  We characterize those (continuously-normed) Banach bundles $\mathcal{E}\to X$
with compact Hausdorff base whose spaces $\Gamma(\mathcal{E})$ 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 $\mathcal{E}$ be locally trivial and of
finite type along locally closed and relatively $F_{\sigma}$ strata in a finite
stratification of $X$; (b) the decomposability of arbitrary elements in
$\ell^p(\Gamma(\mathcal{E}))$, $1\le p<\infty$ as sums of $\le N$ products in
$\ell^p(C(X))\cdot \Gamma(\mathcal{E})$ for some fixed $N$; (c) the analogous
decomposability requirement for maximal Banach-module tensor products
$F\widehat{\otimes}_{C(X)}\Gamma(\mathcal{E})$ or (d) equivalently, only for
$F=\ell^1(C(X))$.
