Tame Topology over Definable Uniform Structures

Abstract

A visceral structure on $\mathfrak{M}$ is given by a definable base for a uniform topology on its universe M in which all basic open sets are infinite and any infinite definable subset $X\subseteq M$ has nonempty interior.

Assuming only viscerality, we show that the definable sets in $\mathfrak{M}$ satisfy some desirable topological tameness conditions. For example, any definable function $f:M\to M$ has a finite set of discontinuities; any definable function $f:M^n\to M^m$ is continuous on a nonnempty 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 some of the 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.