Affine schemes and topological closures in the Zariski-Riemann space of valuation rings

Let F be a field, let D be a subring of F, and let x be the Zariski-Riemann space of valuation rings containing D and having quotient field F. We consider the Zariski, inverse and patch topologies on x when viewed as a projective limit of projective integral schemes having function field contained in F, and we characterize the locally ringed subspaces of x that are affine schemes. (C) 2014 Elsevier B.V. All rights reserved.