Title: Non-compact subsets of the Zariski space of an integral domain,
Abstract: Let $V$ be a minimal valuation overring of an integral domain $D$ and let
$\mathrm{Zar}(D)$ be the Zariski space of the valuation overrings of $D$.
Starting from a result in the theory of semistar operations, we prove a
criterion under which the set $\mathrm{Zar}(D)\setminus\{V\}$ is not compact.
We then use it to prove that, in many cases, $\mathrm{Zar}(D)$ is not a
Noetherian space, and apply it to the study of the spaces of Kronecker function
rings and of Noetherian overrings.