Let $(X,\mathcal{O}\_X)$ be a ringed space with soft structure sheaf. Moreover let $X$ be paracompact.


Let $U$ be an open subset on $X$ and let $E$ be a finite dimensional vector bundle on $U$, i.e. $E$ is a finitely generated locally free sheaf of $\mathcal{O}\_X$-modules on $U$.


$\textbf{My question}$ is: can we always extend $E$ to get a vector bundle on $X$? I believe the statement is true but I cannot find any reference on it.



Can we always extend a vector bundle on an open subset of a ringed space with soft structure sheaf?
No, this is false. Take $X=\mathbb{R}^3$ and $U=\mathbb{R}^3\smallsetminus\{0\} $, 
with $\mathcal{O}\_X$ the sheaf of complex $C^{\infty}$ functions.
Line bundles on $U$ are parametrized by $H^1(U, \mathcal{O}\_U^\*)$, which is isomorphic to $H^2(U,\mathbb{Z})=\mathbb{Z}$ by the exponential exact sequence. Similarly $H^1(X, \mathcal{O}\_X^\*)\cong H^2(X,\mathbb{Z})=0$, so there are nontrivial line bundles on $U$ which do not extend to $X$.


