF̃2(e) (x)} IF̃3(e) (x) = min{IF̃1(e) (x), IF̃2(e) (x)} FF̃3(e) (x) = max{FF̃1(e) (x), FF̃2(e) (x)} Definition 8. Let (F̃1, E) and (F̃2, E) be two complex single-valued neutrosophic soft sets over the universe set x. Then, (F̃1, E) difference (F̃2, E) operation on them is denoted by (F̃1, E) \ (F̃2, E) = (F̃3, E) and is defined by (F̃3, E) = (F̃1, E) ∩ (F̃2, E)c as follows: (F̃3, E) = {( e, ⟨x, TF̃3(e) (x), IF̃3(e) (x), FF̃3(e) (x)⟩ : x ∈ X ) : e ∈ E } , where TF̃3(e) (x) = min{TF̃1(e) (x), FF̃2(e) (x)} IF̃3(e) (x) = min{IF̃1(e) (x), 1− IF̃2(e) (x)} FF̃3(e) (x) = max{FF̃1(e) (x), TF̃2(e) (x)} Definition 9. Let {(F̃i, E) : i ∈ I} be a family of complex single-valued neutrosophic soft sets over the universe set X. Then, ⋃ i∈I (F̃i, E) = {( e, ⟨x, sup i∈I TF̃i(e) (x), sup i∈I IF̃i(e) (x), inf i∈I FF̃i(e) (x)⟩ : x ∈ X ) : e ∈ E } . ⋂ i∈I (F̃i, E) = {( e, ⟨x, inf i∈I TF̃i(e) (x), inf i∈I IF̃i(e) (x), sup i∈I FF̃i(e) ()⟩ : x ∈ X ) : e ∈ E } . Definition 10. Let (F̃1, E) and (F̃2, E) be two complex single-valued neutrosophic soft sets over the universe set X. Then, the AND operation on them is denoted by (F̃1, E) ∧ (F̃2, E) = (F̃3, E × E) and is defined by: (F̃,E×E) = {( (e1, e2), ⟨x, TF̃3(e1,e2) (x), IF̃3(e1,e2) (x), FF̃3(e1,e2) (x)⟩ : x ∈ X ) : (e1, e2) ∈ E×E } . where TF̃3(e1,e2) (x) = min { TF̃1(e1,e2) (x), TF̃2(e1,e2) (x) } , IF̃3(e1,e2) (x) = min { IF̃1(e1,e2) (x), IF̃2(e1,e2) (x) } , FF̃3(e1,e2) (x) = max { FF̃1(e1,e2) (x), FF̃2(e1,e2) (x) } . Definition 11. Let (F̃1, E) and (F̃2, E) be two complex single-valued neutrosophic soft sets over the universe set X. Then, the OR operation on them is denoted by (F̃1, E)∨(F̃2, E) = (F̃3, E×E) and is defined by: M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7147 8 of 41 (F̃ , E×E) = {( (e1, e2), ⟨x, TF̃3(e1,e2) (x), IF̃3(e1,e2) (x), FF̃3(e1,e2) (x)⟩ : x ∈ X ) : (e1, e2) ∈ E×E } . where TF̃3(e1,e2) (x) = max { TF̃1(e1,e2) (x), TF̃2(e1,e2) (x) } , IF̃3(e1,e2) (x) = max { IF̃1(e1,e2) (x), IF̃2(e1,e2) (x) } , FF̃3(e1,e2) (x) = min { FF̃1(e1,e2) (x), FF̃2(e1,e2) (x) } . Definition 12. A complex single-valued neutrosophic soft set (F̃ , E) over the universe set X is said to be a null complex single-valued neutrosophic soft set if TF̃ (e)(x) = 0, IF̃ (e)(x) = 0, FF̃ (e)(x) = 1, ∀ e ∈ E, ∀ x ∈ X. It is denoted by 0(X,E). Definition 13. A complex single-valued neutrosophic soft set (F̃ , E) over the universe set X is said to be a absolute complex single-valued neutrosophic soft set if TF̃ (e)(x) = 1, IF̃ (e)(x) = 1, FF̃ (e)(x) = 0, ∀ e ∈ E, ∀ x ∈ X, Clearly, 0c(X,E) = 1(X,E), 1c(X,E) = 0(X,E). Proposition 1. Let (F̃1, E), (F̃2, E) and (F̃2, E) be three complex single-valued neutrosophic soft sets over the universe set X. Then, (i) (F̃1, E) ∪ [(F̃2, E) ∪ (F̃3, E)] = [(F̃1, E) ∪ (F̃2, E)] ∪ (F̃3, E)and (F̃1, E) ∩ [(F̃2, E) ∩ (F̃3, E)] = [(F̃1, E) ∩ (F̃2, E)] ∩ (F̃3, E); (ii) (F̃1, E) ∪ [(F̃2, E) ∩ (F̃3, E)] = [(F̃1, E) ∪ (F̃2, E)] ∩ [(F̃1, E) ∪ (F̃3, E)]and (F̃1, E) ∩ [(F̃2, E) ∪ (F̃3, E)] = [(F̃1, E) ∩ (F̃2, E)] ∪ [(F̃1, E) ∩ (F̃3, E)]; (iii) (F̃1, E) ∪ 0(X,E) = (F̃1, E)and(F̃1, E) ∩ 0(X,E) = 0(X,E); (iv) (F̃1, E) ∪ 1(X,E) = (F̃1, E)and(F̃1, E) ∩ 1(X,E) = (F̃1, E); Proof. Obvious. Proposition 2. Let (F̃1, E) and (F̃2, E) be two complex single-valued neutrosophic soft sets over the universe set X. Then, (i) [(F̃1, E) ∪ (F̃2, E)]c = (F̃1, E)c ∩ (F̃2, E)c (ii) [(F̃1, E) ∩ (F̃2, E)]c = (F̃1, E)c ∪ (F̃2, E)c Proof. (i). For all e ∈ E and x ∈ X, (F̃1, E) ∪ (F̃2, E) = {⟨x,max{TF̃1(e) (x), TF̃2(e) (x)},max{IF̃1(e) (x), IF̃2(e) (x)}, min{FF̃1(e) (x), FF̃2(e) (x)}⟩} [(F̃1, E) ∪ (F̃2, E)]c = {⟨x,min{FF̃1(e) (x), FF̃2(e) (x)}, 1−max{IF̃1(e) (x), IF̃2(e) (x)}, max{TF̃1(e) (x), TF̃2(e) (x)}⟩} M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7147 9 of 41 Now, (F̃1, E)c = {⟨x, FF̃1(e) (x), 1− IF̃1(e) (x), TF̃1(e) (x)⟩} (F̃2, E)c = {⟨x, FF̃2(e) (x), 1− IF̃2(e) (x), TF̃2(e) (x)⟩} Then, (F̃1, E)c ∩ (F̃2, E)c = {⟨x,min{FF̃1(e) (x), FF̃2(e) (x)},min{1− IF̃1(e) (x), 1− IF̃2(e) (x)}, max{TF̃1(e) (x), TF̃2(e) (x)}⟩} = {⟨x,min{FF̃1(e) (x), FF̃2(e) (x)}, 1−max{IF̃1(e) (x), IF̃2(e) (x)}, max{TF̃1(e) (x), TF̃2(e) (x)}⟩} Therefore, [(F̃1, E) ∪ (F̃2, E)]c = (F̃1, E)c ∩ (F̃2, E)c. (ii). It is obtained in a similar way. Proposition 3. Let (F̃1, E) and (F̃2, E) be two complex single-valued neutrosophic soft sets over the universe set X. Then, (i) [(F̃1, E) ∨ (F̃2, E)]c = (F̃1, E)c ∧ (F̃2, E)c (ii) [(F̃1, E) ∧ (F̃2, E)]c = (F̃1, E)c ∨ (F̃2, E)c Proof. (i). For all e ∈ E and x ∈ X, (F̃1, E) ∨ (F̃2, E) = {⟨x,max{TF̃1 (e)(x), TF̃2 (e)(x)},max{IF̃1 (e)(x), IF̃2 (e)(x)}, min{FF̃1 (e)(x), FF̃2 (e)(x)}⟩} [(F̃1, E) ∨ (F̃2, E)]c = {⟨x,min{FF̃1 (e)(x), FF̃2 (e)(x)}, 1−max{IF̃1 (e)(x), IF̃2 (e)(x)}, max{TF̃1 (e)(x), TF̃2 (e)(x)}⟩}, On the other hand, (F̃1, E)c = {⟨x, FF̃1 (e)(x), 1− IF̃1 (e)(x), RTF̃1 (e)(x), TF̃1 (e)(x)⟩} (F̃2, E)c = {⟨x, FF̃2 (e)(x), 1− IF̃2 (e)(x), RTF̃2 (e)(x), TF̃2 (e)(x)⟩} Then, (F̃1, E)c ∧ (F̃2, E)c = {⟨x,min{FF̃1 (e)(x), FF̃2 (e)(x)},min{1− IF̃1 (e)(x), 1− IF̃2 (e)(x)} max{TF̃1 (e)(x), TF̃2 (e)(x)}⟩} = {⟨x,min{FF̃1 (e)(x), FF̃2 (e)(x)}, 1−max{IF̃1 (e)(x), IF̃2 (e)(x)}, max{TF̃1 (e)(x), TF̃2 (e)(x)}⟩}, Therefore, [(F̃1, E) ∨ (F̃2, E)]c = (F̃1, E)c ∧ (F̃2, E)c. (ii). It is obtained in a similar way. M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7147 10 of 41 Example 1. Suppose that, the universe set X given by X = {x1, x2, x3, x4} and the set of pa- rameters E = (e1, e2). Let us consider complex single-valued neutrosophic soft sets (F̃1, E)and(F̃2, E) over the universe set X as follows (F̃1, E) =  e1 = ⟨x1, 0.3eιπ(0.2), 0.4eιπ(0.3), 0.6eιπ(0.5)⟩, ⟨x2, 0.4eιπ(0.3), 0.2eιπ(0.1), 0.8eιπ(0.7)⟩, ⟨x3, 0.6eιπ(0.5), 0.2eιπ(0.1), 0.5eιπ(0.4)⟩, ⟨x4, 0.2eιπ(0.1), 0.3eιπ(0.2), 0.4eιπ(0.3)⟩ e2 = ⟨x1, 0.4eιπ(0.3), 0.3eιπ(0.2), 0.8eιπ(0.7)⟩, ⟨x2, 0.3eιπ(0.2), 0.4eιπ(0.3), 0.2eιπ(0.1)⟩, ⟨x3, 0.3eιπ(0.2), 0.2eιπ(0.1), 0.7eιπ(0.6)⟩, ⟨x4, 0.1eιπ(0.1), 0.2eιπ(0.1), 0.9eιπ(0.8)⟩  (F̃2, E) =  e1 = ⟨x1, 0.6eιπ(0.5), 0.3eιπ(0.2), 0.8eιπ(0.7)⟩, ⟨x2, 0.2eιπ(0.1), 0.5eιπ(0.4), 0.8eιπ(0.7)⟩, ⟨x3, 0.1eιπ(0.1), 0.1eιπ(0.1), 0.4eιπ(0.3)⟩, ⟨x4, 0.5eιπ(0.4), 0.2eιπ(0.1), 0.3eιπ(0.2)⟩ e2 = ⟨x1, 0.7eιπ(0.6), 0.5eιπ(0.4), 0.6eιπ(0.5)⟩, ⟨x2, 0.4eιπ(0.3), 0.2eιπ(0.1), 0.3eιπ(0.2)⟩, ⟨x3, 0.5eιπ(0.4), 0.3eιπ(0.2), 0.4eιπ(0.3)⟩, ⟨x4, 0.4eιπ(0.3), 0.2eιπ(0.1), 0.6eιπ(0.5)⟩  Then, their union, intersection, AND, and OR operations are given as follows: (F̃1, E)∪(F̃2, E) =  e1 = ⟨x1, 0.6eιπ(0.5), 0.4eιπ(0.3), 0.6eιπ(0.5)⟩, ⟨x2, 0.4eιπ(0.3), 0.5eιπ(0.4), 0.8eιπ(0.7)⟩, ⟨x3, 0.6eιπ(0.5), 0.2eιπ(0.1), 0.4eιπ(0.3)⟩, ⟨x4, 0.5eιπ(0.4), 0.3eιπ(0.2), 0.3eιπ(0.2)⟩ e2 = ⟨x1, 0.7eιπ(0.6), 0.5eιπ(0.4), 0.6eιπ(0.5)⟩, ⟨x2, 0.4eιπ(0.3), 0.4eιπ(0.3), 0.2eιπ(0.1)⟩, ⟨x3, 0.5eιπ(0.4), 0.3eιπ(0.2), 0.4eιπ(0.3)⟩, ⟨x4, 0.4eιπ(0.3), 0.2eιπ(0.1), 0.6eιπ(0.5)⟩  (F̃1, E)∩(F̃ , E) =  e1 = ⟨x1, 0.3eιπ(0.2), 0.3eιπ(0.3), 0.8eιπ(0.7)⟩, ⟨x2, 0.2eιπ(0.1), 0.2eιπ(0.1), 0.8eιπ(0.7)⟩, ⟨x3, 0.1eιπ(0.1), 0.1eιπ(0.1), 0.5eιπ(0.4)⟩, ⟨x4, 0.2eιπ(0.1), 0.2eιπ(0.2), 0.4eιπ(0.4)⟩ e2 = ⟨x1, 0.3eιπ(0.2), 0.3eιπ(0.3), 0.8eιπ(0.7)⟩, ⟨x2, 0.2eιπ(0.1), 0.2eιπ(0.1), 0.8eιπ(0.7)⟩, ⟨x3, 0.1eιπ(0.1), 0.1eιπ(0.1), 0.5eιπ(0.4)⟩, ⟨x4, 0.2eιπ(0.1), 0.2eιπ(0.2), 0.4eιπ(0.3)⟩  (F̃1, E)\(F̃2, E) =  e1 = ⟨x1, 0.3eιπ(0.2), 0.3eιπ(0.2), 0.6eιπ(0.5)⟩, ⟨x2, 0.4eιπ(0.3), 0.2eιπ(0.1), 0.8eιπ(0.7)⟩, ⟨x3, 0.4eιπ(0.3), 0.1eιπ(0.1), 0.5eιπ(0.4)⟩, ⟨x4, 0.2eιπ(0.1), 0.2eιπ(0.1), 0.5eιπ(0.4)⟩ e2 = ⟨x1, 0.4eιπ(0.3), 0.3eιπ(0.2), 0.8eιπ(0.7)⟩, ⟨x2, 0.3eιπ(0.2), 0.2eιπ(0.1), 0.4eιπ(0.3)⟩, ⟨x3, 0.3eιπ(0.2), 0.2eιπ(0.1), 0.7eιπ(0.6)⟩, ⟨x4, 0.1eιπ(0.1), 0.1eιπ(0.1), 0.9eιπ(0.8)⟩  (F̃1, E)∧(F̃ , E) =  (e1, e1) = ⟨x1, 0.3eιπ(0.2), 0.3eιπ(0.2), 0.8eιπ(0.7)⟩, ⟨x2, 0.2eιπ(0.1), 0.2eιπ(0.1), 0.8eιπ(0.7)⟩, ⟨x3, 0.1eιπ(0.1), 0.1eιπ(0.1), 0.5eιπ(0.4)⟩, ⟨x4, 0.2eιπ(0.1), 0.2eιπ(0.1), 0.4eιπ(0.3)⟩ (e1, e2) = ⟨x1, 0.3eιπ(0.2), 0.4eιπ(0.3), 0.6eιπ(0.5)⟩, ⟨x2, 0.4eιπ(0.3), 0.2eιπ(0.1), 0.8eιπ(0.7)⟩, ⟨x3, 0.5eιπ(0.4), 0.2eιπ(0.1), 0.5eιπ(0.4)⟩, ⟨x4, 0.2eιπ(0.1), 0.2eιπ(0.1), 0.6eιπ(0.5)⟩ (e2, e1) = ⟨x1, 0.4eιπ(0.3), 0.3eιπ(0.2), 0.8eιπ(0.7)⟩, ⟨x2, 0.2eιπ(0.1), 0.4eιπ(0.3), 0.8eιπ(0.7)⟩, ⟨x3, 0.1eιπ(0.1), 0.1eιπ(0.1), 0.7eιπ(0.6)⟩, ⟨x4, 0.1eιπ(0.1), 0.2eιπ(0.1), 0.9eιπ(0.8)⟩ (e2, e2) = ⟨x1, 0.3eιπ(0.2), 