U_{x_i})$ are given by projection maps.

Proposition \[prop:poset\_sheaves\] justifies our terminology “sheaf on a poset with the Alexandroff topology.” Throughout this chapter, if $(P,\le)$ is a poset, we will assume it has the Alexandroff topology unless explicitly noted. We will therefore not distinguish between *presheaves* and *sheaves* on $P$ unless a different topology is explicitly specified. For dual sheaves, we will *always* use the Alexandroff topology for the dual poset. Care is needed, if the poset $P$ is ${\bf Open}(X,\col{T})^{op}$ for some topology $\col{T}$, because then the Alexandroff topology on $P$ will generally be different from $\col{T}$!

First, observe that $\shf{R}'$ is a presheaf on $(P,\col{A})$ by construction. Conditions (3) and (4) result in $\shf{R}'$ satisfying the gluing axiom on unions of stars. The gluing axiom leaves the stalks over maximal elements of $P$ unconstrained, so we only need to investigate the other elements addressed by conditions (1) and (2). For a given element $x \in P$, suppose that $y_1, \dotsc$ are the elements strictly greater than $x$ in $P$. Observe that $\bigcup_i U_{y_i}$ cannot cover $U_x$, because in the Alexandroff topology, the only way that $U_{y_1}, \dotsc$ covers $x$ is if one of them contains the star over $x$.

