# Quasi-separatedness as a topological condition on the scheme

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?

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).