A visceral structure on a model 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 of M has non-empty interior. Assuming only viscerality, we show that the definable sets in M satisfy some desirable topological tameness conditions. For example, any definable unary function has a finite set of discontinuities; any definable function from some Cartesian power of M into M is continuous on an 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 two sections, we construct new examples of visceral structures a subclass of which are dp-minimal yet not weakly o-minimal.
We develop tame topology over dp-minimal structures equipped with definable uniformities satisfying certain assumptions. Our assumptions are enough to ensure that definable sets are tame: there is a good notion of dimension on definable sets, definable functions are almost everywhere continuous, and definable sets are finite unions of graphs of definable continuous "multi-valued functions". This generalizes known statements about weakly o-minimal, C-minimal and P-minimal theories.