  One shows for Banach bundles in a certain class that having a second
countable locally compact Hausdorff base space and separable fibers implies the
separability of the Banach space of the all sections that vanish at infinity.
In the reverse direction, it is proved that for a Banach bundle with locally
compact Hausdorff base space the separability of the space of all the sections
that vanish at infinity implies that the base space is second countable. If the
base space is compact and the space of all the sections of the bundle is
generated by a weakly compact subset then the base space is an Eberlein
compact.
