When is the étale open topology a field topology? - Dittmann, P; Walsberg, E; Ye, J (Springer, 2023)
We investigate the following question: Given a field K, when is the ´etale open topology EK induced by a field topology? On the positive side, when K is the fraction field of a local domain R 6= K, using a weak form of resolution of singularities due to Gabber, we show that EK agrees with the R-adic topology when R is quasi-excellent and henselian. Various pathologies appear when dropping the quasi-excellence assumption. For locally bounded field topologies, we introduce the notion of generalized t-henselianity (gthenselianity) following Prestel and Ziegler. We establish the following: For a locally bounded field topology τ, the ´etale open topology is induced by τ if and only if τ is gt-henselian and some non-empty ´etale image is τ-bounded open. On the negative side, we obtain that for a pseudo-algebraically closed field K, EK is never induced by a field topology.