The Stacks project

Situation 18.30.5. Let $\mathcal{C}$ be a site. Let $\mathcal{B} \subset \text{Ob}(\mathcal{C})$ be a set of objects. We consider the following conditions

1. Every object of $\mathcal{C}$ has a covering by elements of $\mathcal{B}$.

2. Every $U \in \mathcal{B}$ is quasi-compact (Sites, Section 7.17).

3. For a covering $\{ U_ i \to U\}$ with $U_ i, U \in \mathcal{B}$ the fibre products $U_ i \times _ U U_ j$ are quasi-compact.