  We consider properties and applications of a compact, Hausdorff topology
called the "ultrafilter topology" defined on an {\sl arbitrary spectral space}
and we observe that this topology coincides with the constructible topology. If
$K$ is a field and $A$ a subring of $K$, we show that the space Zar$(K|A)$ of
all valuation domains, having $K$ as quotient field and containing $A$,
(endowed with the Zariski topology) is a spectral space by giving in this
general setting the explicit construction of a ring whose Zariski spectrum is
homeomorphic to Zar$(K|A)$. We extend results regarding spectral topologies on
the spaces of all valuation domains and apply the theory developed to study
representations of integrally closed domains as intersections of valuation
overrings. As a very particular case, we prove that two collections of
valuation domains of $K$ with the same ultrafilter closure represent, as an
intersection, the same integrally closed domain.
