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?



Quasi-separatedness as a topological condition on the scheme
Yes. A reference is EGA IV, première partie, Proposition (1.2.7), b'), page 228.


[Numdam link](http://www.numdam.org/item?id=PMIHES_1964__20__5_0)


