Thus the gluing axiom requires us to compute the space of sections of $\shf{R}$ over at least the star over $x$, namely $\{x,y_1, \dotsc\}$, which is given by $$\{(a,b_1,\dotsc) \in \shf{R}(x) \times \prod_{i} \shf{R}(y_i) : b_i = \left(\shf{R}(x \le y_i)\right)(a)\},$$ which is evidently in bijective correspondence with $\shf{R}(x)$. Notice that the third axiom in Definition \[df:presheaf\] ensures that the above construction is well-defined.

\[eg:contfunctions2\] The compactly supported continuous functions $C_c(X,\mathbb{R})$ on some topological space $(X,\col{T})$ are best organized in a sheaf $\dshf{C}_c(X,\mathbb{R})$ over the poset ${\bf Open}(X)$, which is a dual sheaf on ${\bf Open}(X)^{op}$. (Notice the use of the fraktur font, and beware that we are using the Alexandroff topology on ${\bf Open}(X)$ in this example!) Similar to the situation in Example \[eg:contfunctions1\], the stalk over $U$ is $C_c(U,\mathbb{R})$. But instead of restricting along $U \le V$, one can *extend* by zero, obtaining a function $C_c(V,\mathbb{R}) \to C_c(U,\mathbb{R})$ as the right frame of Figure \[fig:contfunctions\] shows.

$\dshf{C}_c(X,\mathbb{R})$ has only one global section: the zero function. Observe that the infimum of all of ${\bf Open}(X)$ is the empty set. Thus, the gluing axiom implies that the stalk over the empty set should be the trivial vector space. For finitely many 