Topological cell decomposition and dimension theory in P-minimal fields

by P. Cubides-Kovacsics, L. Darnière and E. Leenknegt

Journal of Symbolic Logic 82 (2017), no. 1, 347-358.

Submitted on December 07, 2015.

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 topological 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 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.

Electronic version of the paper

Version August 2015 (12 pages) pdf