0.3eιπ(0.2), 0.8eιπ(0.7)⟩, ⟨x2, 0.2eιπ(0.1), 0.2eιπ(0.1), 0.8eιπ(0.7)⟩, ⟨x3, 0.1eιπ(0.1), 0.1eιπ(0.1), 0.5eιπ(0.4)⟩, ⟨x4, 0.2eιπ(0.1), 0.2eιπ(0.1), 0.4eιπ(0.3)⟩  M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7147 11 of 41 (F̃1, E)∨(F̃ , E) =  (e1, e1) = ⟨x1, 0.6eιπ(0.5), 0.4eιπ(0.3), 0.6eιπ(0.5)⟩, ⟨x2, 0.4eιπ(0.3), 0.5eιπ(0.4), 0.8eιπ(0.7)⟩, ⟨x3, 0.6eιπ(0.5), 0.2eιπ(0.1), 0.4eιπ(0.3)⟩, ⟨x4, 0.5eιπ(0.4), 0.3eιπ(0.2), 0.3eιπ(0.2)⟩ (e1, e2) = ⟨x1, 0.7eιπ(0.6), 0.5eιπ(0.4), 0.8eιπ(0.7)⟩, ⟨x2, 0.4eιπ(0.3), 0.2eιπ(0.1), 0.3eιπ(0.2)⟩, ⟨x3, 0.6eιπ(0.5), 0.3eιπ(0.2), 0.4eιπ(0.3)⟩, ⟨x4, 0.4eιπ(0.3), 0.3eιπ(0.2), 0.4eιπ(0.3)⟩ (e2, e1) = ⟨x1, 0.6eιπ(0.5), 0.3eιπ(0.2), 0.8eιπ(0.7)⟩, ⟨x2, 0.3eιπ(0.2), 0.5eιπ(0.4), 0.2eιπ(0.1)⟩, ⟨x3, 0.3eιπ(0.2), 0.2eιπ(0.1), 0.4eιπ(0.3)⟩, ⟨x4, 0.5eιπ(0.4), 0.2eιπ(0.1), 0.3eιπ(0.2)⟩ (e2, e2) = ⟨x1, 0.7eιπ(0.6), 0.5eιπ(0.4), 0.6eιπ(0.5)⟩, ⟨x2, 0.4eιπ(0.3), 0.4eιπ(0.3), 0.2eιπ(0.1)⟩, ⟨x3, 0.5eιπ(0.4), 0.3eιπ(0.2), 0.4eιπ(0.3)⟩, ⟨x4, 0.4eιπ(0.3), 0.2eιπ(0.1), 0.6eιπ(0.5)⟩  4. Complex Single-Valued Neutrosophic Soft Topological Spaces In this section, we introduced the complex single-valued neutrosophic soft topology based on the defined operations of the complex single-valued neutrosophic soft union, intersection, the neutrosophic soft null and absolute sets. Some examples are given for better understanding the concepts. Definition 14. Let CSV NSS(X,E) be the family of all complex single-valued neutrosophic soft sets over the universe set X and τ ⊂ CSV NSS(X,E). Then τ is said to be a complex single-valued neutrosophic soft topology on X if (i) 0(X,E) and 1(X,E) ∈ τ , (ii) The union of any number of complex single-valued neutrosophic soft sets in τ ∈ τ , (iii) The intersection of a finite number of complex single-valued neutrosophic soft sets in τ ∈ τ . Then, (X, τ,E) is said to be a complex single-valued neutrosophic soft topological space (CSVN- STS) over X. Each members of τ is said to be complex single-valued neutrosophic soft open set. Definition 15. Let (X, τ,E) be a complex single-valued neutrosophic soft topological space over X and (F̃ , E) be a complex single-valued neutrosophic soft set over X . Then (F̃ , E) is said to be complex single-valued neutrosophic soft closed set iff its complement is a complex single-valued neutrosophic soft open set. Proposition 4. Let (X, τ,E) be a complex single-valued neutrosophic soft topological space over the universe set X. Then, (i) 0(X,E) and 1(X,E) are complex single-valued neutrosophic soft closed sets over X, (ii) The union of any number of complex single-valued neutrosophic soft sets is a complex single-valued neutrosophic soft closed sets over X, (iii) The intersection of a finite number of complex single-valued neutrosophic soft sets is a complex single-valued neutrosophic soft closed sets over X. Definition 16. Let CSV NSS(X,E) be the family of all complex single-valued neutrosophic soft sets over the universe set X. (i) If τ = {0(X,E), 1(X,E)}, then τ is said to be complex single-valued neutrosophic soft in- discrete topology and (X, τ,E) is said to be a complex single-valued neutrosophic soft indiscrete topological space over X. M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7147 12 of 41 (ii) If τ = CSV NSS(X,E), then τ is said to be complex single-valued neutrosophic soft discrete topology and (X, τ,E) is said to be a complex single-valued neutrosophic soft discrete topological space over X. Proposition 5. Let (X, τ1, E) and (X, τ2, E) be two complex single-valued neutrosophic soft topological space over the same universe set X. Then (X, τ1 ∩ τ2, E) is a complex single-valued neutrosophic soft topological space over X. Proof. (i). Since 0(X,E), 1(X,E) ∈ τ1 and 0(X,E), 1(X,E) ∈ τ2, then 0(X,E), 1(X,E) ∈ τ1 ∩ τ2. (ii). Suppose that {(F̃i, E)|i ∈ I} be a family of complex double-valued neutrosophic soft sets in τ1 ∩ τ2. Then (F̃i, E) ∈ τ1 and (F̃i, E) ∈ τ2 for all i ∈ I, so ∪i∈I(F̃i, E) ∈ τ1 and ∪i∈I(F̃i, E) ∈ τ2. Thus ∪i∈I(F̃i, E) ∈ τ1 ∩ τ2. (iii). Let {(F̃i, E)|i = 1, n} be a family of finite number of complex single-valued neutrosophic soft sets in τ1 ∩ τ2. Then (F̃i, E) ∈ τ1 and (F̃i, E) ∈ τ2 for all i = 1, n, so ⋂n i=I(F̃i, E) ∈ τ1 and ⋂n i=I(F̃i, E) ∈ τ2. Thus ⋂n i=I(F̃i, E) ∈ τ1 ∩ τ2. Remark 1. The union of two complex single-valued neutrosophic soft topologies over X may not be a complex single-valued neutrosophic soft topology on X. Example 2. Let X = {x1, x2, x3} be an initial universe set, E = {e1, e2} be a set of parameters and: τ1 = {0(X,E), 1(X,E), (F̃1, E), (F̃2, E), (F̃3, E)}, τ2 = {0(X,E), 1(X,E), (F̃2, E), (F̃4, E)}. be two complex single-valued neutrosophic soft topologies over X. Here, the complex single- valued neutrosophic soft sets (F̃1, E), (F̃2, E), (F̃3, E) and (F̃4, E) are defined as follows: (F̃1, E) =  e1 = ⟨x1, 0.9eιπ(0.9), 0.6eιπ(0.8), 0.3eιπ(0.5)⟩, ⟨x2, 0.7eιπ(0.7), 0.6eιπ(0.5), 0.5eιπ(0.4)⟩ ⟨x3, 0.4eιπ(0.3), 0.5eιπ(0.2), 0.4eιπ(0.2)⟩; e2 = ⟨x1, 0.7eιπ(0.9), 0.3eιπ(0.8), 0.4eιπ(0.7)⟩, ⟨x2, 0.6eιπ(0.5), 0.5eιπ(0.7), 0.4eιπ(0.2)⟩, ⟨x3, 0.7eιπ(0.8), 0.4eιπ(0.4), 0.3eιπ(0.4)⟩.  (F̃2, E) =  e1 = ⟨x1, 0.7eιπ(0.8), 0.5eιπ(0.4), 0.5eιπ(0.5)⟩, ⟨x2, 0.6eιπ(0.4), 0.5eιπ(0.5), 0.7eιπ(0.6)⟩ ⟨x3, 0.3eιπ(0.2), 0.4eιπ(0.2), 0.6eιπ(0.8)⟩; e2 = ⟨x1, 0.6eιπ(0.6), 0.2eιπ(0.6), 0.5eιπ(0.7)⟩, ⟨x2, 0.5eιπ(0.4), 0.4eιπ(0.1), 0.6eιπ(0.7)⟩, ⟨x3, 0.4eιπ(0.3), 0.3eιπ(0.3), 0.5eιπ(0.4)⟩.  (F̃3, E) =  e1 = ⟨x1, 0.5eιπ(0.6), 0.4eιπ(0.2), 0.6eιπ(0.5)⟩, ⟨x2, 0.4eιπ(0.2), 0.3eιπ(0.1), 0.7eιπ(0.7)⟩ ⟨x3, 0.2eιπ(0.1), 0.3eιπ(0.2), 0.8eιπ(0.9)⟩; e2 = ⟨x1, 0.4eιπ(0.3), 0.2eιπ(0.1), 0.6eιπ(0.8)⟩, ⟨x2, 0.4eιπ(0.3), 0.2eιπ(0.1), 0.8eιπ(0.9)⟩, ⟨x3, 0.3eιπ(0.2), 0.3eιπ(0.1), 0.6eιπ(0.5)⟩.  (F̃4, E) =  e1 = ⟨x1, 0.6eιπ(0.7), 0.3eιπ(0.2), 0.2eιπ(0.1)⟩, ⟨x2, 0.5eιπ(0.4), 0.3eιπ(0.3), 0.8eιπ(0.9)⟩ ⟨x3, 0.2eιπ(0.2), 0.3eιπ(0.1), 0.9eιπ(0.9)⟩; e2 = ⟨x1, 0.3eιπ(0.3), 0.2eιπ(0.1), 0.7eιπ(0.8)⟩, ⟨x2, 0.4eιπ(0.2), 0.1eιπ(0.1), 0.9eιπ(0.9)⟩, ⟨x3, 0.2eιπ(0.2), 0.2eιπ(0.1), 0.7eιπ(0.8)⟩.  Since (F̃1, E) ∪ (F̃4, E) /∈ τ1 ∪ τ2, then τ1 ∪ τ2 is not a complex single-valued neutrosophic soft topology over X. M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7147 13 of 41 Proposition 6. Let (X, τ,E) be a complex single-valued neutrosophic soft topological space over X and τ = {(F̃i, E) : (F̃i, E) ∈ CSV NSS(X,E)} = {[e, F̃i(e)]e∈E : (F̃i, E) ∈ CSV NSS(X,E)} where F̃i(e) = {⟨x, TF̃i(e) (x), IF̃i(e) (x), FF̃i(e) (x)⟩ : x ∈ X}. Then τ1 = {[TF̃i(e) (X)]e∈E} τ2 = {[IF̃i(e) (X)]e∈E} τ3 = {[FF̃i(e) (X)]e∈E} define complex fuzzy soft topologies on X. Proof. (i). 0(X,E), 1(X,E) ∈ τ ⇒ 0, 1 ∈ τ1, 0, 1 ∈ τ2, 0, 1 ∈ τ3and0, 1 ∈ τ4. (ii). Sup- pose that (F̃i, E)|i ∈ I be a family of complex single-valued neutrosophic soft sets in τ . Then {[TF̃i(e) (X)]e∈E}i∈I is a family of complex fuzzy soft sets in τ1, {[IF̃i(e) (X)]e∈E}i∈I is a family of complex fuzzy soft sets in τ2, {[FF̃i(e) (X)]e∈E}i∈I is a family of complex fuzzy soft sets in τ3. Since τ is a complex single-valued neutrosophic soft topology, then ∪i∈I(F̃i, E) ∈ τ1. That is, ∪i∈I(F̃i, E) = {⟨sup[TF̃i(e) (X)]e∈E , sup[IF̃i(e) (X)]e∈E , inf [FF̃i(e) (X)]e∈E⟩}i∈I ∈ τ. Therefore, {sup[TF̃i(e) (X)]e∈E}i∈I ∈ τ1 {sup[IF̃i(e) (X)]e∈E}i∈I ∈ τ2 {inf [FF̃i(e) (X)]e∈E}i∈I ∈ τ3. (iii). Suppose {(F̃i, E)|i = 1, n} be a family of finite number of complex single-valued neutro- sophic soft sets in τ . Then {[TF̃i(e) (X)]e∈E}i=1,n is a family of complex fuzzy soft sets in τ1, {[IF̃i(e) (X)]e∈E}i=1,n is a family of complex fuzzy soft sets in τ2, {[FF̃i(e) (X)]e∈E}i=1,n is a family of complex fuzzy soft sets in τ3. Since τ is a complex single-valued neutrosophic soft topology, then ∩n i=1(F̃i, E) ∈ τ1. That is, ∩n i=1(F̃i, E) = {⟨min[TF̃i(e) (X)]e∈E ,min[IF̃i(e) (X)]e∈E , max[FF̃i(e) (X)]e∈E⟩}i=1,n ∈ τ. Therefore, {min[TF̃i(e) (X)]e∈E}i∈I ∈ τ1 {min[IF̃i(e) (X)]e∈E}i∈I ∈ τ2 {min[FF̃i(e) (X)]ce∈E}i∈I ∈ τ3. This completes the proof. Remark 2. Generally, converse of the above proposition is not true. Example 3. Let X = {x1, x2, x3} be an initial universe set, E = {e1, e2} be a set of parameters and: τ1 = {0(X,E), 1(X,E), (F̃1, E), (F̃2, E), (F̃3, E)} be a family of complex single-valued neutrosophic soft sets over X. Here, the complex single- valued neutrosophic soft sets (F̃1, E), (F̃2, E) and (F̃3, E) are defined as follows: M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7147 14 of 41 (F̃1, E) =  e1 = ⟨x1, 0.9eιπ(0.9), 0.6eιπ(0.8), 0.3eιπ(0.7)⟩, ⟨x2, 0.7eιπ(0.7), 0.6eιπ(0.5), 0.5eιπ(0.4)⟩ ⟨x3, 0.4eιπ(0.3), 0.5eιπ(0.2), 0.4eιπ(0.2)⟩; e2 = ⟨x1, 0.7eιπ(0.9), 0.3eιπ(0.8), 0.4eιπ(0.7)⟩, ⟨x2, 0.6eιπ(0.5), 0.5eιπ(0.7), 0.4eιπ(0.2)⟩, ⟨x3, 0.7eιπ(0.8), 0.4eιπ(0.4), 0.3eιπ(0.4)⟩.  (F̃2, E) =  e1 = ⟨x1, 0.7eιπ(0.8), 0.5eιπ(0.4), 0.5eιπ(0.8)⟩, ⟨x2, 0.6eιπ(0.4), 0.5eιπ(0.5), 0.7eιπ(0.6)⟩ ⟨x3, 0.3eιπ(0.2), 0.4eιπ(0.2), 0.6eιπ(0.8)⟩; e2 = ⟨x1, 0.6eιπ(0.6), 0.2eιπ(0.6), 0.5eιπ(0.8)⟩, ⟨x2, 0.5eιπ(0.4), 0.4eιπ(0.1), 0.6eιπ(0.7)⟩, ⟨x3, 0.4eιπ(0.3), 0.3eιπ(0.3), 0.5eιπ(0.4)⟩.  (F̃3, E) =  e1 = ⟨x1, 0.8eιπ(0.8), 0.4eιπ(0.2), 0.6eιπ(0.8)⟩, ⟨x2, 0.4eιπ(0.2), 0.3eιπ(0.1), 0.7eιπ(0.7)⟩ ⟨x3, 0.2eιπ(0.1), 0.3eιπ(0.2), 0.8eιπ(0.9)⟩; e2 = ⟨x1, 0.4eιπ(0.3), 0.2eιπ(0.1), 0.6eιπ(0.8)⟩, ⟨x2, 0.4eιπ(0.3), 0.2eιπ(0.1), 0.8eιπ(0.9)⟩, ⟨x3, 0.3eιπ(0.2), 0.3eιπ(0.1), 0.6eιπ(0.5)⟩.  Then, τ1 = {⟨TF̃0(X,E)(e) (X), TF̃1(X,E)(e) (X), TF̃1(e) (X), TF̃2(e) (X), TF̃3(e) (X)⟩e∈E}, τ2 = {⟨IF̃0(X,E)(e) (X), IF̃1(X,E)(e) (X), IF̃1(e) (X), IF̃2(e) (X), IF̃3(e) (X)⟩e∈E}, τ3 = {⟨FF̃0(X,E)(e) (X), FF̃1(X,E)(e) (X), FF̃1(e) (X), FF̃2(e) (X), FF̃3(e) (X)⟩e∈E}. are complex fuzzy soft topologies on X. For example, τ1 = {⟨(0, 0, 0), (1, 1, 1), (0.9eιπ(0.9), 0.7eιπ(0.7), 0.4eιπ(0.3)), (0.7eιπ(0.8), 0.6eιπ(0.4), 0.3eιπ(0.2)), (0.8eιπ(0.8), 0.4eιπ(0.2), 0.2eιπ(0.1))⟩e1 , ⟨(0, 0, 0), (1, 1, 1), (0.7eιπ(0.9), 0.6eιπ(0.5), 0.7eιπ(0.8)), (0.6eιπ(0.6), 0.5eιπ(0.4), 0.4eιπ(0.3)), (0.4eιπ(0.3), 0.4eιπ(0.3), 0.3eιπ(0.2))⟩e2} τ2 = {⟨(0, 0, 0), (1, 1, 1), (0.6eιπ(0.8), 0.6eιπ(0.5), 0.5eιπ(0.2)), (0.5eιπ(0.4), 0.5eιπ(0.5), 0.4eιπ(0.2)), (0.4eιπ(0.2), 0.3eιπ(0.1), 0.3eιπ(0.2))⟩e1 , ⟨(0, 0, 0), (1, 1, 1), (0.3eιπ(0.8), 0.5eιπ(0.7), 0.4eιπ(0.4)), (0.2eιπ(0.6), 0.4eιπ(0.1), 0.3eιπ(0.3)), (0.2eιπ(0.1), 0.2eιπ(0.1), 0.3eιπ(0.1))⟩e2} τ3 = {⟨(0, 0, 0), (1, 1, 1), (0.3eιπ(0.7), 0.5eιπ(0.4), 0.4eιπ(0.2)), (0.5eιπ(0.8), 0.7eιπ(0.6), 0.6eιπ(0.8)), (0.6eιπ(0.8), 0.7eιπ(0.7), 0.8eιπ(0.9))⟩e1 , ⟨(0, 0, 0), (1, 1, 1), (0.4eιπ(0.7), 0.4eιπ(0.2), 0.3eιπ(0.4)), (0.5eιπ(0.8), 0.6eιπ(0.7), 0.5eιπ(0.4)), (0.6eιπ(0.8), 0.8eιπ(0.9), 0.6eιπ(0.5))⟩e2} Since (F̃2, E) ∩ (F̃3, E) /∈ τ , τ is not a complex single-valued neutrosophic soft topology on X. M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7147 15 of 41 Proposition 7. Let (X, τ,E) be a complex single-valued neutrosophic soft topological space over X. Then τ1e = {[TF̃ (e)(X)] : F̃ , E ∈ τ} τ2e = {[IF̃ (e)(X)] : F̃ , E ∈ τ} τ3e = {[FF̃ (e)(X)] : F̃ , E ∈ τ} for each e ∈ E define complex fuzzy topologies on X. Remark 3. Generally converse of the above proposition is not true. Example 4. Let us consider the Example 3. Then, τ1e1 = {⟨TF̃0(X,E)(e1) (X), TF̃1(X,E)(e1) (X), TF̃1(e1) (X), TF̃2(e1) (X), TF̃3(e1) (X)⟩}, τ2e1 = {⟨IF̃0(X,E)(e1) (X), IF̃1(X,E)(e1) (X), IF̃1(e1) (X), IF̃2(e1) (X), IF̃3(e1) (X)⟩}, τ3e1 = {⟨FF̃0(X,E)(e1) (X), FF̃1(X,E)(e1) (X), FF̃1(e1) (X), FF̃2(e1) (X), FF̃3(e1) (X)⟩}. are complex fuzzy soft topologies on X. For example, τ1e1 = {⟨(0, 0, 0), (1, 1, 1), (0.9eιπ(0.9), 0.7eιπ(0.7), 0.4eιπ(0.3)), (0.7eιπ(0.8), 0.6eιπ(0.4), 0.3eιπ(0.2)), (0.8eιπ(0.8), 0.4eιπ(0.2), 0.2eιπ(0.1))⟩} τ2e1 = {⟨(0, 0, 0), (1, 1, 1), (0.6eιπ(0.8), 0.6eιπ(0.5), 0.5eιπ(0.2)), (0.5eιπ(0.4), 0.5eιπ(0.5), 0.4eιπ(0.2)), (0.4eιπ(0.2), 0.3eιπ(0.1), 0.3eιπ(0.2))⟩} τ3e1 = {⟨(0, 0, 0), (1, 1, 1), (0.3eιπ(0.7), 0.5eιπ(0.4), 0.4eιπ(0.2)), (0.5eιπ(0.8), 0.7eιπ(0.6), 0.6eιπ(0.8)), (0.6eιπ(0.8), 0.7eιπ(0.7), 0.8eιπ(0.9))⟩} Here, {τ1e1 , τ2e1 , τ3e1} and {τ1e2 , τ2e2 , τ3e2} are complex fuzzy quadri-topology on X. But τ is not a complex single-valued neutrosophic soft topology on X. Definition 17. Let (X, τ,E) be a complex single-valued neutrosophic soft topological space over X and (F̃ , E) ∈ CSV NSS(X,E) be a complex single-valued neutrosophic soft set. Then, the complex single-valued neutrosophic soft interior of (F̃ , E), denoted by (F̃ , E)◦ is defined as the complex single-valued neutrosophic soft union of all the complex single-valued neutrosophic soft open subsets of (F̃ , E). Clearly (F̃ , E)◦ is biggest complex single-valued neutrosophic soft open set that is contained by (F̃ , E). Example 5. Let us consider the complex single-valued neutrosophic soft topology τ1 given in Example 2.Suppose that an any (F̃ , E) ∈ CSV NSS(X,E) is defined as following: (F̃ , E) =  e1 = ⟨x1, 0.8eιπ(0.9), 0.6eιπ(0.8), 0.4eιπ(0.3)⟩, ⟨x2, 0.7eιπ(0.6), 0.9eιπ(0.8), 0.3eιπ(0.2)⟩ ⟨x3, 0.5eιπ(0.3), 0.6eιπ(0.7), 0.3eιπ(0.2)⟩; e2 = ⟨x1, 0.8eιπ(0.7), 0.7eιπ(0.6), 0.2eιπ(0.3)⟩, ⟨x2, 0.9eιπ(0.7), 0.7eιπ(0.7), 0.4eιπ(0.3)⟩, ⟨x3, 0.8eιπ(0.7), 0.5eιπ(0.6), 0.4eιπ(0.3)⟩.  Then 0(X,E), (F̃2, E), (F̃3, E) ⊆ (F̃ , E). Therefore, (F̃ , E)◦ = 0(X,E)∪(F̃2, E)∪(F̃3, E) = (F̃ , E). M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7147 16 of 41 Theorem 1. Let (X, τ,E) be a complex single-valued neutrosophic soft topological space over X and (F̃ , E) ∈ CSV NSS(X,E). (F̃ , E) is a complex single-valued neutrosophic soft open set and iff (F̃ , E) = (F̃ , E)◦. Proof. Let (F̃ , E) be a complex single-valued neutrosophic soft open set. Then the biggest complex single-valued neutrosophic soft open set that is contained by (F̃ , E) is equal to (F̃ , E). Hence, (F̃ , E) = (F̃ , E)◦ Conversely, it is known that (F̃ , E)◦ is a complex single-valued neutrosophic soft open set and if (F̃ , E) = (F̃ , E)◦, then (F̃ , E) is a complex single-valued neutrosophic soft open set. Theorem 2. Let (X, τ,E) be a complex single-valued neutrosophic soft topological space over X and (F̃1, E), (F̃2, E) ∈ CSV NSS(X,E). Then, (i) [(F̃ , E)◦]◦ = (F̃ , E)◦, (ii) (0(X,E)) ◦ = 0(X,E) and (1(X,E)) ◦ = 1(X,E), (iii) (F̃1, E) ⊆ (F̃2, E) ⇒ (F̃1, E)◦ ⊆ (F̃2, E)◦, (iv) [(F̃1, E) ∩ (F̃2, E)]◦ = (F̃1, E)◦ ∩ (F̃2, E)◦, (v) (F̃1, E)◦ ∪ (F̃2, E)◦ ⊆ [(F̃1, E) ∪ (F̃2, E)]◦. Proof. (i) Let (F̃1, E)◦ = (F̃2, E) Then (F̃2, E) ∈ τiff(F̃2, E) = (F̃2, E)◦. So [(F̃1, E)◦]◦ = (F̃1, E)◦. (ii) Since 0(X,E) and 1(X,E) are always CSVNS open sets, we have: (0(X,E)) ◦ = 0(X,E), and (1(X,E)) ◦ = 1(X,E). (iii) It is known that (F̃1, E)◦ ⊆ (F̃1, E) ⊆ (F̃2, E) and (F̃2, E)◦ ⊆ (F̃2, E). Since (F̃2, E)◦ is the biggest complex single-valued neutrosophic soft open set contained in (F̃2, E) and so (F̃1, E)◦ ⊆ (F̃2, E)◦. (iv) Since (F̃1, E)∩(F̃2, E) ⊆ (F̃1, E) and (F̃1, E)∩(F̃2, E) ⊆ (F̃2, E), then [(F̃1, E)∩(F̃2, E)]◦ ⊆ (F̃1, E)◦ and [(F̃1, E) ∩ (F̃2, E)]◦ ⊆ (F̃2, E)◦ and so, [(F̃1, E) ∩ (F̃2, E)]◦ ⊆ (F̃1, E)◦ ∩ (F̃2, E)◦. On the other hand, since (F̃1, E)◦ ⊆ (F̃1, E) and (F̃2, E)◦ ⊆ (F̃2, E), then: (F̃1, E)◦ ∩ (F̃2, E)◦ ⊆ (F̃1, E) ∩ (F̃2, E). Besides,[(F̃1, E) ∩ (F̃2, E)]◦ ⊆ (F̃1, E) ∩ (F̃2, E) and is the biggest complex single-valued neutrosophic soft open set. Therefore, (F̃1, E)◦ ∩ (F̃2, E)◦ ⊆ [(F̃1, E) ∩ (F̃2, E)]◦. Thus, [(F̃1, E) ∩ (F̃2, E)]◦ = (F̃1, E)◦ ∩ (F̃2, E)◦. (v) Since (F̃1, E) ⊆ (F̃1, E) ∪ (F̃2, E) and (F̃2, E) ⊆ (F̃1, E) ∪ (F̃2, E), then: (F̃1, E)◦ ⊆ [(F̃1, E) ∪ (F̃2, E)]◦, and (F̃2, E)◦ ⊆ [(F̃1, E) ∪ (F̃2, E)]◦. Therefore, (F̃1, E)◦ ∪ (F̃2, E)◦ ⊆ [(F̃1, E) ∪ (F̃2, E)]◦. M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7147 17 of 41 Definition 18. Let (X, τ,E) be a complex single-valued neutrosophic soft topological space over X and (F̃ , E) ∈ CSV NSS(X,E) be a CSVNSS. Then, the complex single-valued neutrosophic soft closure of (F̃ , E), denoted by (F̃ , E) is defined as the complex single-valued neutrosophic soft intersection of all the complex single-valued neutrosophic soft closed supersets of (F̃ , E). Clearly (F̃ , E) is smallest complex single-valued neutrosophic soft closed set that is containing (F̃ , E). Example 6. Let us consider the complex single-valued neutrosophic soft topology τ1 given in Example 2.Suppose that an any (F̃ , E) ∈ CSV NSS(X,E) is defined as following: (F̃ , E) =  e1 = ⟨x1, 0.2eιπ(0.1), 0.3eιπ(0.2), 0.9eιπ(0.8)⟩, ⟨x2, 0.5eιπ(0.4), 0.2eιπ(0.1), 0.8eιπ(0.8)⟩ ⟨x3, 0.2eιπ(0.1), 0.2eιπ(0.1), 0.7eιπ(0.9)⟩; e2 = ⟨x1, 0.1eιπ(0.1), 0.2eιπ(0.1), 0.8eιπ(0.9)⟩, ⟨x2, 0.1eιπ(0.1), 0.2eιπ(0.1), 0.9eιπ(0.8)⟩, ⟨x3, 0.4eιπ(0.3), 0.2eιπ(0.1), 0.9eιπ(0.9)⟩.  