Grothendieck topology

Remarks

Example: let $X$ be a topological space and $\mathcal{O}(X)$ the poset of open sets as a category. Let $J(U)$ be the set of sieves $S$ on $U$ whose members have union $U$. Clause 1 holds because $U \subseteq U$ is in the maximal sieve. Clause 2 holds because the sets $V \cap W$ with $V \in S$ cover $W$ when $S$ covers $U \supseteq W$. Clause 3 holds because covers of the members of a cover together give a cover. Non-example: on $\mathcal{O}(X)$ with $X$ nonempty, the rule $J(U) = \emptyset$ for all $U$ fails clause 1. On every category, the rule in which only maximal sieves cover is a Grothendieck topology, and so is the rule in which every sieve covers. Many authors start from a pretopology, a choice of covering families of morphisms; it generates a Grothendieck topology by taking the sieves that contain a covering family.