compactly supported functions $\mathbb{R}\to\mathbb{R}$ (right)[]{data-label="fig:contfunctions"}](contfunctions){width="3.5in"}

\[eg:contfunctions1\] Let $(X,\col{T})$ and $(Y,\col{S})$ be topological spaces. The space $C(X,Y)$ of continuous functions $X\to Y$ has the structure of a sheaf $\shf{C}(X,Y)$ on $(X,\col{T})$. As the left frame of Figure \[fig:contfunctions\] shows, the stalk over $U \in \col{T}$ is $C(U,Y)$ and if $U \le V$, then restricting the domain induces a restriction function $C(U,Y)\to C(V,Y)$. The gluing axiom expresses the well-known fact that whenever two continuous functions with overlapping domains are equal on the overlap, then they extend to a common continuous function over the union.

If $(P,\le)$ is a poset with the Alexandroff topology, the distinction between sheaves and presheaves vanishes.

\[prop:poset\_sheaves\] Let $\shf{R}$ be a sheaf on a poset $(P,\le)$ with nonempty stalks (Definition \[df:sheaf\_poset\]). There is a sheaf (Definition \[df:sheaf\]) $\shf{R}'$ on $(P,\col{A})$ where $\col{A}$ is the Alexandroff topology given by

1.  $\shf{R}'(U_x) = \shf{R}(x)$ for each $x\in P$,

2.  $\shf{R}'(U_y \subseteq U_x) = \shf{R}(x \le y) : \shf{R}(x) \to \shf{R}(y)$ for each pair of elements $x \le y$ in $P$,

3.  $\shf{R}'(\bigcup_{i\in I} U_{x_i})$ is the space of sections of $\shf{R}$ over $\{x_i\}_{i\in I}$ for any collection of elements $\{x_i\}_{i\in I}$ in $P$, and

4.  restrictions $\shf{R}'(U_{x_i} \subseteq \bigcup_{i\in I} 