The étale-open topology (suite)

Fix an abstract field K. For each K-variety V, we will define an “étale-open” topology on the set V(K) of rational points of V. This notion uniformly recovers (1) the Zariski topology on V(K) when K is algebraically closed, (2) the analytic topology on V(K) when K is the real numbers, (3) the valuation topology on V(K) when K is almost any henselian field. On pseudo-finite fields, the étale-open topology seems to be new, and has some interesting properties.
The étale-open topology is mostly of interest when Kis large (also known as ample). On non-large fields, theétale-open topology is discrete. In fact, this propertycharacterizes largeness. Using this, one can recover some well-knownfacts about large fields, and classify the model-theoretically stablelarge fields. It may be possible to push these arguments towards aclassification of NIP large fields. Joint work with Chieu-Minh Tran, Erik Walsberg, and Jinhe Ye.