The Stacks project

Example 7.6.4. Let $X$ be a topological space. Let $X_{Zar}$ be the category whose objects consist of all the open sets $U$ in $X$ and whose morphisms are just the inclusion maps. That is, there is at most one morphism between any two objects in $X_{Zar}$. Now define $\{ U_ i \to U\} _{i \in I}\in \text{Cov}(X_{Zar})$ if and only if $\bigcup U_ i = U$. Conditions (1) and (2) above are clear, and (3) is also clear once we realize that in $X_{Zar}$ we have $U \times V = U \cap V$. Note that in particular the empty set has to be an element of $X_{Zar}$ since otherwise this would not work in general. Furthermore, it is equally important, as we will see later, to allow the empty covering of the empty set as a covering! We turn $X_{Zar}$ into a site by choosing a suitable set of coverings $\text{Cov}(X_{Zar})_{\kappa , \alpha }$ as in Sets, Lemma 3.11.1. Presheaves and sheaves (as defined below) on the site $X_{Zar}$ agree exactly with the usual notion of a presheaves and sheaves on a topological space, as defined in Sheaves, Section 6.1.