Tame topology over definable uniform structures

TL;DR

This paper explores topological properties of definable sets in visceral structures, generalizing known results and providing new examples, with implications for understanding tame topology in various mathematical contexts.

Contribution

It introduces a broad class of visceral structures and establishes topological tameness results, extending prior work beyond dp-minimality.

Findings

Definable unary functions have finitely many discontinuities.

Definable functions are continuous on a nonempty open set.

Cell decomposition holds under definable finite choice.

Abstract

A visceral structure on M is given by a definable base for a uniform topology on its universe in which all basic open sets are infinite and any infinite definable subset X of M has non-empty interior. This context includes o-minimal ordered groups, p-adic fields, and other examples. Assuming only viscerality, we show that the definable sets in M satisfy some desirable topological tameness conditions. For example, any definable unary function on M has a finite set of discontinuities; any definable function on a Cartesian power of M is continuous on a nonempty open set; and assuming definable finite choice, we obtain a cell decomposition result for definable sets. Under an additional topological assumption ("no space-filling functions"), we prove that the natural notion of topological dimension is invariant under definable bijections. These results generalize theorems proved by Simon and Walsberg, who assumed dp-minimality in addition to viscerality. In the final section, we construct new examples of visceral structures.