Visceral theories without assumptions

Abstract

Let $T$ be a theory with a definable topology. $T$ is t-minimal in the sense of Mathews if every definable set in one variable has finite boundary. If $T$ is t-minimal, we show that there is a good dimension theory for definable sets, satisfying properties similar to dp-rank in dp-minimal theories, with one key exception: the dimension of $\operatorname{dom}(f)$ can be less than the dimension of $\operatorname{im}(f)$ for a definable function $f$. Using the dimension theory, we show that any definable field in a t-minimal theory is perfect. We then specialize to the case where $T$ is visceral in the sense of Dolich and Goodrick, meaning that $T$ is t-minimal and the definable topology comes from a definable uniformity (i.e., a definable uniform structure). We show that almost all of Dolich and Goodrick's tame topology theorems for visceral theories hold without their additional assumptions of definable finite choice (DFC) and no space-filling functions (NSFF). Lastly, we produce an example of a visceral theory with a space-filling curve, answering a question of Dolich and Goodrick.