Small Banach bundles and modules

We characterize those (continuously-normed) Banach bundles E→ X with compact Hausdorff base whose spaces Γ(E) of global continuous sections are topologically finitely-generated over the function algebra $C(X)$, answering a question of I. Gogi\'c's and extending analogous work for metrizable X. Conditions equivalent to topological finite generation include: (a) the requirement that E be locally trivial and of finite type along locally closed and relatively Fσ strata in a finite stratification of X; (b) the decomposability of arbitrary elements in ᵖ(Γ(E)), 1≤ p<∞ as sums of ≤ N products in ᵖ(C(X))· Γ(E) for some fixed N; (c) the analogous decomposability requirement for maximal Banach-module tensor products F⊗C(X)Γ(E) or (d) equivalently, only for F=¹(C(X)).