This is a followup to this question I made yesterday (disclaimer: I'm new here and I'm not sure if asking a new but related question is the correct procedure).
If $X$ is a connected, compact, Hausdorff space and $E$ is a one-dimensional vector bundle over $X$ then (according to Swan-Serre) the $C(X)$-module $C(X,E)$, of continuous sections of this bundle, is (isomorphic to) a projection of $C(X)$. So $C(E,X)$ is either trivial or isomorphic to $C(X)$. Now, it must be easy to find dim 1 bundles on many such $X$ that cannot have nonvanishing sections (like the Mobius strip) but that still have nontrivial sections, so that $C(X,E)$ is neither $C(X)$ nor trivial. Where is my error?