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.
Simon and Walsberg @cite_25 recently proved some similar results for visceral dp-minimal theories (although they did not call them such; what we call viscerality, they called ()''). For instance, they also proved that definable functions are continuous almost everywhere and that the natural topological dimension function is invariant under definable bijections. We do not assume dp-minimality or even NIP, and in that sense our results are more general; on the other hand, we needed Definable Finite Choice for our cell decomposition theorem and a few other results, whereas Simon and Walsberg compensate for the lack of DFC by decomposing definable sets into graphs of continuous multi-valued functions.''
@cite_25
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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.