Topologically finitely generated Hilbert C(X)-modules

For a Hilbert C(X)-module V, where X is a compact metrizable space, we show that the following conditions are equivalent: (i) V is topologically finitely generated, (ii) there exists K∈N such that every algebraically finitely generated submodule of V can be generated with k≤K generators, (iii) V is canonically isomorphic to the Hilbert C(X)-module Γ(E) of all continuous sections of an (F) Hilbert bundle E=(p,E,X) over X, whose fibres Ex have uniformly finite dimensions, and each restriction bundle of E over a set where dimEx is constant is of finite type, (iv) there exists N∈N such that for every Banach C(X)-module W, each tensor in the C(X)-projective tensor product V⊗πC(X)W is of (finite) rank at most N.