2015/05/24 by Alfred Dolich, Dolich, Alfred, John Goodrick +1
Computer Science · Mathematics · #03C45 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #O3C64
paper · pdf · doi:10.48550/arxiv.1505.06455
openalex publication_date 2015/05/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.