Obviously, (0(X,E)) c, (1(X,E)) c, (F̃1, E)c, (F̃2, E)c, and(F̃3, E)c are all complex single-valued neu- trosophic soft closed sets over (X, τ1, E). They are given as following: (0(X,E)) c = 1(X,E), (1(X,E)) c = 0(X,E) (F̃1, E)c =  e1 = ⟨x1, 0.9eιπ(0.9), 0.4eιπ(0.2), 0.3eιπ(0.5)⟩, ⟨x2, 0.7eιπ(0.7), 0.4eιπ(0.5), 0.5eιπ(0.4)⟩ ⟨x3, 0.4eιπ(0.3), 0.5eιπ(0.8), 0.4eιπ(0.2)⟩; e2 = ⟨x1, 0.7eιπ(0.9), 0.7eιπ(0.2), 0.4eιπ(0.7)⟩, ⟨x2, 0.6eιπ(0.5), 0.5eιπ(0.3), 0.4eιπ(0.2)⟩, ⟨x3, 0.7eιπ(0.8), 0.6eιπ(0.6), 0.3eιπ(0.4)⟩.  (F̃2, E)c =  e1 = ⟨x1, 0.7eιπ(0.8), 0.5eιπ(0.6), 0.5eιπ(0.5)⟩, ⟨x2, 0.6eιπ(0.4), 0.5eιπ(0.5), 0.7eιπ(0.6)⟩ ⟨x3, 0.3eιπ(0.2), 0.6eιπ(0.8), 0.6eιπ(0.8)⟩; e2 = ⟨x1, 0.6eιπ(0.6), 0.8eιπ(0.4), 0.5eιπ(0.7)⟩, ⟨x2, 0.5eιπ(0.4), 0.6eιπ(0.9), 0.6eιπ(0.7)⟩, ⟨x3, 0.4eιπ(0.3), 0.7eιπ(0.7), 0.5eιπ(0.4)⟩.  (F̃3, E)c =  e1 = ⟨x1, 0.5eιπ(0.6), 0.6eιπ(0.8), 0.6eιπ(0.5)⟩, ⟨x2, 0.4eιπ(0.2), 0.7eιπ(0.9), 0.7eιπ(0.7)⟩ ⟨x3, 0.2eιπ(0.1), 0.7eιπ(0.8), 0.8eιπ(0.9)⟩; e2 = ⟨x1, 0.4eιπ(0.3), 0.8eιπ(0.9), 0.6eιπ(0.8)⟩, ⟨x2, 0.4eιπ(0.3), 0.8eιπ(0.9), 0.8eιπ(0.9)⟩, ⟨x3, 0.3eιπ(0.2), 0.7eιπ(0.9), 0.6eιπ(0.5)⟩.  Then (1(X,E)) c, (F̃1, E)c, (F̃2, E)c, (F̃3, E)c ⊇ (F̃ , E). Therefore, (F̃ , E) = (1(X,E)) c ∩ (F̃1, E)c ∩ (F̃3, E)c ∩ (F̃2, E)c = (F̃3, E)c . Theorem 3. Let (X, τ,E) be a complex single-valued neutrosophic soft topological space over X and (F̃ , E) ∈ CSV NSS(X,E). (F̃ , E) is a complex single-valued neutrosophic soft closed set and iff (F̃ , E) = (F̃ , E) Proof. Let (F̃ , E) be a complex single-valued neutrosophic soft closed set. Then the smallest complex single-valued neutrosophic soft open set that containing (F̃ , E) is equal to (F̃ , E). Hence, (F̃ , E) = (F̃ , E) Conversely, it is known that (F̃ , E) is a complex single-valued neutrosophic soft closed set and if (F̃ , E) = (F̃ , E), then (F̃ , E) is a complex single-valued neutrosophic soft closed set. M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7147 18 of 41 Theorem 4. Let (X, τ,E) be a complex single-valued neutrosophic soft topological space over X and (F̃1, E), (F̃2, E) ∈ CSV NSS(X,E). Then, (i) [(F̃ , E)] = (F̃ , E), (ii) (0(X,E)) = 0(X,E) and (1(X,E)) = 1(X,E), (iii) (F̃1, E) ⊆ (F̃2, E) ⇒ (F̃1, E) ⊆ (F̃2, E), (iv) [(F̃1, E) ∪ (F̃2, E)] = (F̃1, E) ∪ (F̃2, E), (v) (F̃1, E) ∩ (F̃2, E) ⊆ [(F̃1, E) ∩ (F̃2, E)]. Proof. (i) Let (F̃1, E) = (F̃2, E) Then (F̃2, E) a complex single-valued neutrosophic soft closed set.Hence (F̃2, E) and (F̃2, E) are equal. Therefore [(F̃1, E)] = (F̃1, E). (ii) Since 0(X,E) and 1(X,E) are always CSVNS closed sets, we have: (0(X,E)) = 0(X,E), and (1(X,E)) = 1(X,E). (iii) It is known that (F̃1, E) ⊆ (F̃1, E) and (F̃2, E) ⊆ (F̃2, E), so (F̃1, E) ⊆ (F̃2, E) ⊆ (F̃2, E). Since (F̃2, E) is the smallest complex single-valued neutrosophic soft closed set containing (F̃1, E), then (F̃1, E) ⊆ (F̃2, E). (iv) Since (F̃1, E) ⊆ (F̃1, E)∪(F̃2, E) and (F̃2, E) ⊆ (F̃1, E)∪(F̃2, E), then (F̃1, E) ⊆ [(F̃1, E) ∪ (F̃2, E)] and (F̃2, E) ⊆ [(F̃1, E) ∪ (F̃2, E)] and so, (F̃1, E) ∪ (F̃2, E) ⊆ [(F̃1, E) ∪ (F̃2, E)]. Conversely, since (F̃1, E) ⊆ (F̃1, E) and (F̃2, E) ⊆ (F̃2, E), then: (F̃1, E) ∪ (F̃2, E) ⊆ (F̃1, E) ∪ (F̃2, E). Besides,[(F̃1, E) ∪ (F̃2, E)] and is the smallest complex single-valued neutrosophic soft closed set that containing (F̃1, E) ∪ (F̃2, E). Therefore, [(F̃1, E) ∪ (F̃2, E)] ⊆ (F̃1, E) ∪ (F̃2, E). Thus, [(F̃1, E) ∪ (F̃2, E)] = (F̃1, E) ∪ (F̃2, E). (v) Since (F̃1, E) ∩ (F̃2, E) ⊆ [(F̃1, E) ∩ (F̃2, E)], and [(F̃1, E) ∩ (F̃2, E)]. is the smallest complex single-valued neutrosophic soft closed set that containing (F̃1, E)∩ (F̃2, E). Then, [(F̃1, E) ∩ (F̃2, E)] ⊆ (F̃1, E) ∩ (F̃2, E). Theorem 5. Let (X, τ,E) be a complex single-valued neutrosophic soft topological space over X, and let (F̃ , E) ∈ CSV NSS(X,E). Then: (i) [(F̃ , E)]c = [(F̃ , E)c]◦, (ii) [(F̃ , E)◦]c = [(F̃ , E)c]. M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7147 19 of 41 Proof. (i) (F̃ , E) = ∩{(G̃, E) ∈ τ c : (G̃, E) ⊇ (F̃ , E)} ⇒ [(F̃ , E)]c = [ ∩{(G̃, E) ∈ τ c : (G̃, E) ⊇ (F̃ , E)} ]c = ∪{(G̃, E)c ∈ τ : (G̃, E)c ⊆ (F̃ , E)c} = [(F̃ , E)c]◦. (ii) (F̃ , E)◦ = ∪{(G̃, E) ∈ τ : (G̃, E) ⊆ (F̃ , E)} ⇒ [(F̃ , E)◦]c = [ ∩{(G̃, E) ∈ τ : (G̃, E) ⊆ (F̃ , E)} ]c = ∩{(G̃, E)c ∈ τ c : (G̃, E)c ⊇ (F̃ , E)c} = [(F̃ , E)c]. 5. Complex Neutrosophic Soft Set-Based Cotangent Similarity Approach for Multi-Source Signal Analysis One such analytical technique for determining how similar two vectors (data) are is the cotangent similarity measure (CSM). It can be used in a wide range of fields that include ma- chine learning, image processing, and data analysis. CSM will be able to identify differences and dependencies that would have gone unnoticed in the absence of pure distance-based measure- ments by taking into account the geometric orientation of the data points. It is frequently used in data analysis, machine learning, and image processing. In the event of military operations, CSM can be used to classify potential targets based on the surveillance data. This is accom- plished by comparing a target’s operational characteristics to those of known classified profiles. The likelihood that the threats will be compared and classed appropriately is then increased by estimating the degree to which a target fits into pre-existing categories of threats. The classification and pattern recognition tasks are aided by a collection of machine learning and visualization methods. Sig: Pearson Correlation, Elbow Method, Spline Interpolation, PCA, K-Means, K-Means++, and t-SNE. 5.1. Cotangent similarity measures Let Ti = (Tik, Iik, Fik), Cj = (Tjk, Ijk, Fjk) are two CNSSs. Each set is represented by three components for each feature k: Tik: Truth-membership of feature kforTi. Iik: Indeterminacy-membership of feature kforTi. Fik: Falsity-membership degree of feature kforTi. The values typically satisfy: Tik, Iik, Fik ∈ [0, 1] Similarly, for object Cj : Tjk: Truth-membership of feature kforCj . Ijk: Indeterminacy-membership of feature kforCj . Fjk: Falsity-membership degree of feature kforCj . The values typically satisfy: Tjk, Ijk, Fjk ∈ [0, 1]. M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7147 20 of 41 The cotangent similarity measures between Ti and Cj is defined as: CotangentSCNSS(Ti, Cj) = 1 n n∑ k=1 {cot[π 4 + π 12 (|Tik − Tjk|+ |Iik − Ijk|+ |Fik − Fjk|)]} 5.2. Real-time military target classification with cotangent similarity and ad- vanced surveillance data integration Using the multi-source signal samples S = {S1, S2, S3} to accurately categorize the opera- tional targets T = {T1, T2, T3} and then applying them to the appropriate operational classes C = {C1, C2, C3} is the primary task in the current surveillance and reconnaissance systems. The available information on each target is inherently ambiguous, imprecise, and diverse; it in- cludes heat emissions, radar signatures, electronic monitoring, and other signal measures. This uncertainty necessitates a mathematical procedure that can simultaneously handle truth, inde- terminacy, and falsity. The relationship between targets, signals, and classes in a multifaceted environment is related in terms of event-based measures (e) and their corresponding measures in the operational classification table (Table 2). However, overlapping values, inconsistent reli- ability, and nonlinear relationships between features make it impossible to directly compare a response of raw signals between targets and classes. This is resolved by proposing a Cotangent similarity measure based on the complex neutrosophic soft set (Cotangent CNSS) framework, which enables the measurement of each target-class pair’s similarity. It considers the joint contri- bution of the truth-membership, indeterminacy-membership, and falsity-membership functions, normalized with respect to n = 3 attributes, in each pair (Ti, Cj). Cotangent similarity scores are generated as a result, and these can be used to rank candidate target profiles in relation to operational class templates. Table 1: Matrix of Target Intelligence Features Based on Cotangent Similarity (T × S) S1 S2 ... Sn T1 T11, I11, F11 T12, I12, F12 ... T1n, I1n, F1n