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A SITE-THEORETIC CHARACTERIZATION OF POINTS IN A TOPOLOGICAL SPACE

We study points in the context of Grothendieck topologies (sites), especially the relationship between “honest” points of a topological space X and site-theoretic points of Xtop. We offer an elementary and self-contained proof of a theorem of Grothendieck-Hakim which gives necessary and sufficient conditions for these two notions to coincide. We also explore analogues of this result for other sites encountered in geometry by giving axioms on a site sufficient to ensure the correspondence between site-theoretic points and the points of a certain topological space. These axioms are satisfied for the étale, fppf, fpqc and crystalline sites. They are not satisfied by the large étale site of SpecC, in which (as we show) the site-theoretic points form a proper class.