Lemma 7.47.7 (00Z5)—The Stacks project

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. Proof of (1). The intersection of the empty family follows from condition (3) of Definition 7.47.6. Let $S, S' \in J(U)$. Consider $S'' = S \cap S'$. For every $V \to U$ in $S(V)$ 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 $S'' \in J(U)$.

Proof of (2). Let $S \in J(U)$ and $S' \supset S$. For every $V \to U$ in $S(V)$ 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. $\square$

Comments (4)

Comment #10010 by Heer on February 26, 2025 at 08:10

In the 1st line of the proof, there seems to be a typo: change " $V\to U$ in $S(U)$ " to " $V\to U$ in $S(V)$ ".

The same happens for the 2nd part of the proof.

Comment #10916 by Laurent Moret-Bailly on October 17, 2025 at 15:05

The proof of (1) should include the case of the empty family, which follows from condition (3) in the definition.

There are also:

5 comment(s) on Section 7.47: Topologies

Comments (4)

Post a comment