such that

3.  For each triple $U \subseteq V \subseteq W$ of open sets, $\shf{S}(U \subseteq W) = \shf{S}(U \le V) \circ \shf{S}(V \le W)$.

When the stalks themselves have structure (they are vector spaces or topological spaces, for instance) one obtains a presheaf *of* that type of object when the restrictions or extensions preserve that structure. For example, a *presheaf of vector spaces* has linear functions for each restriction, while a *presheaf of topological spaces* has continuous functions for each restriction.

A sheaf $\shf{S}$ on a poset $P$ with the Alexandroff topology given by Definition \[df:sheaf\_poset\] is almost a presheaf on $(P,\col{A})$, where $\col{A}$ is the Alexandroff topology on $P$. The only issue is that the stalks on unions of stars are not defined yet, but these will be defined in Proposition \[prop:poset\_sheaves\].

As Definition \[df:presheaf\] makes clear, presheaves on a topological space are only sensitive to the poset of open sets, and *not* to the points in those open sets. Therefore, we can use Definition \[df:section\] to define *sections* of a presheaf on a topological space.

Because of the situation in Example \[ex:same\_poset\], the set of global sections of a presheaf on the whole topological space may be 