In algebraic geometry, a Zariski–Riemann space or Zariski space of a subring k of a field K is a space whose points are valuation rings containing k and properly contained in K. When k is the ring of complex numbers and K the field of rational functions of a complex algebraic curve, then the Zariski–Riemann space has essentially the same underlying set as the usual Riemann surface, a compact nonsingular model of the curve. Zariski–Riemann spaces were introduced by Zariski (1944) who called them Riemann manifolds or (confusingly) Riemann surfaces, and were named Zariski–Riemann spaces after Oscar Zariski and Bernhard Riemann by Nagata (1962) who used them to show that algebraic varieties can be embedded in complete ones.
If S is the Zariski–Riemann space of a subring k of a field K, it has a topology defined by taking a basis of open sets to be the valuation rings containing a given finite subset of K. The space S is quasi-compact.