Éz fields

Let $K$ be a field. The \'etale open topology on the $K$-points $V(K)$ of $K$-variety $V$ was introduced in our previous work. is non-discrete if and only large. If separably, real, $p$-adically closed then agrees with Zariski, order, valuation topology, respectively. We show that existentially definable sets perfect large fields behave well respect to this topology: such are finite unions subsets Zariski sets. This implies arbitrary enjoy some well-known topological properties algebraically, fields. introduce study class \'ez fields: every set union should seen as generalized notion model completeness for Algebraically closed, real bounded $\mathrm{PAC}$ \'ez. (In particular pseudofinite infinite algebraic extensions \'ez.) develop basics theory gives uniform approach across all characteristic zero local new also prominent examples possibly non-model complete model-theoretically tame (characteristic Henselian Frobenius fields)