This is probably not a research level question, I'm sorry if it is inappropriate. I'm reasking here [this question](https://math.stackexchange.com/questions/64179/sheaf-of-sections-and-local-triviality) on math.se.




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Suppose that $\xi: E \to B$ is a bundle (by which I mean simply a continuous mapping), and there exists a manifold $F$ such that $\Gamma(\xi) \cong C(B, F)$ as sheaves over $B$ (here by $\Gamma$ I mean the sheaf of sections, and by $C$ I mean the sheaf of continuous functions from $B$ to $F$). Does this imply that $\xi$ is locally trivial? If the proof is simple, I would like a hint on where to start, otherwise I would like a reference that studies this and related topics in more detail, please.


UPD: I guess it boils down to whether a bundle $\xi: E \to B$ such that $\Gamma(\xi) \cong C(B, \mathbb{R}^n)$ is (locally) trivial, then one can take preimages of charts to obtain the cover of $B$ that I hope can be used in the proof, but not having really worked with sheaves before I'm not sure.


UPD: Please comment if you think this quesion should be retagged, rephrased, or something like that. Having received no comments whatsoever on both math.se and MO, I don't know what to think :(



Sheaf of sections and local triviality
Let $E$ be the union of the coordinate axes in $\mathbb R^2$, and let $\xi:E\to\mathbb R$ be the projection on the first coordinate. Then $\Gamma(\xi)$ is a trivial sheaf, so it is isomorphic to $C(\mathbb R,F)$ with $F$ any one-point space. Yet $\xi$ is not locally trivial.
(This can be modified so that the fiber at $0$ of $\xi$ is pretty much anything...)


If this is *too* trivial an example, let $F$ be any manifold, let $E$ be as before, and let $\xi=E\times F\to\mathbb R$ such that $\xi((x,y),f)=x$. Then $\Gamma(\xi)\cong C(\mathbb R,F)$, just as before.


