Largeness and generalized t-henselianity

Abstract

Let $K$ be a countable field. Then $K$ is large in the sense of Pop if and only if it admits a field topology which is "generalized t-henselian" (gt-henselian) in the sense of Dittmann, Walsberg, and Ye, meaning that the implicit function theorem holds for polynomials. Moreover, the \'etale open topology can be characterized in terms of the gt-henselian topologies on $K$: a subset $U \subseteq K^n$ is open in the \'etale open topology if and only if it is open with respect to every gt-henselian topology on $K$.