Lemma 7.47.9. Let $\mathcal{C}$ be a category. Let $\{ J_ i\} _{i\in I}$ be a set of topologies.

The rule $J(U) = \bigcap J_ i(U)$ defines a topology on $\mathcal{C}$.

There is a coarsest topology finer than all of the topologies $J_ i$.

Lemma 7.47.9. Let $\mathcal{C}$ be a category. Let $\{ J_ i\} _{i\in I}$ be a set of topologies.

The rule $J(U) = \bigcap J_ i(U)$ defines a topology on $\mathcal{C}$.

There is a coarsest topology finer than all of the topologies $J_ i$.

**Proof.**
The first part is direct from the definitions. The second follows by taking the intersection of all topologies finer than all of the $J_ i$.
$\square$

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