Fibrewise Fuzzy Soft Topological Spaces

Fibrewise topology provides a framework for studying topological structures associated with a projection onto a fixed base space, while fuzzy soft topology offers a flexible mathematical setting for handling uncertainty through fuzzy soft sets and their topological properties. However, the interaction between fibrewise structures and fuzzy soft topological spaces has not been sufficiently developed. This paper introduces the concept of fibrewise fuzzy soft topological spaces by considering a fibrewise set over a fuzzy soft topological base space and requiring the associated projection to be fuzzy soft continuous. The induced fibrewise fuzzy soft topology is investigated, and fundamental properties involving fuzzy soft continuous and fuzzy soft open fibrewise mappings are established. The concepts of fibrewise closed fuzzy soft topological spaces and fibrewise open fuzzy soft topological spaces are also introduced through fuzzy soft closed and fuzzy soft open projection mappings, respectively. Several propositions concerning these classes are obtained, including preservation properties under suitable fibrewise mappings, finite coverings, finite products, and subspace constructions. Characterisations associated with fibrewise closed and fibrewise open structures are also examined. These results extend the interaction between classical fibrewise topology and fuzzy soft topology and provide a theoretical framework for further investigation of fibrewise structures in fuzzy soft topological settings.