Lemma 7.47.7. Let $\mathcal{C}$ be a category. Let $J$ be a topology on $\mathcal{C}$. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$.

1. Finite intersections of elements of $J(U)$ are in $J(U)$.

2. 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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