collection of sets of the form $$\label{eq:upset}
U_x = \{y \in P : x \le y\}$$ for each $x \in P$ forms a base for a topology, called the *Alexandroff* topology, shown in Figure \[fig:lattice\_sheaves\].

\[prop:alexandroffness\] Every intersection of open sets in the Alexandroff topology on a poset $P$ is open.

Suppose that $\col{U}$ is a collection of open sets in the Alexandroff topology and that $x \in \cap \col{U}$. This means that $x$ is in every open set of $\col{U}$. Now each of these open sets contains at least $U_x$, since these are the sets of the base. Thus $U_x \subseteq \cap \col{U}$, which therefore shows that $\cap \col{U}$ is a neighborhood of each of its points.

In the Alexandroff topology for a poset, the usual topological notions of closures, interiors, and frontiers[^2] have straightforward interpretations in terms of the poset itself. Additionally, because of Proposition \[prop:alexandroffness\], the concept of a *star* is also available.

If $A \subseteq X$ is a subset of a topological space, the *star* of $A$ is the smallest open set containing $A$. In general, stars need not exist, but in the Alexandroff topology for a poset there is a star of every subset.

It 