Tame Topology over Definable Uniform Structures

A visceral structure on M is given by a definable base for uniform topology its universe in which all basic open sets are infinite and any subset X⊆M has nonempty interior. Assuming only viscerality, we show that the satisfy some desirable topological tameness conditions. For example, function f:M→M finite set of discontinuities; f:Mn→Mm continuous nonnempty set; assuming choice, obtain cell decomposition result sets. Under an additional assumption (“no space-filling functions”), prove natural notion dimension invariant under bijections. These results generalize theorems proved Simon Walsberg, who assumed dp-minimality addition to viscerality. In final section, construct new examples structures.