[Paper Review] Tame topology over definable uniform structures: viscerality and dp-minimality

Alfred Dolich, John Goodrick|arXiv (Cornell University)|May 24, 2015

Advanced Topology and Set Theory19 references3 citations

TL;DR

This paper introduces a framework of visceral structures on models using definable uniform topologies, showing that viscerality alone ensures desirable tameness in definable sets—such as finite discontinuity sets and continuity on open sets—while also establishing topological dimension invariance under definable bijections under an additional no space-filling condition. It generalizes prior results under dp-minimality and constructs new examples of visceral structures, including dp-minimal but not weakly o-minimal ones.

ABSTRACT

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.

Motivation & Objective

• To investigate the topological tameness of definable sets in models equipped with definable uniform topologies.
• To understand the consequences of viscerality—where all basic open sets are infinite and every infinite definable subset has non-empty interior—on the structure of definable functions and sets.
• To generalize results previously obtained under the stronger assumption of dp-minimality by weakening the hypothesis to viscerality alone.
• To construct new examples of visceral structures, particularly those that are dp-minimal but not weakly o-minimal, to better understand the hierarchy of model-theoretic tameness.

Proposed method

• Defining a visceral structure via a definable base for a uniform topology on a model's universe, ensuring all basic open sets are infinite.
• Using the assumption of viscerality to derive topological properties such as finite discontinuity sets for unary definable functions.
• Applying definable finite choice to establish a cell decomposition theorem for definable sets in the structure.
• Introducing the no space-filling functions condition to prove invariance of topological dimension under definable bijections.
• Constructing new examples of visceral structures through model-theoretic and topological constraints, demonstrating that dp-minimality does not imply weak o-minimality.
• Analyzing the interplay between definable topology, uniform structures, and model-theoretic properties like dp-minimality and o-minimality.

Experimental results

Research questions

• RQ1 What topological tameness properties emerge from viscerality alone, without assuming dp-minimality?
• RQ2 How does viscerality ensure that definable functions are continuous on an open set?
• RQ3 Under what conditions is topological dimension invariant under definable bijections in visceral structures?
• RQ4 Can visceral structures be constructed that are dp-minimal but not weakly o-minimal?
• RQ5 What is the relationship between viscerality, dp-minimality, and the absence of space-filling functions in definable topology?

Key findings

• Any definable unary function in a visceral structure has only finitely many discontinuities.
• Any definable function from a Cartesian power of the universe into the universe is continuous on some open subset.
• Assuming definable finite choice, the structure admits a cell decomposition for definable sets.
• Under the no space-filling functions assumption, topological dimension is invariant under definable bijections.
• There exist visceral structures that are dp-minimal but not weakly o-minimal, demonstrating a strict hierarchy in tameness properties.
• The results generalize prior theorems of Simon and Walsberg by removing the need to assume dp-minimality from the outset.