This paper develops a topological framework based on type-1 neutrosophic sets, extending the study of uncertainty beyond classical and intuitionistic fuzzy structures. We investigate fundamental operators—interior, closure, and exterior—on type-1 neutrosophic sets X[I], and establish separation axioms for type-1 topological spaces Ti[I], i = 0, 1, . . . , 4). Furthermore, we analyze continuous and contra type-1 neutrosophic functions, exploring their structural properties within this framework. By isolating type-1 neutrosophic sets and formalizing their topological behavior, our results provide new insights into neutrosophic topology and lay the groundwork for broader applications in mathematical analysis, algebra, and related fields. \
The present manuscript delves into the ideology of best proximity point under the purview of neutrosophic fuzzy metric space. For this purpose, the concept of a new type of mapping named α-proximal admissible mapping has been defined under the realm of the said space. Also for the contractive condition, the pre-existent ideologies of Ciric and rational contraction has been combined. Notably, for the sake of best proximity, concept of α-ψ proximal contraction has also been considered. Here α and ψ can also be regarded as control functions. Additionally, some examples and applications are also deduced in support of the established results, for some particular conditions. To the best of known literatures, the results are established for the first time in the mentioned area.
This paper addresses the problem of utilizing auxiliary information to estimate the population mean under neutrosophic stratified random sampling. Within the framework of neutrosophic statistics, we propose a neutrosophic combined logarithmic type estimator that effectively accounts for the uncertainty and indeterminacy inherent in survey sampling. The suggested estimator’s bias and mean squared error (MSE) are expressed up to the first order of approximation, and ideal circumstances for reducing the MSE are determined. Neutroposophic combined mean, neutrosophic combined ratio, and neutrosophic combined difference estimators are theoretically compared with the suggested estimator.The analytical findings show that in some real-world situations, the suggested estimator performs better than its conventional competitors. Additionally, line and radar chart representations and empirical data analysis are used to assess the performance of the suggested estimator. The results provide useful information for practitioners and survey statisticians who use Neutrosophic stratified random sampling when uncertainty is present.
Decision-making in complex environments often involves ambiguity, inconsistency, and incomplete information—conditions inadequately addressed by classical fuzzy or even interval-valued neutrosophic systems. Building upon the earlier foundations established by Kamal Nasir and Priyanka (2025) in their works on the Multi-Expert, Multi-Criteria Neutrosophic Fuzzy Assignment Problem and its Interval-Valued extension, this study introduces a Generalized Neutrosophic Fuzzy Assignment Model (GNFAM) designed to operate effectively under deep uncertainty.The proposed framework extends the neutrosophic structure to generalized neutrosophic numbers, enabling the representation of varying degrees of truth, indeterminacy, and falsity in flexible forms (single-valued, interval-valued, or polygonal). A multi-expert, multi-criteria evaluation mechanism is integrated through a Priority-Optimized Aggregation Mean, allowing the model to synthesize heterogeneous expert judgments across diverse decision dimensions. The decision process is optimized using a generalized score function, facilitating precise ranking and assignment under uncertain and conflicting data.A real-world case study illustrates the model’s applicability and robustness, demonstrating superior performance compared to conventional fuzzy and interval-valued neutrosophic assignment frameworks. The proposed GNAM framework thus advances neutrosophic decision science by providing a unified, extensible, and computationally efficient tool for multi-criteria decision-making under deep uncertainty.
In this paper, the fourth dimension of neutrosophy, that of "Turiyam," is examined as it symbolizes the "absolute" or "transcendental" essence that preexists and permeates the truth triad, indeterminacy, and the falsity trio. Although the current state of Neutrosophic Sets works perfectly to represent the notion of "ambiguity," they are inefficient in incorporating the essence wherein such truth measures oscillate. Thus, the proposed model, Fuzzy Neutrosophic Turiyam Set (FNTS), will have its corresponding algebraic operator functions and "Law of Turiyam Non-Exclusion." Moreover, models involving distance function in four-dimensional space as well as the hyper-polyhedron will be considered. Quantum computing applications and modeling of socioeconomic "Black Swan" events validate that the proposed FNTS model is significantly more stable in chaotic situations than the current models proposed for neutrosophic settings.
