The Stacks project

Example 7.7.4. Let $X$ be a topological space. Let us consider the site $X'_{Zar}$ which is the same as the site $X_{Zar}$ of Example 7.6.4 except that we disallow the empty covering of the empty set. In other words, we do allow the covering $\{ \emptyset \to \emptyset \} $ but we do not allow the covering whose index set is empty. It is easy to show that this still defines a site. However, we claim that the sheaves on $X'_{Zar}$ are different from the sheaves on $X_{Zar}$. For example, as an extreme case consider the situation where $X = \{ p\} $ is a singleton. Then the objects of $X'_{Zar}$ are $\emptyset , X$ and every covering of $\emptyset $ can be refined by $\{ \emptyset \to \emptyset \} $ and every covering of $X$ by $\{ X \to X\} $. Clearly, a sheaf on this is given by any choice of a set $\mathcal{F}(\emptyset )$ and any choice of a set $\mathcal{F}(X)$, together with any restriction map $\mathcal{F}(X) \to \mathcal{F}(\emptyset )$. Thus sheaves on $X'_{Zar}$ are the same as usual sheaves on the two point space $\{ \eta , p\} $ with open sets $\{ \emptyset , \{ \eta \} , \{ p, \eta \} \} $. In general sheaves on $X'_{Zar}$ are the same as sheaves on the space $X \amalg \{ \eta \} $, with opens given by the empty set and any set of the form $U \cup \{ \eta \} $ for $U \subset X$ open.