T2 T21, I21, F21 T22, I22, F22 ... T2n, I2n, F2n ... ... ... ... ... Tm Tm1, Im1, Fm1 Tm2, Im2, Fm2 ... Tmn, Imn, Fmn The raw surveillance parameters for each target Ti, as gathered from each data source Si, are displayed in this table 1. These are represented by the 3-dimensional feature vectors found in each cell, gives each target-source combination a thorough feature profile for preliminary analysis and threat modeling. M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7147 21 of 41 Table 2: Operational Classification Matrix for Target Surveillance Data (T × S → C) Row-I S1 S2 S3 Row-II C1 C2 C3 T1 0.5eιπ(0.6) 0.7eιπ(0.5) 0.3eιπ(0.3)  0.6eιπ(0.7) 0.4eιπ(0.4) 0.3eιπ(0.4)  0.3eιπ(0.2) 0.5eιπ(0.5) 0.8eιπ(0.3)  S1 0.3eιπ(0.4) 0.5eιπ(0.7) 0.7eιπ(0.7)  0.7eιπ(0.6) 0.6eιπ(0.5) 0.2eιπ(0.3)  0.5eιπ(0.8) 0.4eιπ(0.7) 0.9eιπ(0.3)  T2 0.4eιπ(0.6) 0.7eιπ(0.9) 0.5eιπ(0.7)  0.2eιπ(0.4) 0.4eιπ(0.9) 0.6eιπ(0.3)  0.7eιπ(0.7) 0.4eιπ(0.3) 0.6eιπ(0.6)  S2 0.2eιπ(0.5) 0.3eιπ(0.2) 0.7eιπ(0.3)  0.6eιπ(0.6) 0.7eιπ(0.8) 0.4eιπ(0.3)  0.3eιπ(0.6) 0.6eιπ(0.7) 0.8eιπ(0.9)  T3 0.7eιπ(0.9) 0.6eιπ(0.5) 0.8eιπ(0.8)  0.2eιπ(0.4) 0.8eιπ(0.5) 0.7eιπ(0.6)  0.6eιπ(0.9) 0.3eιπ(0.6) 0.2eιπ(0.3)  S3 0.4eιπ(0.4) 0.7eιπ(0.5) 0.5eιπ(0.9)  0.6eιπ(0.2) 0.3eιπ(0.5) 0.4eιπ(0.4)  0.7eιπ(0.2) 0.3eιπ(0.8) 0.2eιπ(0.3)  This table 2 delineates the cotangent similarity values calculated by comparing the feature vectors of targets with those of established classified operational profiles (e.g., C1: high-value asset, C2: decoy, C3: communication node). The raw cosine similarity between each target (T) and its corresponding surveillance data (S) is presented in Row-I (T × S). Meanwhile, the cotangent similarity between each data source vector and the classified operational profiles is displayed in Row-II (S × C). This method facilitates the prioritization of actions (e.g., attack, monitor, ignore) by classifying or clustering targets according to their correlation with well- defined threat or operational categories. All target-classification pairings are then averaged by adding the cotangent values that were determined for each pair of targets and categorized profiles. The values of the resulting cotangent similarities are: CotangentSCNSS(T1, C1) = 0.4913 CotangentSCNSS(T1, C2) = 0.6653 CotangentSCNSS(T1, C3) = 0.5785 CotangentSCNSS(T2, C1) = 0.5287 CotangentSCNSS(T2, C2) = 0.4937 CotangentSCNSS(T2, C3) = 0.3576 CotangentSCNSS(T3, C1) = 0.4110 CotangentSCNSS(T3, C2) = 0.4604 CotangentSCNSS(T3, C3) = 0.4221 Table 3: Cotangent Similarity scores between signal samples (S1 − S3) and class templates (T1 − T3) Cot SM S1 S2 S3 T1 0.4913 0.6653 0.5785 T2 0.5287 0.4937 0.3576 T3 0.4110 0.4604 0.4221 The degree to which each signal sample resembles the three class templates is shown in the cotangent similarity table 3. The template is most similar to S1, although the score is average in the case of S1, where the maximum similarity with T2 is 0.5287. The highest match in S2 with T1 is 0.6653, which is also the highest value in the entire table. This indicates that S2 is classified under template T1 in a strong and reliable manner. The alignment with T1 in S3 has the highest similarity, at 0.5785, which falls within a moderate range and is not as definitive as S2. Overall, the results show that T1 is capable of drawing in two signals, S2 and S3, indicating M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7147 22 of 41 that this template is a main or special active profile. Only S1 is connected to T2, and their connection is not very strong. T3 may be utilized as a weak or background category because it does not appear to be the best match to any signal and all of its scores are below the best ones. While S1T2 and S3T1 require closer interpretation or further investigation, S2T1 may be regarded as a specific classification in a realistic option. 6. Result and Discussion Targets T1 and T3’s similarity associations are projected onto two principal components (PC1 = 0.65, PC2 = 0.35), whose variability is fully captured in two dimensions, in the PCA clustering plot shown in Fig. 1. The targets are positioned according to how well they align with the centroid clusters of the groups, C1 and C2. It reaches Cluster C2’s location on PC1’s far right, indicating great stability and centrality in the data structure, in the instance of T1. This suggests that T1 is the most appropriate and representative aim. In contrast to both clusters, the point for T2 is located higher along PC2. It is more of an outlier than a member of the cluster because of its high and unique status, which indicates unique behavior.The point in T3 is located on the bottom-left, but both PC1 and PC2 have scaled it out. Given that its alignment with T1 and T2 is weaker and more divergent, it is distantly related to Cluster C1. In conclusion, T1, which is securely attached to Cluster C1, is the strongest target and the most stable one. T3 is the weakest, closely matching Cluster C1, whereas T2 is independent and detached, exhibiting distinct but less coherent behavior. Figure 1 Even though the drawing is shown in three dimensions, Fig. 2 shows that PC1 (65%), PC2 (35%), and PC3 do not contribute to the variance, suggesting that the structure is essentially two-dimensional. The centroid is located near the two apparent clusters that are formed from the points: Cluster 1 on the left and Cluster 2 on the right. The light blue point is more of an outlier or transitional point because it is higher along PC2 rather than between the centroids. Because it is the most central and constant profile, the Cluster 2 point (the right side) is under- stood as a target and is hence the most representative. The point on the left is more divergent, M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7147 23 of 41 indicating less resemblance, whereas the higher light blue point is linked to less cohesive and united conduct. The clustering generally points to a hierarchy where a target is the reference, an outlier is an outlier, and a diverging instance is the last one. Figure 2 Since this would divide the three targets (T1, T2, T − 3), the K-Means plot (Fig. 3) employs k = 2 on the Cot SM data. The color coding indicates cluster membership, while the x-axis (S1) and y-axis (S2) represent signal similarity coordinates. When it comes to T1, the point at the top where S2 = 0.67 is clearly separated and is categorized as belonging to Cluster 1 (yellow). It is the strongest and most distinct target in the dataset since it ranks 1 in terms of similarity. T2’s point, which is comparable to Cluster 0 (purple), is on the right where S1 = 0.53 and S2 = 0.49. While not quite enough to match the dominant T1 cluster, this indicates a moderate level of resemblance. In the example of T3, Cluster 0 (purple), S1 = 0.41, S2 = 0.47, and the point is at the lower left. As a result, T3 is the poorest aligned target overall and exhibits weak similarity. All things considered, T1 forms its high-similarity cluster, highlighting its superiority and distinction from the others. Lower and more similar degrees of similarity are indicated by the T2 and T3 clustering towards one another. This demonstrates that T1 has the greatest influence, while T2 and T3 have a weaker, secondary link. M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7147 24 of 41 Figure 3 The top three maximum similarities are displayed in Fig. 4’s K-Means++ clustering plot. Cluster assignments are coloured, and maxima are circled in red to improve emphasis. Cluster centers are indicated by red X markings. The global maximum level of 0.665 is indicated by the double circle that marks the location on the far right in the instance of T1 (T1, S2). This demonstrates that T1, which is clearly distinguished from the others, is the most powerful and dominant target. Instead of being in the cluster’s centroid, the point in T3 will be in the lower-left quadrant. It is one of the weakest targets, and its inferior position attests to a slight resemblance. The circled maximum in the yellow location (highlighted on the upper left) indi- cates another data value with