  Let $F$ be a field, let $D$ be a subring of $F$, and let ${\mathfrak{X}}$ be
the Zariski-Riemann space of valuation rings containing $D$ and having quotient
field $F$. We consider the Zariski, inverse and patch topologies on
${\mathfrak{X}}$ when viewed as a projective limit of projective integral
schemes having function field contained in $F$, and we characterize the locally
ringed subspaces of ${\mathfrak{X}}$ that are affine schemes.
