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Suppose that for a contractible space $A$ we are given a fiber bundle $p:E\to A$ and denote for $B\subset A$ by $E(B)$ the restricted bundle. I have good reason to believe that in this situation the restriction map $r:\Gamma^0(E)\to\Gamma^0(E(B))$ is a Serre fibration when the section spaces are endowed with the compact-open topology. After a few unsuccessful tries of proving this I decided it might not be a bad idea to ask the denizens of StackExchange for some help. Any hints/ideas/comments are very appreciated.

EDIT: To give a concrete example that bothers me consider $A=D^k\times D^{m-k}$ and $B=D^k_{\frac{1}{2}}\times D^{m-k}$ where $D^k$ denotes the unit disc in $k$ dimensions and $D^k_{\alpha}:=\{x\in D^k\,\big|\,\alpha\leq\|x\|\leq 1\}$.

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    Ah, good ol' Hatcher. I seem to underestimate that book heavily, it's always the last place I consult. Thank you for the help again, figured the appropriate retraction out in the mean time. I'll either write an answer later or just delete the question. Or you might just copy one of your comments into an answer if you're after any of these points they got here, I'll accept gladly.2011-07-03

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In your example, A is contractible, so you can reformulate the problem more concretely. Also, the inclusion of B into A is a cofibration, which implies that the dual map restriction map on mapping spaces is a fibration.