Topological properties of subsets of the Zariski space

We study the properties of some distinguished subspaces of the Zariski space Zar$(K|D)$ of a field $F$ over a domain $D$, in particular the topological properties of subspaces defined through algebraic means. We are mainly interested in two classes of problems: understanding when spaces of the form Zar$(K|D)\setminus \{V\}$ are compact (which is strongly linked to the problem of determining when Zar$(K|D)$ is a Noetherian space), and studying spaces of rings defined through pseudo-convergent sequences on an (arbitrary, but fixed) rank one valuation domain.