The étale-open topology and the stable fields conjecture

For an arbitrary field K and a K -variety V , we introduce the étale-open topology on the set V(K) of K -points of V . This topology agrees with the Zariski topology, Euclidean topology, or valuation topology when K is separably closed, real closed, or p -adically closed, respectively. Topological properties of the étale-open topology correspond to algebraic properties of K . For example, the étale-open topology on A^1(K) is not discrete if and only if K is large. As an application, we show that a large stable field is separably closed.