The constructible topology on spaces of valuation domains

We consider properties and applications of a compact, Hausdorff topology called the "ultrafilter topology" defined on an 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 the 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. © 2013 American Mathematical Society.