Definable groups and fields in t-minimal theories

Abstract

Let $T$ be a theory which is t-minimal, meaning that with respect to some definable topology, a unary definable set $D \subseteq M$ has non-empty interior iff it is infinite. If $K$ is a definable field in $T$, then $K$ is finite or "large" in the sense of Pop: any smooth algebraic curve $C$ over $K$ with at least one $K$-rational point has infinitely many $K$-rational points. We also assign a canonical topology to any abelian definable group $G$ in a t-minimal theory. In the case where the t-minimal theory is "visceral" in the sense of Dolich and Goodrick, meaning that the definable topology is induced by a definable uniformity, we can drop the assumption of abelianity of $G$, and the resulting topology on $G$ is a definable manifold in the style of Acosta L\'opez and Hasson.