suffices to observe that the star over $A$ is the intersection of the collection of all open sets containing $A$.

It is very easy to see that order preserving maps $P\to Q$ between two posets are continuous when both $P$ and $Q$ are given the Alexandroff topology.

Although the Alexandroff topology on $P$ defines a partial order on its open sets, this is both typically larger than $P$ and dual to $P$ in a particular way. Consider the small example shown at left in Figure \[fig:lattice\_sheaves\]. The poset $P$ contains four elements, and is a lattice. The Alexandroff topology consists of five open sets – it includes the empty set – each of which happens to be the star over each original element of $P$. As the Figure shows, the diagram of the Alexandroff topology contains the dual poset, namely $P^{op}$.

\[df:presheaf\] Suppose $(X,\col{T})$ is a topology. A *presheaf $\shf{S}$ of sets on $(X,\col{T})$* consists of the following specification:

1.  For each open set $U \in \col{T}$, a set $\shf{S}(U)$, called the *stalk at* $U$,

2.  For each pair of open sets $U \subseteq V$, there is a function $\shf{S}(U \subseteq V):\shf{S}(V)\to\shf{S}(U)$, called a *restriction function* (or just a *restriction*), 