TITLE: Example of a section $s$ of a presheaf $\mathcal{F}$ which is not determined by $\{ s_x \}_{x}$.
QUESTION [5 upvotes]: One of the advantages of working with a sheaf $\mathcal{F}$ instead of a presheaf is that every section $s \in \mathcal{F}(U)$ is determined by the family of its germs $\{ s_x \}_{x \in U}$. I know this may fail for a presheaf in general, since you don't have the gluing axiom. Can anyone show me an example of this phenomenon?
(Maybe I want my base space $X$ to satisfy some separation axiom, to avoid trivial examples; of course trivial examples are welcome too.)

REPLY [7 votes]: Georges Elencwajg has given a correct example, but here's a simpler one that might also be helpful.  Let $X$ be a discrete space with just two points $p$ and $q$.  Define a presheaf $F$ by letting $F(\{p,q\})=\{0,1\}$ while $F(\{p\})=F(\{q\})=F(\varnothing)=\{0\}$; the restriction maps are the only possible ones.  Then both of the stalks are singletons, so the two gobal sections have the same stalk at each point.