Non-compact subsets of the Zariski space of an integral domain

Let V be a minimal valuation overring of an integral domain D and let 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 Zar(D){V} is not compact. We then use it to prove that, in many cases, Zar(D) is not a Noetherian space, and apply it to the study of the spaces of Kronecker function rings and of Noetherian overrings.