Topological cell decomposition and dimension theory in P-minimal fields

TL;DR

This paper develops a topological cell decomposition for P-minimal structures, establishing dimension properties and continuity results for definable functions, thereby advancing the understanding of dimension theory in these structures.

Contribution

It introduces a cell decomposition method for P-minimal structures and proves new dimension and density properties, answering longstanding questions in the field.

Findings

The frontier of a definable set has smaller dimension than the set itself.

Definable sets can be decomposed into pure-dimensional components.

Any definable function is continuous on a dense open subset.

Abstract

This paper addresses some questions about dimension theory for P-minimal structures. We show that, for any definable set A, the dimension of the frontier of A is strictly smaller than the dimension of A itself, and that A has a decomposition into definable, pure-dimensional components. This is then used to show that the intersection of finitely many definable dense subsets of A is still dense in A. As an application, we obtain that any m-ary definable function is continuous on a dense, relatively open subset of its domain, thereby answering a question that was originally posed by Haskell and Macpherson. In order to obtain these results, we show that P-minimal structures admit a type of cell decomposition, using a topological notion of cells inspired by real algebraic geometry.