The Stacks project

Remark 75.9.5. How to choose the collection $\mathcal{B}$ in Lemma 75.9.4? Here are some examples:

  1. If $X$ is quasi-compact and separated, then we can choose $\mathcal{B}$ to be the set of quasi-compact and separated objects of $X_{spaces, {\acute{e}tale}}$. Then $X \in \mathcal{B}$ and $\mathcal{B}$ satisfies (1), (2), and (3)(a). With this choice of $\mathcal{B}$ Lemma 75.9.4 reproduces Lemma 75.9.3.

  2. If $X$ is quasi-compact with affine diagonal over $\mathbf{Z}$ (as in Properties of Spaces, Definition 66.3.1), then we can choose $\mathcal{B}$ to be the set of objects of $X_{spaces, {\acute{e}tale}}$ which are quasi-compact and have affine diagonal over $\mathbf{Z}$. Again $X \in \mathcal{B}$ and $\mathcal{B}$ satisfies (1), (2), and (3)(a).

  3. If $X$ is quasi-compact and quasi-separated, then the smallest subset $\mathcal{B}$ which contains $X$ and satisfies (1), (2), and (3)(a) is given by the rule $W \in \mathcal{B}$ if and only if either $W$ is a quasi-compact open subspace of $X$, or $W$ is a quasi-compact open of an affine object of $X_{spaces, {\acute{e}tale}}$.

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