A Breakthrough Approach to Quadri-Partitioned Neutrosophic Soft Topological Spaces

Neutrosophic set theory, an advanced framework for error reduction, extends fuzzy sets (FSs) and intuitionistic fuzzy sets (IFSs). It enhances effectiveness by refining the definition of indeterminacy, a concept for situations where values cannot be precisely determined. In this paper, we propose dividing indeterminacy into two components based on membership: relative truth (RT), which leans toward truth, and relative falsehood (RF), which leans toward falsehood. This approach improves accuracy by considering, relative truth and relative falsehood, rather than a single indeterminate value. The modified neutrosophic set, called the quadri-portioned neutrosophic soft set (QPNSS), includes four membership attributes: absolute truth, relative truth, relative falsehood, and absolute falsehood, offering greater clarity in uncertain situations. New operations are introduced on QPNSS, such as the quadri-partitioned neutrosophic soft set, subsets, complement, absolute set, set difference, and null set. AND and OR operations are also defined. Additionally, a quadri-partitioned neutrosophic soft topological space (QPNSTS) is defined, with key results presented. We define and explore new concepts such as pre-open (p-open) sets, interior, and closure. The paper also examines QPNS compactness, reducibility to finite sub-covers, and other important properties like the intersection of QPNS p-closed sets and QPNS p-compact spaces. This work contributes to the theoretical understanding of QPNS spaces, particularly in soft topological spaces.