Lemma 7.47.7. Let \mathcal{C} be a category. Let J be a topology on \mathcal{C}. Let U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}).
Finite intersections of elements of J(U) are in J(U).
If S \in J(U) and S' \supset S, then S' \in J(U).
Proof. Let S, S' \in J(U). Consider S'' = S \cap S'. For every V \to U in S(U) we have
S' \times _ U V = S'' \times _ U V
simply because V \to U already is in S. Hence by the second axiom of the definition we see that S'' \in J(U).
Let S \in J(U) and S' \supset S. For every V \to U in S(U) we have S' \times _ U V = h_ V by Lemma 7.47.5. Thus S' \times _ U V \in J(V) by the third axiom. Hence S' \in J(U) by the second axiom. \square
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