Fibrewise Neutrosophic Homotopy Theory

This monograph develops fibrewise homotopy theory for spaces whose open sets are neutrosophic triples (T, I, F) of truth, indeterminacy and falsity degrees, over a fixed classical base B. The guiding equation is pY H = pX prX: a homotopy deforms a map while every point stays in its fibre. After fixing the lattice of triples, neutrosophic topologies and the crisp embedding of classical topology, we define fibrewise N-homotopies through a product-generated cylinder. We prove that homotopy is an equivalence relation under an explicit pasting convention and that homotopy equivalences satisfy two-out-of-three. We then develop separation conditions, pointed and relative homotopy, neutrosophic fibrations and cofibrations, mapping spaces, Lusternik-Schnirelmann category, local triviality and base change, path objects, mapping cylinders and homotopy pullbacks. Every hypothesis on products, pastings, pullbacks and exponentials is stated explicitly. Two enlargements are treated in parallel: multivalued (upper semicontinuous) correspondences and their homotopies, and ideal modifications of topologies with their fibrewise traces. Explicit examples show that the three theories do not commute. A new final chapter attaches to each neutrosophic space three families of classical level topologies, one for each coordinate. It proves that fibrewise N-homotopies transfer to every level, so that the level profiles are invariants of the neutrosophic homotopy type. It also constructs bitopological neutrosophic spaces in which the indeterminacy coordinate alone obstructs a contraction. A conclusion lists open problems.