The Stacks project

Example 7.33.5. Let $X$ be a topological space. Let $X_{Zar}$ be the site of Example 7.6.4. Let $x \in X$ be a point. Consider the functor

\[ u : X_{Zar} \longrightarrow \textit{Sets}, \quad U \mapsto \left\{ \begin{matrix} \emptyset & \text{if} & x \not\in U \\ \{ *\} & \text{if} & x \in U \end{matrix} \right. \]

This functor commutes with product and fibred products, and turns coverings into surjective families of maps. Hence we obtain a point $p$ of the site $X_{Zar}$. It is immediately verified that the stalk functor agrees with the stalk at $x$ defined in Sheaves, Section 6.11.

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View Example 7.33.5 as pdf