Let D be an integral domain and L be a field containing D. We study the isolated points of the Zariski space Zar(L|D), with respect to the constructible topology. In particular, we completely characterize when L (as a point) is isolated and, under the hypothesis that L is the quotient field of D, when a valuation domain of dimension 1 is isolated; as a consequence, we find all isolated points of Zar(D) when D is a Noetherian domain and, under the hypothesis that D and D' are Noetherian, local and countable, we characterize when Zar(D) and Zar(D') are homeomorphic. We also show that if V is a valuation domain and L is transcendental over V then the set of extensions of V to L has no isolated points.

I would like to thank the referee for pointing out several problems in the first version of the paper and for his or her patience in pointing them out.