the best performance, but it is still less than the T1S2 maximum. Although it does not win out globally, this contributes to the cluster balance. T1 is the most potent target, producing the worldwide maximum (0.665) and focusing intently on its centroid, to sum up. Other emphasized maxima show supplementary substantial contributions that do not dominate T1, and T3 is the weakest and not connected to dominant regions. M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7147 25 of 41 Figure 4 A reduction of the PCA applied to the Iris data set is used in the K-Means clustering plot (Fig. 5), where the data is shown on the two main components. Red X markers, which show the cluster centers, are used to color the individual samples according to the cluster to which they belong. Each sample counts as one point. The samples are highly concentrated when PC1 is negative in the case of the left clustered (teal points). With low levels of within-cluster variance and high levels of cohesiveness, the centroid is located in the center of the cluster. This indi- cates a grouping that is precise and unambiguous. In PC1 and PC2, the middle cluster samples (purple spots) are dispersed more widely. The dispersion suggests that there is less cohesion than with the teal cluster, even if the centroid is near the geometrical center. This suggests that this group is more internally variable. The samples in the most right cluster (yellow points) are moderately dispersed in PC2 and broadly dispersed in PC1’s positive values. In order to balance the wider range of data points, the centroid is likewise positioned near the center of mass. The cluster stands apart from the other two, although selling more broadly. In summary, the yellow cluster is bigger but clearly split, the purple cluster is somewhat cohesive but more variable, and the teal cluster is the smallest and tightest. The centroids’ positions provide for successful separation within the three clusters following PCA reduction. M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7147 26 of 41 Figure 5 The K-Means clustering graphic in Fig. 6 maps the samples onto two main components for visualization and applies PCA reduction to the normalized Iris datasets. With its unique cluster colors and red markers of X that represent the computed centroid, each point represents a single sample. The samples are concentrated along the negative PC1 and PC2 in the case of the left cluster (teal points). The centroid has a tight structure, a high degree of cohesion, and is near the geometric center. The fact that this cluster is the most compact suggests that there is little variation within it. Samples in the middle section of the PC1 and PC2 are widely dispersed in the case of the middle cluster (purple points). In comparison to the teal cluster, the centroid is at the center, and the dispersion is larger, suggesting less cohesion. The members in this category are more diverse. The right cluster (yellow dots) contains samples that are modest in PC2 and broad in positive PC1 values. Despite having a wider spread, the centroid centers the cluster by balancing this distribution. Though not as compact as the teal cluster, this cluster is clearly unique from the others. The yellow cluster is wide and well-separated, suggesting good partitioning of the normalized data set after PCA reduction, the purple cluster is bigger and of moderate cohesion, and the teal cluster is the most compact and internally consistent cluster overall. M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7147 27 of 41 Figure 6 A line represents a sample, and the colors represent the species setosa (red), versicolor (green), and virginica (blue) in the 3D t-SNE representation of the Iris data in parallel coor- dinates shown in Fig. 7. The three dimensions of t-SNE are represented by these axes, and the line patterns show how samples of each class are projected onto the reduced features space. In the case of setosa (red lines), the values in the lower portions of the dimensions are aggre- gated, and the cluster is clearly identified. The general downward inclination of t-SNE1 and t-SNE2 is noteworthy and illustrates the species’ high degree of coherence and distinctiveness. Versicolor (green lines) has more interlacing paths. They overlap with both setosa and virginica and occupy an intermediate position in all dimensions. Versicolor is more intermediate in the structure of the Iris dataset since it is less segregated. Particularly along t-SNE1 and t-SNE3, the values are nearer bigger values in the case of virginica (blue lines). In addition to being more regular than the green, the blue lines exhibit a reasonably compact grouping, despite some versicolor overlap. The traditional structure of the Iris data set is demonstrated by the t-SNE parallel coordinates, which show that the most distinct and well-separated species, setosa, is at the center with versicolor and virginica, which are rather compact and partially overlap with the former, indicating the boundary between the two. M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7147 28 of 41 Figure 7 The tie between X̀ and Y is depicted in Fig. 8, where the bluish line represents the spline interpolation and the red dots represent the initial data points. The noise in the raw data is removed, and the curve emphasizes local variances. The curve climbs through the mid-range X values to a local maximum at zero after beginning at a moderate value of Y 0.26 at low X̀, reaching an early high at 0.73, and a local minimum at -0.54. After this drop, the functional steadily recovers and moves back into the positive area. The green-marked locations of X = 8 and Y = 0.83 represent the world’s maximum point. Higher than the first two peaks, this is the spline’s maximum alignment to the data. Other noteworthy positive numbers, such as 0.59 and 0.61, are also noted, but they fall short of the final highlighted maximum. In sum- mary, the spline-smoothed curve exhibits a cyclic wave-like tendency with alternating peaks and troughs. The dominant maximum (0.83) supports the strongest functional reaction, whereas the successive increases and declines are the intermediate changes that surround the main trend. Figure 8 With normal axes of S1 and S2, Fig. 9 employs a K-Means clustering normalized plot with k = 2 on the Cot SM values. With the scale displayed on the right, the targets (T1 − T3) are M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7147 29 of 41 on normalized coordinates, and the color designates the clusters. T1 is assigned to Cluster 1 (yellow), and the point is located in the upper-right corner (S1 = 1.0, S2 = 1.0). T1 is the strongest and most common target, as evidenced by the fact that it is at the extreme and lies at maximum normalized values. Similar to Cluster 0 (dark blue), the point in T2 is on the far right and in the lower region (S1 = 0.1, S2 = 0.17). This indicates partial but uneven strength due to high performance in S1 and poor fit in S2.Cluster 0 (dark blue) is also responsible for the point in T3, which is located at the bottom-left (S1 = 0.0, S2 = 0.0). T3’s low normalized values confirm that it is the weakest and least aligned target. T1 is the most powerful target, forming its own high-value cluster. T2 and T3 are grouped in the weaker cluster, with T2’s contribution being somewhat greater than T ’s. Figure 9 The Elbow Method (Fig. 10, left) and the Silhouette Method (Fig. 10, right) are used to determine the number of clusters (k) that produce the best result: the evaluation plots are clustering. Elbow Method: As k increases, inertia reduces. The inertia is at its highest (1.627) at k = 1, suggesting that the samples are packed together and that a satisfactory fit is not pos- sible. The inertia value drops significantly (to 0.556) with k = 2, which improves the clustering efficiency significantly. By k = 3, inertia decreases to 0.000, but this is taking into account slight overfitting because each point is really a cluster of its own. The most mild compromise between simplicity and compactness is shown by the strong peak