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Let $X$ be a scheme. Is it true that the morphism $X\rightarrow \mathrm{Spec}\,\mathbb{Z}$ is quasi-separated iff the intersection of two quasi-compact open subspaces of the underlying space of $X$ is quasi-compact (so in particular, quasi-separatedness is a purely topological condition)? If the answer is positive, what is a published reference where this is proved in detail?

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Yes. A reference is EGA IV, première partie, Proposition (1.2.7), b'), page 228.

Numdam link

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Another reference is Tag 01KO in the Stacks project (note that when $S$ is affine the hypothesis that the two opens map into a common affine is empty).

Of course, the condition that the intersection of any two affine is quasi-compact, is equivalent to the condition that the intersection of any two quasi-compact open subsets is quasi-compact.

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