Key Characteristics of Quadri-Partitioned Neutrosophic Riemann integrals and Quadri-Partitioned Neutrosophic Soft Topological Spaces

Neutrosophic Set Theory (NST) is an extension of Intuitionistic Fuzzy Set Theory (IFST). While IFST relies on two possibilities for the complete depiction of a set, neutrosophic set theory familiarizes an additional third possibility, thus providing a more delicate representation. Our research builds upon a further extension of neutrosophic set theory, known as quadri-partitioned neutrosophic set theory (QPNST), which brings in a fourth possibility for a more detailed and complete description of sets. In this study, we define the Riemann Integral Theory (RIT) within the framework of QPNST. This opens new doors for probing the properties and characteristics of the Riemann integral in this extended context. One strategic concept that arises in this work is the level cut. In QPNST, the level cut is defined as a four-tuple (i, j,k, l), which represents the different possibilities inherent in the theory. The notion of the Quadri-Partitioned Neutrosophic Riemann Integral Theory (QPNRIT) is explored numerically inthis study, and the results are systematically presented in tabular form. This numerical approach sheds light on the integral’s properties and facilitates the understanding of its behavior within the QPNST framework. This study explores quadripartitioned neutrosophic soft topological spaces, extending neutrosophic set theory (NST), which incorporates three membership values: true, false, and indeterminacy. The study introduces new concepts such as QPNS semi-open, QPNS pre-open, and QPNS ∗b open sets, and builds on these to define QPNS closure, exterior, boundary, and interior. A key development is the definition of a quadripartitioned neutrosophic soft base, which plays a central role in these topological structures. The paper also explores the concept of a quadripartitioned neutrosophic soft sub-base and discusses local bases, as well as the first- and second-countability axioms. The study further examines hereditary properties of these spaces, distinguishing between inherited and non-inherited properties. Key results include that a quadripartitioned neutrosophic soft subspace of a first-countable space is also first-countable, and a second-countable subspace of a second-countable space remains second-countable. It also highlights the relationship between second countability and separability in these spaces, asserting that a second-countable quadripartitioned neutrosophic soft space is separable, though the converse is not always true. This work lays the foundation for further research in neutrosophic soft topologies.