in k at 2. Plotting the silhouette score with k = 2 in the Silhouette Method yielded a result of 0.154. Despite its tiny size, this score is encouraging because it shows that clusters can be dispersed but are not very firmly linked together. Nonetheless, it favors the partition k = 2 as the data’s most important partition. Overall, k = 2 is determined to be the optimal number of clusters by both evaluation tools. This is demonstrated by the Elbow Method’s reduction of inertia and the Silhouette Method’s maximum score, albeit at a moderate level. M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7147 30 of 41 Figure 10 The Pearson correlation matrix of variables (B1−B4 and Cot SM) is shown in a 3D surface plot in Fig. 11; the correlation value is shown in both surface height and color intensity. The degree of linear association is shown by the scale, which ranges from 0.76 (lower and dark pur- ple) to 1.00 (higher and brilliant yellow/green). When it comes to Cot SM, the self-correlation between Cot SM and Cot SM should be at most 1.00, creating a tall peak at each axis junction. Cot SM has high relationships with these factors, as evidenced by the strong correlations found between it and B1 (0.96) and B3 (0.95). Cot SM (0.96) and the same (1.00) both display the highest values in the case of B1. It shows moderate correspondence between dimensions, with lower but positive correlation coefficients with B2 (0.77) and B4 (0.81). Compared to other variables, B2 exhibits a lower trend. Its values against Cot SM (0.76) and B1 (0.77) are around the bottom spectrum, making it the weakly aligned variable, even though it still has a moder- ate correlation with B3 (0.81) and B4 (0.77). It is discovered that B3 has a strong connection with both Cot SM (0.95) and B3 (1.00). Although B 3 falls into the category of more potent influencers, other combinations, such as those with B2 (0.81) and B4 (0.77), are intermediate. B4 has a strong association with Cot SM (0.85) and B1 (0.81), and its correlation with itself is equal to 1. It has lower and more steady B2 (0.77) and B3 (0.77) values. The most dependable and resilient variables in the summary are Cot SM, B1, and B3, all of which have a constant correlation with one another (0.95). The values on the lower end of the scale ( 0.76 -0.81) indicate that B2 is the weakest contributor, whereas B4 is somewhat aligned. M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7147 31 of 41 Figure 11 The normalized Pearson correlation matrix of the variables (B1−B4 and Cot SM) is shown in Fig. 12 as a 3D surface plot with values scaled from 0.0 to 1.0. The surface’s peaks and color shading (red for lower correlations, blue for higher correlations) show the relative values of associations after normalization. As expected, the diagonal self-correlation in the Cot SM instance is 1.00. Off-diagonal values typically show weak normalized connections, such as 0.00 with B2 and 0.00 with B4. Only a mild peak (0.35) is seen in Cot SM-B1, suggesting a weak but comparatively stronger link. The diagonal (1.00) and Cot SM (0.35), in the instance of B 1, have the highest values. However, its impact on variables is weakly standardized, and its association with B2 (0.03) and B3 (0.19) is not significant. The values for B2 are consistently around the bottom of the scale, with B3 showing the strongest correlation. This yields B2 the lowest normalized correlation, and B3 the highest normalized correlation on its diagonal (1.00) and secondary values of 0.19 to B1 and 0.17 to B4, but virtually zero elsewhere. This suggests that B3 is not dominant but rather moderately orientated. The diagonal maximum in B4 is 1.00, and Cot SM has the strongest off-diagonal value (0.21). The others are weak (e.g., B2 = 0.02, B3 = 0.17), but B4 falls in the middle of the normalized strength range. In summary, Cot SM and B1 are highlighted as somewhat stronger factors by the normalized correlation surface; yet, their peaks are smaller than those of the un-normalized matrix. B2 is always the weakest, followed by B3 and B4, which are average but dispersed. The overall pattern indicates that contrast has decreased after normalization, when the strongest correlations are highlighted. M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7147 32 of 41 Figure 12 7. Aspect-Method Hybrid Analysis of Signal-Template A multifaceted view of signal-template alignment is provided by combining aspect-based interpretation (based on stability, representativeness, and target divergence) with method-based evaluation (PCA, K-Means, t-SNE, spline smoothing, correlation analysis). Because it considers the possibility that the targets may be powerful in one analytical perspective and weak in another, such a hybrid structure does not rely too heavily on any one viewpoint. Since both theoretical and practical approaches can be used to support the ideas, the combination of the elements and approaches gives the outcomes their strength. 7.1. Aspect-based Analysis Three main criteria can be used to view the targets (T1−T3) from an aspectual perspective: stability, divergence, and representativeness. The most representative objective is usually T1. Its proximity to Cluster C2’s centroid in 2D and 3D PCA projections further demonstrates that it is at the center of the data structure. K-Means and K-Means++ plots support this, showing that T1 has the worldwide maximum similarity (0.665 at S2) and is therefore utilized as a cluster formation indication. The extreme values of T1 (0.0,1.0) in normalized clustering with (Fig. 9) further strengthen its hegemony. T2 denotes average yet distinctive behavior. It regularly adopts intermediate positions and does not adhere to any centroid. It appears to be an outlier on PC 2 (i.e., in a high position, showing variability) in PCA (2D and 3D). Its partial divergence, which is not significant enough to be coherently destabilizing but significant enough to be dominating, is shown by K-Means plots, which position it with T3 but align more with S1. The least stable is always T3. It wanders to the negative axes in PCA, which are not at the center but rather weakly connected to C1. It belongs to the same group as T2 in clustering, but its raw and normalized similarity scores are the lowest. Due to its location close to the M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7147 33 of 41 bottom-left quadrant or the divergent and weak alignment, it is either the most unreliable or the most steady target. Thus, aspect-wise hierarchy: T1 = Strongest(stable, representative) → T2 = Intermediate(divergent, transitional) → T3 = Weakest(unstable,marginal). 7.2. Method-based Analysis The information presented by both methods shows that the target relationships are multi- faceted. Method-based comparison makes it easier to see how different approaches both comple- ment and contradict one another rather than handling them independently. PCA does not result in variance loss and reduces dimensional complexity (PC1 = 0.65, PC2 = 0.35). It indicates that T2 is central in C2, that T2 is an outlier, and that T3 is marginal. The 3D image, which shows the intermediate position of T2 and the negative position of T3 along PC1/PC3, helps to narrow this down. PCA works well for structural mapping and visibility. The explicit grouping is the main focus of K-Means and K-Means++. The circled maximum (0.665) indicates that T1 pulls its own odd cluster (Cluster 1, yellow). T2 and T3 are grouped together in weaker groups that show diminished commonality. K-Means++ refines the results by using centroid separation and maxima marking. Structural cohesiveness is justified by the extended clustering of Iris PCA enhancements, which show broader purple/yellow clusters and tight teal clusters. This pertains to the situation where representativeness (T1 even teal) and divergence (T2 T3 purple/yellow) are explained by tight versus loose clusters. Relational overlaps are shown via t-SNE. T2 cuts across groups and fully overlaps T1 and T3, much like versicolor in Iris. T1 is comparable to setosa in that most of them are distinct, stable, and