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
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
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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