The Stacks project

Example 7.33.6. Let $X$ be a topological space. What are the points of the topos $\mathop{\mathit{Sh}}\nolimits (X)$? To see this, let $X_{Zar}$ be the site of Example 7.6.4. By Lemma 7.32.7 a point of $\mathop{\mathit{Sh}}\nolimits (X)$ corresponds to a point of this site. Let $p$ be a point of the site $X_{Zar}$ given by the functor $u : X_{Zar} \to \textit{Sets}$. We are going to use the characterization of such a $u$ in Proposition 7.33.3. This implies immediately that $u(\emptyset ) = \emptyset $ and $u(U \cap V) = u(U) \times u(V)$. In particular we have $u(U) = u(U) \times u(U)$ via the diagonal map which implies that $u(U)$ is either a singleton or empty. Moreover, if $U = \bigcup U_ i$ is an open covering then

\[ u(U) = \emptyset \Rightarrow \forall i, \ u(U_ i) = \emptyset \quad \text{and}\quad u(U) \not= \emptyset \Rightarrow \exists i, \ u(U_ i) \not= \emptyset . \]

We conclude that there is a unique largest open $W \subset X$ with $u(W) = \emptyset $, namely the union of all the opens $U$ with $u(U) = \emptyset $. Let $Z = X \setminus W$. If $Z = Z_1 \cup Z_2$ with $Z_ i \subset Z$ closed, then $W = (X \setminus Z_1) \cap (X \setminus Z_2)$ so $\emptyset = u(W) = u(X \setminus Z_1) \times u(X \setminus Z_2)$ and we conclude that $u(X \setminus Z_1) = \emptyset $ or that $u(X \setminus Z_2) = \emptyset $. This means that $X \setminus Z_1 = W$ or that $X \setminus Z_2 = W$. In other words, $Z$ is irreducible. Now we see that $u$ is described by the rule

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

Note that for any irreducible closed $Z \subset X$ this functor satisfies assumptions (1), (2) of Proposition 7.33.3 and hence defines a point. In other words we see that points of the site $X_{Zar}$ are in one-to-one correspondence with irreducible closed subsets of $X$. In particular, if $X$ is a sober topological space, then points of $X_{Zar}$ and points of $X$ are in one to one correspondence, see Example 7.33.5.

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