free of overlaps. Similar to virginica, T3 is tiny but crosses boundaries. This method brings to light edge effects and transition regions that PCA misses. Spline-smoothed functional curves, which show a wave-like trend with local dips and a global maximum of 0.83, are indicative of functional variability. The idea that sim- ilarity is a dynamic process where T1 predominates with peak alignment rather than a linear phenomenon is supported by this, which shows periodic changes that clustering cannot explain. Relationships between variables and targets are shown by Pearson correlation surfaces. While normalization reduces contrast, leaving only T1B1 reasonably robust, raw correlations (Cot SM of B1 and B3 are over 0.95), on the other hand, have significant stability markers. Because raw views overemphasize strength and normalized views downplay complexity, this dichotomy is harmful to presumptions. Method-wise convergence: All of the approaches concur that T1 is superior and T3 is inferior. The evaluation of T2 is the only area where there is a difference: PCA classifies it as an outlier, K-Means pairs it with T3, and t-SNE classifies it as transitional. The methodology of triangulation shows that T2 is neither totally stable nor entirely marginal; rather, its identity is contingent upon the method employed. 8. Four- Dimensional Evaluation of Analytical Techniques To ascertain the efficacy of each analytical technique, the similarity relationships between targets T1 − T3 are also assessed by combining aspect-based and method-based techniques. The four evaluation dimensions-visibility, associativity, dynamicity, and scalability-are used to standardize comparisons. To provide a quantitative basis, numerical results of PCA loadings, Cot SM maximum, clustering inertia, silhouette results, and correlation coefficients are included. M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7147 34 of 41 Table 4: Aspect-Method Hybrid Evaluation of Analytical Techniques Factors PCA(2D/3D)K- Means/K- Means++ t-SNE(3D, Parallel Coordi- nates) Spline- Smoothed Curves Correlation Sur- faces(Raw/ Normal- ized) Visibility Strong(PC1= 0.65, PC2=0.35;T− 1 → C2, T2 high PC2, T3 bottom left) Very Strong(Global max 0.665 at T1 − S2 circled; cen- troids shown by X) Strong(distinct separation: Setosa- like=T1, transitional overlap=T2, marginal=T3) Strong(global max Y=0.83 at X ≈ 8 clearly high- lighted) Very Strong raw(Cot SM- B1=0.96, B3=0.95); Weak nor- malized(Cot SM-B1=0.35 only) Associativity Strong(T1 ↔ C2;T3 ↔ C1weakly) Strong(T2,T3 in Cluster 0; T1 indepen- dent cluster) Strong(overlap patterns: T2 bridges T1&T3) Medium- Strong(peaks/ troughs re- flect sec- ondary associations) Strong raw(B1&B3 tightly linked to Cot SM,≥ 0.95); Weak normalized(0.00- 0.21 across others) Dynamicity Medium(stable projection, PC3 ≈ 0 adds little variance) Strong(Elbow: inertia drop 1.627 → 0.556 at k=2; Silhou- ette=0.154) Strong( in- terlacing of trajecto- ries shows transitional dynamics) Very Strong(wave cycle: rise 0.26 → 0.73, dip -0.54, peak 0.83) Medium(raw exaggerates highs; nor- malization suppresses contrast, exposing fluctuation sensitivity) Scalability Strong(2D/3D scale inter- pretable with few targets; PC loadings stable) Medium- Strong(works for k=2 but trivializes at k=3 with in- ertia=0.000) Medium (good for small sam- ples; overlaps increase with more classes) Strong( gen- eralizable trend pat- tern) Medium (Raw shows clear peaks; normaliza- tion reduces contrast to near 0-0.35, weaker scal- ing) M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7147 35 of 41 Figure 13 Figure 14 M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7147 36 of 41 Figure 15 Figure 16 9. Conclusion The targets show a consistent hierarchy, according to the relative comparison of PCA, K- Means, K-Means++, t-SNE, correlation surfaces, and spline-smoothed curves. T1 consistently yields the worldwide maximum similarity values and is the strongest and most representative target. It also exactly fits the centroid. T2 is transitional or unsteady because it acts as an intermediary, rotating between clustering with T3 (K-Means) and being an outlier (PCA). T3 will be the least coherent and weakest target; it will be situated on the border between methods M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7147 37 of 41 and be only loosely connected to Cluster C1. T1 is structurally dominating, T2 is method dependent, and T3 is marginal, according to this triangulation methodology. A powerful hybrid perspective on the interpretation of signal-template alignment is provided by combining aspect (representativeness, stability, and divergence) and method-based techniques. 10. Limitations There are several limitations to the analysis. Because it only focuses on three targets (T1−T3) and the signals associated with them, the dataset itself is likewise narrow and low in scope, which restricts the potential to extrapolate the results to more complicated or broad datasets. The third principal component in PCA, PC3 = 0, adds very little variance, which lowers the 3D projections’ informative value and could be deceptive about nonlinear relationships. Since T2 behavior is susceptible to algorithmic assumptions rather than consistent structural identity, it is very method-dependent and alternates between outlier and transitional behavior when PCA, K-Means, or t-SNE are applied. The fact that the Elbow and Silhouette algorithms agree on k = 2, but the silhouette score (0.154) suggests a weak cohesiveness and separation, is another flaw in the algorithm that cluster validation highlights. Because scaling reduces the contrast of correlation matrices, normalization introduces an extra constraint that tends to overemphasize homogeneity and obscure important correlations. Finally, the framework cannot be used to real- time or changing data settings because it is static and requires constant similarity scores and projections without taking into account temporal or dynamic modification of signal behavior. 11. Future work There are several ways that future study can fill in the gaps in this paradigm. Larger, higher- dimensional datasets with more targets and signals to be expanded can be used to demonstrate scalability and stability. To find structural nuances that linear PCA might miss, nonlinear dimensionality reduction techniques like kernel PCA or UMAP may be added. Temporal clus- tering or streaming PCA could be used to improve the way signals evolve over time and get around the framework’s current static nature. It is possible to integrate fuzzy-based or neu- trophilic multi-criteria decision-making techniques to effectively handle uncertainty, particularly when dealing with transitional goals like T2. In order to make the hierarchy more structured and flexible, the hybrid aspect-method model can also be improved by adding the weighted evaluation of visibility, associativity, dynamicity, and scalability. Validation across a range of applications, such as financial data, biomedical signal processing, and military surveillance, would demonstrate robustness and transferability. Finally, to improve the framework’s prac- ticality on a real-time or even large-scale basis, automated visualization pipelines of PCA, K-Means, t-SNE, and spline smoothed curves should be added. This will eliminate manual bias and simplify understanding. Acknowledgements The authors extend their appreciation to the Deanship of Scientific Research at Northern Border University, Arar, KSA for funding this research work through the project number ”NBU- FFR-2025-2727-11”. Conflicts of Interest: The authors declare that there are no conflicts of interest regarding the publication of this paper. Availability of Data and Materials: All the data and materials are provided in the manuscript. M. M. Saeed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7147 38 of 41 References [1] Y. Al-Qudah, N. Hassan, Operations on complex multi-fuzzy sets. Journal of Intelligent and Fuzzy Systems, vol. 33(3), pp. 1527-1540, 2017. [2] H. Garg, D. 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